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THE STRUGGLE
FOR
EXISTENCE
Z 3
THE STRUGGLE
FOR
EXISTENCE
BY
G. F. GAUSE
Zoological Institute of the University of Moscow
BALTIMORE
T;HE WILLIAMS & WILKINS COMPANY
1934
Copyright, 1934
the williams a wilkins company
Made in the United States of America
Published December, 1934
COMPOSED AND PRINTED AT THE
WAVERLT PRESS, INC.
FOR
THE WILLIAMS & WILKINS COMPANY
BALTIMORE, MD., U. S. A.
M»
FOREWORD
For three-quarters of a century past more has been written about
natural selection and the struggle for existence that underlies the
selective process, than perhaps about any other single idea in the
whole realm of biology. We have seen natural selection laid on its
Sterbebett, and subsequently revived again in the most recent times
to a remarkable degree of vigor. There can be no doubt that the
old idea has great survival value.
The odd thing about the case, however, is that during all the years
from 1859, when Darwin assembled in the Origin of Species a masterly
array of concrete evidence for the reality of the struggle for existence
and the process of natural selection, down to the present day, about
all that biologists, by and large, have done regarding the idea is to
talk and write. If ever an idea cried and begged for experimental
testing and development, surely it was this one. Yet the whole array
of experimental and statistical attempts in all these years to produce
some significant new evidence about the nature and consequences of
the struggle for existence is pitifully meager. Such contributions as
those of Bumpus, Weldon, Pearson, and Harris are worthy of all
praise, but there have been so very, very few of them. And there is
surely something comic in the spectacle of laboratories overtly embarking
upon the experimental study of evolution and carefully
thereafter avoiding any direct and purposeful attack upon a pertinent
problem, the fundamental importance of which Darwin surely estab-
lished.
At the present time there is abundant evidence of an altered attitude;
and particularly among the younger generation of biologists.
The problem is being attacked, frontally, vigorously and intelligently.
This renewed and effective activity seems to be due primarily to two
things : first, the recrudescence of general interest in the problems of
population, with the accompanying recognition that population
problems are basically biological problems; and, second, the realization
that the struggle for existence and natural selection are matters
concerning the dynamics of populations, birth rates, death rates,
interactions of mixed populations, etc. These things were recognized
VI FOREWORD
and pointed out by Karl Pearson many years ago. His words, however,
went largely unheeded for a long time. But in the last fifteen
years we have seen more light thrown upon the problems of population
by the work of such mathematicians as Lotka and Volterra, such
statisticians as Yule, and such experimentalists as Allee and Park,
than in the entire previous history of the subject. There can be no
doubt of the fact that population problems now constitute a major
focal point of biological interest and activity.
The author of the present treatise, Dr. G. F. Gause (who stands in
the front rank of young Russian biologists, and is, it gives me great
pleasure and satisfaction to say, a protege of my old student and
friend, Prof. W. W. Alpatov) makes in this book an important contribution
to the literature of evolution. He marshals to the attack
on the old problem of the consequences of the struggle for existence
the ideas and the methods of the modern school of population students.
He brings to the task the unusual and most useful equipment
of a combination in his own person of thorough training and competence
in both mathematics and experimental biology. He breaks
new ground in this book. It will cause discussion, and some will
disagree with its methods and conclusions, but no biologist who
desires to know what the pioneers on the frontiers of knowledge are
doing and thinking can afford not to read it. I hope and believe
that it is but the beginning of a series of significant advances to be
made by its brilliant young author.
Raymond Pearl.
Department of Biology,
School of Hygiene and Public Health,
The Johns Hopkins University.
AUTHOR'S PREFACE
This book is the outcome of a series of experimental investigations
upon which I have been engaged for several years past. In these
experiments an attempt was made to make use of all the advantages
of the controlled study of the struggle for existence in the laboratory
with various organisms low in the evolutionary scale. It became
evident that the processes of competition between different species
of protozoa and yeast cells are sometimes subject to perfectly definite
quantitative laws. But it has also been found that these processes
are extremely complicated and that their trend often do not harmonize
with the predictions of the relatively simple mathematical theory.
There is also a continued need for attack upon the problems of the
struggle for existence along the lines of experimental physiology and
biology, even though the results obtained cannot yet be adequately
expressed in mathematical terminology.
I wish to express my sincere thanks to Professor W. W. Alpatov
for interest in the experimental investigations and for valuable suggestions.
To Professor Raymond Pearl I am deeply indebted for great
assistance in the publication of this book, without which it could
never have appeared before the American reader. I am also grateful
to the Editors of The Journal of Experimental Biology and Archiv
fur Protistenkunde for permission to use material previously published
in these periodicals.
G. F. Gause.
Laboratory of Ecology,
Zoological Institute,
University of Moscow,
Malaia Bronnaia 12, Kv. S3.
November, 1984
vn
CONTENTS
CHAPTER I
The Problem 1
CHAPTER II
The Struggle for Existence in Natural Conditions 12
CHAPTER III
The Struggle for Existence from the Point of View of the Mathematicians
27
CHAPTER IV
On the Mechanism of Competition in Yeast Cells 59
CHAPTER V
Competition for Common Food in Protozoa 90
CHAPTER VI
The Destruction of One Species by Another 114
Appendix 1 143
Appendix II 149
Bibliography 154
Index 161
46027
IX
Chapter I
THE PROBLEM
(1) The struggle for existence is one of those questions which were
very much discussed at the end of the last century, but scarcely any
attempt was made to find out what it really represents. As a result
our knowledge is limited to Darwin's brilliant exposition, and until
quite recently there was nothing that we could add to his words.
Darwin considered the struggle for existence in a wide sense, including
the competition of organisms for a possession of common places
in nature, as well as their destruction of one another. He showed
that animals and plants, remote in the scale of nature, are bound
together by a web of complex relations in the process of their struggle
for existence. "Battle within battle must be continually recurring
with varying success," wrote Darwin, and "probably in no one case
could we precisely say why one species has been victorious over
another in the great battle of life. ... It is good thus to try in imagination
to give to any one species an advantage over another. Probably
in no single instance should we know what to do. This ought to
convince us of our ignorance on the mutual relation of all organic
beings; a conviction as necessary as it is difficult to acquire. All that
we can do, is to keep steadily in mind that each organic being is
striving to increase in a geometrical ratio; that each at some period
of its life, during some season of the year, during each generation or at
intervals, has to struggle for life and to suffer great destruction"
('59, pp. 56-57).
(2) But if our knowledge of the struggle for existence has since
Darwin's era increased to an almost negligible extent, in other domains
of biology a great progress has taken place in recent years.
If we look at genetics, or general physiology, we find that a decisive
advance has been made there, after the investigators had greatly
simplified their problems and taken their stand upon the firm basis
of experimental methods. The latter presents a particularly interesting
example about which we would like to say a few words. We
mean the investigations of the famous Russian physiologist J. P.
Pavlov, who approached the study of the nervous activity of higher
1
2 THE STRUGGLE FOR EXISTENCE
animal by thoroughly objective physiological methods. As Pavlov
('23) himself says, it is "the history of a physiologist's turning from
purely physiological questions to the domain of phenomena usually
termed psychical." The higher nervous activity presents such a
complicated system, that without special experiments it is difficult
to obtain an objective idea of its properties. It is known, firstly, that
there exist constant and unvarying reflexes or responses of the organism
to the external world, which are considered as the especial "elementary
tasks of the nervous system." There exist besides other
reflexes variable to an extreme degree which Pavlov has named
"conditional reflexes." With the aid of carefully arranged quantitative
experiments in which the animal was isolated in a special chamber,
all the complicating circumstances being removed, Pavlov
discovered the laws of the formation, preservation and extinction of
the conditional reflexes, which constitute the basis for an objective
conception of the higher nervous activity. "I am deeply, irrevocably
and ineradicably convinced, says Pavlov, that here, on this way lies
the final triumph of the human mind over its problem—a knowledge
of the mechanism and of the laws of human nature."
(3) The history of the physiological sciences for the last fifty years
is very instructive, and it shows distinctly that in studying the
struggle for existence we must follow the same lines. The complicated
relationships between organisms which take place in nature
have as their foundation definite elementary processes of the struggle for
existence. Such an elementary process is that of one species devouring
another, or when there is a competition for a common place between a
small number of species in a limited microcosm. It is the object of the
present book to bring forward the evidence, firstly, that in studying
the relations between organisms in nature some investigators have
actually succeeded in observing such elementary processes of the
struggle for existence and, secondly, to present in detail the results of
the author's experiments in which the elementary processes have
been investigated in laboratory conditions. The experiments made
it apparent that in the simplest case we can give a clear answer to
Darwin's question: why has one species been victorious over another
in the great battle of life?
(4) It would be incorrect to fall into an extreme and to consider
the complicated phenomena of the struggle for life in nature as simply
a sum of such elementary processes. Leaving aside the existence in
THE PROBLEM 6
nature of climatic factors which undergo rhythmical time-changes,
the elementary processes of the struggle for life take place there amid
a totality of most diverse living beings. This totality presents a
whole, and the separate elementary processes taking place in it are
still insufficient to explain all its properties. It is also probable that
changes of the totality as a whole put an impress on those processes
of the struggle for existence which are going on within it.
Nobody contests the complexity of the phenomena taking place in
the conditions of nature, and we will not enter here into a discussion
of this fact. Let us rather point out all the importance of studying
the elementary processes of the struggle for life. At present our
position is like that of biophysicists in the second half of last century.
First of all it had been necessary to show that separate elementary
phenomena of vision, hearing, etc., can be fruitfully studied by physical
and chemical methods, and thereupon only did the question arise
of studying the organism as a system constituting a whole.
(5) Certain authors at the close of last century occupied themselves
with a purely logical and theoretical discussion of the struggle for
existence. They proposed different schemata for classifying these
phenomena, and we will now examine one of them in order to give
just a general idea of those elementary processes of the struggle for
life with which we will have to deal further on. To the first large
group of these processes belongs the struggle going on between groups
of organisms differing in structure and mode of life. In its turn this
struggle can be divided into a direct and an indirect one. The
struggle for existence is direct when the preservation of life of one
species is connected with the destruction of another, for instance that
of the fox and the hare, of the ichneumon fly and its host larva, of
the tuberculosis bacillus and man. In the chapter devoted to the
experimental analysis of the predator-prey relations we will turn our
attention to this form of the struggle. In plants, as Plate ('13) points
out, the direct form of the struggle for existence is found only in the
case of one plant being a parasite of the other. Among plants it is the
indirect competition, or the struggle for the means of livelihood that
predominates; this has also a wide extension among animals. It
takes place in the case when two forms inhabit the same place, need
the same food, require the same light. We will later give a great deal
of attention to the experimental study of indirect competition. To
the second group of phenomena of the struggle for life belongs the
4 THE STRUGGLE FOR EXISTENCE
intraspecies struggle, between individuals of the same species, which
in its turn can be divided into a direct and an indirect one.
(6) In this book we are interested in the struggle for existence
among animals, and it is just in this domain that exact data are
almost entirely lacking. In large compilative works one may meet
an indication that the struggle for existence "owing to the absence of
special investigations has become transformed into a kind of logical
postulate," and in separate articles one can read that "our data are in
contradiction with the dogma of the struggle for existence." In this
respect zoologists are somewhat behind botanists, who have accumulated
already some rather interesting facts concerning this problem.
What we know at present is so little that it is useless to examine
the questions: what are the features common to the phenomena of
competition in general, and what is the essential distinction between
the competition of plants and that of animals, in connection with the
mobility of the latter and the greater complexity of relations into
which they enter? What interests us more immediately is the practical
question: what are the methods by means of which botanists
study the struggle for existence, and what alterations do these methods
require in the domain of zoology?
First of all botanists have already recognized the necessity of
having recourse to experiment in the investigation of competition
phenomena, and we can quote the following words of Clements ('24,
p. 5) :
'
'The opinions and hypotheses arising from observation are
often interesting and suggestive, and may even have permanent
value, but ecology can be built upon a lasting foundation solely by
means of experiment. ... In fact, the objectivity afforded by comprehensive
and repeated experiment is the paramount reason for its
constant and universal use."
However, the experiments so far made by botanists are devoted to
the analysis of plant competition from the viewpoint of ontogenic
development. The competition began when the young plantlets
came in contact with one another, and all the decisive stages of the
competition took place in the course of development of the same
plants.
In such circumstances the question as to the causes of the victory
of certain forms over others presents itself in the following aspect:
By the aid of what morphological and physiological advantages of
the process of individual development does one plant suppress another
THE PROBLEM 5
under the given conditions of environment? Clements has characterized
this phenomenon in the following manner: "The beginning of
competition is due to reaction when the plants are so spaced that
the reaction of one affects the response of the other by limiting it.
The initial advantage thus gained is increased by cumulation, since
even a slight increase of the amount of energy or raw material is
followed by corresponding growth and this by a further gain in response
and reaction. A larger, deeper or more active root system
enables one plant to secure a larger amount of the chresard, and the
immediate reaction is to reduce the amount obtainable by the other.
The stem and leaves of the former grow in size and number, and thus
require more water, the roots respond by augmenting the absorbing
surface to supply the demand, and automatically reduce the water
content still further and with it the opportunity of a competitor.
At the same time the correlated growth of stems and leaves is producing
a reaction on light by absorption, leaving less energy available
for the leaves of the competitor beneath it, while increasing the
amount of food for the further growth of absorbing roots, taller stems
and overshading leaves" (Clements, '29, p. 318).
(7) It is not difficult to see that for the study of the elementary
processes of the struggle for existence in animals we need experiments
of another type. We are interested in the processes of destruction
and replacing of one species by another in the course of a great number
of generations. We are consequently concerned here with the
problem of an experimental study of the growth of mixed populations,
depending on a very great number of manifold factors. In other
words we have to analyze the properties of the growing groups of
individuals as well as the interaction of these groups. Let us make
for this purpose an artificial microcosm, i.e., let us fill a test tube with
a nutritive medium and introduce into it several species of Protozoa
consuming the same food, or devouring each other. If we then make
numerous observations on the alteration in the number of individuals
of these species during a number of generations, and analyze the factors
that directly control these alterations, we shall be able to form
an objective idea as to the course of the elementary processes of the
struggle for existence. In short, the struggle for existence among
animals is a problem of the relationships between the components in
mixed growing groups of individuals, and ought to be studied from the
viewpoint of the movement of these groups.
6 THE STRUGGLE FOR EXISTENCE
For the study of the elementary processes of the struggle for existence
in animals we can have recourse to experiments of two types.
We can pour some nutritive medium into a test tube, introduce into
it two species of animals, and then neither add any food nor change
the medium. In these conditions there will be a growth of the number
of individuals of the first and second species, and a competition
will arise between them for the common food. However, at a certain
moment the food will have been consumed, or toxic waste products
will have accumulated, and as a result the growth of the population
will cease. In such an experiment a competition will take place
between two species for the utilization of a certain limited amount of
energy. The relation between the species we will have found at the
moment when growth has ceased, will enable us to establish in what
proportion this amount of energy has been distributed between the
populations of the competing species. It is also evident that one
can add to the species "prey" growing in conditions of a limited
amount of energy the species "predator," and trace the process of one
species being devoured by the other. Or, in the experiments of the
second type, we need not fix the total amount of energy as a determined
quantity, and only maintain it at a certain constant level, continually
changing the nutritive medium after fixed intervals of time. In such
an experiment we approach more closely to what takes place in the
conditions of nature, where the inflow of solar energy is maintained
at a fixed level, and we can study the process of competition for common
food, or that of destruction of one species by another, in the
course of time intervals of any duration we may choose.
(8) Experimental researches will enable us to understand the
mechanism of the elementary process of the struggle for existence,
and we can proceed to the next step: to express these processes
mathematically. As a result we shall obtain coefficients of the
struggle for existence which can be exactly measured. The idea of a
mathematical approach to the phenomena of competition is not a
new one, and as far back as 1874 the botanist and philosopher Nageli
attempted to give "a mathematical expression to the suppression of
one plant by another," taking for a starting point the annual increase
of the number of plants and the duration of their life. But this line
of investigation did not find any followers, and the experimental
researches on the competition of plants which have appeared lately
THE PROBLEM 7
are as yet in the stage of nothing but a general analysis of the processes
of ontogenesis.
In past years several eminent men were deeply conscious of the
need for a mathematical theory of the struggle for existence and took
definite steps in this domain. It often happened that one investigator
was ignorant of the work of another but came to the same conclusions
as his predecessor. Apparently every serious thought on
the process of competition obliges one to consider it as a whole, and
this leads inevitably to mathematics. A simple discussion or even
a quantitative expression of data often do not suffice to obtain a
clear idea of the relationships between the competing components
in the process of their growth.
(9) About thirty years ago mathematical investigations of the
struggle for existence would have been premature, or in any case subject
to great difficulties, due to the absence of the needed preliminary
data. Of late years, owing to the publication of a number of investigations,
these difficulties have disappeared of themselves. What is
it that these indispensable preliminary researches represent?
There is no doubt that a rational study of the struggle for existence
among animals can be begun only after the questions of the multiplication
of organisms have undergone a thoroughly exact quantitative
analysis. We have mentioned that the struggle for existence is a
problem of the relationships between species in mixed growing groups
of individuals. We must therefore begin by analyzing the laws of
growth of homogeneous groups consisting of individuals of one and
the same species, and the competition between individuals in such
homogeneous groups. During the second half of the last century and
the beginning of the present much has been said about multiplication,
and "equations of multiplication" have even been proposed of
the following type: the coefficient of reproduction — the coefficient
of destruction = number of adults. (Vermehrungsziffer — Vernichtungsziffer
= Adultenziffer; see Plate ('13) p. 246.) Usually, however,
things did not go any further, and no attempts were made to
formulate exactly all these correlations. Recently the Russian geochemist,
Prof. Vernadsky, has thus characterized from a very wide
viewpoint the phenomena of multiplication of organisms ('26, p. 37
and foil.): "The phenomena of multiplication attracted but little
the attention of biologists. But in it, partly unnoticed by the natu-
8 THE STRUGGLE FOR EXISTENCE
ralists themselves, several empirical generalizations became established
to which we have become so accustomed that they appear to us
almost self-evident.
"Among these generalizations the following must be recorded.
Firstly, the multiplication of all organisms can be expressed by geometric
progressions. This can be evaluated by a uniform formula:
2bf
= Nt
where t is time, b the exponent of progression and Nt the number of
individuals existing owing to multiplication at a certain time t. Parameter
b is characteristic for every kind of living being. In this
formula there are included no limits, no restrictions either for t, for b,
or for Nt . The process is conceived as infinite as the progression is
infinite.
"This infinity of the possible multiplication of organisms can be
considered as the subordination of the increase of living matter in the
biosphere to the rule of inertia. It can be regarded as empirically established
that the process of multiplication is retarded in its manifestation
only by external forces ; it dies off with a low temperature, ceases or
becomes weaker with an insufficiency of food or respiration, with a lack
of room for the organisms that are being newly created. In 1858
Darwin and Wallace expressed this idea in a form that had been long
clear to naturalists who had gone into these phenomena, for instance,
Linnaeus, Buffon, Humboldt, Ehrenberg and von Baer: if there are
no external checks, every organism can, but at a different time, cover
the entire globe by its multiplication, produce a progeny equal in
size to the mass of the ocean or of the earth's crust.
"The rate of multiplication is different for every kind of organisms
in close connection with their size. Small organisms, that is organisms
weighing less, at the same time multiply much more rapidly than large
organisms (i.e., organisms of a great weight).
"In these three empirical generalizations the phenomena of multiplication
are expressed without any consideration of time and space
or, more precisely, in geometrical homogeneous time and space. In
reality life is inseparable from the biosphere, and we must take into
consideration terrestrial time and space. Upon the earth organisms
live in a limited space equal in dimensions for them all. They live
in a space of definite structure, in a gaseous environment or a liquid
environment penetrated by gases. And although to us time appears
THE PROBLEM 9
unlimited, the time taken up by any process which takes place in a
limited space, like the process of multiplication of organisms, cannot
be unlimited. It also will have a limit, different for every kind of
organisms in accordance with the character of its multiplication.
The inevitable consequence of this situation is a limitation of all the
parameters which determine the phenomena of multiplication of
organisms in the biosphere.
'Tor every species or race there is a maximal number of individuals
which can never be surpassed. This maximal number is reached
when the given species occupies entirely the earth's surface, with a
maximal density of its occupation. This number which I will henceforth
call the 'stationary number of the homogeneous living matter'
is of great significance for the evaluation of the geochemical influence
of life. The multiplication of organisms in a given volume or on a
given surface must proceed more and more slowly, as the number of
the individuals already created approaches the stationary number."
These general notions on the multiplication of organisms have
lately received a rational quantitative expression in the form of the
logistic curve discovered by Raymond Pearl and Reed in 1920. The
logistic law mathematically expresses the idea that in the conditions
of a limited microcosm the potentially possible "geometric increase"
of a given group of individuals at every moment of time is realized
only up to a certain degree, depending on the unutilized opportunity
for growth at this moment. As the number of individuals increases,
the unutilized opportunity for the further growth decreases, until
finally the greatest possible or saturating population in the given
conditions is reached. The logistic law has been proved true as
regards populations of different animals experimentally studied in
laboratory conditions. We shall have an opportunity to consider all
these problems more in detail further on. Let us now only note that
the rational quantitative expression of growth of groups consisting
of individuals of the same species represents a firm foundation for a
further fruitful study of competition between species in mixed popu-
lations.
(10) Apart from a great progress as regards the mathematical
expression of the multiplication of organisms, an important advance
has taken place in the theory of competition itself. The first step in
this direction was made in 1911 by Ronald Ross, who at this time
was interested in the propagation of malaria. Considering the
10 THE STRUGGLE FOR EXISTENCE
process of propagation Ross came to the conclusion that he was dealing
with a peculiar case of a struggle for existence between the malaria
Plasmodium and man with a participation of the mosquito. Ross
formulated mathematically an equation of the struggle for existence
for this case, which closely approached in its conception those equations
of the struggle for existence which the Italian mathematician
Volterra proposed in 1926 without knowing the investigations of Ross.
Whilst Ross was working on the propagation of malaria the American
mathematician Lotka ('10, '20a) examined theoretically the
course of certain chemical reactions, and had to deal here with equations
of the same type. Later on Lotka became interested in the
problem of the struggle for existence, and in 1920 he formulated an
equation for the interaction between hosts and parasites ('20b), and
gave a great deal of interesting material in his valuable book, Elements
of Physical Biology ('25). Without being acquainted with
these researches the Italian mathematician Vito Volterra proposed in
1926 somewhat similar equations of the struggle for existence. At
the same time he advanced the entire problem considerably, investigating
for the first time many important questions of the theory of
competition from the theoretical point of view. Thus three distinguished
investigators came to the very same theoretical equations
almost at the same time but by entirely different ways. It is also
interesting that the struggle for existence only began to be experimentally
studied after the ground had been prepared by purely theoretical
researches. The same has already happened many times in
the fields both of physics and of physical chemistry: let us recollect
the mechanical equivalent of heat or Gibbs' investigations.
(11) The study of the struggle for existence will undoubtedly
rapidly progress in the future, but it will have to overcome a certain
gap between the investigations of contemporary biologists and mathematicians.
There is no doubt that the struggle for existence is a
biological problem, and that it ought to be solved by experimentation
and not at the desk of a mathematician. But in order to penetrate
deeper into the nature of these phenomena we must combine the
experimental method with the mathematical theory, a possibility
which has been created by the brilliant researches of Lotka and Volterra.
This combination of the experimental method with the
quantitative theory is in general one of the most powerful tools in
the hands of contemporary science.
THE PROBLEM 11
The gap between the biologists and the mathematicians represents
a significant obstacle to the application of the combined methods of
research. Mathematical investigations independent of experiments
are of but small importance due to the complexity of biological systems,
narrowing the possibilities of theoretical work here as compared
with what can be admitted in physics and chemistry. We are in complete
accord with the following words of Allee ('34): "Mathematical
treatment of population problems is necessary and helpful, particularly
in that it permits the logical arrangement of facts and abbreviates
their expression by the use of a sort of universal shorthand, but
the arrangement and statement may lead to error, since for the sake
of brevity and to avoid cumbersome expressions, variables are omitted
and assumptions made in the mathematical analyses which are not
justified by the biological data. Certainly there is room for the
mathematical attack on population problems, but there is also continued
need for attack along the lines of experimental physiology,
even though the results obtained cannot yet be adequately expressed
in mathematical terminology."
<€,ej^
Chapter II
THE STRUGGLE FOR EXISTENCE IN NATURAL
CONDITIONS
(1) Before beginning any experimental investigation of the elementary
processes of the struggle for existence we must examine what
is the state of our knowledge of the phenomena of competition in
nature. The regularities which it has been possible to ascertain
there, and the ideas which have been expressed in their discussion,
will help us to formulate correctly certain fundamental requirements
for further experimental work.
In thorough field observations the fact which strikes the investigator
most of all is the extreme complexity of the communities of organisms,
and at the same time their possession of a definite structure.
On the one hand they undergo changes under the influence of external
environment, and on the other the slightest changes of some components
produce an alteration of others and lead to a whole chain of consequences.
It is difficult here to arrive at a sufficiently clear understanding
of the processes of the struggle for existence. Elton writes
for instance: "We do not get any clear conception of the exact way
in which one species replaces another. Does it drive the other one
out by competition? and if so, what precisely do we mean by competition?
Or do changing conditions destroy or drive out the first arrival,
making thereby an empty niche for another animal which quietly
replaces it without ever becoming 'red in tooth and claw' at all?
Succession brings the ecologist face to face with the whole problem
of competition among animals, a problem which does not puzzle
most people because they seldom if ever think out its implications at
all carefully. At the present time it is well known that the American
grey squirrel is replacing the native red squirrel in various parts of
England, but it is entirely unknown why this is occurring, and no
good explanation seems to exist. In ecological succession among
animals there are thousands of similar cases cropping up, practically
all of which are as little accounted for as that of the squirrels" ('27,
p. 27-28). All this suggests that an analysis must be made of comparatively
simple desert or Arctic communities where the number of
12
STRUGGLE IN NATURAL CONDITIONS 13
components is small. Such a tendency to examine certain elementary
phenomena is clearly seen in the following words of a Russian
zoologist, N. Severtzov, written as far back as 1855: "It seems to me
that the study of animal groupings in small areas, the study of these
elementary faunas is the firmest point of support for drawing conclusions
about the general laws regulating the distribution of animals
on the globe."
However, besides this first possibility of studying competition phenomena
among a small number of components, an active intervention
into natural conditions by means of biotic experiments may also be
very important. Among such experiments the most frequent ones
consist in the transportation of animals into countries new to them,
which commonly leads to a great number of highly interesting procTABLE
I
Number of fir trunks on a unit of surface under different conditions
From Sukatschev ('28)
14 THE STRUGGLE FOR EXISTENCE
time. At first sight one might think that the better the conditions of
existence the less active is the struggle for life, and the greater the
number of trunks that can survive with age on a unit of surface. Let
us, however, look at the data of the foresters. For an example we
will give in Table I the number of the fir trunks in the government of
Leningrad (Northern Russia) corresponding to five different types
of life conditions (Type I represents the best soil and ground conditions;
V, the worst ones).
These data show, contrary to our expectations, that the better are
the soil and ground conditions, the more active is the struggle for life,
or in other words the smaller the number of trunks remaining on a
unit of surface and, consequently, the greater the percentage of those
which perish. If we think out this phenomenon, it becomes quite
understandable : the more favorable the environment is for the plants'
existence, the more luxuriant will be the development of each plant,
the sooner will the tops of the trees begin to close above, and the
earlier the oppressed individuals become isolated. Also, in better
conditions of existence, every individual in the adult state will be
more developed and occupy a greater space, but the individuals will
be fewer in number. Investigations show that this is a general rule
for all the forest species (Sukatschev, '28, p. 12).
Similar data were obtained by Sukatschev ('28) in experiments
with the chamomile, Matricaria inodora, on fertilized and non-fertilized
soil. In counting up the individuals remaining at the end of
summer (August 17), the following decrease of the original number of
individuals was ascertained (see Table II and Fig. 1).
Here likewise in better conditions of existence competition proceeds
with greater intensity, and the per cent of individuals which
perish is greater.
The results obtained by botanists are certainly characteristic for
the ontogenetic development of plants, but at the same time they
give us an approach to the quantitative appreciation of the intensity
of the struggle for existence, the whole significance of which was
already clearly understood by Darwin. In the next chapter we shall
consider the struggle for life in animals, and there, using entirely
different methods, we shall endeavor to formulate quantitatively the
intensity of this struggle.
(3) In field observations the question often arises as to the struggle
for existence in mixed populations, about which Darwin wrote: "As
STRUGGLE IN NATURAL CONDITIONS 15
the species of the same genus usually have, though by no means invariably,
much similarity in habits and constitution, and always in
structure, the struggle will generally be more severe between them,
if they come into competition with each other, than between the
species of distinct genera." Lately, botanists have tried to approach
this problem experimentally. It became evident that, actually, in a
number of cases competition is keenest when the individuals are most
Percentage, ofperishedindividuals
15.1%
5.8%
good Bad
conditions
Fig. 1. Intensity of the struggle for existence in the chamomile, Matricaria
inodora, on fertilized and non-fertilized soil (dense culture).
TABLE II
Decrease of the number of individuals in the chamomile {Matricaria inodora)
expressed in percentage of the initial number
From Sukatschev ('28)
16 THE STRUGGLE FOR EXISTENCE
data concerning wild-growing plants have been recently published
by Sukatschev ('27) in his "Experimental studies on the struggle for
existence between biotypes of the same species." First of all he
studied the competition between local biotypes of the plant, Taraxacum
officinale Web., from the environs of Leningrad. These biotypes
were cultivated in similar conditions with a fixed distance between
the individuals, and the experiments led Sukatschev to the following
conclusions: (a) One must rigorously distinguish the conditions of
the struggle for existence in a pure population, formed by a single
biotype, and in a mixed population, consisting of various biotypes.
(b) It is to be noted that a biotype which shows itself to be the most
resistant in an intrahiotic, struggle for existence, may turn out to be
the weakest one in an interhiotic, struggle between different biotypes
of the same species, (c) The increase in mutual influence of plants
upon each other with an increase in density of the plant cultures, may
completely reverse the relative stability of separate biotypes in the
process of the struggle for existence. The biotypes yielding the greatest
percentage of survivors under a small density of cultivation may
occupy the last place in this respect in conditions of a dense culture.
