1 Types of stochastic models

Models so far discussed are all deterministic, meaning that, if the presentstate were perfectly known, it would be possible to predict exactly all futurestates.

We have seen instances (like the discrete logistic) of so-called ‘chaotic’systems where the determinism becomes weaker, in the sense that any differ-ence, however small, in the initial state would lead to big changes in futurestates, thus making long-term prediction essentially impossible in such sys-tems: indeed, as any measure of the present state will entail some error, wecannot know exactly the state and thus uncertainty of prediction will growas the prediction horizon gets longer. However, though long-term predic-tion may be impossible, in principle such systems follow the paradigm ofdeterminism.

A stochastic model, instead, assigns only a probability distribution tofuture states. Even if we knew perfectly the present state, we could notpredict future states (except for trivial cases; for instance, if a species isextinct, a stochastic model will generally predict that it will be extinct alsoin the future).

Sometimes the uncertainty may arise from ignorance of many other factorsthat influence the population dynamics: such factors could the density ofother species in the ecosystem, that we choose not to model as there aretoo many of these, the genetic composition of a population that we considerhomogeneous. Otherwise, we may understand the effect of factors, suchas the meteorological conditions, but they remain essentially unpredictablefrom one year to the next. Or we may dwell on the difference between thestatistical regularity that a certain proportion of 50 year old males die everyyear, and the unpredictability of whether any of those will die in the currentyear. Whatever the fundamental reason for this, it is clear that in severalcases the best we can do is to quantify the uncertainty.

In population biology, two differeent forms of stochasticity have been ex-tensively examined in models. In the so-called ‘environmental stochasticity’the demographic rates are allowed to vary unpredictably in time, represent-ing the uncertainty arising, for instance, from climatic factors or variations inthe density of species outside the model. Models with environmental stochas-ticity are generally written in terms of stochastic differential equations, thatare generally based on differential equation models such as those examinedso far in the book, or that will be seen later. We do not discuss at all thismodelling framework in this book, not because we believe there is somethingwrong in this approach, but because we have not seen anything so interestingarising from the use of stochastic differential equations in ecology.

1

The other form of stochasticity is the so-called ‘demographic stochastic-ity’, where the stress is on the fact that biological populations are finite anddiscrete: they can vary only if one (or more) individuals are born or die orimmigrate or emigrate. Hence, while demographic rates represent statisticalaverages, all demographic events are intrinsically stochastic.

This chapter is devoted to this type of modelling and more particularlyto birth-and-death processes, that are (stationary) Markov processes withstate space the natural numbers (representing the number of individuals ina population, but also, in other contexts, people in a queue or machines in aproduction line), and where the only possible transitions (representing birthsor deaths) are one number up or one number down.

Markov processes can be considered corresponding to ordinary (or partial)differential equations, in the sense that probabilities of future states dependonly on present state, and not on past history. A short introduction tothe essential properties of Markov processes with countable state space ispresented in the Appendix, where some references are also given. Anotherexample of a Markov process is examined in the section about stochasticepidemic models.

2 Birth and death models

In a birth-and-death process, the only instantaneous transitions that areallowed are those from a state i to i + 1 ( a birth), and those from a statei to i − 1 ( a death). Precisely, but using a colloquial style, we will assumethat

P(X(t+ h) = i+ 1|X(t) = i) = λih+ o(h)

P(X(t+ h) = i− 1|X(t) = i) = µih+ o(h)∑j:|j−i|>1

P(X(t+ h) = j|X(t) = i) = o(h)(1)

where {λi} and {µi} are sequences of nonnegative coefficients, while o(h)

represents any function such that limh→0

o(h)h

= 0.

As the state space generally is N = 0, 1, . . ., it is necessary to assumethat µ0 = 0. The other coefficients are free to have any value, though wewill discuss how to assign them in a way, similar to what accomplished indeterministic models, that reproduces biological features. Results will befundamentally different according to whether it is assumed λ0 = 0 (when thepopulation reaches size 0, no births are possible; hence the population will

2

be extinct forever) or λ0 > 0 (which may seem odd if one thinks of actualbirths, but may be sensible if immigration from outside is considered).

