Stano Pekár“Populační ekologie živočichů“
dN
= Nr
dt
aim: to simulate (predict) what can happen
model is tested by comparison with observed data
realistic models - complex (many parameters), realistic, used to
simulate real situations
strategic models - simple (few parameters), unrealistic, used for
understanding the model behaviour
a model should be:
1. a satisfactory description of diverse systems
2. an aid to enlighten aspects of population dynamics
3. a system that can be incorporated into more complex models
deterministic models - everything is predictable
stochastic models - including random events
discrete models:
- time is composed of discrete intervals or measured in generations
- used for populations with synchronised reproduction (annual species)
- modelled by difference equations
continuous models:
- time is continual (very short intervals) thus change is instantaneous
- used for populations with asynchronous and continuous overlapping
reproduction
- modelled by differential equations
STABILITY
how population changes in time
stable equilibrium is a state (population
density) to which a population will
move after a perturbation
stable equilibrium
unstable equilibrium
focus on rates of population processes
number of cockroaches in a living room increases:
- influx of cockroaches from adjoining rooms → immigration [I]
- cockroaches were born → birth [B]
number of cockroaches declines:
- dispersal of cockroaches → emigration [E]
- cockroaches died → death [D]
population increases if I + B > E + D
rate of increase is a summary of all events (I + B - E - D)
growth models are based on B and D
spatial models are based on I and E
Blatta orientalisEDBINN tt −−++=+1
Population processes are independent of its density
Assumptions:
immigration and emigration are none or ignored
all individuals are identical
natality and mortality are constant
all individuals are genetically similar
reproduction is asexual
population structure is ignored
resources are infinite
population change is instant, no lags
Used only for
relative short time periods
closed and homogeneous environments (experimental chambers)
for population with discrete generations (annual reproduction), no
generation overlap
time (t) is discrete, equivalent to generation
exponential (geometric) growth
Malthus (1834) realised that any species can potentially increase in
numbers according to a geometric series
N0 .. initial density
b .. birth rate (per capita)
Discrete (difference) model
11 −− −=∆ tt dNbNN
11 )( −− −=− ttt NdbNN
1)1( −−+= tt NdbN
N
B
b =
N
D
d =
d .. death rate (per capita)
λ=−+ db1 Rdb =−
R+=1λ
R .. demographic growth rate
- shows proportional change (in percentage)
λ .. finite growth rate, per capita rate of growth
λ = 1.23 then R = 0.23
.. 23% increase
number of individuals is multiplied each time - the larger the
population the larger the increase
time0
Ntt
t
tt
i
i
1
21
1
1
)...( λλλλλ =





= ∏=
λ < 1
λ > 1
λ = 1
Average of finite growth rates
- estimated as geometric mean
λ1−= tt NN
t
t NN λ0=
λλλ 012 NNN ==
if λ is constant, population number in generations t is equal to
Comparison of discrete and continuous generations
populations that are continuously reproducing, with overlapping
generations
when change in population number is permanent
derived from the discrete model
Nt
time
Continuous (differential) model
t
t NN λ0=
)ln()ln()ln( 0 λtNNt +=
)ln()ln()ln( 0 λtNNt =−
)ln(
1
d
d
λ=
Nt
N
)ln(
d
d
λN
t
N
=
Solution of the differential equation:
- analytical or numerical
at each point it is possible to determine the rate of change by
differentiation (slope of the tangent)
when t is large it is approximated by the exponential function
time
N
r .. intrinsic rate of natural increase,
instantaneous per capita growth rate
r < 0
r > 0
r = 0
Nr
t
N
=
d
d
)ln(λ=rif
Nr
t
N
=
d
d
r
Nt
N
=
1
d
d
∫∫ =
TT
trN
N 00
dd
1
)0()ln()ln( 0 −=− TrNNT
rT
N
NT
=