This can be illustrated by Table III.
If we arrange the biotypes mentioned in Table III according to decreasing
stability, we shall find that in the conditions of a not dense,
pure culture : C > A > B, i.e., the biotype C gives the smallest percentage
of non-survivals and is the most resistant, whilst the biotype B is
the weakest of all. In dense pure cultures the relations are entirely
different: B > A > C, i.e., the biotype B is the most stable one.
Lastly, for dense, but mixed cultures, we have: C > A > B. Almost
similar data have been obtained by Montgomery ('12) in studying
the competition between two races of wheat.
In another series of experiments Sukatschev ('27) studied the
struggle for existence between biotypes of various geographical origin
(from various parts of U. S. S. R.), and inferred the following: (a)
Judging by the percentage of non-survivals, one can say that in pure,
as well as in mixed not-dense cultures, the dying-off is chiefly due to
the influence of physico-geographical factors. Therefore, the biotypes
originating from geographical regions strongly differing from
a given region in their climate, turn out to be less resistant, as compared
with the local biotypes. But these relations can change under
the influence of aggregation, (b) In dense mixed cultures, if we are
STRUGGLE IN NATURAL CONDITIONS 17
to judge by the percentage of perished individuals, it is not the local
biotypes that appear the most resistant in the struggle for existence,
but those introduced from other regions, (c) The struggle for existence
in mixed cultures of various biotypes is not so keen as that in
pure cultures of separate biotypes with the same density.
The analysis of the struggle for existence in mixed populations of
plants is now only at its very beginning. The exact data are few in
number, but there exist numerous observations on the stratified
distribution of plants (see Alechin, '26) , considering these strata as a
result of complex processes of competition and adaptation of the
plants to one another in mixed cultures.
(4) Plants are also very favorable for the study of the influence of
TABLE III
Percentage of eliminated individuals in three biotypes, A, B and C of Taraxacum
officinale
From Sukatschev ('27)
18 THE STRUGGLE FOR EXISTENCE
tions existing here are, however, somewhat complicated (see BraunBlanquet,
'28).
The problem of the influence of environment on competition presents
considerable interest, but as yet what we know is very meager.
In the majority of cases it is observations of a qualitative character,
of which we can give an example here: "The root-systems of the
vegetation in the steppes of Southern Russia form, according to
Patchossky, three strata. The uppermost one consists of short roots
belonging to annual plants which vegetate for a short time. The
second, deeper-lying stratum belongs to the essential plants of the
steppe vegetable covering, the Gramineae. The third, deepest
stratum consists of the vertical stem-like roots of perennial dicotyledons
(among them the steppe Euphorbia). Usually, the second
gramineous stratum dominates. When, however, an immoderate
pasturing takes place in a given locality, the gramineous covering
begins to suffer and does not produce a vigorous root-system. Atmospheric
precipitation can now penetrate to those soil horizons
where roots of the dicotyledons are situated, and the latter begin to
dominate. As a result appears an unbroken vegetable covering consisting
of Euphorbia. Analogous results take place in case of increase
in yearly atmospheric precipitation. In this case, although the water
is energetically absorbed by the second gramineous root stratum, the
rainfall is so considerable that a great part of the water penetrates
deeper, contributing to the development of dicotyledonous plants.
The large dicotyledons act depressingly upon the Gramineae, and
they change places in respect to their domination" (Alechin, '26).
(5) The part which the quantitative relations between species at
the beginning of their struggle play in the outcome of competition
presents an interesting problem. Botanists do not possess exact
quantitative data bearing on this question, and one meets only with
considerations of the following kind: When new soils are colonized,
if the species concerned do not sharply differ in their capacity for
spreading, it mostly depends on chance which species colonizes the
given area first. But this chance determines the further colonizing
Planaria monlenegrina and PI. gonocephala occur in competition with each
other, temperature is the factor which governs the relative success and efficiency
of the two species. PI. monlenegrina is the more successful at temperatures
below 13-14°C. Above these temperatures PI. gonocephala is the more
efficient form."
STRUGGLE IN NATURAL CONDITIONS 19
of the given locality. Even when the species that has first established
itself is somewhat weaker than another species in the same habitat,
it can for a comparatively long time resist its stronger competitor
simply because it was the first to occupy this place. Only in case
of a considerable weakness of the first comer will its domination be
merely a temporary one, and the effect of the first accidental appearance
will be rapidly eliminated (E. Warming ('95), Du-Rietz ('30)).
(6) Let us recapitulate briefly our discussion up to this point.
Botanists have endeavored to investigate the struggle for existence
by experimentation and under simplified conditions, but they are
only beginning to analyze these phenomena. Their experiments are
commonly limited to the process of ontogenetic development, and
in only a few cases, chiefly concerned with competition in cereals, has
displacement of some forms by others been traced through a series of
generations (Montgomery ('12) and others). As concerns animals we
have simply no exact data, and can only mention a few general principles
which have been developed by zoologists in connection with the
phenomena of competition.
One of these ideas is that of the "niche" (see Elton, '27, p. 63). A
niche indicates what place the given species occupies in a community,
i.e., what are its habits, food and mode of life. It is admitted that as
a result of competition two similar species scarcely ever occupy similar
niches, but displace each other in such a manner that each takes
possession of certain peculiar kinds of food and modes of life in which
it has an advantage over its competitor. Curious examples of the
existence of different niches in nearly related species have recently
been obtained by A. N. Formosov ('34). He investigated the ecology
of nearly related species of terns, living together in a definite region,
and it appeared that their interests do not clash at all, as each species
hunts in perfectly determined conditions differing from those of
another. This once more confirms the thought mentioned earlier,
that the intensity of competition is determined not by the systematic
likeness, but by the similarity of the demands of the competitors upon
the environment. Further on we shall endeavor to express all these
relations in a quantitative form.
(7) The above mentioned observations of A. N. Formosov on
different niches in nearly related species of terns can be given here
with more detail, as the author has kindly put at our disposal the
following materials from his unpublished manuscript: According to
20 THE STRUGGLE FOR EXISTENCE
the observations in 1923, the island Jorilgatch (Black Sea) is inhabited
by a nesting colony of terns, consisting of many hundreds of
individuals. The nests of the terns are situated close to one another,
and the colony presents a whole system. The entire mass of individuals
in the colony belongs to four species (sandwich-tern, Sterna
cantiaca; common-tern, S. fluviatilis; blackbeak-tern, S. anglica; and
little-tern, S. minuta), and together they chase away predators (henharriers,
etc.) from the colony. However, as regards the procuring
of food, there is a sharp difference between them, for every species
pursues a definite kind of animal in perfectly definite conditions.
Thus the sandwich-tern flies out into the open sea to hunt certain
species of fish. The blackbeak-tern feeds exclusively on land, and
it can be met in the steppe at a great distance from the sea-shore,
where it destroys locusts and lizards. The common-tern and the
little-tern catch fish not far from the shore, sighting them while flying
and then falling upon the water and plunging to a small depth. The
light little-tern seizes the fish in shallow swampy places, whereas the
common-tern hunts somewhat further from the shore. In this manner
these four similar species of tern living side by side upon a single
small island differ sharply in all their modes of feeding and procuring
food.
(8) Another ecological notion is also important in connection with
our experiments. We have in view the degree of isolation of the
microcosm. The point is that our experimental researches have been
mainly made in isolated microcosms, i.e., in test tubes filled with
nutritive medium and stopped with cotton-wool. It must be remembered
that the degree of isolation of different communities in
natural conditions is very different. Such a system as a lake is almost
isolated, but at times some of the animals inhabiting it go on land.
An oasis in the desert would also seem to be isolated, but for instance
some of the species of birds fly away for the winter, and consequently
there is no real isolation. The habitats not so sharply separated from
the surrounding life-area are, therefore, still less isolated. All this
emphasizes the idea already expressed, that the regularities observed
in isolated microcosms hold true only under certain fixed conditions,
and are not sufficient to explain all the complicated phenomena taking
place in nature. We shall have an opportunity to appreciate the
role of this factor when experimentally studying the predator-prey
relations.
STRUGGLE IN NATURAL CONDITIONS 21
Let us note another important circumstance connected with competition.
This phenomenon can be particularly pronounced during a
periodical food shortage connected with certain seasons, etc., whilst
at another time with an abundance of food it will scarcely take place.
This fact has frequently been pointed out in various discussions of
the struggle for existence.
(9) It remains but to give some examples of the struggle for existence
among animals in order to show with what problems the zoologists
have to deal, and how difficult it is to apply here exact quantitative
methods. An instructive example of competition among
fishes has been recently described by Kashkarov ('28). It concerns
the supplanting of Schizothorax intermedins by wild carp, Cyprinus
albus L., in lakes of Middle Asia. Wild carp were introduced into the
lake Bijly Kul in 1909. Before that only Schizothorax intermedins
with white-fish (Leuciscus sp.) inhabited this lake. Formerly Schizothorax
intermedins were very numerous, but after the introduction of
wild carp their quantity diminished considerably. As an indicator
of the relatively small number of Schizothorax intermedins the following
data on the catch may serve: on May 15, 1926, 19 carp and 1
Schizothorax were caught in two nets; on May 16, 1926, in the same
place the catch was 24 carp and 2 Schizothorax. Schizothorax keeps
chiefly to the south-western part of the lake, where there are stones
making the casting of nets difficult. Now Schizothorax is disappearing
even there, as wild carp devour its spawn. The quantity of whitefish
also decreases because carp devour its young. The particular
interest of this example lies in the fact that a new species not found
in a given microcosm before (Cyprinus albus L.) was introduced, and
in this way a direct proof of one species displacing another was
obtained.
(10) Processes of this kind can often be observed when fish are
introduced into waters to which they are new. Professor G. C.
Embody writes in a letter recently received: "Concerning the competition
between different species of fishes we have two cases in
particular in the eastern United States. The European carp was
introduced in the '70's, and has now in many streams and lakes
multiplied to such an extent that several native species are found in
greatly diminished number. This has probably been due to the high
reproductive capacity of the carp, food competition, destruction of
weed beds by carp, and the fact that very few of them are captured.
22 THE STRUGGLE FOR EXISTENCE
Carp are not used as extensively for food in America as in Europe and
in our smaller lakes are not generally fished for commercially. The
other case is the introduction of the perch (Perca flavescens) into certain
lakes in the Adirondacks and in Maine, which were naturally
populated with the trout (Salvelinus fontinalis). The competition
for food is believed to be one of the causes for the decrease in the number
of the trout."
"These cases are both matters of general observation. I do not
know of any papers describing them nor in fact, dealing with this
subject in American waters."
(11) Another example of competition is the replacing of one species
of cray-fish by another in certain waters of Middle Russia. Some
observations on this were made by Kessler (75) and recently by Birstein
and Vinogradov ('34). Two species of cray-fish inhabit the
waters of European Russia: the broad-legged (Pota?nobius astacus L.)
and long-legged (Potamobius leptodactylus Esch.). The broad-legged
cray-fish is distributed in the western part, and the long-legged in the
south-eastern one, but the areas of their distribution largely overlap
one another. It is observed that the long-legged cray-fish displaces
the broad-legged one and spreads gradually more and more to the
west. It has been possible to establish this replacement with particular
distinctness in White Russia (in the western part of U. S. S. R.).
The cray-fish are found there in lakes isolated from each other, and
most of the lakes are inhabited only by the broad-legged cray-fish.
In some cases long-legged cray-fish were put into such lakes from
other waters. As a result the broad-legged cray-fish began to decrease,
and finally disappeared completely leaving the lake populated
exclusively by the long-legged species. The following examples can
be given. (I) Black lake (White Russia) was populated only by the
broad-legged cray-fish. In 1906, 500 specimens of long-legged crayfish
were introduced, and now (1930) only this species remains. (II)
Forest lake (same region). Up to 1920 there were no long-legged
cray-fish there. Later on they were introduced, and at present (1930)
there is a considerable number of this species. The causes why one
species of cray-fish is replaced by another have scarcely been studied.
(12) Curious are the observations reported by Goldman ('30) on
the competition among predators belonging to different species.
Thus, according to a resident of Telegraph Creek near the Stikine
River, Canada, no coyotes were known in that section prior to 1899.
STRUGGLE IN NATURAL CONDITIONS 23
About that time, however, they came in, apparently following the old
goldrush trail, probably attracted by the hundreds of dead horses
along it. The invasion of Alaska seems destined to continue until
coyotes have extended their range over practically all of the territory.
It has been found in Alaska that the coyotes kill many foxes.
Since the coyotes have increased the foxes have decreased alarmingly.
In some sections practically none are believed to be left. In many
cases a family or entire colony of foxes are run out of their dens or
are both run out and killed by coyotes which then use the dens themselves.
Wolves are well known to have committed similar depredations,
but their killing is not so extensive as is that of the coyotes.
Serious as are the depredations of wolves throughout most of Alaska,
the damage done to game and fur bearing animals by coyotes is not
only far greater but is rapidly increasing in extent. How far the
coyotes will hold back the normal development in the periodical increase
of the snowshoe rabbits and the ptarmigan which are important
items in the food supply of fur bearers, especially the lynx and fox, it
is impossible even to approximate.
(13) The examples just mentioned show that the introduction of
aquatic animals into waters to which they are new, or the penetration
of land animals into new regions, often lead to very interesting processes
of competition. We would now like to say a few words about
the direct struggle for existence, in which one species devours another.
In this case it is very important to ascertain the exact numerical
relation between the population of the devoured species and that of
the devouring one, and this can often be attained by changing their
relative quantities. One of the outstanding facts is the increase of
the deer accompanying the destruction of wolves, foxes, etc., by
early settlers in Illinois which, according to Wood, has been recently
reported by Shelford ('31). Wood depicts a continuous decrease in
wolves and wildcats from the beginning of settlement to their practical
extinction. When the wolf population was reduced to about onehalf,
the deer increased rapidly for a little less than 10 years, reaching
a large maximum of about three times the original number. Unfortunately
we have no exact data on the change of the numerical
relation between the wolves and the deer, but such processes are in
any case of great interest.
The change of the numerical relations between the predator and
the prey sometimes takes place as a consequence of a mass appear-
24 THE STRUGGLE FOR EXISTENCE
ance of the prey in years especially favorable for its multiplication.
This frequently happens with wild mice, and it gives us the possibility
of tracing the process of their being devoured by the predator. In
this connection we may mention the following observations recently
made by Kalabuchov and Raewski ('33) in the North Caucasus:
"The picture of the destruction of mice by different predators is a
curious one. At the beginning of the destruction about the same
number of rodents is devoured daily. But as the density of rodents
diminishes it becomes more and more difficult to catch them, and
the number of mice devoured gradually decreases. Finally a time
comes when the relation between the density of the rodents, the presence
of cover or refuge (burrows, vegetation, etc.) and the biological
«3
-p:
STRUGGLE IN NATURAL CONDITIONS 25
Certain interesting observations have been also recently made by
ecological entomologists (Payne, '33, '34) on the host-parasite balance.
It is possible to trace the process of the destruction of population
of the moth Ephestia by the hymenopterous parasite Microbracon
in the laboratory, and it appears from the observations that a great
many factors are important for the process of their interaction, and
particularly various relations between "susceptible" stage of host and
"effective" stage of parasite. The beginning of the theoretical investigation
of this case has been given by the interesting papers of Bailey
('31, '33) and Nikolson ('33). For such investigations, however, the
populations of unicellular organisms are somewhat more convenient.
(14) The few examples given above show sufficiently that the
processes of the struggle for existence among animals are of extreme
importance from a practical point of view. They are sharply outlined
in isolated microcosms and therefore there is nothing surprising
if they have attracted the particular attention of the workers in the
domain of fishery. Lately the problem of the relations between
predatory and non-predatory fish has been discussed by Italian
authors (D'Ancona ('26, '27), Marchi ('28, '29), Brunelli ('29)).
D'Ancona collected the data of a statistical inspection of the fishmarkets
in Triest, Venice and Fiume for several years. He claims
that the diminished intensity of fishing during the war-period (1915-
1920) has caused a comparative increase of the number of predatory
fish. He therefore reasons that fishing of normal intensity causes a
relative diminution of the number of predatory fish and a comparative
increase of the non-predatory ones. But his data are not convincing
and indeed Bodenheimer ('32) has recently shown that such
variations in the fish population existed before and after the war.
They are apparently not connected with the intensity of fishing but
probably are the results of certain changes of the environment.
However it may be, the material collected by D'Ancona stimulated
the highly interesting mathematical researches on the struggle for
existence of Vito Volterra. Although his mathematical theories are
not confirmed in any way by D'Ancona's statistical data, the importance
of Volterra's methods as a new and powerful tool in the
analysis of biological populations admits of no doubt.
(15) In concluding this descriptive chapter of our book let us note
the following picture of the struggle for existence in nature. It is
only in the domain of botany that these processes are coming to be
26 THE STRUGGLE FOR EXISTENCE
investigated from a certain general viewpoint as (1) intensity of
competition, (2) competition in mixed populations, (3) the influence
of environment upon competition in mixed cultures, and (4) the role
of the quantitative relations between species at the beginning of their
struggle. Among animals the processes of the struggle for existence
are much more complex, and as yet one cannot speak of any general
principles. In this connection an investigation of the elementary
processes of the struggle for life in strictly controlled laboratory conditions
is here particularly desirable, and the material just presented
will be of great help to us in the choice and arrangement of the corresponding
experiments.
Chapter III
THE STRUGGLE FOR EXISTENCE FROM THE POINT OF
VIEW OF THE MATHEMATICIANS
(1) In this chapter we shall make the acquaintance of the astonishing
theories of the struggle for existence developed by mathematicians
at a time when biologists were still far from any investigation of these
phenomena and had but just begun to make observations in the field.
The first attempt at a quantitative study of the struggle for existence
was made by Sir Ronald Ross ('08, '11). He undertook a theoretical
investigation of the propagation of malaria, and came to conclusions
which are of great interest for quantitative epidemiology
and at the same time constitute an important advance in the understanding
of the struggle for existence in general. Let us examine
the fundamental idea of Ross. Our object will be to give an analysis
of the propagation of malaria in a certain locality under somewhat
simplified conditions. We assume that both emigration and immigration
are negligible, and that in the time interval we are studying
there is no increase of population or in other words the birth rate is
compensated by the death rate. In such a locality a healthy person
can be infected with malaria, according to Ross, if all the following
conditions are realized: (1) That a person whose blood contains a
sufficient number of gametocytes (sexual forms) is living in or near
the locality. (2) That an Anopheline capable of carrying the parasites
sucks enough of that person's blood. (3) That this Anopheline
lives for a week or more afterwards under suitable conditions—long
enough to allow the parasites to mature within it, and (4) that it next
succeeds in biting another person who is not immune to the disease
or is not protected by quinine. The propagation of malaria in such
a locality is determined in its general features by two continuous and
simultaneous processes: on the one hand the number of new infections
among people depends on the number and infectivity of the
mosquitoes, and at the same time the infectivity of the mosquitoes is
connected with the number of people in the given locality and the
27
28 THE STRUGGLE FOR EXISTENCE
frequency of the sickness among them. Ross has expressed in mathematical
terms this uninterrupted and simultaneous dependence of
the infection of the first component on the second, and that of the
second on the first, with the aid of what is called in mathematics
simultaneous differential equations. These equations are very simple
and we shall examine them at once. In a quite general form they
can be represented as follows
Rate of increase of
affected individuals
among the human
population
Rate of increase of
infected individuals
among the mosquito
population
> =1
> =\
New infections
per unit of time.
Depends on infected
mosqui-
toes
New infections
per unit of time.
Depends on the
infected humans
[
Recoveries] *
• — \ per unit of
time
)>
(1)
Deaths of
infected
mosquitoes
per unit of
time
t
zl
=
(2) Translating these relations into mathematical language we
shall obtain the simultaneous differential equations of Ross. Let us
introduce the following notation (that of Lotka (51))
total number of human individuals in a given locality.
total number of mosquitoes in a given locality.
total number of people infected with malaria.
total number of mosquitoes containing malaria parasites.
total number of infective malarians (number of persons
with gametocytes in the blood ; a certain fraction of the
total number of malarians).
total number of infective mosquitoes (with matured parasites
; a certain fraction of the total number of mosquitoes
containing parasites).
recovery rate, i.e., fraction of infected population that reverts
to noninfected (healthy) state per unit of time.
r =
* The human death rate is not taken into consideration. To simplify the situation
we assume that it is negligible and counterbalanced by the birth rate,
t We assume that the recovery of the infected mosquitoes does not occur.
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 29
Ml
= mosquito mortality, i.e., death rate per head per unit of
time.
t = time.
If a single mosquito bites a human being on an average bl
times
per unit of time, then the f
x
zx
infective mosquitoes will place bl
p-zl
infective
bites on human beings per unit of time. If the number of
people not infected with malaria is (p — z), than taken in a relative
form as it will show the relative number of healthy people in the
total number of individuals of a given locality. Therefore, out of a
total number of infective bites, equal per unit of time to fr^/V, a
definite fraction equal to - falls to the lot of healthy people, and
the number of infective bites of healthy people per unit of time will
be equal to:
&i/i
2iP_=Li (2)
V
If every infective bite upon a healthy person leads to sickness,
then the expression (2) will show directly the number of new infections
per unit of time, which we can put in the second place of the first
line of the equations (1). By analogous reasoning it follows that if
every person is bitten on an average b times per unit of time, the total
number of infective people fz will be bitten bfz times, and a fraction
-——
of these bites will be made by healthy mosquitoes which will
thus become infected. Consequently the number of new infections
among mosquitoes per unit of time will be
bfz t^L? (3)
V
1
Now, evidently, the total number of mosquito bites on human
beings per unit of time will constitute a certain fixed value, which can
be written either as ¥pl
, i.e., the product of the number of mosquitoes
by the number of bites made by each mosquito per unit of time, or
as bp, i.e., the number of persons multiplied by the number of times
30 THE STRUGGLE FOR EXISTENCE
each human being has been bitten. We have therefore bp = bl
p
l
,
and finally
b = tL (4)
V
Inserting the expression (4) into the formula (3) we obtain
b 1
p
l
fz , .
A b x
fz , , ., /rN
—^f- (p
1
- z
l
) = -i- (p
1
- z
l
) (5)
pp
1
V
The expression (5) fills the second place in the lower line in Ross's
differential equations of malaria (1). It gives the number of new
infections of mosquitoes per unit of time. We can now put down the
rate of increase of infected individuals among the human population
dz
as -j- f
and the rate of increase in the number of infected mosquitoes
Lit
dz1
as —. The number of recoveries per unit of time among human in-
(J/L
dividuals will be rz, as z represents the number of people infected and
r the rate of recovery, i.e., the fraction of the infected population
recovering per unit of time. The number of infected mosquitoes
dying per unit of time can be put down as ik/V, since z1
denotes the
number of mosquitoes infected, and Ml
the death rate in mosquitoes
per head per unit of time. We can now express the equation of Ross
in mathematical symbols instead of words:
dz ,, n .p — z
— = bl
j
l
z
l
rz
dt p
dz1
, , , p
1
— z
l
, ,
— = b l
fz Mdt
J
p
\
1
2
1
(6)
These simultaneous differential equations of the struggle for existence
express in a very simple and clear form the continuous dependence
of the infection of people on the infectivity of the mosquitoes and
vice versa. The increase in the number of sick persons is connected
with the number of bites made by infective mosquitoes on healthy
persons per unit of time, and at the same moment the increase in the
number of infected mosquitoes depends upon the bites made by
healthy mosquitoes on sick people. The equations (6) enable us to
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 31
investigate the change with time (t) of the number of persons infected
with malaria (z).
The equations of Ross were submitted to a detailed analysis by
Lotka ('23, '25) who in his interesting book Elements of Physical
Biology gave examples of some other analogous equations. As Lotka
remarks, a close agreement of the Ross equations with reality is not
to be expected., as this equation deals with a rather idealized case:
that of a constant population both of human beings and mosquitoes.
"There is room here for further analysis along more realistic lines.
It must be admitted that this may lead to considerable mathematical
difficulties" (Lotka ('25, p. 83)).
(3) The equations of Ross point to the important fact that a mathematical
formulation of the struggle for existence is a natural consequence
of simple reasoning about this process, and that it is organically
connected with it. The conditions here are more favorable than
in other fields of experimental biology. In fact if we are engaged in
a study of the influence of temperature or toxic substances on the life
processes, or if we are carrying on investigations on the ionic theory of
excitation, the quantitative method enables us to establish in most
cases only purely empirical relations and the elaboration of a rational
quantitative theory presents considerable difficulties owing to the
great complexity of the material. Often we cannot isolate certain
factors as we should like, and we are constantly compelled to take
into account the existence of complicated and insufficiently known
systems. This produces the well known difficulties in applying
mathematics to the problems of general physiology if we wish to go
further than to establish purely empirical relations. As far as the
rational mathematical theory of the struggle for existence is concerned,
the situation is more favorable, because we can analyze the
properties of our species grown separately in laboratory conditions,
then make various combinations and in this way can formulate
correctly the corresponding theoretical equations of the struggle for
life.
(4) Besides the interest of the equations of Ross as the first attempt
to formulate mathematically the struggle for existence, they allow
us to answer a very common objection of biologists to such equations
in general. It is frequently pointed out that there is no sense in
searching for exact equations of competition as this process is very
inconstant, and as the slightest change in the environmental condi-
32 THE STRUGGLE FOR EXISTENCE
tions or in the quantities of each species can lead to the result that
instead of the second species supplanting the first it is the first species
itself that begins to supplant the second. As Jennings ('33) points
out, there exists a strong strain of uniformitarianism in many biologists.
The idea that we can observe one effect, and then the opposite,
seems to them a negation of science.
In the spreading of malaria something analogous actually takes
place. Ross came to the following interesting conclusion about
this matter: (1) Whatever the original number of malaria cases
in the locality may have been, the ultimate malaria ratio will
tend to settle down to a fixed figure dependent on the number
of Anophelines and the other factors—that is if these factors remain
constant all the time. (2) If the number of Anophelines is sufficiently
high, the ultimate malaria ratio will become fixed at
some figure between per cent and 100 per cent. If the number
of Anophelines is low (say below 40 per person) the ultimate
malaria rate will tend to zero—that is, the disease will tend to
die out. All these relations Ross expressed quantitatively, and later
they were worked out very elegantly by Lotka. This example shows
that a change in the quantitative relations between the components
can change entirely the course of the struggle for existence. Instead
of an increase of the malaria infection and its approach to a certain
fixed value, there may be a decrease reaching an equilibrium with a
complete absence of malaria. In spite of all this there remains a
certain invariable law of the struggle for existence which Ross's equations
express. In this way we see what laws are to be sought in the
investigation of the complex and unstable competition processes.
The laws which exist here are not of the type the biologists are
accustomed to deal with. These laws may be formulated in terms
of certain equations of the struggle for existence. The parameters in
these equations easily undergo various changes and as a result a whole
range of exceedingly dissimilar processes arises.
ii
(1) The material just presented enables one to form a certain idea
as to what constitutes the essence of the mathematical theories of the
struggle for existence. In these theories we start by formulating the
dependence of one competitor on another in a verbal form, then translate
this formulation into mathematical language and obtain differen-
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 33
tial equations of the struggle for existence, which enable us to draw
definite conclusions about the course and the results of competition.
Therefore all the value of the ulterior deductions depends on the
question whether certain fundamental premises have been correctly
formulated. Consequently before proceeding any further to consider
more complicated mathematical equations of the struggle for existence
we must with the greatest attention, relying upon the experimental
data already accumulated, decide the following question:
what are the premises we have a right to introduce into our differential
equations? As the problem of the struggle for existence is a
question of the growth of mixed populations and of the replacement
of some components by others, we ought at once to examine this
problem : what is exactly known about the multiplication of animals
and the growth of their homogeneous populations?
Of late years among ecologists the idea has become very wide
spread that the growth of homogeneous populations is a result of the
interaction of two groups of factors : the biotic potential of the species
and the environmental resistance [Chapman '28, '31]. The biotic
potential1
represents the potential rate of increase of the species
under given conditions. It is realized if there are no restrictions of
food, no toxic waste products, etc. Environmental resistance can be
measured by the difference between the potential number of organisms
which can appear during a fixed time in consequence of the potential
rate of increase, and the actual number of organisms observed
in a given microcosm at a determined time. Environmental resistance
is thus expressed in terms of reduction of some potential rate of
increase, characteristic for the given organisms under given conditions.
This idea is a correct one and it clearly indicates the essential
factors which are operating in the growth of a homogeneous population
of organisms.
However, as yet among ecologists the ideas of biotic potential and
of environmental resistance are not connected with any quantitative
conceptions. Nevertheless Chapman in his interesting book Animal
Ecology arrives at the conclusion that any further progress here can
only be achieved on a quantitative basis, and that in future "this
direction will probably be one of the most important fields of biological
science, which will be highly theoretical, highly quantitative, and
highly practical."
1
What Chapman calls a "partial potential."
34 THE STRUGGLE FOR EXISTENCE
(2) There is no need to search for a quantitative expression of the
potential rate of increase and of the environmental resistance, as
this problem had already been solved by Verhulst in 1838 and quite
independently by Raymond Pearl and Reed in 1920. However,
ecologists did not connect their idea of biotic potential with these
classical works. The logistic curve, discovered by Verhulst and
Pearl, expresses quantitatively the idea that the growth of a population
of organisms is at every moment of time determined by the relation
between the potential rate of increase and "environmental resistance."
The rate of multiplication or the increase of the number
of organisms (N) per unit of time (t) can be expressed as —=-. The
rate of multiplication depends first on the potential rate of multiplication
of each organism (6), i.e., on the potential number of offspring
which the organism can produce per unit of time. The total potential
number of offspring that can be produced by all the organisms per
unit of time can be expressed as the product of the number of organisms
(N) and the potential increase (6) from each one of them, i.e.,
bN. Therefore the potential increase of the population in a certain
infinitesimal unit of time will be expressed thus:
dN
This expression represents a differential equation of the population
growth which would exist if all the offspring potentially possible were
produced and actually living. It is an equation of geometric increase,
as at every given moment the rate of growth is equal to the number
of organisms (N) multiplied by a certain constant (b).
