182    Population genetics and Markov chains
(a) Without doing any calculations, is this a regular Markov chain? Why?
(b) If the answer to (a) is yes, compute the equilibrium probability distribution p.
(c) If X0 = 3, what is the eventual probability that the position of the particle is 3?
19. In the Markov chain model of random mating with mutation in a population of size .V, find P if ai = x2 - « ¥■ 0. Given an arbitrary initial probability distribution p(0), find pfl) and deduce that the stationary distribution is attained in one generation.
20. What will happen in the Markov chain model of random mating with mutation if a, ^ 0 but a2 = 0?
Population growth I: birth and death processes
9.1 INTRODUCTION
j
! It is clearly desirable that governments and some businesses be able to predict I future human population numbers. Not only are the total numbers of male t and female individuals of interest but also the numbers in certain categories | such as age groups. The subject which deals with population numbers and I movements is called demography.
Some of the data of concern to human demographers is obtained from our ifiUing out census forms. The type of data is exemplified by that in Tables 9.1 land 9.2, In Tabic 9.1 is given the total population of Australia at various times since 1881 and Table 9.2 contains some data on births and deaths and their ■rates. Notice the drastic fall in the birth rate in the last few decades compared With an almost steady death rate.
Table 9.1
Time	Population of Australia (thousands)*
3 April 1881	2250.2
5 April 1891	3 177.8
31 March 1901	3 773.8
3 April 1911	4455.0
4 April 1921	5435.7
30 June 1933	6629.8
30 June 1947	7 579.4
30 June 1954	8 986.5
30 June 1961	10548.3
30 June 1966	11599.5
30 June 197!	12937.2
30 June 1979	14417.2
30 Juns 1981	14923.3
♦Obtained from Cameron (1982} and Austral Bureau of Statistics.
184 Birth and death processes Table 9.2
	Annual (average) number		Annual rates per thousand	
Time period	Births	Deaths	Births	Deaths
1956-60	222459	86488	22.59	8.78
1961-65	232 952	95465	21.34	8.75
1964-70	240325	107 263	19,95	8.90
1971-75	253438	111216	18.99	8.32
1976	227810	112662	16.37	8.10
1977	226291	108 790	16.08	7.73
1978	224 181	108425	15.73	7.61
Deterministic model
An accurate mathematical model for the growth of a population would be a very useful thing to have. There is a substantial literature on various models, as exemplified by the books of Bartlett (I960), Kcyfitz (1968), Pollard (1973) and Ludwig (1978). A first division of such models is into deterministic versus stochastic ones. In the former category there are no chance effects.
Let N(t) be the population size at time t>0 and assume that the initial population size W(0) > 0 is given. The number of individuals at t is a non-negative integer, but it is convenient to assume that Nit) is a differentiable function of time. A simple differential equation for A' is obtained by letting there be bAt births and dAt deaths per individual in (f, t + At] where 6, d ^ 0 are the per capita birth and death rates. Then
N(t + At) = N(t) + N(l)bAt - N(t)dAt
and it follows, upon rearranging and taking the limit At -»0, that
d.V
cr
= {b~d)N, t>0.
This differential equation is called the Malthusian growth law (after Thomas Malthus, whose essay on population appeared in 1798). Its solution is
|_iV(l) = N{oy-m,
and once we specify JV(0) the population size N(t) is determined for all r > 0. Three qualitatively different behaviours are possible, depending on the relative magnitudes of 6 and d. These are illustrated in Fig. 9.1. It is seen that
Itm N(t) •
0,     b<d(exponential decayl W(0),   b = d (constant population) oo,     b > d (exponen tial growth).
Simple Poisson processes 18S
	/ b>d
	
	
Figure 9.1 Population size as a function of lime for various relative magnitudes of the birth and death rates in the Malthusian growth model.
Experience tells us that the times of occurrence of births and deaths are not predictable in real populations. Hence the population sizes iV = {N{l), t> 0} constitute a random process. As observed above, the number of individuals is a non-negative integer so N is a random process with a discrete stale space {0,1,2,...} and a continuous parameter set {t»0}. The models we will consider do not contain very many elements of reality. They are nevertheless interesting because they may be solved exactly and provide a starting point for more complicated models.