0
ln
rt
t eNN 0=
rTT
e
N
N
=
0
t
t NN λ0= rt
t eNN 0=
rtt
e=λ
)ln(λ=r
r versus λ
r is symmetric around 0, λ is not
r = 0.5 ... λ = 1.65
r = -0.5 ... λ = 0.61
doubling time: time required for a
population to double
r
t
)2ln(
=
Demography - study of organisms with special attention to stage
or age structure
processes are associated to age, stage or size
x .. age/stage/size category
px .. age/stage/size specific survival
mx .. reproductive rate (expected average number of offspring per
female)
x
x
x
S
S
p 1+
=
main focus on births and deaths
immigration & emigration is
ignored
no adult survive
one (not overlapping)
generation per year
egg pods over-winter
despite high fecundity they just
replace themselves
Chorthippus
Richards & Waloff (1954)
Annual speciesAnnual species
breed at discrete
periods
no overlapping
generations
BBiennaliennal speciesspecies
breed at discrete periods
adult generation may overlap
adults
adults
0
birth
t0
t1
adults
pre-adults
0
birth
t0
t1
adults t2
p
pre-adults
birth
t0
t1adults
t2
0
pre-adults
p
breed at discrete periods
breeding adults consist of
individuals of various ages (1-5 years)
adults of different generations are
equivalent
overlapping generations
PerennialPerennial speciesspecies
Parus major
Perins (1965)
age/stage
classification is based
on developmental time
size may be more
appropriate than age
(fish, sedentary
animals)
Hughes (1984) used
combination of
age/stage and size for
the description of coral
growth
AgeAge--sizesize--stage lifestage life--tabletable
Agaricia agaricites
show organisms‘ mortality and reproduction as a function of age
examination of a population during
one segment (time interval)
- segment = group of individuals of
different cohorts
- designed for long-lived organisms
ASSUMPTIONS:
- Birth rate and survival are constant
over time
- population does not grow
DRAWBACKS: confuses age-specific changes in e.g.
mortality with temporal variation
Static (vertical) life-tables
Cervus elaphus
Sx .. number of survivors
Dx .. number of dead individuals
lx .. standardised number of
survivors
qx .. age-specific mortality
px .. age-specific survival
Lowe (1969)
0S
S
l x
x =
x
x
x
S
D
q =
x Sx Dx lx px qx mx
1 129 15 1.000 0.884 0.116 0.000
2 114 1 0.884 0.991 0.009 0.000
3 113 32 0.876 0.717 0.283 0.310
4 81 3 0.628 0.963 0.037 0.280
5 78 19 0.605 0.756 0.244 0.300
6 59 -6 0.457 1.102 -0.102 0.400
7 65 10 0.504 0.846 0.154 0.480
8 55 30 0.426 0.455 0.545 0.360
9 25 16 0.194 0.360 0.640 0.450
10 9 1 0.070 0.889 0.111 0.290
11 8 1 0.062 0.875 0.125 0.280
12 7 5 0.054 0.286 0.714 0.290
13 2 1 0.016 0.500 0.500 0.280
14 1 -3 0.008 4.000 -3.000 0.280
15 4 2 0.031 0.500 0.500 0.290
16 2 2 0.016 0.000 1.000 0.280
1+−= xxx SSD
x
x
x
l
l
p 1+
=
examination of a population in a cohort = a group of individuals born
at the same period
followed from birth to death
provide reliable information
designed for short-lived organisms
only females are included
Cohort (horizontal) life-table
Vulpes vulpes
x Sx Dx lx px qx mx
0 250 50 1.000 0.800 0.200 0.000
1 200 120 0.800 0.400 0.600 0.000
2 80 50 0.320 0.375 0.625 2.000
3 30 15 0.120 0.500 0.500 2.100
4 15 9 0.060 0.400 0.600 2.300
5 6 6 0.024 0.000 1.000 2.400
6 0 0 0.000
survival and reproduction depend on stage / size rather than age
age-distribution is of no interest
used for invertebrates (insects, invertebrates)
time spent in a stage / size can differ
Lymantria dispar
Campbell (1981)
x Sx Dx lx px qx mx
Egg 450 68 1.000 0.849 0.151 0
Larva I 382 67 0.849 0.825 0.175 0
Larva II 315 158 0.700 0.498 0.502 0
Larva III 157 118 0.349 0.248 0.752 0
Larva IV 39 7 0.087 0.821 0.179 0
Larva V 32 9 0.071 0.719 0.281 0
Larva VI 23 1 0.051 0.957 0.043 0
Pre-pupa 22 4 0.049 0.818 0.182 0
Pupa 18 2 0.040 0.889 0.111 0
Adult 16 16 0.036 0.000 1.000 185
display change in survival by plotting log(lx) against age (x)
sheep mortality increases with age
survivorship of lapwing (Vanellus) is independent of age
but survival of sheep is age-dependent
Pearls (1928) classified hypothetical age-specific mortality:
Type I .. mortality is concentrated at the end of life span (humans)
Type II .. mortality is constant over age (seeds, birds)
Type III .. mortality is highest in the beginning of life (invertebrates,
fish, reptiles)
ln(Survivorship)
0
Type I
Type II
Type III
1
Time
fecundity - potential number of offspring
fertility - real number of offspring
semelparous .. reproducing once a life
iteroparous .. reproducing several times during life
birth pulse .. discrete reproduction
(seasonal reproduction)
birth flow .. continuous
reproduction
Numberofoffsprings
0
Time
reproductivepre-reproductive post-reproductive
0
0.1
0.2
0.3
0.4
0.5
0.6
0 20 40 60 80 100 120 140
Time [Days]
Fecundity
Triaeris stenapsis
Geospiza scandens
number.ofbirths/individual
0
0.4
age 16
Cervus elaphus
Odocoileus
numberofbirths/individual
0 6age
0.8
k-value - killing power - another measure of mortality
k-values are additive unlike q
Key-factor analysis - a method to identify the most important
factors that regulates population dynamics
k-values are estimated for a number of years
important factors are identified by regressing kx on log(N)
)log(pk −=
x
kK ∑=
over-wintering adults emerge in June → eggs are laid in
clusters on the lower side of leafs → larvae pass through 4 instars
→ form pupal cells in the soil → summer adults emerge in August
→ begin to hibernate in September
mortality factors overlap
Leptinotarsa decemlineata
Harcourt (1971)
highest k-value indicates the role of a factor in each generation
profile of a factor parallel with the K profile reveals the key
factor
emigration is the key-factor
Summary over 10 years
model of Leslie (1945) uses parameters (survival and fecundity) from
life-tables
where populations are composed of individuals of different age, stage
or size with specific natality and mortality
generations are not overlapping
reproduction is asexual
used for modelling of density-independent processes (exponential
growth)
Nx,t .. no. of organisms in age x and time t
Gx .. probability of persistence in the same size/stage
Fx .. age/stage specific fertility (average no. of offspring per female)
px .. age/stage specific survival
class 0 is omitted
number of individuals in the first age class
number of individuals in the remaining age class
∑=
+ ++==
n
x
ttxtxt FNFNFNN
1
2,21,1,1,1 ...
xtxtx pNN ,1,1 =++
N1 N2 N3 N4
Age-structured
p12 p23 p34
F4
F3
F2
F1