As has been already stated, the potential geometrical rate of population
growth is not realized, and its reduction is due to the environmental
resistance. This idea was quantitatively expressed by Pearl
in such a form that the potential geometric increase at every moment
of time is only partially realized, depending on how near the already
accumulated size of the population (N) approaches the maximal popution
(K) that can exist in the given microcosm with the given level
of food resources, etc. The difference between the maximally possible
and the already accumulated population (K — N), taken in a relative
form, i.e., divided by the maximal population (
——— J,
shows
m
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 35
the relative number of the "still vacant places" for definite species in a
given microcosm at a definite moment of time. According to the
number of the still vacant places only a definite part of the potential
rate of increase can be realized. At the beginning of the population
growth when the relative number of unoccupied places is considerable
the potential increase is realized to a great extent, but when the already
accumulated population approaches the maximally possible or
saturating one, only an insignificant part of the biotic potential will
be realized (Fig. 3). Multiplying the biotic potential of the population
(bN) by the relative number of still vacant places or its "degree
K — Nof realization" —=—, we shall have the increase of population per
infinitesimal unit of time:
Geometric increase
Saturating population
Holistic curve
Unutilised opportunity
for growth
Time
Fig. 3. The curve of geometric increase and the logistic curve
Rate of growth
Potential
increase of
or increase perf = ^population
unit of time
Degree of realization
of the potential in"
X i crease. Depends on
per unit of the number of still
time J vacant places.
Expressing this mathematically we have
_ = W_-
..(8)
(9)
This is the differential equation of the Verhulst-Pearl logistic
curve.2
(3) Before going further we shall examine the differential form
2
It is to be noted that we have to do in all the cases with numbers of individuals
per unit of volume or area, e.g., with population densities (N).
36 THE STRUGGLE FOR EXISTENCE
of the logistic curve in a numerical example. Let us turn our attention
to the growth of a number of individuals of an infusorian, Paramecium
caudatum, in a small test tube containing 0.5 cm3
of nutritive
medium (with the sediment; see Chapter V). The technique of
experimentation will be described in detail further on. Five individuals
of Paramecium (from a pure culture) were placed in such a
microcosm, and for six days the number of individuals in every tube
was counted daily. The average data of 63 separate counts are
given in Figure 4. This figure shows that the number of individuals
in the tube increases, rapidly at first and then more slowly, until
towards the fourth day it attains a certain maximal level saturating
2 3 V S 6
Days
Fig. 4. The growth of population of Paramecium caudatum
the given microcosm. The character of the curve should be the same
if we took only one mother cell at the start. Indeed, if one Paramecium
is isolated and its products segregated as a pure culture, the
generation time of each cell is not identically the same as that of its
neighbors, and consequently at any given moment some cells are
dividing, whereas the others are at various intermediate stages of the
reproductive cycle; it is, however, no longer possible to divide the
population up into permanent categories, since a Paramecium which
divides rapidly tends to give rise to daughter cells which divide slowly,
and vice versa. The rate of increase of such a population will be
determined by the percentage of cells actually dividing at any instant,
and the actual growth of the population can be plotted as a smooth
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 37
curve, instead of a series of points restricted to the end of each reproductive
period. The smooth curve of Figure 4 is drawn according
to the equation of the logistic curve, and its close coincidence with
the results of the observations shows that the logistic curve represents
a good empirical description of the growth of the population. The
practical method of fitting such an empirical curve will also be considered
further (Appendix II). The question that interests us just
now is this: what is, according to the logistic curve, the potential
rate of increase of Paramecium under our conditions, and how does it
become reduced in the process of growth as the environmental resistance
increases?
According to Figure 4, the maximal possible number of Paramecia
in a microcosm of our type, or the saturating population, K = 375
individuals. As a result of the very simple operation of fitting the
logistic curve to the empirical observations, the coefficient of multiplication
or the biotic potential of one Paramecium (b) was found.
It is equal to 2.309. This means that per unit of time (one day)
under our conditions of cultivation every Paramecium can potentially
give 2.309 new Paramecia. It is understood that the coefficient b is
taken from a differential equation and therefore its value automatically
obtained for a time interval equal to one day is extrapolated from
a consideration of infinitesimal sections of time. This value would
be realized if the conditions of an unoccupied microcosm, i.e., the
absence of environmental resistance existing only at the initial moment
of time, existed during the entire 24 hours. It is automatically
taken into account here that if at the initial moment the population
increases by a certain infinitesimal quantity proportional to this
population, at the next moment the population plus the increment
will increase again by a certain infinitesimal quantity proportional
no longer to the initial population, but to that of the preceding moment.
The coefficient b represents the rate of increase in the absence
of environmental resistance under certain fixed conditions. At
another temperature and under other conditions of cultivation the
value b will be different. Table IV gives the constants of growth of
the population of Paramecia calculated on the basis of the logistic
curve. There is shown N or the number of Paramecia on the first,
second, third and fourth days of growth. These numbers represent
the ordinates of the logistic curve which passes near the empirical
observations and smoothes certain insignificant deviations. The
38 THE STRUGGLE FOR EXISTENCE
values bN given in Table IV express the potential rate of increase of
the whole population at different moments of growth, or the number
of offspring which a given population of Paramecia can potentially
produce within 24 hours at these moments. We must repeat here
what has been already said in calculating the value b. The potential
rate bN exists only within an infinitesimal time and should these conditions
exist during 24 hours the values shown in Table IV would be
TABLE IV
The growth of population of Paramecium caudatum
b (coefficient of multiplication; potential progeny per individual per day)
2.309.
K (maximal population) = 375.
AT (number of individuals according to the
logistic curve)
bN (potential increase of the population per
day)
K-N
(degree of realization of the potential
K
increase)
K-N . , .
1 — —=- (environmental resistance)
dN K-N r ,
'
,
— = bN —77~ (rate of growth of the popudt
A
lation)
dN
hN
~Jt
dN
dt
istence)
(intensity of the struggle for exTIME
IN DATS
20.4
47.1
0.945
0.055
44.5
0.058
137.2
316.8
0.633
0.367
200.0
0.584
319.0
736.6
0.149
0.851
109.7
5.72
369.0
852.0
0.016
0.984
13.6
61.7
K-Nobtained. The expression —=— shows the relative number of yet
unoccupied places. At the beginning of the population growth when
K — NN is very small the value —r-— approaches unity. In other words
the potential rate of growth is almost completely realized. As the
K M
population grows, —^— approaches zero. The environmental re-
K
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 39
sistance can be measured by that part of the potential increase which
has not been realized—the greater the resistance the larger the unK
— N
realized part. This value can be obtained by subtracting —=— from
unity. At the beginning of growth the environmental resistance is
mm* K — Nsmall and 1 ^— approaches zero. As the population increases
A
MaY*A*
&
Rate ofgrowths <p/ Environmental
y
A/Degree of realization of
S thepotential fncreai
Intensityofthe
struggle for
existence
Ddys
Fig. 5. The characteristics'of competitiorfin'ajhomogeneous population of
Paraviecium caudatum.
K - Nthe environmental resistance increases also and 1— ==— ap-
K
proaches unity. This means that the potential increase remains
K — N
almost entirely unrealized. Multiplying bN by =— for a given
dN
moment we obtain a rate of population growth -=—, which increases
at first and then decreases. The corresponding numerical data are
given in Table IV and Figure 5.
40 THE STRUGGLE FOR EXISTENCE
(4) We must now analyze a very important principle which was
clearly understood by Darwin, but which is still waiting for its rational
quantitative expression. I mean the intensity of the struggle
for existence between individuals of a given group.3
The intensity of
the struggle for existence is measured by the resistance which must be
overcome in order to increase the number of individuals by a unit at a
given moment of time. As we measure the environmental resistance
by the eliminated part of the potential increase, our idea can be
formulated thus : what amount of the eliminated fraction of the potential
increase falls upon a unit of the realized part of the increase at
a given moment of time? The intensity of the struggle for existence
keeps constant only for an infinitesimal time and its value shows
with what losses of the potentially possible increment the establishment
of a new unit in the population is connected. The realized
value of increase at a given moment is equal to
dN ,„K-N
Tt=
hN
-ir~
and the unrealized one:
,,,/, K-N\ ... .
AT K-N ...
bN ( 1 =— 1 = bN — bN —=— = bN — dN
dt
'
Then the amount of unrealized potential increase per unit of realized
increase, or the intensity of the struggle for existence (i), will be
expressed thus:
unrealized part of the . .. dN
potential increase dt ,„^
i = —- = (10)
realized part of the dN
potential increase dt
3
As we have seen in Chapter II, botanists are beginning to deal with the
intensity of the struggle for existence, simply characterizing it by the per cent
of destroyed individuals. Haldane investigating the connection of the intensity
of competition with the intensity of selection ('31) and in his interesting
book The Causes of Evolution ('32) specifies the intensity of competition by Z
and determines it as the proportion of the number of eliminated individuals to
that of the surviving ones. Thus if the mortality is equal to 9 per cent, Z =
9/91, i.e., approximately 0.1.
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 41
The values of i for a population of Paramecia are given in Table
IV and we see that at the beginning of the population growth the
intensity of the struggle for existence is not great, but that afterwards
it increases considerably. Thus on the first day there are 0.058 "unrealized"
Paramecia for every one realized, but on the fourth day 61.7
"unrealized" ones are lost for one realized (Fig. 6). Figure 5 shows
graphically the changes of all the discussed characteristics in the
course of the population growth of Paramecia.
(5) The intensity of the struggle for existence can evidently be expressed
in this form only in case the population grows, i.e., if the number
of individuals increases continually. If growth ceases the popu-
dN
lation is in a state of equilibrium, and the rate of growth -^- = 0;
X0.0S8 per
one
First day
Fourth day
Fig. 6. Intensity of the struggle for existence in Paramecium caudatum. On
the first day of the growth of the population 0.058 "unrealized" Paramecia are
lost per one realized, but on the fourth day 61.7 per one.
in this case the expression of intensity will take another form. Population
in a state of equilibrium represents a stream moving with a
certain rapidity: per unit of time a definite number of individuals
perishes, and new ones take their places. The number of these
liberated places is not large if compared with the number of organisms
that the population can produce in the same unit of time according
to the potential coefficients of multiplication. Therefore a considerable
part of the potentially possible increase of the population
will not be realized and liberated places will be occupied only by a
very small fraction of it. If the potential increase of a population in
the state of equilibrium per infinitesimal unit of time is &iiVi, and a
42 THE STRUGGLE FOR EXISTENCE
certain part (77) of this increase takes up the liberated places, it is
evident that a part (1 — 77) will remain unrealized. The mechanism
of this "nonrealization" is of course different in different animals.
Then as before the intensity of the struggle for existence, or the proportion
of the unrealized part of the potential increase to the realized
part, will be:
(1 - 7?) bi Ni 1 - V nnn
77 Oi iVi 77
(6) We may say that the Verhulst-Pearl logistic curve expresses
quantitatively and very simply the struggle for existence which takes
place between individuals of a homogeneous group. Further on we
shall see how complicated the matter becomes when there is competition
between individuals belonging to two different species. But
one must not suppose that an intragroup competition is a very simple
thing even among unicellular organisms. Though in the majority of
cases the symmetrical logistic curve with which we are now concerned
expresses satisfactorily the growth of a homogeneous population,
certain complicating factors often appear and for some species the
curves are asymmetrical, i.e., their concave and convex parts are not
similar. Though this does not alter our reasoning, we have to take
into account a greater number of variables which complicate the
situation. Here we agree completely with the Russian biophysicist
P. P. Lasareff ('23) who has expressed on this subject, but in connection
with other problems, the following words: "For the development
of a theory it is particularly advantageous if experimental methods
and observations do not at once furnish data possessing a great degree
of accuracy and in this way enable us to ignore a number of secondary
accompanying phenomena which make difficult the establishment of
simple quantitative laws. In this respect, e.g., the observations of
Tycho Brahe which gave to Kepler the materials for formulating his
laws, were in their precision just sufficient to characterize the movement
of the planets round the sun to a first approximation. If on the
contrary Kepler had had at his disposal the highly precise observations
which we have at present, then certainly his attempt to find an
empirical law owing to the complexity of the whole phenomenon
could not have led him to simple and sufficiently clear results, and
would not have given to Newton the material out of which the theory
of universal gravitation has been elaborated.
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 43
"The position of sciences in which the methods of experimentation
and the theory develop hand in hand is thus more favorable than the
position of those fields of knowledge where experimental methods far
outstrip the theory, as is, e.g., observed in certain domains of experimental
biology, and then the development of the theory becomes more
difficult and complicated" (p. 6).
Therefore we must not be afraid of the simplicity of the logistic
curve for the population of unicellular organisms and criticize it from
this point of view. At the present stage of our knowledge it is just
sufficient for the rational construction of a theory of the struggle for
existence, and the secondary accompanying circumstances investigators
will discover in their later work.4
(7) The application of quantitative methods to experimental
biology presents such difficulties, and has more than once led to such
erroneous results that the reader would have the right to consider
very sceptically the material of this chapter. It is very well known
that the differential equations derived from the curves observed in
an experiment can be only regarded as empirical expressions and they
do not throw any real light on the underlying factors which control
the growth of the population. The only right way to go about the
investigation is, as Professor Gray ('29) says, a direct study of factors
which control the growth rate of the population and the expression of
these factors in a quantitative form. In this way real differential
equations will be obtained and in their integrated form they will
harmonize with the results obtained by observation. In our experimental
work described in the next chapter special attention has been
given to a direct study of the factors controlling growth in the simplest
populations of yeast cells. It has become evident that the value
4
The usual objection to the differential logistic equation is that it is too
simple and does not reflect all the "complexity" of growth of a population of
lower organisms. In Chapter V we shall see that this remark is to a certain
extent true for some populations of Protozoa. The realization of the biotic
potential in certain cases actually does not gradually diminish with the decrease
in the unutilized opportunity for growth. But this ought not to
frighten any one acquainted with the methodology of modern physics: it is
K — Nevident that the expression ——— is but a first approximation to what actually
K.
exists, and, if necessary, it can be easily generalized by introducing before N a
certain coefficient, which would change with the growth of the culture (for
further discussion see Chapter V).
44 THE STRUGGLE FOR EXISTENCE
of the environmental resistance which we have determined on the basis of
a purely 'physiological investigation coincides completely with the value
of the environmental resistance calculated according to the logistic equation,
using the latter as an empirical expression of growth. In this
way we have proved that the logistic equation actually expresses the
mechanism of the growth of the number of unicellular organisms
within a limited microcosm. All this will be described in detail in
the next chapter.
in
(1) We are now sufficiently prepared for the acquaintance with the
mathematical equations of the struggle for existence, and for a critical
consideration of the premises implied in them. Let us consider first
of all the case of competition between two species for the possession
of a common place in the microcosm. This case was considered
theoretically for the first time by Vito Volterra in 1926. An experimental
investigation of this case was made by Gause ('32b), and at
the same time Lotka ('32b) submitted it to a further analysis along
theoretical lines.
If there is competition between two species for a common place in
a limited microcosm, we can quite naturally extend the premises implied
in the logistic equation. The rate of growth of each of the
competing species in a mixed population will depend on (1) the potential
rate of population increase of a given species (hNi or b2 N2 )
and (2) on the unutilized opportunity for growth of this species, just
as in the case of a population of the first and second species growing
separately. But unutilized opportunity for growth of a given species
in a mixed population is a complex variable. It measures the number
of places which are still vacant for the given species in spite of
the presence of another species, which is consuming the common food,
excreting waste products and thereby depriving the first one of some
of the places. Let us denote as before by Ni the number of individuals
of the first species, though, as we shall see further on, we shall
have to deal in many cases not with the numbers of individuals but
with masses of species ( = the weight of the organisms present or its
equivalent) and we shall introduce corresponding alterations. The
unutilized opportunity for growth, or the degree of realization of the
potential increase for the first species in a mixed population, may be
expressed thus: — *
—-, where Ky is the maximal possible
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 45
number of individuals of this species when grown separately under
given conditions, Ni is the already accumulated number of individuals
of the first species at a given moment in the mixed population, and m
is the number of the places of the first species in terms of the number
of individuals of this species, which are taken up by the second species
at a given moment. The unutilized opportunity for growth of the
first species in the mixed population can be better understood if we
compare it with the value of the unutilized opportunity for the separate
growth of the same species. In the latter case the unutilized
opportunity for growth is expressed by the difference (expressed in a
relative form) between the maximal number of places and the number
of places already occupied by the given species. Instead of this for
the mixed population we write the difference between the maximal
number of places and that of the places already taken up by our
species together with the second species growing simultaneously.
(2) An attempt may be made to express the value m directly by
the number of individuals of the second species at a given moment,
which can be measured in the experiment. But it is of course unlikely
that in nature two species would utilize their environment in
an absolutely identical way, or in other words that equal numbers of
individuals would consume (on an average) equal quantities of food
and excrete equal quantities of metabolic products of the same chemical
composition. Even if such cases do exist, as a rule different species
do not utilize the environment in the same way. Therefore the
number of individuals of the second species accumulated at a given
moment of time in a mixed population in respect to the place it
occupies, which might be suitable for the first species, is by no means
equivalent to the same number of individuals of the first species.
The individuals of the second species have taken up a certain larger
or smaller place. If N2 expresses the number of individuals of the
second species in a mixed population at a given moment, than the
places of the first species which they occupy in terms of the number of
individuals of the first species, will be m = aN2 . Thus, the coefficient
a is the coefficient reducing the number of the individuals of the
second species to the number of places of the first species which they
occupy. This coefficient a shows the degree of influence of one
species upon the unutilized opportunity for growth of another. In
fact, if the interests of the different species do not clash and if in the
microcosm they occupy places of a different type or different "niches"
then the degree of influence of one species on the opportunity for
46 THE STRUGGLE FOR EXISTENCE
growth of another, or the coefficient a, will be equal to zero. But if
the species lay claim to the very same "niche," and are more or less
equivalent as concerns the utilization of the medium, then the coefficient
a will approach unity. And finally if one of the species utilizes
the environment very unproductively, i.e., if each individual consumes
a great amount of food or excretes a great quantity of waste
products, then it follows that an individual of this species occupies as
large a place in the microcosm as would permit another species to produce
many individuals, and the coefficient a will be large. In other
words an individual of this species will occupy the place of many
individuals of the other species. If we remember here the specificity
of the metabolic products, and all the very complex relations which
can exist between the species, we shall understand how useful we
may find the coefficient of the struggle for existence a, which objectively
shows how many places suitable for the first species are occupied
by one individual of the second.
Taking the coefficient a, we can now express in the following manner
the unutilized opportunity for growth of the first species in a
mixed population : — *
—. The unutilized opportunity
for growth of the second species in a mixed population will have a
similar expression :
— -J
—• The coefficient of the strug-
A2
gle for existence |8 indicates the degree of influence of every individual
of the first species on the number of places suitable for the life of the
second species. These two expressions enable one to judge in what
degree the potential increase of each species is realized in a mixed
population.
(3) As we have already mentioned in analyzing the Ross equations,
an important feature of mixed populations is the simultaneous influence
upon each other of the species constituting them. The rate of
growth of the first species depends upon the number of places already
occupied by it as well as by the second species at a given moment.
As growth proceeds the first species increases the number of places
already occupied, and thus affects the growth of the second species as
well as its own. We can introduce the following notation:
—^, —r^- = rates of growth of the number of individuals of the
first and second species in a mixed population at a
given moment.
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 47
JVi, N2 = number of individuals of the first and second species
in a mixed population at a given moment.
bi, b2 = potential coefficients of increase in the number of individuals
of the first and second species.
Ki, K2 = maximal numbers of individuals of the first and second
species under the given conditions when separately
grown.
a, 13 = coefficients of the struggle for existence.
The rate of growth of the number of individuals of the first species
in a mixed population is proportional to its potential rate (&iiVi),
which in every infinitesimal time interval is realized in greater or less
degree depending on the relative number of the still vacant places:
— ^ —. An analogous relationship holds true for the
second species. The growth of the first and second species is simultaneous.
It can be expressed by the following system of simultaneous
differential equations:
Rate of growth
of the first species
in a mixed
population
> = <
Potential increase
of the
population of
the first species.
>X
Rate of growth
of the second
species in a
mixed popula-
tion
Potential increase
of the
population of
the second spe-
cies
X<
Degree of realization
of the potential
increase.
Depends on the
number of still
vacant places.
Degree of realization
of the potential
increase.
Depends on the
number of still
vacant places j )
• (ID
Translating this into mathematical language we have:
dN1
= lhNi
Kt-(N1 + aNd)
dt
dN2
= b2 N2
K2 - (Nt + ffli)
(12)
dt K2
The equations of the struggle for existence which we have written
48 THE STRUGGLE FOR EXISTENCE
express quantitatively the process of competition between two species
for the possession of a certain common place in the microcosm. They
are founded on the idea that every species possesses a definite potential
coefficient of multiplication but that the realization of these
potentialities (biNi and b2N2 ) of two species is impeded by four processes
hindering growth: (1) in increasing the first species diminishes
its own opportunity for growth (accumulation of Ni), (2) in increasing
the second species decreases the opportunity for growth of the
first species (aNz), (3) in increasing the second species decreases its
own opportunity for growth (accumulation of N2), and (4) the increase
of the first species diminishes the opportunity for growth of
the second species (/3iVi). Whether the first species will be victorious
over the second, or whether it will be displaced by the second
depends, first, on the properties of each of the species taken separately,
i.e., on the potential coefficients of increase in the given conditions
(&i, 62), and on the maximal numbers of individuals (K\, K2).
But when two species enter into contact with one another, new coefficients
of the struggle for existence a and /3 begin to operate. They
characterize the degree of influence of one species upon the growth of
another, and participate in accordance with the equation (12) in
producing this or that outcome of the competition.
(4) It is the place to note here that the equation (12) as it is written
does not permit of any equilibrium between the competing species
occupying the same "niche," and leads to the entire displacing of one
of them by another. This has been pointed out by Volterra ('26),
Lotka ('32b) and even earlier by Haldane ('24), and for the experimental
confirmation and a further analysis of this problem the reader
is referred to Chapter V. We can only remark here that this is
immediately evident from the equation (12). The stationary state
dNi idN2 ., u .,, ., (dNx dN2 \
occurs whenever —z— and —3— both vanish together I —z— = —7— = CM,
and the mathematical considerations show that with usual a and /3
there cannot simultaneously exist positive values for both Nitao
and N2 , w. One of the species must eventually disappear. This
apparently harmonizes with the biological observations. As we
have pointed out in Chapter II, both species survive indefinitely only
when they occupy different niches in the microcosm in which they have
an advantage over their competitors. Experimental investigations
of such complicated systems are in progress at the time of this writing.
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 49
(5) We have just discussed a very important set of equations of
the competition of two species for a common place in the microcosm,
and it remains to make in this connection a few historical remarks.
Analogous equations dealing with a more special case of competition
between two species for a common food were for the first time given
in 1926 by the Italian mathematician Vito Volterra who was not
acquainted with the investigations of Ross and of Pearl.
Volterra assumed that the increase in the number of individuals
dN
obeys the law of geometric increase : —r- = bN, but as the number of
individuals (N) accumulates, the coefficient of increase (6) diminishes
to a first approximation proportionally to this accumulation (b — X2V),
where X is the coefficient of proportionality. Thus we obtain
d
-£ =(b-\N)N (13)
at
It can be easily shown, as Lotka ('32) remarks, that the equation
of Volterra (13) coincides with the equation of the logistic curve of
Verhulst-Pearl (9). In fact, if we call the rate of growth per individual
a relative rate of growth and denote it as : -^ -r-, then the
equation (13) will have the following form:
i.^ = 6_XiV (14)
N dt
This enables us to formulate the equation (13) in this manner: the
relative rate of growth represents a linear function of the number
of individuals N, as b — X N is the equation of a straight line. If
we now take the equation of the Verhulst-Pearl logistic curve (9):
dN K — N— = bN —=—, and make the following transformations:
dt K.
dN .»
(
l
-*"} rf = ^(i-^4 weshaIlhave:
1 dN , & ,
7
,
1ex
N-Tt= b
~K N (15)
50 THE STRUGGLE FOR EXISTENCE
In other words the logistic curve possesses the property that with
an increase in the number of individuals the relative rate of growth
decreases linearly (this has been recently mentioned by Winsor ('32)).
Consequently the expression (13) according to which we must subtract
from the coefficient of increase b sl certain value proportional to
the accumulated number of individuals in order to obtain the rate of
growth, and the expression (9), according to which we must multiply
the geometric increase bN by a certain "degree of its realization,"
coincide with one another. Both are based on a broad mathematical
assumption of a linear relation between the relative rate of growth and
the number of individuals. Volterra extended the equation (13) to
the competition of two species for common food, assuming that the
presence of a certain number of individuals of the first species (Ni)
decreases the quantity of food by hiNi, and the presence of N2 individuals
of the second species decreases the quantity of food by h2 N2 .
Therefore, both species together decrease the quantity of food by
hiNi + h2 N2 , and the coefficient of multiplication of the first species
decreases in connection with the diminution of food
6i - Xi (hiNt + h2 N2 ) (16)
But for the second species the degree of influence of the decrease of
food on the coefficient of multiplication b2 will be different (X2 ), and
we shall obtain:
b2 - X2 (hNi + h2N2) (17)
Starting from these expressions Volterra ('26) wrote the following
simultaneous differential equations of the competition between
two species for common food:
d
^ = fo-lufcNi + hNMNi
dN2
dt
= [b2 - XtQitNt + faN^Ns
(18)
These equations represent, therefore, a natural extension of the principle
of the logistic curve, and the equation (12) written by Gause
('32b) coincides with them. Indeed the equation (12) can be transformed
in this manner:
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 51
dt
eft
[*-*(£ n2 + |i
2 A2
(19)
The result of the transformation shows that the equation (12) coincides
with Volterra's equation (18), but it does not include any parameters
dealing with the food consumption, and simply expresses the
competition between species in terms of the growing populations
themselves. As will be seen in the next chapter, the equation (12)
is actually realized in the experiment.
IV
(1) In the present book our attention will be concentrated on an
experimental study of the struggle for existence. In this connection
we are interested only in those initial stages of mathematical researches
which have already undergone an experimental verification.
At the same time we are writing for biological readers and we would
not encumber them by too numerous mathematical material. All
this leads us to restrict ourselves to an examination of only a few
fundamental equations of the struggle for existence, referring those
who are interested in mathematical questions to the original investigations
of Volterra, Lotka and others.
We shall now consider the second important set of equations of the
struggle for existence, which deals with the destruction of one species
by another. The idea of these equations is very near to those of
Ross which we have already analyzed. They were given for the
first time by Lotka ('20b) and independently by Volterra ('26).
After the previous discussion these equations ought not to present
any difficulties. Let us consider the process of the prey iVi being
devoured by another species, the predator iV2 . We can put it in a
general form:
52 THE STRUGGLE FOR EXISTENCE
Change in the) fNatural increase)
number of prey > = < of prey per unit f
per
unit of time.
Change in the]
I number of prejdators
per unit
[of time
of time J
Increase in the
number of predators
per unit
of time resulting
from the devouring
of the prey,
Destruction of
the prey by the
predators per
unit of time
Deaths of the
predators per
unit of time
(20)
We can introduce here the following notation:
dNt
dt
= rate of increase of the number of prey.
61 = coefficient of natural increase of prey (birth rate
minus death rate).
biNi = natural increase of the number of prey at a given
moment.
/1 (Ni, N2 ) = the function characterizing the consumption of prey
by predators per unit of time. This is the greater
the larger is the number of predators (N2 ) and the
larger is the number of the prey themselves (iVi).
dN2
dt
= rate of increase of the number of predators.
F (Ni, N2) = the function characterizing simultaneously the natality
and the mortality of predators.
We can now translate the equations (20) into mathematical language
by writing:
dNi
dt
dNj
dt
= biNi - MNlf #2)
= F(Nl ,NJ
(21)
In a particular case investigated by Volterra in detail, the functions
in these equations have been somewhat simplified. He put /1 (JYi, N2)
= kiNzNi, e.g., the consumption of prey by predators is directly proportional
to the product of their concentrations. Also F(Ni, N2 )
=
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 53
k2N2 Ni — ckN2 . Here k2N2Ni is the increase in the number of predators
resulting from the devouring of the prey per unit of time, and
d2N2
— number of predators dying per unit of time {d2 is the coefficient
of mortality). This translation of (20) gives
*£ = b1 N1 -k1 N2 N1
at
= k2 N2 Ni — d2 N2
dN,
dt
(21a)
These equations have a very interesting property, namely the
periodic solution, which has been discovered by both Lotka ('20)
and Volterra ('26). As the number of predators increases the prey
diminish in number,5
but when the concentration of the latter becomes
small, the predators owing to an insufficiency of food begin to
decrease.6
This produces an opportunity for growth of the prey,
which again increases in number.
(2) In our discussion up to this point we have noted how the process
of interaction between predators and prey can be expressed in a
general form covering a great many special cases (equation 21), and
how this general expression can be made more concrete by introducing
certain simple assumptions (equation 21a). There is no doubt
that we shall not obtain any real insight into the nature of these
processes by further abstract calculations, and the reader will have
to wait for Chapter VI where the discussion is continued on the
sound basis of experimental data.
Let us better devote the remainder of this chapter to two rather
special problems of the natural increase of both predators and prey
in a mixed culture simply in order to show how the biological reasoning
can be translated into mathematical terms.
In the general form the rate of increase in the number of individuals
of the predatory species resulting from the devouring of the prey
dN2
dt
B
When the number of predators (N2 ) is considerable the number of the prey
devoured per unit of time (k l NiN2 ) is greater than the natural increase of the
prey during the same time (+&i2Vi), and (biNi — kiNiN2 ) becomes a negative
value.
6
With a small number of prey (Ni) the increase of the predators owing to
the consumption of the prey (+k2 NiN2 ) is smaller than the mortality of the
predators ( — dN2 ), and (k 2 NiN2 — dN2 ) becomes a negative value.
54 THE STRUGGLE FOR EXISTENCE
can be represented by means of a certain geometrical increase which
is realized in proportion to the unutilized opportunity of growth.