9.2 SIMPLE POISSON PROCESSES
The random processes discussed in this section are of fundamental importance even though, as will be seen, their use as population growth models is very limited. The following process has a different parameterization from that defined in Chapter 3.
Definition A collection of random variables N = {A(r), r > 0} is called a simple Poisson process with intensity J. if the following hold:
(I) A'(0) = 0.
(ii) For any collection of times 0 «S t0 < t ,<...< r„_ i < t„ < <x> the random variables A'(tt) - N(lk _k = 1,2,..., n, are mutually independent.
(iii) For any pair of times OsStj < f2, the random variable A'(i2) —/V^) is Poisson distributed with parameter X(/2 - f,).
Condition (i) is a convenient initialization. The quantity ANk — fV(tk) - N(tk-1) is the change or increment in the process in the time interval
186    Birth and death processes
('k-i> fJ- Condition (ii) says that the increments in disjoint (non-overlappjng) time intervals are independent. Condition (iii) gives the probability law of the increments. Then
Markov chains in continuous time 187
Pr{Jr(t2)-W(t1) = li} =
q(t2-tijye-*"-
k = 0,l,2,...
and in particular, choosing tL = 0 and t2 = r>0 we have, since W(0)-0.
fe = 0,l,2...
Pr{JV(ŕ) = ŕ)
(9.1)
The mean and variance of N(t) are therefore
E(N((f) = yax(N{t)) = Xt.
Sample paths
What does a typical realization of a simple Poisson process look like? To get some insight, let Af be a very small time increment and consider the increment
&N(t)-N(t + &t)-N(t).
The probability law of this increment is
Pr {AA'(t) = k]
1 - Mi + o(A£), it = 0 Mr + o(Ar), k-1
o(Aí), k >Z
(9.2)
We see therefore that when Ar is very small, N(t + Ar) is most likely to be the same as A'(r), with probability iAt that it is one larger, and there is a negligible chance that it differs by more than one from N(t). We may conclude that sample paths are right-continuous step functions with discontinuities of magnitude unity - sec Fig. 9.2. Note that equation (9.2) can be used as a definition and the probability law of the increments derived from it-see Fxercise 2.
The simple Poisson process is called simple because all of its jumps have the same magnitude - unity in the above case. This contrasts with compound Poisson processes in which jumps may be any of several magnitudes (see for example Parzen, 1962).
To relate the Poisson process to a growth model wc imagine that each time a new individual is born, N(t) jumps up by unity. Thus N(t) records the number of births in (0,t], i.e. up to and including time t and {N(t), t>0} may be regarded as a birth process, ff in Fig. 9.2 we place a cross on the f-axis at each
N(t|
Figure- 9.2 Two representative sample paths for a simple Poisson process. The mean value function At is also indicated.
jump of W(t) we obtain a collection of points. Thus there is a close relationship between the simple Poisson processes of this section and the Poisson point processes of Section 3.4. This is elaborated on in the exercises.
Poisson processes, though dearly limited as population growth models, find many other applications. Some were mentioned in Chapter 3. Furthermore they may result when several sparse point processes, not themselves Poisson, are pooled together. This makes them useful in diverse areas (TuckweU, 1981).
9.3 MARKOV CHAINS IN CONTINUOUS TIMF
In Chapter 8 we encountered Markov random processes which take on discrete values and have a discrete parameter set. Markov processes of that kind are called Markov chains. However, Markov processes in continuous time and with discrete state space are also called Markov chains. Usually we mention that they have continuous parameter set when we refer to them in order to distinguish them from the former kind of process. If X = [X(i], t S 0} is such a process, the Markov properly may be written, with t0<ti<
<2-<t.-,<t.,
PT{Jf(t,) = J„|Jf((„   ,) = *„-„   X(t„.1) = s„-2.....X(t1) = sl,X(t0)=s0}
= Pr{y(0-a.l*(t.-1)-s.-i}.
where {s0, .....s,} is any set in the state space of the process. The quantities
p(5„,f„|s„.„t.-1) = Pr [XftJ = !.|X(t,.,) = s„-,},0< (,_! < t, < co, (9.3) are called the transition probabilities of the continuous time Markov chain. If
188    Birth and death processes
the transition probabilities depend only on time differences they are said to be stationary and the corresponding process is called temporally homogeneous.