From assumptions (1), one can obtain the (infinite) linear systems ofdifferential equations for the matrices P (t). The derivation is based on theso-called Chapman-Kolmogorov equations :

Pij(t+ s) =∑k∈E

Pik(t)Pkj(s) for all i, j ∈ N, t, s ≥ 0. (2)

Intuitively the equations state that for going from i in j in time t + s, theprocess will move to some k in time t, and then from k to j in time s.

After some steps (outlined in the Appendix) one obtains the backwardKolmogorov equations:

P ′ij(t) = λiPi+1,j(t) + µiPi−1,j(t)− (λi + µi)Pi,j(t) i, j ∈ N. (3)

(3) can be seen, for each j (that can be considered as a parameter), as a(finite or infinite) system of equations in the unknowns Pij, i ∈ E. Formallyit can be written in matrix notation as P ′(t) = QP (t) with Q a tri-diagonalmatrix with Qi,i+1 = λi, Qi,i−1 = µi, Qi,i = −(λi + µi).

In an analogous way to the backward equation, one can derive a differentsystem of equations, known as Kolmogorov forward equations:

P ′ij(t) = λj−1Pi,j−1(t) + µi+1Pi,j+1(t)− (λi + µi)Pi,j(t) i, j ∈ N (4)

or, in matrix notation, P ′(t) = P (t)Q. Now i can be taken as a parameterand the unknowns are Pij, j ∈ N.

However, at this level of generality, it is not possible to prove that (4)actually holds. For instance, in the Appendix it is shown that in the purebirth process (µi ≡ 0), if λi = i2, with probability 1 realizations of the processreach infinity in a finite time, and one can construct different processes havngthe same infinitesimal transitions; only one of them satisfies (4). The validityof (4), as well as the existence and unqiueness (for each initial condition) ofsolutions of (3) and (4) can be rigorously proved only under some conditionson the parameters. The condition

λi ≤ a+ bi, i ∈ N

where a and b are nonnegative constants avoids “explosions in finite time”and guarantees well-posedness for (3) and (4).

3

3 Stationary distribution

A fundamental difference in the behaviour of birth-and-death processes hingeson whether λ0 > 0 or λ0 = 0.

In the first case, that implies that immigration from outside occurs, allstates are in the same communicating class and it is possible that a non-trivialstationary distribution exists.

In the second case, when the population is extinct, it will be extinctforever. In mathematical terms 0 is an absorbing state. It can be provedthat all other states are in the same transient class, and an important ques-tion is determining the probability of extinction, and the mean time beforeextinction.

Looking for stationary solutions, we assume λi > 0 for all i ≥ 0 and µi > 0for all i ≥ 1. As shown in the Appendix, the most convenient condition tocheck whether, for a Markov process with states in E, {πi}i∈E is a stationarysolution is ∑

i∈E

πiqij = 0 ∀ j ∈ E. (5)

For a f birth-and-death process, equations (5) become

λi−1πi−1 − (λi + µi)πi + µi+1πi+1 = 0 i ≥ 1

−λ0π0 + µ1π1 = 0.(6)

It is easier finding the solution of (6) through the so-called “detailed balanceequations” (7):

Lemma 1. A solution of (6) satisfies the equations

λiπi = µi+1πi+1. (7)

Proof. By induction. For i = 0, (7) is the last of (6). Now assume that (7)holds for i− 1. From the first of (6), we obtain

µi+1πi+1 = (λi + µi)πi − λi−1πi−1 = λiπi

where the last equality comes from the inductive hypothesis.

Note that the detailed balance equations (7) can be interpreted as sayingthat, at the stationary distribution, the rate at which the process moves(through births) from i to i + 1 must be equal to the rate at which theprocess moves (through deaths) from i+ 1 to i.

4

From (7), one immediately has

π1 =λ0µ1

π0 and iteratively πn =λ0 · · ·λn−1µ1 · · ·µn

π0.

One can then find π0 from the condition that∑∞

n=0 πn = 1.Setting

ρn =λ0 · · ·λn−1µ1 · · ·µn

with ρ0 = 1

the condition becomes π0∑∞

n=0 ρn = 1.Hence there are two possibilities:

• if∑∞

n=0 ρn <∞, then

π0 =1∑∞

n=0 ρnπi =

ρi∑∞n=0 ρn

is the unique stationary distribution.

• if∑∞

n=0 ρn =∞, there are no stationary distributions.