This unutilized opportunity is a function of the number of prey at a
given moment: f(Ni). Therefore,
dt
= b2 N2 f(Nd (22)
The simplest assumption would be that the geometric increase in
the number of predators is realized in direct proportion to the number
of prey (XiVi) . Were our system a simple one we could say with
Lotka and Volterra that the rate of growth might be directly connected
with the number of encounters of the second species with the
first. The number of these encounters is proportional to the number
of individuals of the second species multiplied by the number of individuals
of the first (aNiN2 ), where a is the coefficient of proportionality.
If it were so, the increase of the number of predators would
be in direct proportion to the number of the prey. Indeed, if the
number of the prey iV\ has doubled and is 2Ni, the number of their
encounters with the predators has also doubled, and instead of being
aNiNz is equal to aN22Ni. Consequently the increase in the number
of predators instead of the former b2N2 \Ni would become equal to
dN2
dt
= b2N2 \ 2Ni, and the relative increase (per predator) would be
therefore: -=—r— = b2\ 2Ni. In other words, the relative increase
N2 dt
— -j^- would be a rectilinear function of the number of prey Nh i.e.,
JS 2 dt
with a rise of the concentration of the prey the corresponding values
of the relative increase of the predators could be placed on a straight
line (ab in Fig. 7) . But experience shows the following : If we study
the influence of the increase in the number of the prey per unit of
volume upon the increase from one predator per unit of time, we will
find that this increase rises at first rapidly and then slowly, approaching
a certain fixed value. A further change in density of the prey
does not call forth any rise in the increase per predator. In the limits
which interest us we can express this relationship with the aid of a
curve rapidly increasing at first and then approaching a certain
asymptote. Such a curve is represented in Figure 7 (ac). The concentration
of prey (Ni) is marked on the abscissae, and the relative
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 55
'
1 dN2\
increase in the number of predators ( ^ ^^ J, or the rate of growth
VV2 at /
per predator at different densities of prey, is marked on the ordinates.
The curve connecting the relative increase of the predators with
the concentration of the prey can be expressed by the equation:
y = a (1 — e~iz
),
7
which in our case takes the following form:
N2 dt
= b2 (l - e~™i) (23)
The properties of this curve are such that as the prey becomes
Site
56 THE STRUGGLE FOR EXISTENCE
was justified in the case of the competition for common food discussed
before, we now take a further step and express a non-linear
relation of the increase per predator with the concentration of the
prey. All this will be easier to understand when in Chapter VI we
pass on the analysis of the experimental material. We shall then
explain also the meaning of the coefficient X in the equation (23).
(3) The question now arises as to how to express the natural increase
of the prey. As noticed already, the growth of the prey in a
limited microcosm in the absence of predators can be expressed in the
form of a potential geometric increase biNi, which at every moment
of time is realized in dependence on the unutilized opportunity for
growth — ?
—-. Therefore, the natural increase in the number of
prey per unit of time can be expressed thus
_ = 61
Ar1 _^__.
It is easy to see that in the presence of the predator devouring the
prey the expression of the unutilized opportunity for growth of the
latter will take a more complex form. The unutilized opportunity
for growth will, as before, be expressed by the difference (taken in a
relative form) between the maximal number of places which is possible
under given conditions (K{), and the number of places already
occupied. But the number of prey (Ar
i) which in the presence of the
predator exist at the given moment, does not reflect the number of
the "already occupied places." In fact the prey which have been
devoured by the predator have together with the actually existing
prey participated in the utilization of the environment, i.e., consumed
the food and excreted waste products. Therefore, the degree of
utilization of the environment is determined by the total of the present
population (JVi) and that which has been devoured (n). 8
The
expression of the unutilized opportunity for growth of the prey will
have then the following form:
8
These calculations are true only in the case of the competition for a certain
limited amount of energy (see Chapter V). In other words it is assumed that
the nutritive medium is not changed in the course of the experiment. With
change of the medium at short intervals (as discussed in Chapter V) the term
n disappears, and the situation becomes more simple.
STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 57
K, - (Nr + n)
Kx
(25)
Here we arrive at a very interesting conclusion, namely, that the
development of a definite biological system is conditioned not only by its
state at a given moment, but that the past history of the system exerts a
powerful influence together with its present state. This fact is apparently
very widespread, and one can read about it in a general form
in almost every manual of ecology. Lotka and Volterra expressed
mathematically the role of this circumstance in the processes of the
struggle for existence.
In order to complete the consideration we have to express the number
of the devoured prey from the moment the predator is introduced
up to the present time. If in every infinitesimal time interval the
predator devours a certain definite number of prey then the total
number of the devoured prey will be equal to the sum of the elementary
quantities devoured from the moment the predator is introduced
up to the given time (t) . This total can be apparently expressed by
a definite integral.
(4) We cannot at present ignore the difficulties existing in the field
of the mathematical investigation of the struggle for life. We began
this chapter with comparatively simple equations dealing with an
idealized situation. Then we had to introduce one complication
after another, and finally arrived at rather complicated expressions.
But we had in view populations of unicellular organisms with an
immense number of individuals, a short duration of generations and a
practically uniform rate of natality. The phenomena of competition
are reduced here to their simplest. What enormous difficulties we
shall, therefore, encounter in attempts to find rational expressions for
the growth of more complicated systems.9
Is it worth while, on
the whole, to follow this direction of investigation any further?
There is but one answer to this question. We have at present no
other alternative than an analysis of the elementary processes of the
struggle for life under very simple conditions. Nothing but a veryactive
investigation will be able to decide in the future the problem
of the behavior of the complicated systems. Now we can only point
9
We can have an idea of this from the recent papers of Stanley ('32) and
Bailey ('33) who try to formulate the equations of the struggle for existence
for various insect populations.
58 THE STRUGGLE FOR EXISTENCE
out two principal conditions which must be realized in a mathematical
investigation of the struggle for existence in order to avoid serious
errors and the consequent disillusionment as to the very direction of
work. These conditions are: (1) The equations of the growth of
populations must be expressed in terms of the populations themselves,
i.e., in terms of the number of individuals, or rather of the biomass,
constituting a definite population. It must always be kept in mind
that even in such a science as physical chemistry it is only after the
course of chemical reactions has been quantitatively formulated in
terms of the reactions themselves, that the attempt has been made
to explain some of them on the ground of the kinetic theory of gases.
(2) The quantitative expression of the growth of population must go
hand in hand with a direct study of the factors which control growth.
Only in those cases, where the results deduced from equations are
confirmed by the data obtained through entirely different methods, by a
direct study of the factors limiting growth, can we be sure of the correctness
of the quantitative theories.
Chapter IV
ON THE MECHANISM OF COMPETITION IN YEAST CELLS
(1) No mathematical theories can be accepted by biologists without
a most careful experimental verification. We can but agree with
the following remarks made in Nature (H.T.H. P. '31) concerning the
mathematical theory of the struggle for existence developed by Vito
Volterra: "This work is connected with Prof. Volterra's researches
on integro-differential equations and their applications to mechanics.
In view of the simplifying hypothesis adopted, the results are not
likely to be accepted by biologists until they have been confirmed experimentally,
but this work has as yet scarcely begun." First of all,
very reasonable doubts may arise whether the equations of the
struggle for existence given in the preceding chapter express the
essence of the processes of competition, or whether they are merely
empirical expressions. Everybody remembers the attempt to study
from a purely formalistic viewpoint the phenomena of heredity by
calculating the likeness between ancestors and descendants. This
method did not give the means of penetrating into the mechanism of
the corresponding processes and was consequently entirely abandoned.
In order to dissipate these doubts and to show that the
above-given equations actually express the mechanism of competition,
we shall now turn to an experimental analysis of a comparatively
simple case. It has been possible to measure directly the factors
regulating the struggle for existence in this case, and thus to verify
some of the mathematical theories.
Generally speaking, biologists usually have to deal with empirical
equations. The essence of such equations is admirably expressed in
the following words of Raymond Pearl ('30) : "The worker in practically
any branch of science is more or less frequently confronted with
this sort of problem: he has a series of observations in which there
is clear evidence of a certain orderliness, on the one hand, and evident
fluctuations from this order, on the other hand. What he obviously
wishes to do ... is to emphasize the orderliness and minimize the
59
60 THE STRUGGLE FOR EXISTENCE
fluctuations about it. . . . He would like an expression, exact if
possible, or, failing that, approximate, of the law if there be one.
This means a mathematical expression of the functional relation
between the variables. . . .
"It should be made clear at the start that there is, unfortunately,
no method known to mathematics which will tell anyone in advance
of the trial what is either the correct or even the best mathematical
function with which to graduate a particular set of data. The choice
of the proper mathematical function is essentially, at its very best,
only a combination of good judgment and good luck. In this realm,
as in every other, good judgment depends in the main only upon
extensive experience. What we call good luck in this sort of connection
has also about the same basis. The experienced person in this
branch of applied mathematics knows at a glance what general class
of mathematical expression will take a course, when plotted, on the
whole like that followed by the observations. He furthermore knows
that by putting as many constants into his equation as there are
observations in the data he can make his curve hit all the observed
points exactly, but in so doing will have defeated the very purpose
with which he started, which was to emphasize the law (if any) and
minimize the fluctuations, because actually if he does what has been
described he emphasizes the fluctuations and probably loses completely
any chance of discovering a law.
"Of mathematical functions involving a small number of constants
there are but relatively few. ... In short, we live in a world which
appears to be organized in accordance with relatively few and relatively
simple mathematical functions. Which of these one will
choose in starting off to fit empirically a group of observations depends
fundamentally, as has been said, only on good judgment and
experience. There is no higher guide" (pp. 407-408).
(2) We are now confronted by an entirely different problem which
has often arisen in other domains of exact science and which represents
the next step after establishing the first empirical relations without
any mathematical theory. The problem is that from clearly
formulated hypotheses which appear probable on the ground of collected
experimental material certain mathematical consequences are deduced,
connecting the experimental values in equations accessible to experimental
verification. As a result a mathematical theory of the phenomena
observed in a given field of science is obtained. The equations of the
MECHANISM OF COMPETITION IN YEAST CELLS 61
struggle for existence are just such theoretical equations that have
been deduced from hypotheses about potential coefficients of multiplication
of species and the participation of these species in the utilization
of a limited opportunity for growth. The verification of such a
theoretical equation of the struggle for existence may be reduced to
the following: (1) we must determine experimentally the potential
coefficients of multiplication of the species; (2) by means of a direct
study of the factors limiting growth we must evaluate the degree of
influence of one species on the opportunity for growth of another,
i.e., the coefficients of the struggle for existence; (3) by inserting all
these values into a theoretical equation we must obtain a complete
agreement with the experimental data, if our mathematical theory
connects correctly the coefficients furnished by experimentation. It
seems to us that these three steps of verifying our theoretical equations
must be somewhat modified, taking into account the complicated
situation in the competition between two species for a common
place in the microcosm. We proceed as follows: (1) having determined
the potential coefficients of multiplication b\, b2 and the maximal
biomasses K\, K2 we pass on at once to (3), i.e., on the basis of the
experimental data, taking our equations as purely empirical expressions
or, in other terms, considering that they must describe the values
observed, we calculate those empirical coefficients of the struggle for
existence with which the equations actually describe the experimental
data. It is only then that we pass to (2), and compare these empirically
found coefficients of the struggle for existence with those which are
to be expected from a direct study of the factors limiting growth. If the
empirical coefficients coincide with the theoretical ones, the correctness of
the mathematical theory will be proved.
This mode of verification of the mathematical theory has been
adopted by us because the coincidence of theoretical coefficients with
the empirical ones is but rarely to be expected. Such a rare case
representing, most likely, rather an exception than a rule is described
in this chapter. This small probability of a coincidence of the coefficients
is connected with the fact that usually the growth of populations
depends on numerous factors, many of which (e.g., waste-products)
we often cannot specify exactly, and the influence of one species
on the opportunity of growth of another under these conditions is
realized in a very complicated manner. Hence the empirical coefficients
of the struggle for existence, calculated by an equation which
62 THE STRUGGLE FOR EXISTENCE
in certain cases has already been verified, can serve as a guide for the
study of the very mechanism of the influence of one species on the
growth of another.
ii
(1) To verify our differential equations of the struggle for existence
we had recourse to populations of yeast cells. Yeast cells were
cultivated in a liquid nutritive medium, where they were nourished
by various substances dissolved in water and excreted certain wasteproducts
into the surrounding medium. Owing to the considerable
practical importance of yeast for the food industry a great number of
papers has been devoted to investigation of its growth, and although
the majority deals with purely practical questions that do not at
present interest us, nevertheless it is pretty well ascertained what
substances yeast requires for its growth, and what is the chemical
composition of the waste-products it excretes.
For the study of competition we took two species of yeast: (1) a
pure line of common yeast, Saccharomyces cerevisiae stock XII,
received from the Berliner Gahrungsinstitut, and (2) a pure line of
the yeast Schizosaccharomyces kephir, cultivated in the Moscow Institute
of the Alcohol Industry and obtained from Dr. Pervozvansky.1
Both these species can grow under anaerobic conditions as well as
when oxygen is accessible. It is very well known that the processes
of life activity are connected with a continuous consumption of
energy which is supplied by certain chemical reactions. In the case
when the growth of yeast proceeds in the absence of oxygen it is the
decomposition of sugar into alcohol and carbon dioxide which fur-
1
We began our experiments with yeast in 1930. The first group of experiments
on competition between species was made in September-December, 1931,
and appeared in the Journal of Experimental Biology (Gause, '32b). These
experiments were extended and repeated in September-December, 1932. Their
results coincided completely with the data of 1931. Later it appeared that the
yeast culture kept in the Museum of the Institute of the Alcohol Industry
under the name of "Schizosaccharomyces kephir" and used under the same
name in our experiments, has been incorrectly determined by the specialists of
the Museum and that it belonged to another species. The culture consists of
oval, budding yeast cells much more minute than Saccharomyces cerevisiae and
producing an alcoholic fermentation. An exact systematic determination
presented extreme difficulty and seemed not to be indispensable, as this culture
is kept in the Museum and can be obtained thence under the name of "Schizosaccharomyces
kephir."
MECHANISM OF COMPETITION IN YEAST CELLS 63
nishes the available energy, and in the nutritive medium there takes
place a considerable accumulation of the waste product—ethyl alcohol.
If we alter the conditions of cultivation and allow a direct access
of oxygen to the growing yeast cells, although fermentation will still
continue, a part of the available energy (different for different species)
will be furnished by oxidation of sugar into carbon dioxide. In the
commercial utilization of yeast, when it is desirable to accumulate
alcohol in the culture, yeast is grown nearly without oxygen. But if
alcohol is not needed and the object is to obtain a great quantity of
yeast cells themselves, an intensive aeration of the growing culture
is carried on, which leads to an enormous increase of oxidation processes.
The yeast Saccharomyces cerevisiae as well as Schizosaccharomyces
kephir produces alcoholic fermentation, and both can obtain a
part of the available energy by oxidation, but they differ from one
another in the relative intensities of the oxidation and fermentation
processes. Common yeast, Saccharomyces cerevisiae, develops well
in the absence of oxygen as for it fermentation is a powerful source of
energy. It continues mainly to ferment even in the presence of
oxygen (when cultivated in Erlenmeyer flasks without aeration) and
utilizes the oxidation process only to a very small extent. As regards
our species of Schizosaccharomyces, it grows very slowly under anaerobic
conditions. However, when oxygen is available it has recourse
to this source of energy; its rapidity of growth increases and it approaches
Saccharomyces in its properties. Hence, Saccharomyces
represents a species with distinctly expressed fermentative capacities,
whilst Schizosaccharomyces is a species of a more oxidizing type. By
mixing these species we obtain a very interesting situation for studying
the competition between species in different conditions of environ-
ment.
(2) We cultivated yeast in a sterilized nutritive medium which was
prepared in the following manner: 20 gr. of dry pressed beer-yeast
were mixed with 1 liter of distilled water, boiled for half an hour in a
Kochs boiler, and then filtered through infusorial earth. Five per
cent of sugar was added to this mixture, and then the medium was
sterilized in an autoclave. A medium of such a type is very favorable
for the growth of yeast, because the decoction contains all the nutritive
substances required. The only disadvantage is our ignorance of
the exact chemical composition of this medium. Therefore each
series of experiments must be made with a solution of the very same
64 THE STRUGGLE FOR EXISTENCE
preparation. But on the whole this method enables one to have
sufficiently standardized conditions for cultivation.
The nutritive medium was sterilized in a large flask and then aseptically
poured into small vessels for cultivation. These vessels were
previously sterilized by dry heat (by heating to 180° for three hours).
This method has many advantages as compared with the direct
sterilization of the nutritive medium in small culture vessels. The
fact is that when a liquid is heated in glass vessels in an autoclave,
even if the best kind of glass be used, the latter can somewhat alter
the composition of the nutritive liquid. This produces a considerable
/0cm3
/0cm3
«m*v
Fig. 8. The vessels for cultivation of yeast: (a) test tube, (b) Erlenmeyer's
flask.
variation in the initial conditions of separate microcosms. The vessels
used for cultivation belonged to two types: (1) in experiments
with the deficiency in oxygen we used common test tubes with a
diameter of 13 mm. Ten cm3
of nutritive medium were poured into
such a tube, the depth of the liquid being about 80 mm. (2) To
obtain better aeration, cultures were made in small Erlenmeyer
flasks of about 50 mm in diameter, and when 10 cm3
of nutritive
medium were poured in, the liquid reached a depth of 7-8 mm. In
these conditions the layer of the liquid was almost ten times thinner
than in the test tubes (Fig. 8). The test tubes as well as Erlenmeyer
MECHANISM OF COMPETITION IN YEAST CELLS 65
flasks were closed by cotton wool stoppers. The experiments made
in the flasks will be described in this book as "aerobic" and those in
test tubes as "anaerobic."
(3) An inoculation of yeast cells was made into the sterilized nutritive
medium. Special attention was given to the standardization
of the inoculating material, for in order to obtain exact and comparable
results the inoculating cells had to be in a certain fixed physiological
condition. Cells for inoculation were always taken from test
tubes where the growth was just finished. For an anaerobic inoculation
of Saccharomyces cultures 48 hours old (at 28°C.) were used,
whilst the slow-growing Schizosaccharomyces for an anaerobic inoculation
was taken at the age of five days at 28°C. Before inoculation
the contents of the test tube was shaken, and a fixed number of drops
of the liquid was introduced into the nutritive medium by means of
a sterilized pipette. It was also necessary that an equal initial
quantity of each species or, in other words, equal initial masses should
be inoculated. It was found that in anaerobic test tubes intended for
inoculation a mass of yeast in a unit of volume of the nutritive liquid
is two and a half times smaller in Schizosaccharomyces than in Saccharomyces.
Therefore in order to inoculate an equal initial quantity
two drops of uniform suspension of Saccharomyces and five drops of
Schizosaccharomyces were always introduced. In the case of a mixed
culture, two drops of the first species plus five drops of the second
were taken. 2
We must prepare a perfectly uniform suspension of
seed-yeast and the inoculation itself must be carried out rapidly so as
to avoid possible errors from a settling of yeast cells in the inoculating
pipette. This circumstance was pointed out by Richards ('32) and
Klem ('33). All the experiments were carried out in a thermostat at
a temperature of 28°C.
(4) After inoculation it was necessary to study the growth of
number and mass of yeast cells, and on the other hand to trace and
to evaluate the changes in the factors of the medium. The counting
of the number of yeast cells per unit of volume does not present any
difficulty and for this purpose the Thoma counting chamber is usually
employed. In our experiments three test tubes (or flasks) of the
2
A very strict equality of the masses of two species sown is not absolutely
necessary. It is only important that the very same quantity of each species
should be introduced into the mixed population and into the separately grown
culture. This is very easy to do with our mode of inoculation.
66 THE STRUGGLE FOR EXISTENCE
same age were taken and a uniform suspension of yeast was made byshaking.
One cm3
of liquid was taken by a pipette from every tube
and poured into another clean tube, where the three cm3
obtained
from three tubes were fixed by three cm3
of 20 per cent solution of
H2SO4. Individual fluctuations of separate cultures were thus neutralized,
and a certain "average suspension" from three test tubes was
obtained. The material fixed was more or less diluted with water,
and then the number of cells per unit of volume was counted in the
Thoma chamber. Quite recently Richards ('32) in his interesting
paper describes in detail the methods of studying the growth of yeast,
where he points out that the counting of the number of yeast cells
is a very satisfactory method. As regards the possible sources of
error, he indicates the following: (1) the sample placed in the counting
chamber is not truly representative of the population sampled;
(2) the cells do not settle evenly in the counting chamber. To eliminate
these errors it is necessary to take several sample groups from
the "average suspension," and to count a great number of squares in
the chamber. In our experiments the fixed suspension was carefully
mixed before the taking of the sample, a few drops were taken with a
pipette, placed in the chamber, and ten squares were counted. Six
such sample groups were successively taken, and the total number of
counted squares amounted to sixty. Sometimes a lesser number of
squares sufficed.
The average number of cells in one large square of a Thoma chamber
at the dilution corresponding to the material fixed (i.e., twice
thinner than the initial suspension) is given in our tables. It is
understood that the counts sometimes were made with considerably
stronger dilutions, and they were correspondingly reduced to the
accepted standard. A few words must be added concerning the
counting of cells in mixed cultures. After a certain amount of practice
it is quite easy to distinguish the two species of yeast, as the cells
of Saccharomyces are much larger than those of Schizosaccharomyces
and their structure is different.
(5) The numbers of yeast cells belonging to two different species
do not allow us to form an idea as to their masses. But it is just
the masses of the species that are of particular importance in the
processes of the struggle for life. This is because a unit of mass of a
given species is usually connected by definite relations with the
amount of food consumed or that of the waste-products excreted or,
MECHANISM OF COMPETITION IN YEAST CELLS 67
generally speaking, with the factors limiting growth. Therefore the
equations of the struggle for existence ought to be expressed in tern^s of
masses of the species concerned and not in terms of the numbers of
individuals, which are connected by more complex relations with the
factors limiting growth.
In order to pass on from the number of yeast cells of the first and
second species counted at a definite moment to the masses of these
species, we must take into account that: (1) the cells of the first
species differ in their average volume from those of the second, (2)
this average volume of the cell in each species can change in the course
of growth of the culture. (Richards ('28b) showed that the average
size of a cell of Saccharomyces cerevisiae is different at different stages
of growth), and (3) the species can be of different specific weight.
Therefore, by multiplying the volume of all the cells of a definite
species at a given moment of time by their specific weight, we shall
obtain the weight of the given organisms enabling us to judge of their
mass. Assuming for the sake of simplification that the cells of our
yeast species are near to one another in their specific weight, we can
measure the volumes occupied by each species of yeast cells in order
to obtain an idea of the masses of these cells.
(6) The volume of yeast was determined by the method of centrifugation.
The fluid from the test tubes or flasks with the counted number
of yeast cells was centrifuged for one minute in a special tube placed
in an electric centrifuge making 4000 revolutions per minute (usually
in portions of 10 cm3
each). The liquid was then poured off and the
yeast cells that had settled on the bottom were shaken up with the
small quantity of the remaining liquid. The mixture thus obtained
was transferred by means of a pipette into a short graduate glass tube
of 3.5 mm in diameter. The mixture in the graduated tube was
again centrifuged for 1.5 minutes, and then the volume of the sediment
was rapidly measured with the aid of a magnifying glass. To
avoid errors connected with the different degree of compression of the
yeast in different cases, the quantity of the mixture poured into the
short graduated tube was always such that the sediment did not
exceed ten divisions of the graduated tube and, if necessary, the secondary
centrifugation was made by several doses. The volume of
yeast occupying one division of the graduated tube was taken for a
unit.
The centrifugation method may be criticized as, according to
LjIubrary
ysra*
68 THE STRUGGLE FOR EXISTENCE
Richards ('32), even in employing the super-centrifuge of Harvey one
can not succeed in obtaining a solid packing of the cells, and interstices
remain between them. If we draw our attention to the fact
that the size of the cells changes in the process of the growth of the
culture, and that in mixed populations of the two species we have to
deal with cells of different sizes then, theoretically, this must lead to
a very different degree of packing of the cells in different cases, and
the volume of the cells determined by centrifugation apparently does
not yet allow us to judge of their mass. However, the measurements,
some of which will be given further on, show that the errors which
actually arise are small, and that the centrifugation method is perfectly
reliable for our purposes.
In the study of the population growth of yeast it is difficult to carry
on observations upon the very same culture, as it is urgent to strictly
maintain the sterility of the medium and to avoid injury to the cells.
For this reason a great number of test tubes were inoculated at the
beginning of the experiment ; at certain fixed moments determinations
were made upon a group of test tubes which were then put aside and
further determinations were made upon new tubes.
in
(1) Having examined the technical details of cultivation of yeast
cells we can now pass to the problem which interests us first of all:
how does the multiplication of the yeast proceed in a microcosm with
a limited amount of energy, and what are the factors which check the
growth of the population? Let us begin by examining the kinetics
of growth under anaerobic conditions. Figure 9 represents the
growth of volume of the yeast Saccharomyces cerevisiae, according to
the data of one of our experiments in 1930. It is clearly seen that the
volume increases slowly at first, then faster, and finally slows down
on approaching a certain fixed value. The curve of growth is asymmetrical,
i.e., its concave part does not represent a reverse reflection
of the convex one (Richards, '28c, Gause, '32a). The first of them is
somewhat steep but the second comparatively inclined. This asymmetry
is, however, not sharply expressed, and it can be neglected if
we analyze the growth in a first approximation to reality.
In experiments of this type immediately after the yeast cells are
inoculated an intensive multiplication begins. There is scarcely any
lag-period, or period of an extremely slow initial growth, while the
MECHANISM OF COMPETITION IN YEAST CELLS 69
cells adapt themselves to the medium. This is because we used for
inoculation fresh yeast cells developed in a medium of an identical
composition with those used in the experiment. This circumstance
has been pointed out by Richards ('32).
(2) An investigation of the shape of the curve which represents
the accumulation of the yeast volume in the population of yeast cells
does not enable us to judge what factors control the growth of the
population and limit the accumulation of the biomass. The fact
that the growth curve is S-shaped and resembles the well-known autocatalytic
curve does not prove at all that the phenomenon we are
Fig. 9. Growth in volume of the yeast, Saccharomyces cerevisiae. From
Gause ('32a).
studying has anything in common with autocatalysis. The question
of the basic nature of the yeast growth in a limited microcosm can be
elucidated only by means of specially arranged experiments. Such
experiments were recently carried out by Richards ('28a) and confirmed
by Klem ('33).
We have already mentioned that the process of multiplication of
organisms is potentially unlimited. It follows the law of geometric
increase, and limitations are here introduced only by the external
forces. In the case of yeast this circumstance was noted by Slator
('13), and recently Richards carefully verified it in the following
manner. A control culture after the inoculation of yeast was left
70 THE STRUGGLE FOR EXISTENCE
to itself, and the growth of the number of cells in this culture followed
a common S-shaped curve and then stopped. In an experimental
culture a change of the medium was made at very short intervals of
time (every 3 hours). Here the conditions were all the time maintained
constant and favorable for growth. Under these conditions
the multiplication of yeast followed the law of geometric increase:
in every moment of time the increase of the population constituted a
certain definite portion of the size of the population. The relative
rate of growth (i.e., the rate of growth per unit of population) remained
constant all the time, or in other words there was no autocatalysis
here. Figure 10 represents the data of Richards. To the
left are shown the growth curves of the number of cells per unit of
volume: the S-shaped curve in the control culture, and the exponen-
300
tiedi urn changed
every 3nrj
every nhYevery 2Vtfr'
jConirof
=5?
v.
o
/ipJium changed
every 3t>rr
t
'
eyery I2hn
^^ifvtryVth
Control
rt>)
no hours o to so
Time
no
Fig. 10. Growth curves of the yeast Saccharomyces cerevisiae. (a) Growth
of the number of cells, (b) The same, plotted on logarithmic scale. From
Richards ('28a).
tially increasing one with continuously renewed medium. One can
in the following manner be easily convinced that the exponentially
increasing curve corresponds to the geometric increase: if against
the absolute values of time we plot the logarithms of cell numbers, a
straight line will be obtained (see the right part of Figure 10 taken
from Richards). As is well known this is a characteristic property
of a geometric increase. Nearly the same results were recently
obtained by Klem ('33).
The experiments made by Richards show clearly that the growth
of the yeast population is founded on a potential geometric multiplication
of yeast cells (&iiVi), but the latter can not be completely realized
owing to the limited dimensions of the microcosm and consequently
to the limited number of places (K). As a result the geometric
increase becomes S-shaped. It is easy to see that the experimenta-
MECHANISM OF COMPETITION IN YEAST CELLS 71
tion has led us to the very same assumptions that are at the bottom
of Pearl's logistic equation of growth (see Chapter III, equations (8)
and (9)) . This equation is one that gives us the S-shaped curve starting
from the point that growth depends on a certain potential geometric
increase which at every moment of time is realized only in a
certain degree depending on the unutilized opportunity for growth
at that moment.
In the equation of Pearl the unutilized opportunity for growth is
expressed in terms of the population itself, i.e., as the relative number
of the still vacant places. This presents a great advantage as we
shall see later on. The unutilized opportunity of growth often depends
on various factors, and to translate the number of "still vacant
places" into the language of these factors may become a very difficult
task.
(3) Let us now analyze this problem. What is the nature of those
factors of the environment which depress the growth of the yeast
population and finally stop it? Of course they may be different in
various cases, and we have in view only our conditions of cultivation.
The nature of the factors limiting growth in such an environment has
been explained mainly by the investigations of Richards. When the
growth of yeast ceases in a test tube under almost anaerobic conditions,
there still exists in the nutritive medium a considerable amount
of sugar and other substances necessary for growth. A simple experiment
made by Richards ('28a) is convincing: if at the moment
when the growth ceases in the microcosm yeast cells from young
cultures are introduced, they will give a certain increment and the
population will somewhat increase. Consequently, there is no lack
of substances required for growth. The presence of a considerable
quantity of sugar at the moment when the growth ceases has been
chemically established, and in our experiments this is even more
apparent than in those of Richards, as our initial concentration of
sugar was 5 per cent and his only 2 per cent.