Simple Poisson processes are examples of continuous time Markov chains with stationary transition probabilities which may be written
p(k,tV,5) = -Pr{N(t) = k\N(s)=j}
=—(jt=j)i—• *-J=au.-
where 0 O < f and/ < k are two non-negative integers. Further examples are discussed in the following sections.
9.4 THE YULE PROCESS
Among the deficiencies of the Poisson process as a model for the growth of populations is that no matter how large She existing population, the chance of a birth in any time interval is always the same. The simple birth process we are about to describe was proposed by Yule (1924) as a model for the appearance of new species. Its applicability as a population growth model is limited except perhaps in a few cases such as algae undergoing relatively unchecked reproduction in large lakes (Pielou, 1969). The Yule process is sometimes called the Yule-Furry Process due to a related application in physics by Furry (1937). Again we denote by N(t) the number of individuals in existence at time t.
Assumptions on births to individuals
We begin with the following assumptions concerning the births which occur to the individual members of the population.
(i) Births occur to any individual independently of those to any other individual.
(ii) In any small time interval of length At, the probability of an offspring to any individual is Mt + o(A(), the probability of no offspring is 1 — KAt + o(At) and the probability of more than one offspring is o(At).
Notice that there are no deaths individuals persist indefinitely and are forever capable of producing offspring.
Population birth probabilities
Suppose that there are known to be n individuals at time I so that A'(r) = n. Under the above assumptions (i) and (ii), the number of births in the whole population in (t. t T At] is a binomial random variable with parameters n and AAr. We can drop reference to o(At) as such terms eventually make no
The Yule process 189 contribution. Then
Pr {it births in((, (+ At] | JV(r) = n]
= ^"jtMiftl - lAif-\     k - 0,1.....n.
If k = 0 we have
Pr {0 births in (t, t + At] | N(t) = n} = (1 - XAtf
= t - XnAt + o(A(), (9.4)
whereas for k — 1,
Pr{l birth in (t,i + Ar]|iV(t) =n) -AnAl(l - AAt)""1
= XnAi + o(Ar). (9.5)
Also, for k   2 we find
Pr {it births in (I, t + At] | N(t) = n) = o(At). (9.6)
Differential-difference' equations satisfied by the transition probabilities
We consider a continuous time Markov chain JV = {N(t), t > 0} with initial value
lV(0) = no>0,
subject to the evolutionary laws (9.4)-(9.6). Define the transition probabilities
p„(l) = Pr(JV(t) = «|N(0) = no},      B0>0, t>0,
and note that these arc stationary. Our aim is first to find equations governin g the evolution in time of p, and then to solve them.
To obtain a differential equation for p„ we seek a relation between p„(t) and P,(t -+ At). If N(t + At) = n>n0 then, ignoring the possibility of more than one birth, we must have
W) =
\n and no births in (t,l + At], \n - 1 and one birth in (t,t +- At].
Dropping reference to the initial state, the law of total probability gives
Pr {N(t + At) •= n] = Pr {JV(t + At) = n|JV(t) - "}
x Pr{JV(t) = n} + Pr{JV(f + Ar) = n|N(t) = n-l}
xPr{AT(i) = n-l}. <9-7)
In symbols this is written, using (9.4) and (9.5),
p„(t + At) = (1 - AnAf)p„(t) I* ■ l)Atp„ - ,(t) + o(At),  n > na.