4 Probability of extinction

We assume λ0 = 0, so that if X(T ) = 0, X(t) = 0 for all t ≥ T . In otherwords, 0 is an absorbing state; we assume λi + µi > 0 for all i ≥ 1, so thatthere are no other absorbing states.

We want to compute

ui = P(X(T ) = 0 for some T > 0|X(0) = i).

We compute this through the jump Markov chain Zn, i.e.

ui = P(Zn = 0 for some n > 0|Z0 = i).

When X(t) is a birth-and-death process, Zn can jump only 1 upwards ordownwards; in other words Zn is a random walk with

P(Zn+1 = i+1|Zn = i) =λi

λi + µiP(Zn+1 = i−1|Zn = i) =

µiλi + µi

i ≥ 1.

5

4.0.1 Hitting probabilities for random walks

Let us consider a random walk Zn such that

P(Zn+1 = i+ 1|Zn = i) = pi P(Zn+1 = i−1|Zn = i) = qi = 1−pi i ≥ 1.

The classical random walk occurs when pi ≡ p and necessarily qi ≡ q = 1−p.Let us first compute

umi = P(Zn hits 0 before m|Z0 = i)

= P(exists n ≥ 0 s.t. Zn = 0 and 0 < Zk < m ∀0 ≤ k < n|Z0 = i).

By conditioning on the first step, and using the Markov property one has

umi = piP(Zn hits 0 before m|Z1 = i+1)+qiP(Zn hits 0 before m|Z1 = i−1).

By shifting the time origin, it is clear that

P(Zn hits 0 before m|Z1 = i+ 1) = umi+1

so that the equation reduces to

umi = piumi+1 + qiu

mi−1, i = 1, . . .m− 1. (8)

These are m− 1 equations to which the obvious conditions

um0 = 1 umm = 0

have to be added, leaving m− 1 unknowns.The resulting linear system (8) has a unique solution, that we go ahead

computing: set wi = umi −umi+1−, i = 0, . . . ,m−1. Then (8) can be rewrittenas

piwi = qiwi−1 =⇒ wi =qi · · · q1pi · · · p1

w0 =i∏

j=1

qjpjw0. (9)

From wj’s, we can compute

1−umi = (um0 −um1 ) + (um1 −um2 ) + · · ·+ (umi−2−umi−1) + (umi−1−umi ) =i−1∑j=0

wj.

Setting i = m and using umm = 0, we get

1 =m−1∑i=0

wi =m−1∑i=0

i∏j=1

qjpjw0

6

where the final step is due to (9).This gives

1− um1 = w0 =1

m−1∑i=0

i∏j=1

qjpj

,

so that one obtains

um1 = 1− 1m−1∑i=0

i∏j=1

qjpj

(10)

and

umi = 1−i−1∑j=0

wj = 1−i−1∑j=0

j∏k=1

qkpkw0 = 1−

i−1∑j=0

j∏k=1

qkpk

m−1∑i=0

i∏j=1

qjpj

. (11)

Expressions (10) and (11) appear very cumbersome, but allow for anexplicit computation of the probability of hitting 0 before m for an arbitraryrandom walk.

The simplest case is that of a standard random walk, pi ≡ p, in whichcase

um1 = 1−1− q

p

1−(qp

)m =

qp−(qp

)m1−

(qp

)m , umi = 1−1−

(qp

)i1−

(qp

)m =

(qp

)i−(qp

)m1−

(qp

)m .

(12)These expressions hold, as long as q 6= p. Instead if p = q = 1/2, then

umi =m− im

.

We are not so interested in umi as in its limit, as m → ∞, which (usingthe rules of σ-additivity) can be interpreted as

limm→∞

umi = ui = P(Zn = 0 for some n > 0|Z0 = i).

From (12), one sees that

ui =

1 if q ≥ p(q

p

)iif q < p.

In words, the probability of ever hitting 0 is 1, if the probability of movingto the left is greater or equal than the probability of moving to the right.

7

Instead, if the probability of moving to the right is greater than the proba-bility of moving to the left, there is a positive probability that the randomwalk will drift to infinity without ever hitting 0.