If the growth ceases before the reserves of food and energy have
been exhausted we must eyidently seek an explanation in some kind
of changes in the environment. This question has been studied by
Richards and led him to conclude that the decisive influence here is
the accumulation of ethyl alcohol. As has already been mentioned,
when yeast cells grow in test tubes under almost anaerobic conditions
the decomposition of sugar into alcohol and carbon dioxide serves
72 THE STRUGGLE FOR EXISTENCE
them as a source of energy. Sugar is almost entirely utilized to
obtain the available energy, and serves as food only in a very slight
degree. As a result a considerable amount of alcohol accumulates in
the nutritive medium, which corresponds pretty well to the amount
of sugar consumed. Curves of such an accumulation of alcohol,
16 -
>»
V. 10
o
w 8
c
o b
First experiment
16-
*/2
curve
Growth curve
30 io
Hours
so 60
Second experiment
/
_ <s Alcohol curve
Growth curve
16
H
*,*
06-%
02
70
IS
2 0^
o
-10
-OS
50 60 1030 VO
Hours
Fig. 11. The growth in volume and accumulation of alcohol in Saccharomyces
cerevisiae in test tubes. From Gause ('32b).
taken from the paper of Gause ('32b), are represented in Figure 11.
Here are given the results of two experiments made in test tubes, but
on a nutritive medium of somewhat different concentration. In both
cases a certain time after the experiment was begun the accumulation
of alcohol (and, consequently, the consumption of sugar) proceeds
MECHANISM OF COMPETITION IN YEAST CELLS 73
almost in proportion to the increase of the volume of yeast. In other
terms, a proportionality exists between the metabolism of the yeast
cells and the growth of their volume. Later on, conditions arise in
which the growth of the yeast ceases, but alcohol continues to accumulate.
Therefore, at the moment when the growth ceases there are
still unutilized resources of sugar in the medium. The life activity
of the yeast cells and the accumulation of alcohol continue after the
biomass has ceased growing.
The microscopical study of the population of yeast cells made by
Richards at the moment when growth was ceasing, has shown the
following facts. The yeast cells continue to bud actively, but as soon
0% 1% z% 3»/o yo/o
Additional alcohol
Fig. 12. The effect of additional alcohol upon the level of saturating population
in Saccharorriyces cerevisiae in test tubes.
as a bud separates from the mother cell it perishes. In this way,
unfavorable chemical changes in the medium destroy the most sensitive
link in the population, and lead to a cessation of its growth.
According to Richards ('28a) the accumulating ethyl alcohol is just
the factor which kills the young buds and inhibits the growth of the
population. He showed this experimentally: with an addition of 1.2
per cent of ethyl alcohol to the nutritive medium, the maximal yield
of population was 65 per cent from that of the control population
(acidity kept constant). Therefore, with the additional alcohol the
critical concentration of waste products at which growth ceases was
reached with a smaller quantity of accumulated yeast volume.
74 THE STRUGGLE FOR EXISTENCE
These data were criticized by Klem ('33) who carried out experiments
with wort and not with William's synthetic medium, which
Richards worked upon. Klem did not obtain any depression of
growth by adding a small quantity of alcohol corresponding to the
quantity which is usually accumulated in his cultures at the moment
when the growth ceases. According to Klem, it is only at a concentration
above 3 per cent that alcohol begins to depress growth,
and only concentrations of about 7 per cent have a distinctly
hindering influence. The experiments which I have made with
yeast decoction and 5 per cent sugar confirm the data of Richards
and not those of Klem. Figure 12 presents the results of several
experiments. The level of the maximal population in the control
was taken as ,100, and the levels of the maximal populations in the
cultures with this or that per cent of alcohol (added before the yeast
was sown, all other conditions being equal) were expressed in per cent
from the population level in the control. This figure shows that even
1 per cent of alcohol in our conditions lowers the maximal level of
population considerably. As we have already seen (Fig. 11, bottom)
at the moment the growth ceases in our cultures the concentration of
alcohol is near to 2 per cent (with the usual composition of medium).
This concentration is undoubtedly sufficiently high to be
responsible for the cessation of growth.
Klem expressed an interesting idea, namely that the cessation of
growth is connected with the reaching of a definite relation between
the concentration of the waste-products and the nutritive substances,
i.e., alcohol and sugar. In other terms, the critical concentration of
alcohol checking growth is by no means of an absolute character.
With a small concentration of sugar, a comparatively weak concentration
of alcohol hinders growth. But if the quantity of sugar be
increased, this concentration of alcohol will no longer be sufficient
for checking growth which will continue. Klem's opinion is perfectly
justified and many experimental data confirm it. But, as he
himself remarks, the ratio alcohol/sugar left at the moment growth
ceases, also varies within rather wide limits. (A critical analysis of
Figs. 53-54 on pp. 80-81 of his paper ('33) shows that even with
concentrations of sugar from 1 to 5 per cent the ratio alcohol/sugar
left does not remain constant, and that Klem's calculations are
not quite exact.)
(4) All we have said may be resumed thus: under our conditions
MECHANISM OF COMPETITION IN YEAST CELLS 75
of cultivation the cessation of growth of the population of yeast cells
begins before the exhaustion of the nutritive and energetic resources
of the medium. The direct cause of this cessation is the accumulation
of ethyl alcohol which kills the most sensitive members of the
population—the young buds. This critical concentration of alcohol
is not of an absolute character, and in a first approximation we can
say that the cessation of growth is connected with the establishment
of a definite ratio between the concentrations of waste-products
(alcohol) and the nutritive substances (sugar). We now have to
answer the question raised earlier: what factors will furnish us with
the terminology for expressing the "number of vacant places" or "the
unutilized opportunity for growth" in the population of yeast cells
under our conditions of cultivation? Since the growth of population
ceases with the establishment of a certain ratio alcohol/sugar a
thought might appear that we ought to connect the unutilized opportunity
for growth somehow with the ratio. However this would be a
false deduction from correct premises. We can see at once that we
have to deal here with two different things. (1) Should we wish to
make a purely theoretical calculation of the level of saturating population
in our microcosm, we would certainly be obliged to take into
consideration the ratio between the concentrations of alcohol and
sugar, and to try to calculate the moment when this ratio attains a
definite value. But certainly we should at once have to introduce
numerous corrections, as various other factors have also an influence
here. (2) The conditions of the problem before us are quite different.
We know beforehand at what level the population ceases to grow, and
what is the corresponding value of different factors of the environment.
We wish only for different moments of time preceding the
cessation of growth to translate "the unutilized opportunity for
growth" into terms of the limiting factor. Such limiting factor is
always alcohol destroying the young buds. However considerably
other factors of the environment and the condition of the cells themselves
should alter the absolute value of the critical alcohol concentration,
this does not essentially change the matter. Consequently
"the unutilized opportunity for growth" or "the number of still vacant
places" can simply be determined by the difference between the critical
concentration of alcohol at the moment of cessation of growth, which is
characteristic for the given conditions and established experimentally in
every case, and the concentration of alcohol at a given moment of time.
76 THE STRUGGLE FOR EXISTENCE
The accumulation of the yeast volume at the moment of the cessation
of growth is everywhere marked by K, and the amount of volume
at a given moment is N. Alcohol production per unit of yeast volume
is rather constant, and increases somewhat only before growth is
checked (see Fig. 11). Taking the alcohol production per unit of
yeast volume as a constant for the entire process of growth of the
population as the very first approximation to reality, we can easily
pass from the given (N) and maximal (K) amount of yeast, through
multiplying them by certain coefficients, to the given and critical
concentrations of alcohol.
(5) It is easy to see that, while we give up any attempt to discover
a certain universal growth equation forecasting the level of the saturating
population under any conditions, if we use the logistic equation
we express rationally, very simply, and in complete agreement with
experimental data, the mechanism of growth of a homogeneous
population of yeast cells. The attempts to find universal equations
will scarcely lead to satisfactory results, and in any case all this would
be too complicated for a mathematical theory of the struggle for existence
in a mixed population of two species. One of the leading ideas
of this book is that all the quantitative theories of population growth
must be only constructed for strictly determined cycles or epochs of
growth, within which the same limiting factors dominate and a certain
regulating mechanism remains invariable.
Experiments with yeast point also to a very important circumstance
in the experimental analysis of populations. All the conditions
of cultivation ought to be so arranged that the growth depends
distinctly on only one limiting factor. In the case of yeast we must
have a sufficiently high concentration of sugar and other necessary
substances in the nutritive medium so that the alcohol can in full
measure manifest its inhibitory action. As we shall see in the next
chapter in experimenting with Protozoa, it is very easy to arrange
experiments under such complicated conditions and with the interference
of such a great number of various factors that the attempts to
discover certain fundamental quantitative relations in the struggle
for existence will never have any success.
IV
(1) Our study of the growth of homogeneous populations of yeast
cells was only a preparation before we pass on to the investigation of
MECHANISM OF COMPETITION IN YEAST CELLS 77
the struggle for existence between two species in a mixed culture.
The simplest way to do this is again to begin by an analysis of the
kinetics of growth. Let us examine the experiments of 1931. In
Table 1 (Appendix) data are given on the anaerobic growth of the
volume and of the number of cells in the two species of yeast : Saccharomyces
and Schizosaccharomyces, cultivated separately and in a
mixed population in two independent series of experiments. One
hundred and eleven separate microcosms were studied in these two
series, and every figure in Table 1 (Appendix) is founded on three
observations. Figure 13 represents graphically the growth of the
yeast volume. We can see that the growth of Schizosaccharomyces
under anaerobic conditions is exceedingly slow. Let us note also
that its population attains a much lower level than that of Saccharo-
vr
Saccharomyces
a
afa\ Mixed population
u
Schtzosaccfiaromycer
so wo no no
Hours
Fig. 13. The growth in volume of Saccharomyces cerevisiae, Schizosaccharomyces
kephir and mixed population in two series of experiments. Anaerobic
conditions. From Gause ('32b).
myces. The volume of the mixed population is also smaller than the
volume of the pure culture of Saccharomyces.
The parts taken up by each of the species in the yeast volume of a
mixed culture have been evaluated in the following manner. First
of all, a calculation was made of the average number of cells per unit
of yeast volume for the separate growth of Saccharomyces and Schizosaccharomyces
(see Appendix, Table 1). It appears that the mean
number of cells occupying a unit of yeast volume varies in the course
of the growth of the culture, as Richards has already established.
However, these variations are not great, and for further calculations
average values for the entire cycle of growth can be taken. According
to the first series of experiments, in Saccharomyces 16.59 cells in a
square of a Thoma counting chamber correspond to one unit of yeast
78 THE STRUGGLE FOR EXISTENCE
volume; in the smaller species Schizosaccharomyces there are 57.70
cells in one unit of yeast volume. Starting from these averages, we
have calculated the volumes occupied by each species in the mixed
population at a given moment, according to the number of cells of
each species observed in the mixed population. (In the experiments
of 1932 which are given further on we did not use such general averages
for our calculations, but started every time from the average
number of cells observed at a given moment of time.)
The sum of the calculated volumes of both species in the mixed
culture at a given moment should agree with the actual volume of
mixed population at this moment determined by the method of centrifugation.
In the first series the totals of calculated volumes are
somewhat smaller than the volumes actually observed, and we know
the causes of this disagreement. In the second series these causes
have been eliminated, and the coincidence between the totals of the
volumes calculated and the volumes actually observed is a satisfactory
one.
(2) Figures 14 and 15 give the curves of the growth of the yeast
volume in Saccharomyces and Schizosaccharomyces cultivated separately
and in a mixed population. The curves of the separate growth
of each species are expressed with the aid of simple logistic curves of
the following type (the details of these calculations are to be found in
the Appendix)
dN ,, T
K-N
-dt=
hN
-K->
where N is yeast volume, t is time, b and K are constants. The fitting
of the logistic curves has given us the following values of the parameters
for the separate growth of our species (Sp. No. 1 is Saccharomyces,
No. 2 is Schizosaccharomyces) :
Maximal volumes: Kx = 13.0; K2 = 5.8
Coefficients of geometric increase:
&i = 0.21827; b2 = 0.06069
The calculated coefficients of geometric increase show that per
unit of time (one hour) every unit of volume of Saccharomyces can
potentially give an increase equal to 0.21827 of this unit, and in
Schizosaccharomyces equal to only 0.06069.
MECHANISM OF COMPETITION IN YEAST CELLS 79
K,=/3
g
<*> u°o
Saccharomyces separately
SaccharomycQS in mixed population.
20 30
Hours
HO so €0>
Fig. 14. The growth in volume of Saccharomyces cerevisiae cultivated separately
and in the mixed population in two series of experiments. Anaerobic
conditions. From Gause ('32b).
2 j+Q2'f7SSO-0-06069t
Schizosaccharomyces separately
K2
~5.80
o_
Schizosaccharomyces in mixed population
so
Hours
too no HO 160
Fig. 15. The growth in volume of Schizosaccharomyces kephir cultivated
separately and in the mixed population in two series of experiments. Anaerobic
conditions. From Gause ('32b).
Having obtained in this way the potential coefficients of multiplication
of our species (or, which means the same, the coefficients of geometric
increase) we must now according to the general plan given at
the beginning of this chapter pass on to a calculation of the empirical
80 THE STRUGGLE FOR EXISTENCE
coefficients of the struggle for existence. In this we start by assuming
that the system of equations of competition (see Chap. 3, equations
(11) and (12)):
dNr K, - Qh + aN2 )
-g.-MT,
m
dt
2 2
K2
actually describes the experimental data. All the values in these
equations except the coefficients of the struggle for existence a and
/3, are known to us. To find the latter let us solve this system of
two equations with two unknown values in respect to a and /3. We
obtain
v dNJdt-K, dN2 /dt-K2 ..
a =
ni
;
*
=
m
The values on the right side of both expressions can easily be calculated
from experimental data. Thus in the case of the coefficient
a: (1) 6i and Ki are known from the curve of separate growth of the
first species, (2) JVi and N2 , or the volumes of the first and second
species in a mixed population at a given moment of time (t), can be
taken from the graph by measuring the ordinates of the corresponding
curves of growth, (3) —1
represents the rate of growth of the
dt
first species in the mixed population, or the increase of volume per
unit of time, and can also be easily determined from the graph. It
will be sufficient for this to draw a tangent at a given point and to
measure —- graphically or, better, to use a Richards-Roope ('30)
dt
tangent meter for graphical differentiation.3
As a result we shall
obtain the values of the coefficients of the struggle for existence (a
and /3) for different points of the curve, i.e., for different moments of
growth: t ht2 , etc. The values of the coefficients calculated for
different moments are subject to fluctuations, but by using the middle
zone of growth sufficiently constant values will be obtained. Thus,
3
Made by Bausch and Lomb Optical Co.
MECHANISM OF COMPETITION IN YEAST CELLS 81
the coefficient /3 in the experiments of 1931 was equal to: 0.501, 0.349,
0.467, with an average of 0.439. The fluctuations of the coefficient
a were more considerable, but the experiments of 1932 give more
constant values for a also: 3.11, 3.06, 2.85, etc.
The fluctuations in the values of the coefficients of the struggle for
existence are due in this case in a considerable measure to an imperfect
method of their calculation.4
However, this is of no serious
consequence, as we have a good method for verifying the average
values of the coefficients of competition. This method consists in
constructing a curve corresponding to the differential equation of
competition (the details of this calculation are to be found in the
Appendix). A close agreement of the calculated curve of growth of
each species in a mixed population with experimental observations
represents a good proof of the correctness of the numerical values of
the coefficients of the struggle for existence. As regards the yeasts
Saccharomyces and Schizosaccharomyces here concerned, their calculated
curves of growth are given in Figures 14 and 15.
In a mixed population of Saccharotnyces and Schizosaccharomyces
under anaerobic conditions the coefficients of the struggle for existence
have the following values : a (showing the intensity of the influence
of Schizosaccharomyces on Saccharomyces) = 3.15; /3 (intensity
of the influence of Saccharomyces on Schizosaccharomyces) = 0.439.
In other words, one unit of volume of Schizosaccharomyces decreases
the unutilized opportunity for growth of Saccharomyces 3.15 times as
much as an equal unit of volume of Saccharomyces itself. The species
Schizosaccharomyces with its comparatively small volume takes up
"a great number of places" in the microcosm. The reverse action of
Saccharomyces on Schizosaccharomyces is comparatively weak. One
unit of volume of Saccharomyces decreases the unutilized opportunity
for growth of Schizosaccharomyces as much as 0.439 unit of the latter
species' own volume.
(3) We now pass on to the most important part of this chapter, i.e.,
to the comparison of the empirically established coefficients of the
struggle for existence with those which are to be expected on the
basis of a direct study of the factors controlling growth. The values
of the coefficients of the struggle for existence mentioned above are
founded upon an analysis of the kinetics of growth of a mixed popula-
4
In Chapter V we shall meet a more complicated situation.
82 THE STRUGGLE FOR EXISTENCE
tion. Let us at present leave them aside and endeavor to calculate
the values of the coefficients of competition starting from the alcohol
production. As mentioned above, the cessation of growth is connected
with the reaching of a certain critical concentration of alcohol
(characteristic for the given species under given conditions) . Let us
now assume that it is mainly alcohol that matters and that other byproducts
of fermentation are but of subordinate importance. Consequently,
every unit of volume in each species produces a determined
amount of alcohol, and when the latter reaches a certain threshold
concentration the growth is checked. It follows that when a unit of
volume of the first species produces an amount of alcohol considerably
surpassing that produced by a unit of volume of the other species
TABLE V
Alcohol production in Saccharomyces cerevisiae and Schizosaccharomyces kephir
From Gause ('32b)
MECHANISM OF COMPETITION IN YEAST CELLS 83
volume. At the same time this is an explanation of the low level
of the accumulation of biomass in the separate cultures of Schizosaccharomyces,
and the diminished volume of the mixed population in
comparison with the volume of Saccharomyces cultivated separately.
We can now calculate approximately the critical concentrations of
alcohol for the separate growth of each species of yeast if we multiply
the maximal volumes of these species (K) by the alcohol production
per unit of yeast volume. For Saccharomyces we shall have:
13.0 X 0.113 = 1.47, and for Schizosaccharomyces: 5.8 X 0.247 =
1.43. In other words, the critical alcohol concentrations for both
species are about equal.
Let us now calculate the degree of influence of one species upon the
unutilized opportunity for growth of another in a mixed population,
or the coefficients of the struggle for existence. If we take as a unit
the degree of decrease of the unutilized opportunity for growth of
Saccharomyces by a unit of its own yeast volume, we have then to
answer the following question: how much more or less does a unit
of the yeast volume of Schizosaccharomyces decrease the unutilized
opportunity for growth of Saccharomyces in the mixed population,
in comparison with the effect of a unit of the volume of the latter
species? Then, taking the ratio of the alcohol production per unit
of yeast volume in Schizosaccharomyces to the alcohol production of
Saccharomyces we shall find the coefficient of the struggle for existence
247
according to the alcohol production: a =
'
= 2.186. CorreU.l
lo
.. .
a 0.113 „,,„
spondingly: = ^-^ = 0.457.
(4) Comparing the results of the examination of the kinetics of
growth of a mixed population with the data on the alcohol production,
we observe a certain agreement in the general features. A very
strong influence of Schizosaccharomyces upon Saccharomyces made
apparent in the analysis of the kinetics of growth proved itself to be
connected with the great alcohol production per unit of yeast volume
in the former species. However, a strict coincidence of the data of
these two independent methods of investigation does not occur here.
Thus Schizosaccharomyces excretes a quantity of alcohol per unit of
yeast volume 2.186 times as great as Saccharomyces, but influences
the growth of the latter 3.15 times as much. Consequently, Schizosaccharomyces
not only produces a greater amount of alcohol, but the
84 THE STRUGGLE FOR EXISTENCE
alcohol produced by it is so to say "more toxic" for Saccharomyces
than the alcohol produced by the latter itself. All this tends to
imply that the situation is here complicated by the influence of certain
other waste products getting into the surrounding medium in small
quantities. The relations between species in these experiments are
therefore not so simple as has been supposed at the beginning of this
section.
(1) The above described experiments of 1931 were repeated in
1932, and the new data confirmed all the observed regularities. In
these new experiments the influence of oxygen upon the growth of a
mixed population of the same two species of yeast was investigated,
and this enabled us to further somewhat our understanding of the
nature of the competitive process.
The experimental data given in the preceding section have to do
with the growth of a yeast population under "anaerobic conditions,"
i.e., in test tubes. In order to study the influence of oxygen on the
growth of the yeast population, together with experiments in test
tubes we arranged other experiments under conditions of somewhat
better aeration. The technique of such "aerobic" and "anaerobic"
experiments has already been described at the beginning of this
chapter. Here it must only be remarked that in the "aerobic" series
the access of oxygen was very limited, and a part of the available
energy was, as before, obtained by our species through alcoholic
fermentation. As a result, a considerable amount of alcohol accumulated
in the nutritive medium (as will be seen in the corresponding
tables), and in its essential features the mechanism limiting the
growth of the yeast population remained the same. The experiments
of 1932 consisted of two aerobic and two anaerobic series. In them
168 separate microcosms were studied.
In all the experiments of 1932 nutritive medium of the same preparation
was used. It was made according to the usual method, but
the dry beer yeast was of another origin. As a result, the absolute
values of growth were somewhat different. It must also be remarked
that in all the new experiments the centrifuged volume of yeast was
always reduced to 10 cm3
of nutritive medium.
(2) Figure 16 represents the growth curves of Saccharomyces,
Schizosaceharomyces and of the mixed population according to two
MECHANISM OF COMPETITION IN YEAST CELLS 85
series of experiments in conditions analogous to the former anaerobic
ones. The general character of these curves coincides with that of
Figure 13. A more careful comparison of the anaerobic series of 1932
with that of 1931 shows that the first is characterized by considerably
smaller absolute values of growth (Table VI) . At the same time
Schizosaccharomyces grown separately attains a somewhat higher
level in comparison with Saccharomyces than formerly. Thus, the
volume of the saturating population of the separately growing Schizo-
5 8
saccharomyces represented in older experiments —^— = 44.6 per cent
fa ccharomyces Aerobic conditions
Mixedpopulation
a a.
TSchizosaccharomyces
.— Saccharomyces
•* % dnaerobic conditions
_ v-Mxed population
SO 60
Hours
^ Fig. 16. The growth in volume of Saccharomyces cerevisiae, Schizosaccharomyces
kephir and mixed population. Above: Aerobic conditions. Below:
Anaerobic conditions (1932).
of that of Saccharomyces (1931), but in the new experiments it is
3
^— = 48.0 per cent (1932). In the experiments of 1932 the relative
b.25
volume of Schizosaccharomyces in the mixed population increased
also. As a result the decrease of the volume of the mixed population
in comparison with the volume of separately growing Saccharomyces
is more pronounced in 1932 than in 1931.
In spite of the alterations in the absolute values of growth and a
certain change in the relative quantities of species, the coefficients
of the struggle for existence which we had calculated for the anaerobic
86 THE STRUGGLE FOR EXISTENCE
experiments of 1932 coincided almost completely with those of the
year before. A similar coincidence exists in the ratio of the alcohol
production of one species to that of another, which is to be found in
Table VII. In this manner the coefficients of the struggle for existence
remain invariable under definite conditions in spite of the changing
absolute values of growth.
TABLE VI
Parameters of the logistic curves for separate growth of Saccharomyces cerevisiae
and Schizosaccharomyces kephir under aerobic and anaerobic
conditions {1932)
Saccharomyces anaerobic
Saccharomyces aerobic
Schizosaccharomyces anaerobic
Schizosaccharomyces aerobic . . .
K
(MAXIMAL
volume)
MECHANISM OF COMPETITION IN YEAST CELLS 87
species under anaerobic conditions, with a low level of biomass, it
begins to grow rapidly with an access of oxygen and in its properties
approaches Saccharomyces. The maximal volumes and
coefficients of geometric increase given in Table VI show these
regularities in a quantitative form. When there is no oxygen and
fermentation is the only source of available energy, the coefficient
of geometric increase in Schizosaccharomyces is very low and equal to
0.04375. Under the influence of oxygen this coefficient increases
Saccharomyces aerobic ,
Saccharomyces anaerobic
K=6 9
Schizosaccharomycef aerobic
j i_ J •IO
20 30 VO So
Schizosaccharomyces anaerobic
K*3.Q o
10 20 30 VO SO
Hours
160
Fig. 17. The growth in volume of Saccharomyces cerevisiae and Schizosaccharomyces
kephir cultivated separately and in the mixed population under
aerobic and anaerobic conditions (1932). All curves are drawn according to
equations.
4.3 times and attains 0.18939, whereas in Saccharomyces the coefficient
of geometric increase under the same conditions rises but slightly
(from 0.21529 to 0.28769).
The sharp changes in the properties of our species under aerobic
conditions produce a completely new situation for the growth of a
mixed population (see Fig. 17). As before, we have calculated the
coefficients of the struggle for existence and Table VII shows that
they differ considerably from the anaerobic ones. If in anaerobic
experiments the coefficient a, which characterizes the intensity of in-
88 THE STRUGGLE FOR EXISTENCE
fluence of Schizosaccharomyces upon Saccharomyces, was equal to
3.05-3.15, than under aerobic conditions it is equal to 1.25. In
other terms the influence of Schizosaccharomyces on Saccharomyces
is no longer 3.05, but only 1.25 times as strong as the influence of the
latter upon itself.
(4) Let us now examine the production of alcohol under aerobic
conditions. The corresponding data are given in Table 2 (Appendix).
As was to be expected, in aerobic conditions the amount of
alcohol per unit of yeast volume is smaller than in anaerobic ones,
because a part of the available energy is furnished by oxidation. It
is interesting to compare the critical concentration of alcohol at which
growth ceases, in aerobic and anaerobic conditions. Let us multiply
as before the production of alcohol per unit of yeast volume by the
maximal volume. For the anaerobic experiments of 1932 we shall
obtain: Saccharomyces, 6.25 X 0.245 = 1.53; Schizosaccharomyces,
3.0 X 0.510 = 1.53. These threshold concentrations of alcohol
coincide in both species, and they are sufficiently near to those with
which we have had to deal in the anaerobic experiments of 1931. As
to the threshold concentrations of alcohol in aerobic conditions, they
prove to be higher than in the anaerobic ones, and in Saccharomyces
the threshold lies somewhat higher than in Schizosaccharomyces:
Saccharomyces, 9.80 X 0.207 = 2.03; Schizosaccharomyces,
6.9 X 0.258 = 1.78.
If we now calculate for aerobic conditions the degree of influence
of Schizosaccharomyces upon Saccharomyces starting from the production
of alcohol per unit of yeast volume, we shall obtain
0.258
ai =
O207
= L25 Correspondingly
the coefficient
* = °oi-s = «
Comparing these results with the data of the kinetics of growth, we
see (Table VII) that in aerobic conditions the degree of influence of
one species upon another calculated according to the system of equations
of the struggle for existence fully coincides with the coefficients
of the relative alcohol production. Therefore, the process of com-
MECHANISM OF COMPETITION IN YEAST CELLS 89
petition between our species in aerobic conditions is entirely regulated
by alcohol, and there is scarcely any interference of other
factors.
(5) We can now appreciate from a more general viewpoint the
results of the aerobic experiments as well as those of this chapter. It
has been shown that under aerobic conditions the theoretical equation
of competition between two species of yeast for a common place
in the microcosm given for the first time by Vito Volterra is completely
realized. In other words, if we know the properties of two
species growing separately, i.e., their coefficients of geometric increase,
their maximal volumes, and alcohol production per unit of volume
when alcohol limits the growth, then connecting these values into a
theoretical equation of the struggle for existence we can calculate in what
proportion a certain limited amount of energy will be distributed between
the populations of two competing species. This means that we can
calculate theoretically the growth of species and their maximal
volumes in a mixed population. The equation of the struggle for
existence expresses the idea that a potential geometric increase of each
species in every infinitesimal interval of time is only realized up to
a certain degree depending on the unutilized opportunity for growth
at that moment, and that the species possesses certain coefficients of
seizing this unutilized opportunity. Such theoretical calculations
agree completely with the experimental data only under aerobic
conditions, where the limitation of growth in both species depends
almost completely on the ethyl alcohol. In the case of anaerobic
conditions the situation becomes more complicated as a result of the
influence of certain other waste products. This shows that extreme
care is necessary in the investigation of biological systems, because
various and often unexpected factors may participate in the process
of interaction between two species.
Chapter V
COMPETITION FOR COMMON FOOD IN PROTOZOA
i
(1) At the end of the last century Boltzmann, considering the
struggle for existence in the biosphere as a whole, remarked that there
exists a considerable quantity of essential mineral substances needed
by all living beings, but that the resources of available solar energy
are comparatively more restricted and they constitute the narrow
link representing the principal object of competition. This circumstance
has since been pointed out by many biophysicists, and we will
quote the words of Boltzmann himself ('05): "The general struggle
for existence of all living beings is not the struggle for the fundamental
substances, for these fundamental substances indispensable for all
living creatures exist abundantly in the air, the water and the soil.
This struggle is not a struggle for the energy which in the form of heat,
unfortunately not utilizable, is present in a great quantity in every
object, but it is a struggle for entropy, which is available when energy
passes from the hot sun to the cold earth. In order to utilize in the
best manner this passage, the plants spread under the rays of the
sun the immense surface of their leaves, and cause the solar energy
before reaching the temperature level of the earth to make syntheses
of which as yet we have no idea in our laboratories. The products
of this chemical kitchen are the object of the struggle in the animal
world." This idea of Boltzmann that the available solar energy
represents the narrow link for the living matter in the biosphere taken
as a whole is in a certain agreement with the data of the modern
geochemists. Thus Professor Vernadsky ('26) points out that a
part of the solar energy which is capable of producing chemical work
on the earth is to the very end utilized in the mechanism of the biosphere.
In other words, the transforming surface of the green living
matter utilizes entirely the rays of a definite wave-length in the process
of photosynthesis.
(2) However that may be, the energetic side of the struggle for
existence in the biosphere as a whole has as yet been little studied,
90
COMPETITION FOR COMMON FOOD IN PROTOZOA 91
and at the present level of our knowledge we will have to undertake a
detailed analysis of the most simple cases only. But the words of
Boltzmann compel us to turn our particular attention to those narrow
links in the conditions of our microcosm which constitute the real
objects of the competition. In the foregoing chapter we had to deal
with a competition of the yeast cells for the utilization of a certain
limited amount of energy in the test tube. The limit for the growth
of the biomass in these organisms was connected with the accumulation
of the waste product (alcohol), and this factor stopped the growth
before the exhaustion of the energetic resources of the microcosm.