=














×












+
+
+
+
1,4
1,3
1,2
1,1
,4
,3
,2
,1
34
23
12
4321
000
000
000
t
t
t
t
t
t
t
t
N
N
N
N
N
N
N
N
p
p
p
FFFF
each column in A specifies fate of an organism in a specific age:
3rd column: organism in age 2 produces F2 offspring and goes to age 3
with probability p23
A is always a square matrix
Nt is always one column matrix = a vector
transition matrix A age distribution vectors Nt
1+= tt NAN
fertilities/fecundities (F) and survivals (p) depend on census and
reproduction
- populations with discrete pulses post-reproductive census
- populations with discrete pulses pre-reproductive census
- for pre-reproductive census 0 age is omitted
- for populations with continuous reproduction
x
x
x
l
l
p 1+
=








+
+
=
−
+
xx
xx
x
ll
ll
p
1
1 ( )
2
11 ++
= xxx
x
mpml
F
1+= xxx mpF
x
x
x
l
l
p 1+
= 10 += xx mpF
includes p of
reproductive stages
includes p of the
youngest stage
Egg Larva Pupa Imago
Stage-structured
p2 p3p1
F4












000
000
000
000
3
2
1
4
p
p
p
F
only imagoes reproduce thus F1,2,3 = 0
no imago survives to another reproduction period: p4 = 0
Size-structured
Tiny Small Medium Large
p1
p2 p3
F4
F3
G11
G22 G33 G44












443
332
221
43211
00
00
00
Gp
Gp
Gp
FFFG
model of Lefkovitch (1965) uses 3 parameters (mortality, fecundity
and persistence)
F1 = 0
F2
multiplication by a vector
by a scalar
Matrix operations
determinant
eigenvalue (λ)
λ1 = 2.41
λ2 = -0.41a
acbb
2
42
2,1
−±−
=λ






=×





2115
96
3
75
32






=





×+×
×+×
=





×





55
23
5745
5342
5
4
75
32
23472
74
32
=×−×=





012)425.0()0()2(
025.0
42 2
=−−=×−−×−=





−
−
λλλλ
λ
λ
0)det( =− IA λuAu λ=






025.0
42
0
t
t NAN =
12 ANN =
23 ANN =
ttt NAAANN 2
2 ==+
parameters are constant over
time and independent of population
density
follows constant exponential
growth after initial damped
oscillations