For a general random walk, one can use the same idea in (11), obtainingthe following

Proposition 1. If∞∑i=0

i∏j=1

qjpj

= +∞ (13)

then ui ≡ 1 for all i ∈ N.If the sum in (13) is finite, then

ui = 1−

i−1∑j=0

j∏k=1

qjpj

∞∑i=0

i∏j=1

qjpj

.

4.0.2 Application to birth-and-death processes

Consider the linear birth-and-death process

λi = λi µi = µi (14)

corresponding to the Malthus’s deterministic model.As seen above, Zn is a random walk with

pi =λi

λi + µi=

λ

λ+ µqi =

µiλi + µi

=µ

λ+ µ.

Hence from the previous subsection, we see that the probability of extinction

ui =

1 if µ ≥ λ(µλ

)iif µ < λ.

(15)

The property that ui = ui1 can be justified intuitively on the basis that,since in this model birth and death rate of each individual are not influ-enced by how many other individuals are present, the probability ui that apopulation starting from i individuals gets extinct is equal to the productof the probability that each family-tree descending from one of the initialindividuals gets extinct: um1 .

8

Consider now an equivalent of the logistic model:

λi = λi µi = µi+ νi2

for some constants 0 < λ, µ, ν. Then Zn has probabilities

pi =λ

λ+ µ+ νiqi =

µ+ νi

λ+ µ+ νi.

It is then easy to see that condition (13) holds, so that extinction is certainstarting from any initial population i.

A similar property holds for any model in which the growth rate becomesnegative at high densities: a sufficient condition for this is λi ≤ ηµi for ilarge enough and η < 1.

4.1 Time to extinction

For processes in which extinction is certain, one can compute the mean timeto extinction. This can be obtained following the same arguments as in theprevious Section, but the time to transitions has to be taken into account.

Let τ the random variable denoting the time to extinction, i.e.

τ = inf{t : X(t) = 0}.

Let Wi = E(τ |X(0) = i). Consider now T1 as defined previously the time ofthe first transition, and condition on the value of X(T1). We obtain

Wi = E(T1|X(0) = i) + piE(τ − T1|X(0) = i,X(T1) = i+ 1)

+ qiE(τ − T1|X(0) = i,X(T1) = i− 1)

where pi is the probability (already computed) that the first transition is tothe right, while qi = 1 − pi is the probability that the first transition is tothe left

By the Markov property and the time-homogeneity of transitions, X(t−T1) is distributed like X(t) conditional on the initial condition, hence

E(τ − T1|X(0) = i,X(T1) = i+ 1) = E(τ |X(0) = i+ 1) = Wi+1.

The previous equation can then be written as

Wi =1

λi + µi+

λiλi + µi

Wi+1 +µi

λi + µiWi−1, i ≥ 1. (16)

A boundary condition is clearly W0 = 0, but we are now left with an infinitesystem of linear equation. A way out, similar to the analysis of the previous

9

section, could be to define the expected time to reach 0 or m, and thenconsider the limit for m → ∞. Since this would lead to rather lengthycomputations, I prefer to refer to the following

Theorem 2 (Lemma 6.1.5 in Anderson(1991)). Wi is the minimal nonneg-ative solution of (16).

Let us apply this result to the Malthusian case where λi = λi and µi = µi,and λ ≤ µ (we saw before that for λ > µ extinction is not certain). Equation(16) becomes

(λ+ µ)Wi =1

i+ λWi+1 + µWi−1.

Introducing Ui = Wi+1 −Wi, this becomes

Ui =µ

λUi−1 −

1

λi(17)

from which recursively one obtains

Ui =(µλ

)iU0 −

i∑j=1

(µλ

)i−j 1

λj=(µλ

)i(U0 −

i∑j=1

(λ

µ

)j1

λj

), i ≥ 1.

(18)We still don’t know U0 = W1. Letting i go to infinity in the term in

brackets in the rightmost term in (18), we see the following:

• if∑+∞

j=1

(λµ

)j1λj

= +∞, then for any choice of U0 > 0, we would obtain∑ij=1

(λµ

)j1λj

> U0 for i large enough, and thus Ui < 0 for i large.

This is inconsistent probabilistically (it cannot be E(τ |X(0) = i+ 1) <E(τ |X(0) = i)) and would also lead to have Wk < 0 for k large enough.