As a result the entire process of competition could be expressed in
terms of the narrow link—alcohol production. Thus the situation
in these experiments was a peculiar one.
In the present chapter we will try to approach the regularities
which, as Boltzmann supposed, are characteristic for the biosphere
as a whole. We will examine the struggle for existence in carefully
controlled populations of Protozoa. Here the growth will be limited
by an insufficiency of organic nutritive substances, a factor analogous
to an insufficiency of available energy. The second peculiarity is
that the energetical resources of the microcosm will be maintained
continuously at a certain fixed level in the course of the experiment.
This approaches somewhat to what exists in nature, where the level
of energy is maintained by the uninterrupted influx of solar energy.
As before we will be concerned in these experiments with the problem
in what proportion the energy of the microcosm will be distributed
between the populations of the two competing species. But besides
this first stage we shall be enabled to examine here the following
fundamental question: Will one species drive out the other after all
the available energy of the microcosm has been already taken hold off
And if so, will one species in these conditions drive the other one out
completely, or will a certain equilibrium become established between
them?
(3) It has been already tried more than once to use Protozoa for the
study of the communities of organisms and their succession under
laboratory conditions. But as an ecologist has recently remarked,
the mere fact of a community set up in a laboratory dish does not
mean at all that it is simple. Interesting observations have been
made on the succession of communities of Protozoa in a hay infusion
by Woodruff ('12) Skadowsky ('15) and more recently by Eddy ('28).
92 THE STRUGGLE FOR EXISTENCE
However, in experiments of this type there exists a great number of
different factors not exactly controlled, and a considerable difficulty
for the study of the struggle for existence is presented by the continuous
and regular changes in the environment. It is often mentioned
that one species usually prepares the way for the coming of another
species. Recollecting what we have said in Chapter II it is easy to
see that in such a complicated environment it is quite impossible to
decide how far the supplanting of one species by another depends on
the varying conditions of the microcosm which oppress the first
species, and in what degree this is due to direct competition between
them. In this connection one of the main problems of our experiments
with Protozoa has been to eliminate the complicating influence
of numerous secondary factors, and to apply such a technique of
cultivation as would enable one to form a perfectly clear idea as to
the nature of the factor limiting growth. This could not be done at
once and the technique of our first experiments presented all the usual
defects. Only later, taking into account certain suggestions of American
authors, we made use of a synthetic medium for cultivating the
Protozoa, and the result furnished exceedingly clear data to a detailed
description of which we will soon pass.
(4) A new property of the infusorian population distinguishing it
from that of yeast cells is that the infusorian population constitutes a
secondary population living at the expense of bacteria which it devours.
Thus here appears an elementary food chain: bacteria —
infusoria. In our initial experiments the standardization of the
conditions of cultivation was only a quite superficial one. Without
taking any precautions as to an exact control of the physicochemical
properties of the medium and the number of growing bacteria, we
prepared the nutritive medium in the following manner: to 100 c.c.
of tap water 0.5 gr. of oatmeal was added; the whole was boiled for 10
minutes, left to stand and then the upper liquid was carefully poured
off, diluted 1^ times by water, and sterilized in an autoclave. After
this an inoculation of Bacillus subtilis was made, and the medium was
put into the thermostat at 32° for seven days in order to obtain an
abundant growth of bacteria. Before using, the medium was diluted
twice by tap-water, and without any further sterilization was put
into test tubes. (This was the so-called "oaten medium without
sediment." The "oaten medium with sediment" mentioned in
Chapter VI differs in its not being diluted by water before using, and
COMPETITION FOR COMMON FOOD IN PROTOZOA 93
a small quantity of sediment originating from the oatmeal was
allowed to remain.) The cultivation was made in tubes with a flat
bottom (about 1 cm. in diameter and 5-6 cm. high) of the nutritive
solution. The tubes were closed by cotton wool stoppers and kept
in a moist thermostat at 26°C. Close paramnized cork stoppers were
not found convenient because if we use them the population begins
to die off immediately after cessation of growth, and the curves take
the form described by Myers ('27). At the same time under optimal
conditions after the growth of the population has ceased the level of
the population is maintained unchanged for a certain time, and only
later Paramecia begin to die off.
In the initial experiments no change of the medium in the course
of growth of the population was made, and the increase in the number
of individuals was studied according to the average values for the test
tubes of a definite age. The contents of the tube was destroyed every
time after examination just as in the experiments with yeast. The
counting was made under a magnifying glass on a slide plate. Figure
18 represents the growth of the number of individuals in pure lines
of Paramecium caudatum and Stylonychia mytilus cultivated separately
and in a mixed population. These data are founded on two
experiments which gave similar results. At the beginning of the
experiment into each tube were placed five Paramecium, or five
Stylonychia, or five Paramecium plus five Stylonychia in the case of a
mixed population. Stylonychia for inoculation must be taken from
young cultures to avoid an inoculation of degenerating individuals.
(5) The growth curves of the number of individuals in Figure 18
are S-shaped and resemble our well known yeast curves. After
growth has ceased the level of the saturating population is maintained
for a short time, and then begins the dying off of the population which
is particularly distinct in Stylonychia. It is evident that this dying
off is regulated by factors quite different from those which regulate
growth, and that a new system of relations comes into play here.
Therefore there is no reason to look for rational equations expressing
both the growth and dying off of the populations.
Figure 18 shows that Stylonychia, and especially Paramecium, in a
mixed culture attain lower levels than separately. The calculated
coefficients of the struggle for existence have the following values:
a (influence of Stylonychia on Paramecium) = 5.5 and /3 (influence of
Paramecium on Stylonychia) = 0.12. This means that Stylonychia
94 THE STRUGGLE FOR EXISTENCE
influences Paramecium very strongly, and that every individual of the
former occupies a place available for 5.5 Paramecia. With our
technique of cultivation it is difficult to decide on what causes this
depends. As a supposition only one can point to food consumption.
to
6t>-
>/0
%20
I*
n
P. caudatum separate/y
n+d=7? o
d*r v
P. cat/datum in mixed population.
2
S myii/us separately
K+d'is.s
S.myiilus in mixed population
\
a
o I Z 3 t/
Days
Fig. 18. The growth in number of individuals of Paramecium caudatum and
Stylonychia mytilus cultivated separately and in the mixed population, d
denotes lower asymptote. From Gause ('34b).
(6) We have but to change slightly the conditions of cultivation
and we shall obtain entirely different results. Figure 19 represents
the growth of populations of the same species on a dense "oaten
medium with sediment" sown with various wild bacteria. Here
COMPETITION FOR COMMON FOOD IN PROTOZOA 95
owing to an increase in the density of food the absolute values of the
maximal population in both species have considerably increased.
The character of growth of the mixed population now essentially
differs from the former one : Paramecium strongly influences Stylonychia,
while Stylonychia has almost no influence upon Paramecium.
We simply have here an "alchemical stage" of investigation, and the
absence of an exact control of the conditions of the medium creates
the impression of a complete arbitrariness of the results of our experiments.
Unfortunately many protozoological researches are still in
this stage, and the idea is very widespread that "Protozoa are not
oe
*oo-
J.myttlus
Jn mixed population
P. caudatum
Separately *-&sls~~~
o f
Jn mixedpopulation
V J
-
6 7
Days
Fig. 19. The growth in number of individuals of Paramecium caudatum and
Stylonychia mytilus cultivated separately and in the mixed population. Medium
contains wild bacteria.
entirely satisfactory for the study of populations as they require
bacteria for food, and it is very difficult to measure accurately and to
analyze the relations between protozoan population and the bacterial
population."
In order to draw any reliable conclusions as to the quantitative
laws of the struggle for existence in Protozoa we must begin by elaborating
the technique of cultivation, keeping in mind the following excellent
words of Raymond Pearl: "When the biologist exercises something
approaching the same precision and infinitely painstaking care,
over all the most trivial details of a biological experiment that the
96 THE STRUGGLE FOR EXISTENCE
physicist does over his, the results tend to take on a degree of precision
and uniformity not so far short of that usual in the older science, as
we are accustomed to expect" ('28, p. 35).
ii
(1) Jennings pointed out the necessity of a careful control of bacteria
in the cultures of Protozoa in 1908, and one of the first attempts
to grow Paramecia in pure cultures of bacteria was made by Hargitt
and Fray ('17), Oehler ('20) and Jollos ('21). Since then numerous researches
have appeared and a good review of them can be found in the
recently published book of Sandon ('32) The Food of Protozoa as well
as in Hartmann ('27) and Belar ('28). It can be noted in a quite
general form that in order to standardize the conditions of cultivation
TABLE VIII
Balanced physiological salt solution of Osterhout
NaCl 2 .35 gr.
MgCl2 0. 184 gr.
MgS04 0.089 gr.
KC1 0.050 gr.
CaCl2 0.027 gr.
Bidistilled water to 100 c.c.
This solution is diluted with bidistilled water 225 times.
of Protozoa it is necessary: (1) to standardize the quality of food
to cultivate the Protozoa on bacteria of a definite species, (2) to standardize
the quantity of food—the number of bacteria per unit of
volume must have a fixed value, and (3) to standardize the physicochemical
conditions of the medium. These difficult and as one may
think hardly realizable problems have been solved very simply for
Oxytrichia by Johnson ('33), who has been partly preceded by Barker
and Taylor ('31). The method is this: a culture of a certain bacterium
is made on a solid medium and then a fixed quantity of bacteria
is taken off the solid medium and transferred into a balanced
physiological salt solution, where these bacteria do not multiply and
serve as food for the Protozoa. Like Johnson we used Osterhout's
salt solution, the composition of which is given in Table VIII. As
will be shown further on, in certain experiments this medium was
COMPETITION FOR COMMON FOOD IN PROTOZOA 97
buffered and kept at a definite hydrogen ion concentration (pH).
Special experiments made by Johnson showed that the bacteria do
not multiply in this medium, and that their number scarcely changes
within 24 hours.
Beginning our experiments on the growth of pure and mixed populations
of Paramecium caudatum, Paramecium aurelia and Stylonychia
pustulata, we devoted a certain time to finding a culture of bacteria
suitable as food for all three species of Protozoa. Using the data
published by Philpot ('28) we chose finally the pathogenic bacterium
Bacillus pyocyaneus, which was cultivated in Petri dishes at 37°C. on
a solid medium of the following composition : peptone, 1 gr. ;
glucose,
2 gr.; K2HPO4, 0.02 gr.; agar-agar, 2 gr., per 100 cm3
of tap-water.
One standardized uniformly filled platinum loop of fresh Bacillus
pyocyaneus taken off the solid medium was placed in 10 cm3
of Osterhout's
salt solution. This mixture was prepared anew every day,
and we will speak of it as the "one-loop" medium.
(2) Such a standard and convenient technique of cultivation enables
us to approach the experimental investigation of an important
problem: the course of the process of competition for a source of
energy kept continually at a certain level. With this object the
cultivation was carried on in graduate tubes for centrifugation of
10 c.c. capacity, which were filled with nutritive medium up to 5
c.c. and closed with cotton wool stoppers. Twenty individuals of the
corresponding species were placed in every tube, or 20 plus 20 in
case of a mixed culture. The medium was changed daily in the
following manner. The tube was placed in a centrifuge, and after
two minutes of centrifugation with 3500 revolutions per minute the
infusoria fell to the bottom, the liquid above was very gently drawn
off by means of a pipette with a caoutchouc ball and a freshly made
nutritive medium was poured in. Besides this, every other day each
culture was washed with the salt solution free of bacteria, in order to
prevent the accumulation of waste products in the few drops of liquid
remaining at the bottom of the tube with the Paramecia at the moment
when the medium was changed. For this purpose after the
pouring off of the old medium the tubes were filled with a pure salt
solution, centrifuged and the liquid was drawn off a second time.
Every day before the medium was changed each culture was carefully
stirred up, 0.5 c.c. of the liquid was taken out and the number of infusoria
in it counted. After counting the sample was destroyed. All
98 THE STRUGGLE FOR EXISTENCE
the experiments were made in a moist thermostat at 26°C. with pure
lines of infusoria.
(3) In an experiment of such a type all the properties of the medium
are brought to a certain invariable "standard state" at the end
of every 24 hours. Hence, we acquire the possibility of investigating
the following problem : can two species exist together for a long
time in such a microcosm, or will one species be displaced by the
other entirely? This question has already been investigated theoretically
by Haldane ('24), Volterra ('26) and Lotka ('32b). It appears
that the properties of the corresponding equation of the struggle for
existence are such that if one species has any advantage over the other
it will inevitably drive it out completely (Chapter III). It must be
noted here that it is very difficult to verify these conclusions under
natural conditions. For example, in the case of competition between
TABLE IX
Contents of the microcosms in the experiments with Osterhout' s medium
CONTENTS OF THE MICROCOSM
(1) Paramecium caudatum separately.
(2) Stylonychia pustulata separately
(3) Paramecium aurelia separately...
(4) P. caudatum + P. aurelia
(5) P. caudatum + S. pustulata
(6) P. aurelia + S. pustulata
NUMBER OP
MICROCOSMS
4
5
3
3
3
3
two species of crayfish (Chapter II) a complete supplanting of one
species by another actually takes place. However, there is in nature
a great diversity of "niches" with different conditions, and in one
niche the first competitor possessing advantages over the second will
displace him, but in another niche with different conditions the advantages
will belong to the second species which will completely displace
the first. Therefore side by side in one community, but occupying
somewhat different niches, two or more nearly related species
(e.g., the community of terns, Chapter II) will continue to live in a
certain state of equilibrium. There being but a single niche in the
conditions of the experiment it is very easy to investigate the course
of the displacement of one species by another.
(4) Two series of experiments were arranged by us in which the
COMPETITION FOR COMMON FOOD IN PROTOZOA 99
process of competition was studied in 21 microcosms for a period of
25 days. Table IX shows the combinations of separate species of
Protozoa which were used. Let us first of all analyze the competition
between Paramecium caudatum and Paramecium aurelia. The data
on the growth of pure and mixed populations of these species are
presented in Table 3 (Appendix) which gives the number of individuals
in a sample of 0.5 cm3
taken from a culture of 5 cm3
in volume.
(A separate counting of the number of individuals in every culture
was discontinued from the twentieth day, and we began to take
average samples from the similar cultures.)
Fig. 20. Paramecium caudatum (1) and Paramecium aurelia (2) according to
Kalmus ('31). (3) Measurements for the calculation of volume of Para-
mecium.
In order to investigate the process of competition, we had to pass
from the number of individuals of P. caudatum and P. aurelia to their
biomasses, as these species differ rather strongly in size (see Fig. 20).
In order to obtain an idea of the biomass we had recourse to the volumes
of these species. P. caudatum and P. aurelia were measured
under the conditions of our experiments (Table X) and on the basis
of these measurements the volumes were calculated. As in shape
Paramecium after fixation approaches somewhat closely to an ellipsoid
of rotation with the half-axes: -, -, - (see Fig. 20), the calculaa
Ji a
tion of the volumes was made according to the formula for this body.
100 THE STRUGGLE FOR EXISTENCE
Taking the volume of the larger P. caudatum equal to unity, the
volume of P. aurelia can be easily expressed in a relative form. Under
different conditions the relative volume of P. aurelia varies somewhat,
but for Osterhout's medium it can be taken as equal on an
average to 0.39 of the volume of P. caudatum. In this way, in order
to pass from the growth of the number of individuals of the two species
of Paramecia to the growth of their volumes, we can leave without
alteration the number of individuals of P. caudatum, and only diminish
the number of individuals of the small P. aurelia by multiplying
it in every case by 0.39.
(5) Figure 21 represents graphically the growth in the number of
TABLE X
Measurements of Paramecium caudatum and Paramecium aurelia (after fixation)
Specification of measurements is taken from Figure 20
ORIGIN OF PARAMECIA
Growing culture with Osterhout's
medium
Old culture with Osterhout's medium
Culture with the buffered medium. . .
AVERAGE
VALUES FOR
P. CAUDATUM
IN DIVISIONS
OF OCULAR-
MICROMETER
18.3
6.0
17.1
5.1
a = 18.0
b = 6.2
AVERAGE
VALUES FOR
P. AURELIA
IN DIVISIONS
OF OCULAR-
MICROMETER
13.8
4.2
12.6
3.8
a = 12.9
b = 4.8
CALCULATED
VOLUME OF
P. AURELIA
(VOLUME OF
P. CAUDATUM
= 1)
0.39
0.37
0.41
0.429
individuals and in the volumes of P. caudatum and P. aurelia cultivated
separately in a medium changed daily for 25 days. The
general character of the curves shows that the growth of population
under these conditions has an S-shaped form. At a certain moment
the possibility of growth in a given microcosm is apparently exhausted,
and with a continuously maintained level of nutritive resources
a certain equilibrium of population is established. The oscillations
of population round this state of equilibrium are not governed
by any apparent law, and depend on various accidental causes (variation
in temperature of the thermostat, a slight variability in the
composition of the synthetic medium, etc.). A comparison of the
COMPETITION FOR COMMON FOOD IN PROTOZOA 101
curves of growth of P. caudatum and P. aurelia shows that as regards
the number of individuals the level of the saturating population of
P. aurelia is considerably higher than that of P. caudatum. Nevertheless,
the comparison of the volumes shows something completelydifferent;
in this respect P. aurelia only slightly surpasses P. caudatum,
accumulating at the expense of a certain definite level of food
resources a scarcely larger biomass. As will be shown further on,
the Osterhout salts medium is not quite favorable in its properties
fM
5»»
>
"6 Jeo-
200J
"x>
P. caudatum
o-eI
i
Oayt
Fig. 21. The growth of the number of individuals and of the "volume" in
Paramecium caudatum and Paramecium aurelia cultivated separately on the
medium of Osterhout. From Gause ('34d).
for the Paramecia, and this complicates the question as to the factors
limiting growth. On the one hand, the insufficiency of food plays
a part here which we can judge of by a direct observation of the cultures
: with a population in equilibrium the turbid bacterial medium
introduced daily becomes quite transparent after a certain time, as
the bacteria are entirely devoured by the Paramecia. However, owing
to a comparatively high concentration of bacteria and a somewhat
unoptimal reaction of the medium a depressory action of certain
other influences plays also a role here.
102 THE STRUGGLE FOR EXISTENCE
(6) The data on the growth of the volumes of P. caudatum and P.
aurelia in a mixed population are given in Figure 22. The curves of
growth of each species in a mixed culture are presented here on the
background of control curves corresponding to the free growth of the
same species. It is easy to see that the growth of a mixed population
consists of two periods: (a) during the first period (till the eighth day),
the species grow and compete for the seizing of the still unutilized
energy (food resources). But the moment approaches gradually
P. aurelia
100
cr Jn mixed population
P. caudatum
Fig. 22. The growth of the "volume" in Paramecium caudatum and Paramecium
aurelia cultivated separately and in the mixed population on the
medium of Osterhout. From Gause ('34d).
when all the utilizable energy is already taken hold of, and the total
of the biomasses of the two species tends to reach the maximal possible
biomass under given conditions. (This happens on the eighth
day ; the total biomass is equal to about 210.) This first period corresponds
to what we have already observed in yeast cells, (b) After
this there can only arise the redistribution of the already seized energy
between the two species, i.e. the displacement of one species by another.
Figure 22 shows that such a displacement is actually observed in the
experiment: the number of P. caudatum gradually diminishes as a
COMPETITION FOR COMMON FOOD IN PROTOZOA 103
result of its being driven out by P. aurelia. As several further experiments
have shown (see Fig. 24), the process of competition under our
conditions has always resulted in one species being entirely displaced
by another, in complete agreement with the predictions of the mathematical
theory.
If we consider the curves in Figure 22 more in detail, we shall note
that they generally are of a rather complicated character. It is
interesting to note that P. caudatum in a mixed culture at the beginning
of the experiment grows even better than separately. This is
apparently a consequence of the more nearly optimal relationships
between the density of the Paramecia and that of the bacterial food,
in accordance with the observations of Johnson ('33).
in
(1) Although the situation in our experiments with Osterhout's
medium has been considerably simpler than in the case of the "oaten
medium," it is still too complicated for a clear understanding of the
mechanism of competition. In fact, why has one species been victorious
over another? In the case of yeast cells we answered that
the success of the species during the first stage of competition depends
on definite relations between the coefficients of multiplication and
the alcohol production, and that it can be exactly predicted with the
aid of an equation of the struggle for existence. What will be our
answer for the population of Paramecia?
To investigate this problem we made the conditions of the experimentation
the next step in the simplification. We endeavored to
make a medium with a very small concentration of nutritive bacteria
and optimal in its physicochemical properties for Paramecia. Under
such conditions the competition for common food between two species
of Protozoa has been reduced to its simplest form.
(2) As Woodruff has shown ('11, '14), the waste products of Paramecia
can depress the multiplication and be specific for a given
species. In any case we are very far from an exact knowledge of
their role and chemical composition. Therefore first of all we must
eliminate the complicating influence of these substances. This
problem is the reverse of the one we had to do with in the preceding
chapter. There in the experiments with yeast we tried to set up
conditions under which the food resources of the medium should be
very considerable at the time when the concentration of the waste
104 THE STRUGGLE FOR EXISTENCE
products had already attained a critical value. Now with Paramecia
our object is that the concentration of the waste products
should still be very far from the critical threshold at the moment when
the food is exhausted.
First of all we turned our attention to the hydrogen ion concentration
(pH), which in the light of the researches of Darby ('29) can
be of great importance for our species. When Paramecia are cultivated
in Osterhout's medium, pH is near to 6.8 and unstable, where-
no
ISO
no
so
"Half- loop"medium
K=ior
*—* a— a
K'6V
Faureha jeparate/y
i
- T1 *• • • • •
P.caudatum separately
~ One loop medium
K=I9? D
^ Raureha separately
• K'/3Z
*
P.caudatum jeparate/y
• •
10 /v it 18 20
Days
Fig. 23. The growth of the "volume" in Paramecium caudatum and Paramecium
aurelia cultivated separately on the buffered medium ("half-loop"
and "one-loop" concentrations of bacteria). From Gause (\34d).
as the reaction in our wild cultures is commonly near to 8.0. Therefore
we, like Johnson, buffered Osterhout's medium by adding 1 cm3
mof — KH2 P04 to 30 cm3
of diluted salt solution, and bringing the
reaction of the medium with the aid of — KOH to pH = 8.0. At
the same time we isolated new pure lines of Paramecia out of our wild
culture, as the Paramecia which had been cultivated for a long time
on Osterhout's medium could not stand a sudden transfer into a
buffered medium.
COMPETITION FOR COMMON FOOD IN PROTOZOA 105
In order to dimmish the concentration of the bacteria we made a
new smaller standard loop for preparing the "one-loop medium," and
also arranged experiments in which the one loop medium was diluted
twice ("half-loop medium") . The data obtained are given in Table 4
(Appendix) where every figure represents a mean value from the observations
of two microcosms. This material is represented graphically
in Figures 23, 24 and 25.
Let us examine Figure 23. The curves of growth of pure populations
of P. caudatum and P. aurelia with different concentrations of
the bacterial food show that the lack of food is actually a factor limiting
growth in these experiments. With the double concentration of
food the volumes of the populations of the separately growing species
also increase about twice (from 64 up to 137 in P. caudatum;^ X 2 =
128; from 105 up to 195 in P. aurelia; 105 X 2 = 210). Under these
TABLE XI
Parameters of the logistic curves for separate growth of Paramecium caudatum
and Paramecium aurelia
Buffered medium with the "half-loop"
106 THE STRUGGLE FOR EXISTENCE
Figures 22, 24 and 25 is almost the same, but there are certain differences
concerning secondary peculiarities. For a detailed acquaintance
with the properties of a mixed population we will consider the
growth with a half-loop concentration of bacteria (Fig. 24). First of
all we see that as in the case examined before the competition between
our species can be divided into two separate stages: up to the fifth
80 . P. caudatum.
9*m W
K^€V
• f
Separately
v,aw«, &&&L
Jn mixedpopulation
a
Fig. 24. The growth of the "volume" in Paramecium caudatum and Paramecium
aurelia cultivated separately and in the mixed population on the buffered
medium with the "half-loop" concentration of bacteria. From Gause
C34d).
day there is a competition between the species for seizing the so far
unutilized food energy; then after the fifth day of growth begins the
redistribution of the completely seized resources of energy between
the two components, which leads to a complete displacement of one
of them by another. The following simple calculations can convince
one that on the fifth day all the energy is already seized upon. At
COMPETITION FOR COMMON FOOD IN PROTOZOA 107
the expense of a certain level of food resources which is a constant one
in all "half-loop" experiments and may be taken as unity, P. aurelia
growing separately produces a biomass equal to 105 volume units,
and P. caudatum 64 such units. Therefore, one unit of volume of
P. caudatum consumes ^T = 0.01562 of food, and one unit of volume
Zoo
P.aure/fa
Separately
Jn mixed population
P. caudatum
19
Days
Fig. 25. The growth of the "volume" in Paramecium caudatum and Paramecium
aurelia cultivated separately and in the mixed population on the
buffered medium with the "one-loop" concentration of bacteria. From Gause
C34d).
of P. aurelia -jfe = 0.00952. In other words, one unit of volume of
P. caudatum consumes 1.64 times as much food as P. aurelia, and
the food consumption of one unit of volume in the latter species constitutes
but 0.61 of that of P. caudatum. These coefficients enable
us to recalculate the volume of one species into an equivalent in
respect to the food consumption volume of another species.
108 THE STRUGGLE FOR EXISTENCE
On the fifth day of growth of a mixed population the biomass of
P. caudatum (in volume units) is equal to about 25, and of P. aurelia
to about 65. If we calculate the total of these biomasses in equivalents
of P. aurelia, we shall have: (25 X 1.64) -f- 65 = 106 (maximal
free growth of P. aurelia is equal to 105). The total of the biomasses
expressed in equivalents of P. caudatum will be (65 X 0.61) + 25 =
65 (with the free growth 64). This means that on the fifth day of
growth of the mixed population the food resources of the microcosm
are indeed completely taken hold of.
(4) The first period of competition up to the fifth day is not all
so simple as we considered it in the theoretical discussion of the third
chapter, or when examining the population of yeast cells. The nature
of the influence of one species on the growth of another does not
remain invariable in the course of the entire first stage of competition,
and in its turn may be divided into two periods. At the very beginning
P. caudatum grows even somewhat better in a mixed population
than separately (analogous to Fig. 22), apparently in connection
with more nearly optimal relations between the density of Paramecia
and that of the bacteria in accordance with the already mentioned
data of Johnson ('33). At the same time P. aurelia is but very
slightly oppressed by P. caudatum. As the food resources are used
up, the Johnson effect disappears, and the species begin to depress
each other as a result of competition for common food.
It is easy to see that all this does not alter in the least the essence
of the mathematical theory of the struggle for existence, but only
introduces into it a certain natural complication: the coefficients of
the struggle for existence, which characterize the influence of one
species on the growth of another, do not remain constant but in their
turn undergo regular alterations as the culture grows. The curves
of growth of every species in a mixed population in Figure 24 up to
the fifth day of growth have been calculated according to the system
of differential equations of competition with such varying coefficients.
In the first days of growth the coefficient 13 is negative and near to
— 1, i.e., instead of —(3Ni we obtain -J-JVi- In other words, the
presence of P. aurelia does not diminish, but increases the possibility
of growth of P. caudatum, which proceeds for a certain time
with a potential geometrical rate, outrunning the control culture
(
2
—1
remains near to unity I. At this time the coefficient
COMPETITION FOR COMMON FOOD IN PROTOZOA 109
a is equal to about +0.5; in other words, P. aurelia suffers from a
slight depressing influence of P. caudatum. Later the inhibitoryaction
of one species upon the growth of another begins to manifest
itself more and more in proportion to the quantity of food consumed,
because the larger is the part of the food resources already consumed
the less is the unutilized opportunity for growth. In our calculations
for P. caudatum from the second and for P. aurelia from the fourth
days of growth we have identified the coefficients of competition with
the coefficients of the relative food consumption, i.e., a = 1.64, /3 =
0.61. It is obvious that this is but a first approximation to the actual
state of things where the coefficients gradually pass from one value
to another. The entire problem of the changes in the coefficients of
the struggle for existence in the course of the growth of a mixed population
(which apparently are in a great measure connected with the
fact that the Paramecia feed upon living bacteria) needs further
detailed investigations on more extensive experimental material than
we possess at present.
(5) It remains to examine the second stage of the competition,
i.e., the direct displacement of one species by another. An analysis
of this phenomenon can no longer be reduced to the examination of
the coefficients of multiplication and of the coefficients of the struggle
for existence, and we have to do in the process of displacement with a
quite new qualitative factor: the rate of the stream which is represented
by population having completely seized the food resources.
As we have already mentioned in Chapter III, after the cessation of
growth a population does not remain motionless and in every unit of
time a definite number of newly formed individuals fills the place of
those which have disappeared during the same time. Among different
animals this can take place in various ways, and a careful biological
analysis of every separate case is here absolutely necessary. In
our experiments the principal factor regulating the rapidity of this
movement of the population that had ceased growing was the following
technical measure: a sample equal to yu of the population was
taken every day and then destroyed. In this way a regular decrease
in the density of the population was produced and followed by the
subsequent growth up to the saturating level to fill in the loss.
During these elementary movements of thinning the population
and filling the loss, the displacement of one species by another took
place. The biomass of every species was decreased by ro daily.
110 THE STRUGGLE FOR EXISTENCE
Were the species similar in their properties, each one of them would
again increase by to, and there would not be any alteration in the
relative quantities of the two species. However, as one species grows
quicker than another, it succeeds not only in regaining what it has
lost but also in seizing part of the food resources of the other species.
Therefore, every elementary movement of the population leads to a
diminution in the biomass of the slowly growing species, and produces
its entire disappearance after a certain time.
(6) The recovery of the population loss in every elementary movement
is subordinate to a system of the differential equations of competition.
In the present stage of our researches we can make use of
these equations for only a qualitative analysis of the process of displacement.