190 Birth and death processes This rearranges to
pjjt + &t)-pjjt)
A[(»-l)p,-1(t)-«p,(t)] +
o(Ar)
At ■ At
Taking the limit Ar->0 we obtain the required equation
dp, it
= 4.(n - 1)p,
n = na + 1, n0 + 2,.
since o(A[)/A( -»0 as At -> 0 by definition. When n - n0 there is no possibility that the population was n0- 1 so (9.7) becomes
Pr {N(t + At) = n0} - Pr {JV(< + At) = n„ | AT(f) = fl0} Pr {JV(t) = »„}
which leads to
IT"-*1*
(9.9)
Initial conditions
Equations (9.8) and (9.9) are first-order differential equations in time. To solve them the values of p„(0) are needed and since an initial population of n0 individuals was assumed, we have
P„(0) =
fl, n = n0, (0,  n > n„.
Solutions of the differential-difference equations The solution of (9.9) with initial value unity is
Armed with this knowledge of pn(, we can now find p„a + 1. The differential equation (9.8) is, with n = n0 + 1,
dr
(9.10)
This will be recognized as a linear first-order differential equation in standard form (see any first-year calculus text). Its integrating factor is eM"°+111 and in Exercise 10 it is shown that
p„o + 1(0 = "o?"""n'(i   e-*^. (9.11) Having obtained pBo+1 we can solve the equation for p„0+2, etc. In general we obtain for the probability that there have been a total number of* births at t,
Pro **M =
h„ + k - 1
k = 0,1,1.
(9.12)
as will be verified in Exercise 11.
The Yule process 191
It can be seen that p„„(r) is an exponentially decaying function of time. If we define as the time of the first birth and observe that pB(r) is the probability of no birth in (0,1] wc get
Pr{T, >t}=e-'h°',
or equivalenlly
Pr{r1<f} = l-e"
Hence Tt is exponentially distributed and has mean
We notice that the larger rt0 is, the faster does p„0(t) decay towards zero - as it must because the larger the population, the greater the chance for a birth. Note also that p„0(t) is never zero for t < oo. Thus there is always a non-zero probability that the population will remain uncharged in any finite time interval.
A plot of p„0(!) versus t when k = n0 — 1 is shown in Fig. 9.3. Also shown are the graphs of p„0, j(f) and p„0 + 2(e) which rise from zero to achieve maxima before declining to zero at t = oo. In Exercise 12 it is shown that for these parameter values Pi.n(r) has a maximum at t = ln(l +k).
1.0
Figure 9.3 Probabilities of no births, one birth and two births as (unctions of time in the Yule process with /.= 1 and initially n„ = 1 individual.
192    Birth and death processes
9.5 MEAN AND VARIANCE FOR THE YULE PROCESS
For the simple birth process described in the previous section we will show lhat the mean population size ,u(r) = E(N(r)|N(0) = n0) at time t is
nit) - r,y.
This is the same as the Malthusian growth law with no deaths and a birth rate b = L
Proof By definition
Differentiate to get
Substituting from the differential and differential-difference equations (9.9) and (9.8),
^7=-*&» + Jl   1 [(»2-«)P„-i-«2PJ-
at n-i»+l
The coefficient of pn _ 1 is now rewritten to contain a perfect square: dft
dr.
f! = Flo+ 1
Put m = n — I in the first two sums on the right and get if
dt
The terms in brackets cancel to leave the first-order differential equation which has the initial condition
u(0) = m„.
(9.14)
Integrating (9.13) and using (9.14) gives the required result. Similarly, the second moment of N[t) may be found and hence the variance. However, it is quicker to use the properties of the negative binomial distribution.
Mean and variance for the Yule process 193
Mean and variance from the negative binomial distribution
Consider a sequence of Bernoulli trials with probability p of success and probability q — 1   j> of failure at each trial. Let the random variable X, be the
number of trials up to and including the rth success, r = 1,2.....Then it is
easily seen that the distribution of Xr is given by
-o
fc = r,r+ l,r + 2,.
(9.15)
the smallest possible value ofjf, being ?■ since there must be at least r (rials to obtain r successes. The mean and variance of Xr are found to be (see Exercise 13),
E(JQ = Var(XJ =
We now put j — n0 + k in (9.12) to get '/-I
Pr{AT(r) = j} =
This is seen to be a negative binomial distribution as in (9.15) with parameters r = n0,
p = e~",q = 1 — e~u.