The only possibility is that U0 = +∞, i.e. W1 = +∞. This means thatthe expected time to extinction starting with 1 individual (and thuswith more than 1 individual) is infinite. The series diverges for λ ≥ µ;since this analysis assumes λ ≤ µ, this case occurs for λ = µ.

• if∑+∞

j=1

(λµ

)j1λj< +∞, any choice of U0 ≥

∑+∞j=1

(λµ

)j1λj

would lead to

a positive value of Ui and thus of Wi for all i. However, we had statedbefore that the expected time is given by the minimal nonnegative

solution of (16), that is attained when U0 =∑+∞

j=1

(λµ

)j1λj

.

10

In this case (λ < µ) we thus have

E(τ |X(0) = 1) =+∞∑j=1

(λ

µ

)j1

λj=

1

λ

+∞∑j=1

∫ λ/µ

0

xj−1 dx

=1

λ

∫ λ/µ

0

+∞∑j=1

xj−1 dx =1

λ

∫ λ/µ

0

1

1− xdx = −1

λlog

(1− λ

µ

)=

1

λlog

(µ

µ− λ

). (19)

In the book by Karlin and Taylor, one can find similar computations in thecase of a general birth-and-death process.

In principle, one could compute mean extinction time, even when extinc-tion is not certain. In that case, in order to obtain relevant results, one wouldhave to condition on the fact that extinction does occur: E(τ |X(0) = i, τ <infty). The results can be obtained in a way similar to above, but keepinginto account the conditioning.

4.1.1 Extinction time with a bound on population size

Finally, let us consider a variation of the Malthusian case where there existsan upper barrier that cannot be passed, i.e.

λi =

{λi if i < K

0 if i ≥ Kµi = µi.

Once the process reaches state K it will stay there until a transition bringsit back to i− 1.

Now (16) becomes

(λ+ µ)Wi =1

i+ λWi+1 + µWi−1 1 ≤ i ≤ K − 1

to which one must add W0 = 0 and WK =1

µK+WK−1.

Passing to the variables Ui, the last condition means UK−1 =1

µK. Since

(18) is still valid for 1 ≤ i ≤ K − 1, one obtains

1

µK=(µλ

)K−1U0 −

K−1∑j=1

(µλ

)K−1−j 1

λj

11

i.e.

W1 = U0 =K−1∑j=1

(λ

µ

)j1

λj+

(λ

µ

)K−11

µK=

K∑j=1

(λ

µ

)j1

λj. (20)

Thus, E(τ |X(0) = 1) = W1 = U0 is given by the first K terms of a seriesthat is convergent if λ < µ and divergent for λ ≥ µ; this observations meansthat, when λ ≥ µ, W1 grows to infinity as K is increased. An asymptoticexpansion makes it possible to quantify this statement.

In fact, when λ > µ, one obtains

W1 =

(λ

µ

)K K∑j=1

(λ

µ

)j−K1

λj=

(λ

µ

)K K−1∑l=0

(µλ

)l 1

λ(K − l)

≈ 1

λK

(λ

µ

)K ∞∑l=0

(µλ

)l=

1

(λ− µ)K

(λ

µ

)K. (21)

This can be written saying that the mean extinction time grows exponentiallywith K, precisely W1 grows like eαK/K where α = log(λ/µ). Thus it becomesastronomically large for moderate K.

On the other hand, for λ = µ, W1 is given by the first terms of theharmonic series. Hence, its asymptotic expansion is well known:

W1 ≈1

λ(log(K) + γ) (22)

where γ ≈ 0.5772 is Euler’s constant.Finally, for λ < µ, (20) are the first terms of a convergent series, so that

W1 can be approximated by its sum, i.e. (19).

4.2 Relations with deterministic processes

In the case of the linear birth-and-death process (14), consider

mi(t) = E(X(t)|X(0) = i) =∞∑j=0

jpij(t). (23)

Using Kolmogorov forward equations (4), and interchanging (this could berigorously justified by first showing that the series converge absolutely) deriva-

12

tive and (infinite) sums, one obtains

m′i(t) =∞∑j=0

jp′ij(t) =∞∑j=0

j[pi,j+1(j + 1)µ+ pi,j−1(j − 1)λ− pi,jj(λ+ µ)]

=∞∑k=1

µ(k − 1)kpi,k +∞∑k=0

λ(k + 1)kpi,k −∞∑j=0

(λ+ µ)j2pi,j =∞∑k=1

(λ− µ)kpi,k

= (λ− µ)mi(t). (24)

(24) shows that the expected value of the process follows Malthus equationwith parameter r = λ− µ, so that mi(t) = ie(λ−µ)t.