They will show us exactly what particular species in the
population will be displaced. However, the quantitative side of the
problem, i.e., the rate of the displacement, still requires further experimental
and mathematical researches and we will not consider it at
present.
The qualitative analysis consists in the following. Let us assume
that the biomass of each component of the saturating population is
decreased by yu- Then according to the system of differential equations,
inserting the values of the coefficients of multiplication and of
the coefficients of food consumption, we shall be able to say how each
one of the components can utilize the now created possibility for
growth. The result of the calculations shows that P. aurelia, primarily
owing to its high coefficient of multiplication, has an advantage
and increases every time comparatively more than P. caudatum*
In summing up we can say that in spite of the complexity of the
process of competition between two species of infusoria, and as one
may think a complete change of conditions in passing from one period
of growth to another, a certain law of the struggle for existence which
may be expressed by a system of differential equations of competition
remains invariable all the time. The law is that the species possess
definite potential coefficients of multiplication, which are realized at
2
It is obvious that in these calculations it is necessary to introduce varying
coefficients of the struggle for existence. At the same time with our technique
of cultivation corrections to the "elementary movements" must be also included
in an analysis of the first stage of growth of a mixed population (an
approximation to the asymptote). But at the present stage of our researches
we have neglected them.
COMPETITION FOR COMMON FOOD IN PROTOZOA 111
every moment of time according to the unutilized opportunity for
growth. We have only had to change the interpretation of this unutilized
opportunity.
(7) It seems reasonable at this point to coordinate our data with
the ideas of the modern theory of natural selection. It is recognized
that fluctuations in numbers resembling the dilutions we have artificially
produced in our microcosms play in general a decisive role in
the removal of the less fitted species and mutations (Ford, '30). An
interesting mathematical expression of this process proposed by
Haldane ('24, '32) can be formulated thus: how does the rate of
increase of the favorable type in the population depend on the
value of the coefficient of selection k? In its turn the coefficient of
selection characterizes an elementary displacement in the relation
between the two types per unit of time—one generation. Therefore
the problem resolves itself into a determination of the functional relationship
between the increase of concentration of the favorable type
and the elementary displacement in its concentration. A recent
theoretical paper by Ludwig ('33) clearly shows how the fluctuation
in the population density alters the relation between the two types
owing to the fact that one of them has a somewhat higher probability
of multiplication than the other. It seems to us that there is a great
future for the Volterra method here, because it enables us not to
begin the theory by the coefficient of selection but to calculate theoretically
the coefficient itself starting from the process of interaction
between the two species or mutations.
IV
(1) How complicated are processes of competition under the conditions
approaching those of nature can be seen from the experiments
made by Gause, Nastukova and Alpatov ('35). They studied the influence
of biologically conditioned media on the growth of a mixed
population of Paramecium caudatum and P. aurelia. The analysis
of the relative adaptation of the two species at different stages of
population growth has shown that P. caudatum has an advantage
over P. aurelia in the coefficient of geometric increase (in the absence
of Bacillus pyocyaneus in the nutritive medium which in the experiments
described above inhibited P. caudatum by its waste products)
whilst P. aurelia surpasses P. caudatum in the resistance to waste
products. Therefore if the decisive factor of competition is a rapid
112 THE STRUGGLE FOR EXISTENCE
utilization of the food resources, P. caudatum has an advantage over
P. aurelia; but if the resistance to waste products is the essential
point, then P. aurelia will take place of P. caudatum.
It is interesting to note also that in the complicated situation of
these experiments the superiority of one species over another in competition
did not simply reflect the properties of these species taken
independently, but was often essentially modified by the process of their
interaction.
P.cauditum
(tS.pust) Q
pays
Fig. 26. The growth of the number of individuals of Sttjlonychia pustulata
cultivated separately, and in the mixed populations with Paramecium caudatum
and Paramecium aurelia (on the medium of Osterhout).
(2) If we turn to the population growth of Stylonychia pustulata
and its competition with two species of Paramecium, we shall encounter
extremely complicated processes. The corresponding data
are given in Table 5 (Appendix) and Figure 26. These experiments
were made with Osterhout's medium containing Bacillus pyocyaneus,
simultaneously with those mentioned above. Therefore, the data
on the separate growth of P. caudatum and P. aurelia given in Appendix
Table 3 serve as a control for these experiments.
First of all, the separate growth of Stylonychia pustulata is very
COMPETITION FOR COMMON FOOD IN PROTOZOA 113
peculiar: having attained a certain maximum, the density of population
decreases and remains stationary at a lower level. Direct
observation shows that the bacteria at the close of the twenty-four
hour intervals between the changes of medium remain partly unconsumed,
and the limiting factor here is apparently an accumulation
of waste products and not an insufficiency of food. The fluctuations
in the density of population of the separately growing S. pustulata
are probably connected with some complex processes of the influence
of metabolic products on growth. As to the mixed populations, the
same regularity with which we had to deal previously repeats itself
here : one species finally completely displaces another, and the species
displaced is always Stylonychia pustulata.
(3) Let us summarize the data of this chapter. We have studied
the competition between two species for a source of energy kept continually
at a certain level. This process may be divided into two
periods. In the first period the two species compete for the still unutilized
resources of energy. In what proportion this energy will be
distributed between the two species is determined by the system of
Vito Volterra's differential equations of competition, but the coefficients
of the struggle for existence in these equations change in
the course of the growth of the population and are therefore more
complicated than in the preceding chapter. In the second period there
is but a redistribution of the completely seized energy between the two
species, which is again controlled by the differential equations of competition.
Owing to its advantages, mainly a greater value of the
coefficient of multiplication, one of the species in a mixed population
drives out the other entirely.
Chapter VI
THE DESTRUCTION OF ONE SPECIES BY ANOTHER
(1) In the two preceding chapters our attention has been concentrated
on the indirect competition, and we have to turn now to an
entirely new group of phenomena of the struggle for existence, that
of one species being directly devoured by another. The experimental
investigation of just this case is particularly interesting in connection
with the mathematical theory of the struggle for existence developed
on broad lines by Vito Volterra. Mathematical investigations have
shown that the process of interaction between the predator and the
prey leads to periodic oscillations in numbers of both species, and
all this of course ought to be verified under carefully controlled laboratory
conditions. At the same time we approach closely in this chapter
to the fundamental problems of modern experimental epidemiology,
which have been recently discussed from a wide viewpoint by
Greenwood in his Herter lectures of 1931. The epidemiologists feel
that the spread of microbial infection presents a particular case of
the struggle for existence between the bacteria and the organisms
they attack, and that the entire problem must pass from the strictly
medical to the general biological field.
(2) As the material for investigation we have taken two infusoria
of which one, Didinium nasutum, devours the other, Paramecium
caudatum (Fig. 27). Here, therefore, exists the following food chain:
bacteria —* Paramecium —> Didinium. This case presents a considerable
interest from a purely biological viewpoint, and it has more
than once been studied in detail (Mast ('09), Reukauf ('30), and
others). The amount of food required by Didinium is very great
and, as Mast has shown, it demands a fresh Paramecium every three
hours. Observation of the hunting of Didinium after the Paramecia
has shown that Didinium attacks all the objects coming into contact
with its seizing organ, and the collision with suitable food is simply
due to chance (Calcins '33). Putting it into the words of Jennings
('15) Didinium simply "proves all things and holds fast to that which
is good."
114
DESTRUCTION OF ONE SPECIES BY ANOTHER 115
All the experiments described further on were made with pure
lines of Didinium ("summer line") and Paramecium. In most of
the experiments the nutritive medium was the oaten decoction, "with
sediment" or "without sediment," described in the preceding chapter.
Attempts were also made to cultivate these infusoria on a synthetic
medium with an exactly controlled number of bacteria for the Paramecia,
but here we encountered great difficulties in connection with
differences in the optimal physicochemical conditions for our lines
of Paramecium and Didinium. The introduction of a phosphate
buffer and the increase of the alkalinity of the medium above pH =
Fig. 27. Didinium nasutum devouring Paramecium caudatum
6.8-7.0 has invariably favored the growth of Paramecium, but hindered
that of Didinium. Satisfactory results have been obtained on
Osterhout's medium, but here also Didinium has grown worse than
on the oaten medium. Therefore, absolute values of growth under
different conditions can not be compared with one another though all
the fundamental laws of the struggle for existence remained the same.
The experiments were made in a moist thermostat at a temperature
of 26°C.
(3) Let us first of all analyze the process of interaction between
the predator and the prey from a qualitative point of view. It is well
known that under natural conditions periodic oscillations in the numbers
of both take place but in connection with the complexity of the
116 THE STRUGGLE FOR EXISTENCE
situation it is difficult to draw any reliable conclusions concerning
the causes of these oscillations. However, quite recently Lotka
(1920) and Volterra (1926) have noted on the basis of a purely mathematical
investigation that the properties of a biological system consisting
of two species one of which devours the other are such that
they lead to periodic oscillations in numbers (see Chapter III).
These oscillations should exist when all the external factors are invariable,
because they are due to the properties of the biological
system itself. The periods of these oscillations are determined by
certain initial conditions and coefficients of multiplication of the
species. Mathematicians arrived at this conclusion by studying the
properties of the differential equation for the predator-prey relations
which has already been discussed in detail in Chapter III (equation
21a). Let us now repeat in short this argument in a verbal
form. When in a limited microcosm we have a certain number of
prey (Ni), and if we introduce predators (N2), 1
there will begin a
decrease in the number of prey and an increase in that of the predators.
But as the concentration of the prey diminishes the increase
of the predators slows down, and later there even begins a certain
dying off of the latter resulting from a lack of food. As a result of
this diminution in the number of predators the conditions for the
growth of the surviving prey are getting more and more favorable,
and their population increases, but then again predators begin to
multiply. Such periodic oscillations can continue for a long time.
The analysis of the properties of the corresponding differential equation
shows that one species will never be capable of completely destroying
another: the diminished prey will not be entirely devoured by the
predators, and the starving predators will not die out completely,
because when their density is low the prey multiply intensely and in a
certain time favorable conditions for hunting them arise. Thus a
population consisting of homogeneous prey and homogeneous predators
in a limited microcosm, all the external factors being constant, must
according to the predictions of the mathematical theory possess periodic
oscillations in the numbers of both species. 2
These oscillations may be
1
It is assumed that all individuals of prey and predator are identical in their
properties, in other words, we have to do with homogeneous populations.
2
According to the theory, such oscillations must exist in the case of one
component depending on the state of another at the same moment of time, as
well as in the case of a certain delay in the responses of one species to the
changes of the other.
DESTRUCTION OF ONE SPECIES BY ANOTHER 117
called "innate periodic oscillations," because they depend on the
properties of the predator-prey relations themselves, but besides
these under the influence of periodic oscillations of external factors
there generally arise "induced periodic oscillations" in numbers depending
on these external causes. The classic example of a system
which is subject to innate and induced oscillations is presented by the
pendulum. Thus the ideal pendulum the equilibrium of which has
been disturbed will oscillate owing to the properties of this system
during an indefinitely long time, if its motion is not impeded. But
in addition to that we may act upon the pendulum by external forces,
and thereby cause induced oscillations of the pendulum.
If we are asked what proof there is of the fact that the biological
system consisting of predator-prey actually possesses "innate"
periodic oscillations in numbers of both species, or in other terms that
the equation (21a) holds true, we can give but one answer: observations
under natural conditions are here of no use, as in the extremely
complex natural environment we do not succeed in eliminating "induced"
oscillations depending on cyclic changes in climatic factors
and on other causes. Investigations under constant and exactly
controlled laboratory conditions are here indispensable. Therefore,
in experimentation with two species of infusoria one of which devours
the other the following question arose at the very beginning:
does this system possess "innate" periodic oscillations in numbers,
which are to be expected according to the mathematical theory?
(4) The first experiments were set up in small test tubes with
0.5 cm3
of oaten medium (see Chapter V). If we take an oaten medium
without sediment, place in it five individuals of Paramecium
caudatum, and after two days introduce three predators Didinium
nasutum, we shall have the picture shown in Figure 28. After the
predators are put with the Paramecia, the number of the latter
begins to decrease, the predators multiply intensely, devouring all
the Paramecia, and thereupon perish themselves. This experiment
was repeated many times, being sometimes made in a large vessel
in which there were many hundreds of thousands of infusoria. The
predator was introduced at different moments of the growth of population
of the prey, but nevertheless the same result was always produced.
Figure 29 gives the curves of the devouring of Paramecia by
Didinium when the latter are introduced at different moments of the
growth of the prey population (in 0.5 cm3
of oaten medium without
118 THE STRUGGLE FOR EXISTENCE
sediment). This figure shows the decrease in the number of Paramecia
as well as the simultaneous increase in number and in volume
of the population of Didinium. (We did not continue these curves
beyond the point where Didinium attained its maximal volume.) It
is evident that the Paramecia are devoured to the very end. As it is
necessary that the nutritive medium should contain a sufficient
quantity of bacteria in order to have an intense multiplication of
Paramecia, we arranged also experiments in the test tubes on a daily
changed Osterhout's medium containing Bacillus pyocyaneus (see
Chapter V). In Figure 30 are given the results of such an experiment
which has led up, as before, to the complete disappearance of
both Paramecium and Didinium. Thus we see that in a homogeneous
P. caudatum
D. nasutum
P>-.
Days
Fig. 28. The elementary interaction between Didinium nasutum and Paramecium
caudatum (oat medium without sediment). Numbers of individuals
pro 0.5 c.c. From Gause ('35a).
nutritive medium under constant external conditions the system
Paramecium-Didinium has no innate periodic oscillations in numbers.
In other words, the food chain: bacteria —> Paramecium —» Didinium
placed in a limited microcosm, with the concentration of the first link
of the chain kept artificially at a definite level, changes in such a direction
that the two latter components disappear entirely and the food resources
of the first component of the chain remain without being utilized
by any one.
We have yet to point out that the study of the properties of the
predator-prey relations must be carried out under conditions favorable
for the multiplication of both prey and predator. In our case,
there should be an abundance of bacteria for the multiplication of
DESTRUCTION OF ONE SPECIES BY ANOTHER 119
Paramecia, and suitable physicochemical conditions for the verysensitive
Didinium. It is self-evident that if at the very beginning
!o
a)
JS Number of Didinium
f Volume ofDidinium
zo
-jL_ b
Number of Paramecia
/
So
<t>
to
^3o
J"
lot
to
€0
*o .
Zo .
Number of Paramecin
I
(c)
Numter of Didinium , ,- —
y °
/A // Volume of
/ Oidimunx
-Jt—s_
Number of
/ Paramecia
Number of
Paramecia
Paramecium
separately
Days
Fig. 29. The destruction of Paramecium caudatum by Didinium nasutum.
(a) Growth of P. caudatum alone, (b) Didinium is introduced at the very
beginning of growth of Paramecia population, (c) Didinium is introduced
after 24 hours, (d) Didinium is introduced after 36 hours, (e) Didinium is
introduced after 48 hours. Numbers of individuals pro 0.5 c.c.
we set up unfavorable conditions under which Didinium begins to
degenerate, and as a result is unable to destroy all the prey, or if the
120 THE STRUGGLE FOR EXISTENCE
diminishing prey should perish not in consequence of their having
been devoured by the predators but from other causes, we could not
be entitled to draw any conclusions in respect to the properties of the
predator-prey relations in the given chain.
(5) We may be told that after we have "snatched" two components
out of a complex natural community and placed them under "artificial"
conditions, we shall certainly not obtain anything valuable
and shall come to absurd conclusions. We will therefore point out
beforehand that under such conditions it is nevertheless possible to
obtain periodic oscillations in the numbers of the predators and prey,
if we but introduce some complications into the arrangement of the
"^1200
DESTRUCTION OF ONE SPECIES BY ANOTHER 121
a vast decrease in the size of the individual. The following data give
an idea of the diminution in size of Didinium : three normal individuals
of this species placed in a medium free of Paramecia continue
to multiply intensely, and in an interval of 24 hours give on an average
7.1 small individuals able to attack the prey. This vast increase
of the "seizing surface" represents, metaphorically speaking, those
"tentacles by means of which the predators suck out the prey completely."
Translating all this into mathematical language, we can
say: the function characterizing the consumption of prey by predators
[fi(Ni, N2)], as well as the natality and the mortality of predators
[F(Ni, N2)],* are apparently more complicated than Lotka and Volterra
have assumed in the equation (21a), and as a result the corresponding
process of the struggle for existence has no periodic properties.
We shall soon return to a further analysis of this problem along
mathematical lines.
11
(1) We have but to introduce a slight complication into the conditions
of the experiment, and all the characteristic properties of our
biological system will be altogether changed. In order to somewhat
approach natural conditions we have introduced into the microcosm
a "refuge" where Paramecia could cover themselves. For this purpose
a dense oaten medium "with sediment" was taken (see Chapter
V). Direct observations have shown that while the Paramecia are
covered in this sediment they are safe from the attack of predators.
It must be noted that the taxis causing the hiding of Paramecia in this
"refuge" manifests itself in a like manner in the presence of the predators
as in their absence. We
must have a clear idea of the role which a refuge plays in the
struggle for existence of the species under observation, as a lack of
clearness can lead further on to serious misunderstandings. If
Didinium actively pursued a definite Paramecium which escaping
from it hid in the refuge, the presence of the refuge would be a definite
parameter in every elementary case of one species devouring another.
In other words, the nature and the distribution of refuges would
constitute an integral part of the expressions fi(Ni, iV2 ) and F(Nh N2 )
of the corresponding differential equation of the struggle for existence.
* See Chapter III, equation (21).
122 THE STRUGGLE FOR EXISTENCE
Such a situation has recency been analyzed by Lotka ('32a). We
might be told in this case that in experimenting with a homogeneous
microcosm without refuges we have sharply disturbed the process of
elementary interaction of two species. Instead of investigating "in
a pure form" the properties of the differential equation of the struggle
for existence we obtain a thoroughly unnatural phenomenon, and
all the conclusions concerning the absence of innate oscillations in
numbers will be entirely unconvincing. But for our case this is not
true. We have already mentioned that Dininium does not actively
hunt for Paramecia but simply seizes everything that comes in its
way. In its turn Paramecium fights with the predator by throwing
out trichocysts and developing an intense rapidity of motion, but
never hiding in this connection in the refuge of our type. In this manner,
we have actually isolated and studied "in a pure form" the elementary
phenomenon of interaction between the prey and the predators
in a homogeneous microcosm. The refuge in our experiment
presents a peculiar "semipermeable membrane," separating off a part
of the microcosm into which Paramecium can penetrate owing to its
taxis, in general quite independently of any pursuit of the predator,
and which is impenetrable for Didinium.
When the microcosm contains a refuge the following picture can
be observed (see Fig. 31): if Paramecium and Didinium are simultaneously
introduced into the microcosm, the number of predators
increases somewhat and they devour a certain number of Paramecia,
but a considerable amount of the prey is in the refuge and the predators
cannot attain them. Finally the predators die out entirely owing
to the lack of food, and then in the microcosm begins an intense multiplication
of the Paramecia (no encystment of Didinium has been
observed in our experiments) . We must make here a technical note
the microcosm under observation ought not to be shaken in any way,
as any shock might easily destroy the refuge and cause the Paramecia
to fall out. On the whole it may be noted that when there appears a
refuge in a microcosm, a certain threshold quantity of the prey cannot
be destroyed by the predators. The elementary process of predatorprey
interaction goes on to the very end, but the presence of a certain
number of undestroyed prey in the refuge creates the possibility of
the microcosm becoming later populated by the prey alone.
(2) Having in the experiment with the refuge made the microcosm
a heterogeneous one, we have acquired an essential difference of the
DESTRUCTION OF ONE SPECIES BY ANOTHER 123
corresponding process of the struggle for existence from all the elementary
interactions between two species which we have so far
examined. In the case of an elementary interaction between predator
and prey in a homogeneous microcosm very similar results were
obtained in various analogous experiments (see Table 6, Appendix).
In any case the more attention we give to the technique of experimentation,
the greater will be this similarity. In other terms, in a
homogeneous microcosm the process of the struggle for existence in
every individual test tube was exactly determined by a certain law,
and this could be expressed by more or less complex differential equa-
St>
124 THE STRUGGLE FOR EXISTENCE
Or else (2) as shown in Figure 31a certain number of prey might have
in the refuge been entirely out of reach of the predators, and the
latter will perish finally from lack of food. (3) Lastly, prey may
from time to time leave the refuge and be taken by the predators; as a
result a mixed population consisting of prey and predators will
continue to exist for a certain time. All this depends on the circumstance
that in our experiments the absolute numbers of individuals
were not large, and the amplitude of fluctuations connected with
multiplicity of causes proved to be wider than these numbers.
(3) Let us consider the corresponding data. In one of the experiments
30 microcosms were taken (tubes with 0.5 cm3
of oaten medium
with sediment), in each of them five Paramecium and three Didinium
were placed, and two days after the population was counted . It turned
out that in four microcosms the predators had entirely destroyed
the prey whilst in the other 26 there were predators as well as prey.
The number of prey fluctuated from two to thirty-eight. In another
experiment 25 microcosms were examined after six days; in eight of
them the predators had died out entirely and prey alone remained.
Therefore, in the initial stage for every individual microcosm we can
only affirm with a probability of -£5 that it will develop in the direction
indicated in Figure 31. Certain data on the variability of populations
in individual microcosms are to be found in Table 7 (Appendix).
Further experimental investigations are here necessary. First of all
we had to do with too complicated conditions in the microcosms owing
to variability of refuges themselves. It is not difficult to standardize
this factor and to analyze its role more closely.
In concluding let us make the following general remarks. When
the microcosm approaches the natural conditions (variable refuges)
in its properties, the struggle for existence begins to be controlled by
such a multiplicity of causes that we are unable to predict exactly
the course of development of each individual microcosm. 3
From the
language of rational differential equations we are compelled to pass
on to the language of probabilities, and there is no doubt that the
corresponding mathematical theory of the struggle for existence may
be developed in these terms. The physicists have already had to
3
This means only that the development of each individual microcosm is influenced
by a multiplicity of causes, and it would be totally fallacious to conclude
that it is not definitely "caused." All our data have of course no relation
to the concept of phenomenal indeterminism.
DESTRUCTION OF ONE SPECIES BY ANOTHER 125
face a similar situation, and it may be of interest to quote their usual
remarks on this subject: "Chance does not confine itself here to introducing
small, practically vanishing corrections into the course of
the phenomenon; it entirely destroys the picture constructed upon
the theory and substitutes for it a new one subordinated to laws of
its own. In fact, if at a given moment an extremely small external
factor has caused a molecule to deviate very slightly from the way
planned for it theoretically, the fate of this molecule will be changed
in a most radical manner: our molecule will come on its way across
a great number of other molecules which should not encounter it, and
at the same time it will elude a series of collisions which should have
taken place theoretically. All these 'occasional' circumstances in
their essence are regular and determined, but as they do not enter
into our theory they have in respect to it the character of chance"
(Chinchin, '29, pp. 164-165).
(4) If we take a microcosm without any refuge wherein an elementary
process of interaction between Paramecium and Didinium is
realized, and if we introduce an artificial immigration of both predator
and prey at equal intervals of time, there will appear periodic oscillations
in the numbers of both species. Such experiments were made
in glass dishes with a flat bottom into which 2 cm3
of nutritive liquid
were poured. The latter consisted of Osterhout's medium with a
two-loop concentration of Bacillus pyocyaneus, which was changed
from time to time. The observations in every experiment were made
on the very same culture, without any interference from without (except
immigration) into the composition of its contents. At the
beginning of the experiment and every third day thereafter one Paramecium
-f- one Didinium were introduced into the microcosm. The
predator was always taken when already considerably diminished in
size; if it did not find any prey within the next 12 hours, it usually
degenerated and perished. Figure 32 represents the results of one of
the experiments. Let us note the following peculiarities : (1) At the
first immigration into the microcosm containing but few Paramecia
the predator did not find any prey and perished. An intense growth
of the prey began. (2) At the time of the second immigration the
concentration of the prey is already rather high, and a growth of the
population of the predator begins. (3) The third immigration took
place at the moment of an intense destruction of the prey by the
predators, and it did not cause any essential changes. (4) Towards
126 THE STRUGGLE FOR EXISTENCE
the time of the fourth immigration the predator had already devoured
all the prey, had become reduced in size and degenerated. The prey
introduced into the microcosm originates a new cycle of growth of the
prey population. Such periodic changes repeat themselves further on.
Comparing the results of different similar experiments with immigration
made in a homogeneous microcosm, we come to the same
conclusions as in the preceding paragraph. Within the limits of each
cycle when there is a great number of both Paramecium and Didinium
it is possible by means of certain differential equations to predict the
course of the process of the struggle for existence for some time to
|
I
DESTRUCTION OF ONE SPECIES BY ANOTHER 127
33. In a homogeneous microcosm the process of elementary interaction
between the predator and the prey led up to the disappearance
of both the components. By making the microcosm heterogeneous
»»
£
Homogeneous microcosm without immigrations
Bacteria +Pa£&*f?oa *^TsU<itunx
Heterogeneous microcosm without immigrations
Bacteria* Parametia •* 0£ibk(t7m
Homogeneous microcosm with immigrations
Bacteria -» Paramecin + Pidinium
prey
Time —r
Fig. 33. A schematic representation of the results of a qualitative analysis
of the predator-prey relations in the case of two Infusoria.
(refuge) and thus approaching the natural conditions we began to
deal with a "probability" of change in various directions. The predator
sometimes dies out and only prey populate the microcosm. By
128 THE STRUGGLE FOR EXISTENCE
introducing immigration into a homogeneous microcosm we obtain
periodic oscillations in the numbers of both species.
in
(1) The above given example shows that in Paramecium and
Didinium the periodic oscillations in the numbers of the predators
and of the prey are not a property of the predator-prey interaction
itself, as the mathematicians suspected, but apparently occur as a
result of constant interferences from without in the development of
this interaction. There is evidence for believing that this is characteristic
for more than our special case. Jensen ('33) in his monograph
on periodic fluctuations in size of various stocks of fish concluded
that he has not found any periodicity identic to those treated
by Volterra. The same conclusion was arrived at by S. Severtzov
('33) dealing with vertebrates. There are also plenty of entomological
observations showing the possibility of a complete local extermination
of hosts by parasites. We may according to Cockerell ('34) recall
some observations on Coccidae (scale insects) made in New
Mexico. Certain species occur on the mesquite and other shrubs
which exist in great abundance over many thousands of square miles
of country. Yet the coccids are only found in isolated patches here
and there. They are destroyed by their natural enemies, but the
young larvae can be blown by the wind or carried on the feet of birds,
and so start new colonies which flourish until discovered by predators
and parasites. This game of hide-and-seek results in frequent local
exterminations, but the species are sufficiently widespread to survive
in parts of their range, and so continue indefinitely. Such local exterminations
in grayfish have been recently observed by Duffield
C33).
(2) Experimental epidemiology is the one domain where the problems
of a direct struggle for existence have already been submitted to
an exact analysis in laboratory conditions. Therefore let us consider
in brief the results there obtained. If we took a microcosm of any
size populated by homogeneous organisms, not allowing any immigration
or emigration, and if we caused it to be fatally infected, we should
obtain a complete dying out of the organisms (if among them there
were no immune ones, and if they were unable to acquire any immunity).
In other terms, we should have before us the well-known
elementary interaction of two species. However, the process of dying
DESTRUCTION OF ONE SPECIES BY ANOTHER 129
out does not usually go on to the end owing to the heterogeneity of
population and presence of immune individuals, which are in a certain
degree equivalent to the individuals protected in the refuge:
they are not carried away by the process of destruction which goes on
to the end among the non-immune ones. The nature of this "refuge"
is very complicated, and it is interesting to quote here the following
words of Topley ('26, pp. 531-532) : "Most of us who have been concerned
at all with the problem of immunity have been accustomed
to take the individual as our unit. When we take as our unit not the
individual but the herd, entirely new factors are introduced. Herdresistance
must be studied as a problem sui generis. One factor
peculiar to the development of communal as opposed to individual
immunity may be referred to here. A herd may clearly increase its
average resistance by a process of simple selection, by the elimination
through death of its more susceptible members. . . . That some process
of active immunization will be associated with the occurrence of
non-fatal infection may safely be assumed, though its degree and
importance may be very difficult to assess, so that we must allow for
the possibility that the spread of infection which is killing some of
our hosts is immunizing others.
"A very imperfect analogy may help to depict the position. Suppose
we take a number of stakes of different thickness, plant them
in the ground and expose them to bombardment with stones of varying
size from catapults of varying strengths. After a certain time
we shall find that a number of the stakes have been broken. This
will not have happened to many of the thicker stakes, but other survivors
will consist of thinner stakes, around which ineffective missiles
have formed a protective armour. Survivors of the latter class are
in a precarious state; subsequent bombardment may displace the
protective heap, and perhaps add its impetus to that of the new missile.
Survivors of the former class may eventually be destroyed by
a missile of sufficient momentum."
(3) The experiments of epidemiologists dealing with the influence
of immigration on the course of an epidemic among mice in a limited
microcosm are also very interesting. One can distinctly see here
that a continuous interference from without acting upon a definite
population causes periodic oscillations in the epidemic which disappear
immediately as the interference ceases. Let us quote Topley
again: "When susceptible mice gain access to the cage at a steady
130 THE STRUGGLE FOR EXISTENCE
rate the deaths are not uniformly distributed in time, nor do they
occur in a purely random fashion. They are grouped in a series of
waves, each wave showing minor fluctuation. The equilibrium
between parasite and host seems to be a shifting one. As the result
of some series of changes, the parasite appears to obtain a temporary
mastery, so that a considerable proportion of the mice at risk fall
victims to a fatal infection. This is followed by a phase in which
there is a decreased tendency for the occurrence of fatal infection, and
the death-rate falls. As fresh susceptibles accumulate this succession
of events is repeated, and the deaths increase to a fresh maximum,
only to fall again when this maximum is passed." But if only "no
such immigration occur the epidemic gradually dies down, leaving a
varying number of survivors."
We can conclude that the process of elementary interaction between
the homogeneous hosts and the homogeneous bacterial population possesses
no "classical" periodic variations. Without wishing to adopt at once
the preconceived opinion that such a phenomenon is generally impossible,
we ought in any case require a clear demonstration of its
possibility. This demonstration will be really given below.