Thus we quickly see that the mean and variance of the population in the Yule
process arc
E(N(t)) = n0e>' Vai(W(r)) = Ho<l-
Approximations
(i) Large t For large t the variance is asymptotically Var(JV(r)) ~ n„e™
and the standard deviation of the population is
<r(f) - Jn~0eM.
Thus the mean and standard deviation grow, as indicated in Fig. 9.4, and eventually their ratio is constant.
194    Birth and death processes Hi
Figure 9.4 Mean and standard deviation of the population as functions of time in a Yule process.
(ii) Large n0 Let us suppose that there is just one individual to start with. The probability that there is still only one individual at time t, an event we will call a 'failure', is
Pr {population is unchanged at t} =e~Xl- Pr {'failure'}.
Now, if there are n0 > 1 individuals to start with, since individuals act independently,
Pr {population is unchanged at t) = Pr{n0 failures in n0 trials} = {e~xJa = e~Xno',
this being an exact result. Also, when n0 ■ 1 let the mean and variance of the population be denoted by /j, and o\ which are given by
px=e"'(i-«■■")•
When n0 > 1 we may at any time divide the population into n0 groups, those in each group being descendants of one of the original individuals. The number in each group is a random variable and it follows that the population at time f is the sum of n0 independent and identically distributed random variables each having mean /z, and variance a\. By the central limit theorem we see that for large ;i0 and any t, the random variable N(t) is approximately normal with mean n0pt and variance n0a\. Hence we may estimate with reasonable accuracy the probability that the population lies within prescribed limits (see exercises).
9.6 A SIMPLE DEATH PROCESS
In this rather macabre continuous time Markov chain, individuals persist only until they die and there are no replacements. The assumptions are similar to
A simple death process 195 those in the birth process of the previous two sections, but now each individual, if still alive at time t, is removed in (t,t + AQ with probability pAt + o(Af). Again we are interested in finding the transition probabilities
p„(l) = Pr {N(t) - n\N(0) = n0],     n = »0t n0 - 1,..., 2,1,0.
We could proceed via differential-difference equations for />„ but there is a more expeditious method.
The case of one individual
Let us assume n0 = I. Now p^t) is the probability that this single individual is still alive at I and we see that
Pl(t + At) = p,(0(l - pAt) + o(At)
(9.16)
since 1 — ^Af is the probability that the individual did not die in (t,t + At]. From (9.16) it quickly follows that
dp,
t>0.
The solution with initial value p,(0) = 1 is just
The initial population size is A'(0) = n0 > 1
If there are n0 individuals at t = 0, the number alive at t is a binomial random variable with parameters n0 and p,(t). Therefore we have immediately
n = n0, no-l,...,l,0.
Also
E(N(t)) = n0e-'u,
which corresponds to a Malthusian growth law with d - p and b = 0, and
Var(JV(t)) = n0e-"(l -«-<").
Extinction
In this pure death process the population either remains constant or it decreases. It may eventually reach zero in which case we say that the population has gone extinct. The probability that the population is extinct at time f is
Pr{JV(0-0|JV(0) = Ho}=(l-e T>^1   as t-»oo.
196    Birth and death processes 1.0
Prob. exrincl-
0.5
Figure 9.5 Probabilities that the population is extinct at t, versus t for various initial populations with n = 0.5.
Thus extinction is inevitable in this model. In Fig. 9.5 are shown the probabilities of extinction versus time for various initial populations.
9.7 SIMPLE BIRTH AND DEATH PROCESS
We now combine the ideas of the Yule process and the simple death process of the previous section. Let there be n0 individuals initially and N(t) at time t. In [t,t + At] an individual has an offspring with probability AAt + o(Af) and dies with probability nAt + o(At). Using the same kind of reasoning as in Section 9.4 for the population birth probabilities we find that
Pr {one birth in (t, r + At] | AT(t) = n} = XnAt + o(At) Pr {one death in (t, t + Ai]\N(t) = n} = finAt + o(At) Pr {no change in population size in (t, t + At]\N(t) - n}
= 1 — {X + n)nAt + o(Ar).