This formula can be contrasted with (15). If λ > µ, the expected valueof the population grows exponentially; still extinction may be likely.

To give a numerical example, if λ = 1 and µ = 0.9, and the initial valueis 1 individual, the expected value of the population at time t = 100 ise10 ≈ 22 × 103, but the probability of extinction is 9/10, and we are quitesure that, if extinction occurs, it has occurred before. Thus, there is 90 %probability that the population will be extinct, but its expected values isvery large, meaning that, conditional on non-extinction, we expect it to bearound 220× 103.

Note that the Malthusian growth rate depends on the difference λ − µ,while the extinction probability depends on the ratio µ/λ. Thus popula-tion with the same expected growth rate may have very different extinctionprobability.

The fact that the expected value of the stochastic model coincides with thevalue of the deterministic model holds only for the linear case (14). In general,they differ, and the fundamental reason for this is that E(f(X)) 6= f(E(X))unless f is linear.

One may wonder whether there is then any relation between stochasticand deterministic models. Indeed, there is one and basically it follows fromthe law of large numbers: as the number of trials grows to infinity, the factionof successes converges to the expected value.

In this context, the problem requires mathematical techniques much be-yond the level of this text. However, it is possible to quote a result, dueespecially to Kurtz and his co-workers, that have actually handled muchmore general cases and obtained much more detailed results. We assumethat there exists a typical scale N of the population (it may represent habi-tat size), and that the parameters of the birth-and-death process depend onN as

λ(N)i = Nb

(i

N

)µ(N)i = Nm

(i

N

)(25)

13

where b and m are given functions.For instance, logistic growth could be represented by

λ(N)i = λi, µ

(N)i = i(µ+ ν

i

N) (26)

where ν iN

represents the extra mortality to crowding. In this case, we wouldhave b(x) = λx and m(x) = x(µ+ νx).

The following theorem (a special case of Theorem 8.8.1 from Kurtz, 1981)represents a law of large number for this case.

Theorem 3. Let X(N)(t) be a family of birth-and-death process, with ratesgiven by (25). Let F (x) = b(x)− µ(x) be a Lipschitz function on R+. Let

limN→∞

X(N)(0)

N= x0 ≥ 0 w.p. 1.

Then for all T > 0,

limN→∞

supt∈[0,T ]

|X(N)(t)

N− x(t)| = 0 w.p. 1 (27)

where x(t) is the solution of the Cauchy problem{x′(t) = F (x(t))x(0) = x0

. (28)

In words, we can say that, as the scale of the population grows to infin-ity, the stochastic model converges (technically, it is an almost-everywhereconvergence) to the deterministic model uniformly in any finite interval [0, T ].

Several aspects of (27) can be noted. First of all, if x0 > 0, this means thatthe initial condition X(N)(0) = O(N), i.e. very large. The approximation ofthe stochastic model with equation (28) works well when N is large, as wellas the initial condition X(N)(0).

If instead X(N)(0) is kept fixed (as N increases) at some (small) value(the situation considered when looking at extinction probabilities), we have

limN→∞X(N)(0)

N= 0. Then, if we assume λ

(N)0 = 0 (extinction is possible),

x(t) ≡ 0. This is not a useful information, because it does not allow todistinguish between the cases in which X(N)(T ) = 0 and those in whichX(N)(T ) > 0 and the population will continue growing. The techniques ofSection 4 are required to study the probability of extinction.

Moreover, in that Section, we have shown that, when birth and deathrates follow the logistic-like rule (26), then limt→+∞X

(N)(t) = 0, however

14

large isN . On the other hand, it is easy to show that (if λ > µ) limt→+∞ x(t) =λ−µν

that represent the carrying capacity for the deterministic equation. Thisshows that we cannot interchange limits:

0 = limN→∞

limt→+∞

X(N)(t)

N6= lim

t→+∞limN→∞

X(N)(t)

N=λ− µν

.