IV
(1) Turning back again from empirical observations to the general
principles let us note that there can exist two different types of innate
periodic oscillations in the systems, as Hill ('33) has noted recently in
connection with physiological problems. One of them which was
assumed by Lotka-Volterra and which we have searched above must
be called a "classical" fluctuation and it is entirely analogous to wellknown
oscillations in physics arising as the consequence of the reaction
with one another of properties analogous to inertia and elasticity.
A changing system tends, on one hand, to maintain its state of motion
because it possesses mass, whilst on the other the force of elasticity
increases according to the removal from equilibrium and ultimately
reverses the motion or change. In the classical theory of
biological population the predator tends to multiply indefinitely, but
by a removal in this way from an equilibrium with the prey the change
in the predator population becomes reversed, later again replaced by
an increase, and so on (equation 21a).
There is, however, another type of oscillation with which physiologists
are concerned and to which apparently belong the spread of
DESTRUCTION OF ONE SPECIES BY ANOTHER 131
epidemics and fluctuations in our protozoan population. A certain
potential or a certain state is here built up by a continuous process
and the conditions become less and less stable until a state is reached
at which a discharge (or epidemics) must take place. It is evident
that the interaction between the two components instead of periodicity
leads here to an interruption of contact (depending from specific
biological conditions in the case of epidemics and from a disappearance
of predators and prey in our Protozoa), and then ceases until the
next critical threshold. Such oscillations with an interruption of contact
bear in physics the name of "relaxation oscillations."
AJ (concentration of the predator
Fig. 34. Diagrams illustrating two types of innate periodic fluctuations in
numbers of animals. 1. "Classical" fluctuation of Lotka-Volterra. 2. "Relaxation
fluctuations."
It is easy to visualize the difference between these two types of
oscillations employing the illuminating graphs so often used by Lotka.
On the coordinate paper we usually plot time on the abscissae and
densities of predators (iV2) and prey (iVi) on the ordinates. But if
we abstract from time and plot iV2 on the abscissae and Ni on the
ordinates we obtain a clear idea of the nature of interspecific interaction.
As Figure 34 shows in the case of classical oscillation we must
have a closed curve.4
4
The transformation of the usual time-curves into such graphs is illustrated
by a numerical example reproduced in Fig. 35. The upper part of it presents a
132 THE STRUGGLE FOR EXISTENCE
Let us now turn our attention to the graph for the relaxation interaction.
Suppose we introduce a definite amount of the predator, iV2 ,
at different densities of the prey (Ni). Then, before the critical
threshold of the latter is reached (N°, Fig. 34), an epidemic of Didinium
cannot start and the curves return on the ordinate. After the
60
.S. 70
t^ io
^
SO
SO
<0
5 V*
% 3*
/V2 (predator)
20(30)
O 10 20 ZO VO SO tO
Individuals (Mi)-*
Fig. 35. Diagram illustrating the transformation of the usual time-curves
(1) into relative graphs of interaction (2) for the "classical" Lotka-Volterra
fluctuation in numbers.
critical threshold is reached there appears a relaxation which leads to
the destruction of the prey—the curves cross the abscissa.
(2) This little amount of theory enables us to formulate our problem
thus: How do the biological adaptations consisting of a very
active consumption of Paramecium by Didinium disturb the conditheoretical
case of the classical Volterra's oscillation in the usual form. If we
note the values of Ni and JV2 at different moments of time, and then plot Ni
against the corresponding N2, we shall obtain the closed curve reproduced
below.
DESTRUCTION OF ONE SPECIES BY ANOTHER 133
tions of the classical equation (21a) and transform it into that of an
elementary relaxation? For all the technical details the reader is
referred to the original paper (Gause and Witt, '35), and we will discuss
here only its essential ideas.
In a first approximation to the actual state of affairs we can write
an elementary equation of relaxation. It can be admitted [on the
basis of the observations on Paramecium (Ni) and Didinium (Nz)]
that if iV2 is large the mortality of the predators is negligible when
Ni > 0. In addition, the increase of predators only slightly depends
on Ni (with an insufficiency of prey the predators continue to multiply
at the expense of a decrease in size of the individuals; in this connection
the consumption of prey but slightly depends on iVi).
Introducing these assumptions into the equation (21) we write
_ d#?
= where N ^ and - ^r- = *#« where Nt = 0.* To
dt dt
reduce the dependence upon Nx of the members characterizing the
interaction of species, we substitute y/Ni to iVi.f Then
dNj
dt
dN2
If
= biNi - fciiV2 VNl
= b«Nt \/Wx Ctfi * 0)
= - (kN* (Ni = 0) J
(21b)
Figure 36 shows that the solution of the equation 21b (the integral
curves on the graph Ni,Ni) actually coincides with biological observations
on Paramecium and Didinium. It is therefore safe to assume
that the general equation of the destruction of one species by another
(21) takes in our special case the form (21b) instead of the classical
expression of Lotka-Volterra (21a).
(3) The equation of relaxation (21 b) represents but a first approximation
to the actual state of things, and is true only if Nx or iV2 are
large. Looking at the trend of the experimental curves on the surface
Ni, N2 with small densities (Fig. 37) we notice that they pass from
the right to the left and cross the ordinate (Fig. 37, a). This means
* This condition is already sufficient for an exclusion of the "classical"
periodic fluctuations.
f Special experiments show that this substitution is satisfactory (Gause
and Witt, '35).
134 THE STRUGGLE FOR EXISTENCE
that "an epidemic" of predators cannot break out if the concentration
of the prey has not attained the threshold value ah . Below it
predators disappear5
and leave a pure population of prey, but above
it we find usual relaxations.
Taking into account all these features we can write for Paramecium
and Didinium a complicated equation of relaxation representing an
adequate expression of what actually exists. We admit that the mortality
of predators appears not only with Ni = 0, but that a slight
mortality generally exists increasing with a diminution of the concenTheoreticz/
curves calculated
from the equation fiib)
^(Didi nilim) pro 0,S~cc -S*
Fig. 36. The solution of the equation 21b (to the left) and empirical observations
on Paramecium and Didinium (to the right). No "residual growth" of
the population of predators (in the absence of the prey) is taken into account
in the theoretical equation. From Gause and Witt, '35.
tration of N2 , and that the intensity of hunting also increases with an
insufficiency of prey.
6
In these experiments were used predators possessing no 'residual growth,'
e.g. already diminished in size. See experiments with immigrations where
predators usually did not find any prey in the microcosms containing very few
Paramecia, and consequently perished.
DESTRUCTION OF ONE SPECIES BY ANOTHER 135
The solution of this complicated equation6
is represented on Figure
38. It is a further concretization for Paramecium and Didinium of
the principle of relaxation represented on Figure 34. An epidemic
Nl(Otdiniuny)Fig.
37. The interaction between Paramecium and Didinium at different
densities of population. The absence of the "residual growth" (comp. Fig. 36,
right) as well as the differences between both curves are connected with slightly
unfavorable conditions of the medium.
of predators cannot start below the threshold in the concentration of
the prey, but above it we find usual relaxations. A characteristic
feature of our food-chain is an extraordinarily low value of the
threshold.
6
The equation given by Gause and Witt ('35) is:
dNi
dt
dm.
dt
= biNi - MNi) NiNt
= b2 Ni VNl -f(N2 ) iVi^O
d2 N2 Ni = Oj
(21c)
136 THE STRUGGLE FOR EXISTENCE
A mathematical analysis of the properties of the complicated equation
of relaxation given in Fig. 38 shows that there is a point on the
map of the curves of interaction which is usually called "singular
point." The powers of mortality, natality and interaction of prey
and predators are so balanced that the "classical" oscillations in
numbers are theoretically possible around it. But in the case of
Paramecium and Didinium the coordinates of this singular point are
exceedingly small. In other terms the zone of possible classical oscillations
is displaced here to such small densities that these oscillations
Line of Aortgo/rda/ tangents
1/
A/z (Didinium) -^
Fig. 38. The solution of the complicated equation of relaxation (21c).
ah represents the threshold concentration of the prey.
are completely annihilated by the statistical factors which are much
more powerful in this zone.
(4) In conclusion let us consider the appearance of periodic variations
in numbers under the influence of immigrations (a slight and
synchronous inflow of Ni and N2 after intervals of time t). In other
terms we have to deal here with the problem of the influence of small
impulses. At the origin (a, Fig. 37) they lead to a return of the curve
to the ordinate. Relaxations arise when the concentration of Ni rises
above the threshold. From Figure 38 it is easy to calculate how a
DESTRUCTION OF ONE SPECIES BY ANOTHER 137
delay of the inflow after the threshold has been reached increases the
dimensions of the relaxations (the importance of this problem for
epidemiology has been pointed out by Kermack and McKendrick
('27)). When relaxation is going on, slight impulses do not disturb it
seriously until crossing of the abscissa by the integral curves, and later
on up to their intersection with the line of horizontal tangents (Fig.
38). After this the impulses lead to a return on the ordinate and the
process begins again.
(1) The theory of the preceding section shows that the consumption
of one species by another in the population studied is so active
that the classical oscillations in numbers are transformed into an
elementary relaxation and the coordinates of the singular point
around which such oscillations could be theoretically expected are
exceedingly small. This fact except its independent interest enables
us to predict that were we in a position to reduce the intensity of
consumption we could increase the coordinates of the singular point,
and in this way observe the classical oscillations of Lotka-Volterra.
The situation is entirely analogous to that of classical physiology.
The rate of propagation of nervous impulses is under usual conditions
too high and it is sometimes desirable to decrease it, to cool the nerve,
in order to be able to observe certain phenomena. How can one decrease
the intensity of consumption of one species by another?
The simplest way is to investigate a system where this intensity is
naturally low. This has been recently made by Gause ('35b) who
analyzed the properties of the food chain consisting of Paramecium
bursaria and Paramecium aurelia devouring small yeast cells, Schizosaccharomyces
pombe and Saccharomyces exiguus. Special arrangements
allowed of controlling artificially the mortality of predators by
rarefying them, and of avoiding the settling of yeast cells on the
bottom by a slow mixing of the medium. Figure 39 shows that under
such specialized conditions fluctuations of the Lotka-Volterra type
actually take place, and in this manner the conditions of the equation
(21a) are realized in general features.
It must be remarked that the equation (21a) does not hold absolutely
true because the oscillations do not apparently belong to the
"conservative" type. In other terms they do not keep the magnitude
initially given them but tend to an inherent magnitude of their own
138 THE STRUGGLE FOR EXISTENCE
(compare the first and second cycles on Figure 39, i and 2 ). This
problem, however, requires further investigations.
(2) It is interesting to compare our data with some observations
A/z
—P.bunaria (predator)
N, — S.pomSz (pre.y)
no noo Vo So
1^2.- Pzurelia (predator) -=9
Fig. 39. "Classical" periodic fluctuations in a mixed population of Paramecium
bursaria and Schizosaccharomyces pombe in the usual (1) and in the
relative form (2). (3) Periodic fluctuations in a population of Paramecium
aurelia and Saccharomyces exiguus in the relative form. According to Gause,
'35b.
made at the Rothamsted Station (Cutler, '23, Russell, '27) which also
attracted the attention of Nikolson ('33). There is reason to suspect
that the fluctuations in the numbers of soil organisms observed byCutler
and referred to by Russell are interspecific oscillations. "Bac-
DESTRUCTION OF ONE SPECIES BY ANOTHER 139
teria do not fluctuate in numbers when grown by themselves in
sterilized soil; they rise to high numbers and remain at approximately
a constant level. Their numbers fall, however, as soon as the soil
amoebae are introduced, but no constant level is reached; instead
there are continuous fluctuations as in normal soils. There is a
sharp inverse relationship between the numbers of bacteria and those
of active amoebae; when the numbers of amoebae rise, those of
bacteria fall." It is hardly possible therefore to avoid the conclusion
that Cutler had to deal here (in a complicated form) with classical
periodic variations of the Volterra type.
In conclusion let us note that this final demonstration of the possibility
of "classical" oscillations showed that very specialized conditions
are required for their realization, and it is therefore easy to
understand why in real biological systems with their typical adaptations
leading to very intensive attacks of one species on another the
so much discussed "relaxation interaction" between the species apparently
predominates.
VI
(1) In Chapter III we have pointed out that the connection between
the relative increase of the predator and the number of prey is
not a linear one, and that this is of significance for the processes of one
species devouring another. We can now be convinced that this
connection is actually non-linear.
Recently Smirnov and Wladimirow ('34) have investigated under
laboratory conditions the connection between the density of the hosts
Ni (pupae of the fly Phormia groenlandica) and the relative increase
of the parasite — —^ [the progeny of one pair ( c? + 9 ) of a parai\'
2 dt
sitic wasp, Mormoniella vitripennis, per generation]. The experimental
data they obtained are represented in Figure 40. As the
density of the hosts increases, the relative increase of the parasite
increases also until it reaches the maximal possible or "potential increase"
(62) from one pair under given conditions. That the curve
showing the connection between — —^- and iVi can actually be exJ\
2 dt
pressed by the equation [Chapter III, (23)]:
N2 at
140 THE STRUGGLE FOR EXISTENCE
can be seen in the following manner: by plotting as ordinates the
values log ( 62 — j^- —j^ ) corresponding to different values of abscissae
Ni, we should obtain a straight line which, as Figure 40 shows, is actually
observed. The slope of this straight line is characterized by the
coefficient X, which thus expresses the rate at which the relative increase
of the parasites approaches its maximal value with the increase
of the density of hosts.
(2) To summarize: We expected at the beginning of this chapter
to find "classical" oscillations in numbers arising in consequence of the
continuous interaction between predators and prey as was assumed
o
I
APPENDICES
APPENDIX I 143
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TABLE 2
Alcohol production in Saccharomyces cerevisiae and Schizosaccharornyces kephir
under aerobic and anaerobic conditions (1932)
146 THE STRUGGLE FOR EXISTENCE
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APPENDIX I 147
TABLE 6
The growth of a mixed population consisting of Didinium nasutum and
Paramecium caudatum in the medium without sediment
Variations of individual cultures
Number of individuals of
P.c.
D.n.
in the tube with 0.5 c.c. of the medium. Each
II. APPENDIX TO CHAPTER IV
CALCULATION OF THE COEFFICIENTS OF MULTIPLICATION AND
OF THE EQUATIONS OF THE STRUGGLE FOR EXISTENCE
We have had to deal frequently with the coefficients of geometric increase or
coefficients of multiplication as well as with the equations of competition.
For calculating the latter a very simple approximate integration by the method
of Runge-Kutta is necessary, with which, however, most biologists are not
familiar. Therefore we will briefly give here the technical details of the calculations
and examine a concrete example.
/. We possess experimental data on the separate growth of the first and second
species. We wish to calculate the maximal masses, Ki and K2 , and the coefficients
of multiplication, bi and b2 .
Example: The growth of Schizosaccharomyces kephir under anaerobic conditions
(1931). Placing the experimental data on the graph we determine
approximately the maximal mass K (Fig. 41). For calculation of the parameter
b let us apply the method elaborated by Pearl and Reed. For every
K - N
individual observation (Nu N2 , etc.) we take K — N, and then ——— and log
K-N K-N———-. Placing the calculated log ——— against the corresponding values of
time (t) we should obtain a straight line in the case of a symmetrical S-shaped
curve (Fig. 41, bottom). We draw this straight line approximately, approaching
the experimental observations.
Some brief theoretical explanations are needed here. The differential equadN
K — N
tion of Pearl's logistic curve :
= bN ——— after integration takes this
if zr at
form: N = . Hence it follows that ea ~bt
= ———, and therefore
1 + ea ~bt
N
. K-N .
, .
K-Nlog ——— = (a — bt) loge. Taking natural logarithms we obtain: In ——— =
P
K- N
a — bt. Now it is clear that if we place the Napierian logarithms of ——
against the absolute values of time (t) we must obtain a straight line whose
tangent will be the required coefficient of multiplication b, whilst a characterizes
the position of the zero ordinate. Their calculation is reduced to
determining the equation of the straight line we have drawn approximately.
For this purpose we measure the ordinates of two points of the straight line,
for instance t = 0, logio y = m, and t = 80, logio y = n. Passing on to the natural
logarithms, i.e. multiplying m by 2.3026 we obtain directly the coefficient a
(a — Ob = m X 2.3026). We can then easily calculate b in this way : a — 806 =
149
150 THE STRUGGLE FOR EXISTENCE
n X 2.3026. In order to verify the calculations it is necessary to insert all the
-pr
parameters into the logistic equation: N = -and by substituting differ-
1 + e ° bt
ent values in place of t to find the "calculated curve," which must approach
closely the observations. A more detailed account is to be found in the book
of Raymond Pearl: An introduction to medical biometry and statistics, and in
the paper of Gause and Alpatov ('31).
2. Calculation of the equations of the struggle for existence. We have deterzo
VO €o So
Hoursft)
too no
Fig. 41. Calculation of the logistic curve for the separate growth of the
yeast, Schizosaccharomyces kephir, grown under anaerobic conditions (1931).
mined the maximal masses and the coefficients of multiplication : 6i, b2 ,
K\, K2 ,
and following the instructions of Chapter IV we have calculated the coefficients
of the struggle for existence a and /3. The curves corresponding to the system
of the differential equations of the struggle for existence must now be found
for a verification of the numerical values of a and 13.
We have a system of equations:
dNi t xr
A', - (Ni + aN2 )
-aT
= blNl
K,
dm . __ K% - (AT2 + ffli)
—- = b2 N2 =;
dt A2
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+
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151
152 THE STRUGGLE FOR EXISTENCE
where the numerical values of the parameters are already known, and where it
is necessary to calculate the values Ni and 2V2 corresponding to different
moments of time. As this system of differential equations cannot be solved at
present, we will use the method of approximate numerical integration of
Runge-Kutta.
§1. Ordinates at origin. The system of differential equations does not include
the special parameters characterizing the values N\ and N2 with t =
(corresponding to parameter a of the logistic curve). We know these values
from the experiment, and they correspond to those quantities of the first and
second species which are introduced into a mixed population at the very beginning.
The same quantities are also introduced into the separate populations.
Therefore, according to the logistic curves of separate growth we calculate
the value Ni with t = 0, and N2 with t = 0. The quantities obtained are
taken for the initial ordinates of the system of differential equations.
§2. Calculations. The meaning of the calculations is that knowing the
value of Ni and iV2 at t = 0, we give to time a certain increment h and for this
new moment (t + h) we calculate according to the system of the equations of
competition the corresponding increments of the populations of the first and
second species: (Ni + k) and (iV2 + I). These calculations are made with the
aid of the formulae of Runge-Kutta, which for our case take the form given in
Table 8.
dN\ , dN2
The initial values of the ordinates enable us to calculate —— and ——
at at
Multiplying them by h we obtain ki, and h. Thereupon in the second line we
calculate again and starting no longer from the initial ordinates but
dt dt
with (
Ni H
—-J
and ( N2 + - ) . Multiplying the result by h we obtain k2 and
h, and continue to calculate until we obtain fci, k 2 , k 3 ,
k A andZi, l 2 , h, h. We then
pass on to the right side of the table. Here we calculate § (fci + fc 4 ) and then
k2 + k 3 . Summing and taking § of the sum we obtain the required increment
of the first species equal to k. The increment of the second species is calculated
in the same way.
The ordinates thus calculated (Ni + k) and (N2 + 1) can in their turn be considered
as already known initial values, and following the same plan we can
calculate the new increments corresponding to the moment of time (t + 2h).
Continuing these calculations we will find the growth curves of the first and
second species in a mixed population. Table 9 presents a numerical example
of the calculation of such curves for Saccharomyces (No. 1) and Schizosaccharomyces
(No. 2) in a mixed anaerobic culture according to the experiments of 1931.
§3. Final values ofNi and N2 . The process of calculating will bring us to the
moment when and — will be near to zero, and we will approach the final
dt dt
values of Ni and N2 (the experiments described in Chapter IV were made in
conditions of limited resources of energy, where on the disappearance of the
unutilized opportunity growth simply ceased). If in the approximation to
APPENDIX II 153
TABLE 9
The numerical example of the approximate integration of the system of equations
of the struggle for existence with the aid of the method of Runge-Kutta
a = 3.15 Ki = 13.0 units of volume b 1
= 0.21827 units per hour h = 10
= 0.439 K2 = 5.8 units of volume b2 = 0.06069 units per hour hours
t
BIBLIOGRAPHY1
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1
This list of literature includes mainly references cited in the text.
154
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78. L'Heritier, Ph. et Teissier, G. 1933 Etude d'une population de
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91. Mast, S. O. 1909 The reactions of Didinium nasutum with special
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94. Nageli, C. 1874 Verdrangung der Pflanzenformen durch ihre Mitbewerber.
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97. Osterhout, W. J. V. 1906 On the importance of physiological balanced
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98. Pavlov, J. P. 1923 Twenty years of objective study of the higher nervous
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99. Payne, N. M. 1933 The differential effect of environmental factors
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100. Pearl, R. and Reed, L. J. 1920 On the rate of growth of the population
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101. Pearl, R. 1928 The rate of living. New York.
102. Pearl, R. 1930 Introduction to medical biometry and statistics. 2nd
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103. Philpot, C. H. 1928 Growth of Paramecium in pure cultures of pathogenic
bacteria and in the presence of soluble products of such bacteria.
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104. Plate, L. 1913 Selektionsprinzip und Probleme der Artbildung.
Vierte Auflage. Leipzig und Berlin.
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Vergl. Physiol. 11, p. 689.
106. Richards, O. W. 1928a Potentially unlimited multiplication of yeast
with constant environment, and the limiting of growth by changing
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108. Richards, O. W. 1928c The growth of the yeast Saccharomyces cerevisiae.
I. Ann. Bot. 42, p. 271.
109. Richards, O. W. and Roope, P. L. 1930 A tangent meter for graphical
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111. Ross, R. 1908 Report on the prevention of malaria in Mauritius.
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112. Ross, R. 1911 The prevention of malaria. London.
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vol. 2.
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pour la vie. Paris.
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134. Winsor, C. P. 1932 The Gompertz curve as a growth curve. Proc.
Nat. Acad. Sci. 18, p. 1.
160 THE STRUGGLE FOR EXISTENCE
135. Wood, F. E. 1910 A study of the mammals of Champaign Country,
Illinois. Bull. 111. Nat. Hist. Surv. 8, p. 501.
136. Woodruff, L. L. 1911 The effect of excretion products of Paramecium
on its rate of reproduction. Journ. Exp. Zool. 10, p. 551.
137. Woodruff, L. L. 1912 Observations on the origin and sequence of the
protozoan fauna of hay infusions. Journ. Exp. Zool. 12, p. 205.
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INDEX
Adaptation, in plant communities,
17.
Adaptation, in protozoan populations,
132 et seq., 140.
Alechin, W. W., 17, 18.
Allee, W. C, vi, 11.
Alpatov, W. W., 105, 111, 150.
Bacillus pyocyaneus, 97, 105, 118, 125.
Bacillus subtilis, 92.
Baer, von, 8.
Bailey, V. A., 25, 57.
Barker, H. A., 96.
Beauchamp, R., 17.
Beers, C. D., 154.
Belar, A., 96.
Biotic experiments, 13.
Birstein, J. A., 22.
Bodenheimer, F. S., 25.
Boltzmann, L., 90, 91.
Braun-Blanquet, J., 18.
Brunelli, G., 25.
Buffon, 8.
Bumpus, v.
Calcins, G. N., 114.
Chance, its role in the struggle for
existence, 18, 19, 124, 125.
Chapman, R. N., 33.
Chinchin, A. J., 125.
Classification of the struggle for
existence, 3.
Clements, F. E., 4, 5, 15.
Cockerell, T. D. A., 128.
Coefficient of selection, 111.
Coefficients of the struggle for existence,
their calculation, 80.
Competition in field conditions, its
complexity, 3, 12, 13, 111.
Cutler, D. W., 138, 139.
Cyprinus albus, 21.
D'Ancona, U., 25.
Darby, H. H., 104.
Darwin, Ch., v, 1, 2, 8, 14, 40.
De Candolle, A. P., 17.
Dewar, D., 155.
Didinium nasutum, 114 et seq., 132
et seq.
Direct struggle for existence, definition,
3.
Drosophila funebris, 17.
Drosophila melanogaster, 17.
Duffield, J. E., 128.
Du-Rietz, G. E., 19.
Eddy, S., 91.
Ehrenberg, 8.
Elementary process of the struggle
for existence, definition, 2.
Elton, Ch., 12, 19.
Embody, G. C, 21.
Empirical equations, their meaning,
59, 60.
Ephestia, 25.
Euphorbia, 18.
Equation of competition, 44 et seq.
Experimental study of the struggle
for existence, methods of, 6,
62 et seq., 92, 96 et seq.
Fisher, R. A., 155.
Ford, E. B., 111.
Formosov, A. N., 19.
Fray, W., 96.
Gause, G. F., 44, 50, 62, 68, 72, 77 et
seq., 94, 101 et seq., Ill, 118 et
seq., 133 et seq., 137 et seq., 150.
Gibbs, 10.
Goldman, E. A., 22.
Gray, J., 43.
Greenwood, M., 114.
161
162 INDEX
Haldane, J. B. S., 40, 48, 98, 111.
Hanson, H. C, 155.
Hargitt, G. T., 96.
Harris, v.
Hartmann, M., 96.
Hill, A. V., 130.
Humboldt, 8.
Immigration, its role in the struggle
for existence, 20, 125 et seq.
Indirect competition, definition, 3.
Intensity of the struggle for existence,
13, 14, 15, 16, 38, 39, 40,
41, 42.
Jennings, H. S., 32, 96, 114.
Jensen, A. J. C, 128.
Johnson, W. H., 96, 97, 103, 104, 108.
Jollos, V., 96.
Kalabuchov, N. J., 24.
Kalmus, H., 99.
Kashkarov, D. N., 21.
Kepler, 42.
Kermack, W. O., 137.
Kessler, J. Th., 22.
Klem, A., 65, 69, 70, 74.
Konig, H., 153.
Lasareff, P. P., 42.
Leuciscus, 21.
L'Heritier, Ph., 157.
Linnaeus, 8.
Logistic curve, 34 et seq., 149, 150.
Lotka, A. J., vi, 10, 28, 31, 32, 44,
48, 51, 53, 54, 57, 98, 116, 121, 122,
130 et seq., 140.
Ludwig, W., 111.
Malaria equation, 27, 28, 29, 30, 31.
Marchi, C, 25.
Mast, S. O., 114.
Mathematical investigation of the
struggle for existence, essence of,
32, 33, 60, 61.
Matricaria, 14, 15.
McKendrick, A. G., 137.
Microbracon, 25.
Montgomery, E. G., 15, 19.
Mormoniclla vitripennis, 139, 140.
Multiplication of organisms, laws of,
7, 8, 9.
Myers, E. C, 93.
Nageli, C, 6.
Nastukova, O., 105, 111.
Newton, 42.
Niche, its role in the struggle for
existence, 19, 20, 48.
Nicholson, A. J., 25, 138.
Oehler, R., 96.
Oscillations in numbers, their nature,
115 et seq., 130 et seq.
Osterhout, W. J. V., 96.
Paramecium aurelia, 97 et seq., 137,
138.
Paramecium bursaria, 137, 138.
Paramecium caudatum, 36, 37, 38, 39,
41, 93 et seq., 97 et seq., 114 et
seq., 132 et seq.
Park, T., vi.
Past history of the system, its role
in competition, 57.
Patchossky, 18.
Pavlov, J. P., 1, 2.
Payne, N. M., 25.
Pearl, R., 9, 34, 35, 42, 49, 59, 71, 95,
149, 150.
Pearson, K., v, vi.
Perca flavescens, 22.
Pervozvansky, 62.
Philpot, C. H., 97.
Phormia groenlandica, 139, 140.
Planaria gonocephala, 18.
Planaria montenegrina, 18.
Plate, L., 3, 7.
Potamobius astacus, 22.
Potamobius leptodactylus, 22.
Raewski, W., 24.
Reed, L. J., 9, 34, 149.
INDEX 163
Refuge, its r61e in the struggle for
existence, 121, 122, 123, 124.
Relaxation fluctuations, 131 et seq.
Reukauf, E., 114.
Richards, O. W., 65, 66, 67, 68, 69, 70,
71, 73, 74, 77.
Ross, R., 9, 10, 27, 28, 30, 31, 32, 46,
49.
Runge, C, 149, 152, 153.
Russell, E. J., 138.
Saccharomyces cerevisiae, 62 et seq.,
77 et seq., 152.
Saccharomyces exiguus, 137, 138.
Salvelinus fontinalis, 22.
Sandon, H., 96.
Schizosaccharomyces kephir, 62 et
seq., 77 et seq., 149 et seq.
Schizosaccharomyces pombe, 137, 138.
Schizothorax, 21.
Severtzov, N. A., 13.
Severtzov, S. A., 128.
Shelford, V. E., 23.
Skadovsky, S. N., 91.
Slator, A., 69.
Smirnov, E., 139, 140.
Stanley, J., 57.
Sterna anglica, 20.
Sterna cantiaca, 20.
Sterna fluviatilis, 20.
Sterna minuta, 20.
Stylonychia mytilus, 93 et seq.
Stylonychia pustulata, 97, 98, 112, 113.
Sukatschev, W. N., 13, 14, 15, 16, 17.
Tansley, A. G., 17.
Taraxacum, 16, 17.
Taylor, C. V., 96.
Teissier, G., 157.
Thomson, 13.
Timofeeff-Ressovsky, N. W., 17.
Topley, W. W. C., 129.
Tycho-Brahe, 42.
Ullyott, P., 17.
Verhulst, P. E., 34, 35, 42.
Vernadsky, V. J., 7, 90.
Vinogradov, L. G., 22.
Volterra, V., vi, 10, 25, 44, 48, 49,
50, 51, 52, 53, 54, 57, 59, 89, 98,
111, 114, 116, 121, 128, 130 et
seq., 140.
Wallace, 8.
Warming, E., 19.
Weaver, J. E., 155.
Weldon, v.
Winsor, C. P., 50.
Witt, A. A., 133 et seq.
Wladimirow, M., 139, 140.
Wood, F. E., 23.
Woodruff, L. L., 91, 103.
Yule, G. U., vi.
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