The ways to obtain a population size n at time t + At are, if n> 1,
N{t) = n-\ and one birth in (t,l + At] N(t) = n and no change in (t, t + At] N(t) = n + 1 and one death in (t, t + At].
Hence
pjf + At) = Pn-^Mn - l)At + p„(i)ri - (A + n)nAt\
+ P»-iW/<(" + l)Ai + o(Af).
Simple birth and death process 197
It quickly follows that
= M.n - l)pn-1 - (* + (ti»P* + Mn + l)Pn +:
R>1.
If n = 0 we have simply
and the initial conditions are
P„(0)
dpa dt
(9.17)
(9.18)
• f'l, n = n0, ~[0, n#n„.
The system of equations (9.17) and (9.18) cannot be solved recursively as could the equations for the simple birth (Yule) process as there is no place to get started.
The probability generating function of N(t)
By definition, the probability generating function of N(t) is
<£(s,t)= £ pMf.
n = 0
This can be shown (see Exercise 16) to satisfy the first-order partial differential equation
dt os
which is to be solved with the initial condition
-JHs.O^s"0. (9-20)
It may be shown (see, for example, Pollard, 1973; Bailey, 1964) and it will be verified in Exercise 17, that the solution of (9.19) and (9.20) is
where
Ás — n
Ms)-
s-1
(9.21)
(9.22)
The probability of extinction
A few sample paths of the simple birth and death process are shown in Fig. 9.6. The state space is the set of all non-negative integers {(), 1,2,...} and the state 0
198    Birth and death processes
N(t)
Realization ol the extinction —*- 1 time
Figure 9.6 Two representative sample paths of a birth and death process. Here N(0) = 6 and one path is shown on which the population goes extinct
is clearly absorbing. A sample path may terminate at 0 which corresponds to extinction of the population.
We can easily find the probability that extinction has occurred at or before time t from the probability generating function. This is just
Pr {N(t) - 01 N(0) - n„} = p0(f) = 0(0, r).
From (9.22) we have ij/(0) = ji and thence, from (9.21),
a — fie
k^p..
(9.23)
When /. = p. the following expression is obtained by taking the appropriate limit in (9.23):
fX kt + 1
X-ii.
(9.24)
In the limit t-»oo, <p(0,t) approaches the probability that the population ever goes extinct. Denote this quantity by p„,. Then from (9.23) and (9.24) we find
	\	
pc«=<		
	si) •	
Thus extinction is inevitable if the probability of a birth to any individual is less than or equal to the probability of death in any small time interval. It may seem surprising that extinction is certain when k = p. To understand this we note that 0 is an absorbing barrier which is always a finite distance from the
Mean and variance for the birth and death process 199
value of N(t). The situation is similar in the random walk on [0.1.2,...) with p = q where we found that absorption at 0 is certain (see Section 7.6).
In the cases /. ^ p where extinction is certain, we may define the random variable T which is the extinction time. Evidently the distribution function of T is
Pr{rsSt} = #), t)
since this is the probability that extinction occurs at or before i. When X = p the expected extinction time is infinite (sec Exercise 18) but it is finite when k < p. When I > p. we may still talk of the random variable T, the extinction time, However, we then have
Pr{r <»}-(£)",
so we must also have
Clearly in these cases T has no finite moments and, because its probability mass is not all concentrated on (0, oo) we say it is not a 'proper' random variable.
9.8 MEAN AND VARIANCE FOR THE BIRTH AND DEATH PROCESS
The expected number of individuals at time t is
m(t) = £[/V(r)[rV(0) = no]= £ np„(t) = £ np„(t)-
We will find a differential equation for m(t). We have dm    A dp„
dr=„?,"^
and on substituting from the differential-difference equation (9.17) we get
-JT = £ "M" _ 1 >P« - 1 ~ (>- + f)"Pn + f(» + 1)P. * 11
which rearranges to dm
dt
= /.[(»-l)V,+'K«- l)P.-i -U +P) I "2P«
n-l »-1 "-1
+n £ (» +1)2/>„+1-/< £ ('i+!)/>„+,.