Equation (28) describes accurately the stochastic model (under the assump-tions of the previous Theorem) only for finite intervals of time, while it iscertain that sooner or later the stochastic model will randomly drift to ex-tinction. However, the time scale of these fluctuations may be so large (seethe computation in Subsection 4.1.1) to be irrelevant from a practical pointof view.

0 5 10 15 20

010

2030

40

T

x

0 20 40 60 80 100

010

2030

4050

60

T

x

Figure 1: Some simulations of a birth-and-death process X(t) correspondingto the logistic differential equations: λi = bi, µi = (d + i(b − d)/K)i. Hereb = 1, d = 0.5, K = 30, X(0) = 1. On the left panel 10 simulations on ashort interval: 4 of those undergo early extinction. On the right panel 2 ofthe simulations not going extinct on a longer time interval. In both cases,the solid line represents the solution of the differential equation X ′(t) =rX(t)(1−X(t)/K with r = 0.5, K = 30, X(0) = 1.

Fig. 1 shows some simulations of the model that exhibit some typicalfeatures. Some simulations undergo early extinction; the probability of thisevent can be approximated very well by 15: for the parameter values used inthe Figure, this amounts to 0.5.

Those that do not undergo early extinction fluctuate around the equilib-rium of the differential equations. The simulations are not very close to the

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solution of the differential equation, as the previous Theorem suggests, asthat is a limiting Theorem as a scale parameter (which in this case could beK) goes to infinity, while here K = 30 definitely is not very large. Other re-sults obtained by Kurtz and co-workers provide central limit theorems for theconvergence X(N)(t). These can be used to analyse the fluctuations aroundthe equilibrium of the trajectories, but this is definitely beyond the level ofthis book.

Finally, as discussed above, all these realizations of the process will even-tually reach 0, leading to population extinction, but this is very difficult tosee on the time scale at which simulations are run.

Exercises

1. Let us consider a birth-and-death process in which the rate of transi-tions from m to m+ 1 is β(m+ 1), m ≥ 0; the rate of transitions fromm to m− 1 is γm, m ≥ 1.

(a) Write down the Kolmogorov backward and forward equations forthis model.

(b) Show that there are no absorbing states.

(c) Find under which conditions on the parameter β and µ there existsa stationary probability distribution.

(d) Compute the stationary probability distribution (when it exists);is it one of the distributions that are considered in introductorycourses in probability theory?

(e) Intuitively, what will the trajectories of the birth-and-death pro-cess will do when there is no stationary probability distribution?

2. The release of sterile males is a technique has sometimes been appliedin the attempt to eradicate pests. The idea is that a certain proportionof females will mate with the released sterile males and will not produceoffspring, leading to a reduction of the population. Clearly, this can beeffective only if sterile males are quite abundant compared to normalmales.

Repeating this process for a few generations (while normal males be-come less and less abundant) could lead to a strong reduction of thepopulation, and possibly to extinction.

We make extreme assumptions, in order to be able to build a verysimplified model of this mechanism in the form of a birth-and-deathprocess.

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First, assume that the number of females and ‘normal’ males is at alltimes equal: a male dies when and only when a female dies (at rate µ,independently of population size); offspring are born in pairs (one maleand one female).

Second, assume that the number of sterile males is kept constant at thevalue S (as soon as one dies, it is replaced by a newly released one).

Finally, assume that each females mates at rate λ (independently ofpopulation size) with a male chosen at random among the normal andsterile ones present in the population: if the male chosen is normal,it produces one female and one male; if it sterile, it does not produceoffspring.

(a) Write the infinitesimal transition rates for this process (i.e., therates at which the number of females changes from j at a differentvalue).

(b) Write down the corresponding Kolmogorov differential equations.

(c) Noting that 0 is an absorbing state for the process, write down asystem for the probabilities of the population to become extinctsooner or later, conditional to the initial number of females (andmales) being equal to j. Intuitively, will these probabilities bealways equal to 1?

(d) Modify the model by assuming that there exists a level K > 0,such that when the number of females reaches the number K themating rate drops to 0, while being given by the model above forj < K. Write down a system of equation for the mean time toextinction, conditional on the number of females (and males) attime 0.

(e) Assume K = 3, λ = 1.2, µ = 1. Find the value of T1, the meantime to extinction, conditional on 1 being the number of females(and males) at time 0 [I believe that a simple expression can beobtained using a generic value for S; if this seems too difficult,use S = 2]

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