200    Birth and death processes
A relabeling of indices with n' = n — 1 in sums involving p„_i and with n" = ii -r 1 in sums involving p„+ ( yields
dm
dt 1
In the first three sums here, terms from it, n', n" = 2 and onward cancel and leave — (X + ujp; + Xpl    - npx. Thus
dm dr :
= - Wi + X £ "'P.--f I n"P»-
n' = 0 v - 2
or simply
dm
(X — pi)m.
With initial condition m(0) — n0 the solution is
m(t)-nc,e'A-""7|
This is the same as the deterministic result (Malthusian law) of Section 9.1 with the birth rate b replaced by I and the death rate d replaced by pi. The second moment of N(t),
M(f)= £ n-p„(f)
can be shown to satisfy
d;\J
— = 2(X-u)M + (l + u)M,  M(0) = rc02, (9.25)
as will be seen in Exercise 19. The variance of the population in the birth and death process may then be shown to be
Vai (JV(() | iV(0) = «„) = «,
(* + M)
X fit.
In the special case /. = p.,
Var(Af(f)|/V(0) = n0) = 2^n„t.
An alternative method of finding the moments of N(t) is to use the moment generating function (sec Exercise 20).
Exercises 201
Birth and death processes have recently become very important in studies of how ions move across cell membranes. In the simplest model there are just two states for an ion channel - open and closed. The channel slays in each state for an exponentially distributed time before making a transition to the other stale. It is hoped that a study of such continuous lime Markov chain models will elucidate the mechanisms by which molecules of the membrane interact with various drugs. For details of this fascinating application see Colquhoun and Hawkes (1977), Hille (1984) and Tuckwell (1988).
REFERENCES
Bailey, N.T.J. (1964). Tlte Elements of Stochastic Processes. Wiley, New York. Bartlett, M. S. (1960). Stochastic Population Models. Methuen, London. Cameron, R.J. (cd.) (1982). Year Book Australia. Australian Bureau of Statistics, Canberra.
Colquhoun, D. and Hawkes, A.G. (1977). Relaxations and fluctuations of membrane currents that flow through drug operated channels. Proc. R. Soc. Land. B., 199, 231-62.
Furry, W.H. (193 7). On fluctuation phenomena in the passage of high-energy electrons
through lead. Phys. Rev., 52, 569-81. Hille, B. (1984). Ionic Channels of Excitable Membranes. Sinauer, Sunderland, Mass. Keyfitz, N. (1968). Introduction to the Mathematics of Populations. Addison-Wesley.
Reading, Mass.
Ludwig, D. (1978). Stochastic Papulation Jheories. Springer, New York, Parzen, E. (1962). Stochastic Processes. HoJden-Day, San Francisco. Pielou, E. C. (1969). An Introduction to Mathematical Ecology. Wiley, New York. Pollard, J.H, (1973). Mathematical Models for the Growth of Human Populations.
Cambridge University Press, London. Tuckwell, H.C. (1981). Poisson processes in biology. In Stochastic Nonlinear Systems.
Springer-Vcrlag, Berlin, pp. 162-71. Tuckwell, H.C. (1988). Stochastic Processes in the Newosciences. SIAM, Philadelphia. Yule, G.U. (1924). A mathematical theory of evolution based upon the conclusions of
Dr J.C. Willis, F.R.S. Phil. Trans. Roy. Soc. Land. B., 213. 21-87.
EXERCISES
1. Using the birth and death rates for 1966 given in Table 9.2 and the 1966 population of Australia given in Table 9.1, estimate the 1971 population, Compare with the actual population in 1971. Is the discrepancy in the direction you would expect? Why?
2. In a simple Poisson process, let p„(t)-Pr{N{l) = n\N{0)= 0}. Use the relations
Pr{AiV(t) = fc} -
1 - Mr + o(At), XAt + o(Af), o(At),
k = 0. k = l. k>2.