EXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS

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1 (February 25, 2004) EXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS BEN CAIRNS, University of Queensland PHIL POLLETT, University of Queensland Abstract The birth, death and catastrophe process is an extension of the birth-death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model, in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction and the distribution of the population size conditional on non-extinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We addresses this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times. CATASTROPHE PROCESSES; PERSISTENCE TIMES; HITTING TIMES. AMS 2000 SUBJECT CLASSIFICATION: PRIMARY 60J27 SECONDARY 92B05;60J80 1. Introduction Accounting for catastrophic events has become an important part of stochastic population modelling, particularly in ecology, but also in an array of other fields, including economics, chemistry and telecommunications. In the context of population processes, catastrophes are sudden declines in the population and, according to Shaffer [12] and others, they are one of the primary sources of variation in the abundance of species. The use of catastrophe processes, such as the ones described here, is not limited to studying the numbers of individuals in a population. Mangel and Tier [7], for example, have used a catastrophe process to model the number of occupied habitat patches in a metapopulation. For a review of the significance of catastrophes in ecological modelling, see [11]. Of primary importance is the effect of catastrophes on the persistence of a population, and in particular their effect on the expected time to extinction. Recent work, beginning with Brockwell et al. [3], examines extinction probabilities, conditions for eventual extinction Postal address: Department of Mathematics, The University of Queensland, Queensland 4072, Australia. address: bjc@maths.uq.edu.au Postal address: Department of Mathematics,The University of Queensland, Queensland 4072, Australia. address: pkp@maths.uq.edu.au 1

2 2 Ben Cairns and Phil Pollett and expected extinction times, in a variety of different cases. Brockwell [2] laid out a programme for evaluating the probability of extinction and the Laplace transform of the distribution of the time to extinction for a general birth, death and catastrophe process. As Anderson points out (see Section 9.2 of [1]), Brockwell s argument can be used to evaluate expected extinction times, and indeed his entire programme extends, at least in principle, to any upwardly skip-free Markov chain. However, explicit results can only be obtained in particular cases. Here we examine one such case in which the transition rates are allowed to depend on the current population size. Our main result gives an explicit expression for the expected extinction times. We illustrate our result with several examples. 2. The model Let X(t) be the number in the population at time t, and suppose that (X(t), t 0) is a continuous-time Markov chain taking values in S = {0, 1,... }. Let f i (> 0) be the rate at which the population size changes when there are i individuals present, and suppose that, when a change occurs, it is a birth with probability a (> 0) or catastrophe of size k (the removal of k individuals) with probability d k, k 1. (Simple death events are to be interpretted as catastrophes of size 1.) Assume that d k > 0 for at least one k 1 and that a + k 1 d k = 1. Thus, the process has transition rates Q given by f i k i d k, j = 0, i 1, f i d i j, j = 1, 2,... i 1, i 2, q ij = f i, j = i, i 1, (1) f i a, j = i + 1, i 1, 0, otherwise. Notice, in particular, that q 0j = 0, j 0, and that q i0 > 0 for at least one i 1. Thus, the sole absorbing state 0, corresponding to population extinction, is accessible from {1, 2,... } (an irreducible class). The special case f i = ρi, where ρ (> 0) is a per-capita transition rate, was studied by Brockwell [2], Pakes [8] and Pollett [9]. Our purpose here is to evaluate the expected time to extinction for the present general model, thus extending Brockwell s results for the linear case. Whilst this model is quite general, it does have limitations. Firstly, it is frequently useful to separate death and catastrophe events, and to assign different rate functions to births, deaths and catastrophes, as in Mangel and Tier [7]. Another drawback is that the catastrophe size distribution does not depend on the number of individuals present. For example, it rules out two important cases: where the size of a catastrophe has (i) a binomial B(i, p) distribution (each of the i individuals present is removed with probability p) and (ii) a uniform distribution on the set {1, 2,..., i}; see for example [3]. 3. Extinction probabilities The probability of extinction does not depend on the event rates (f i, i 1), because the jump chain is the same in all cases. It was shown by Pakes [8] that the probability of

3 Extinction times for a catastrophe process 3 extinction α i, starting with i individuals, is 1 for all i 1 if and only if the drift D (drift away from 0), given by D = a id i = 1 (i + 1)d i, is less than or equal to 0. Note that the process is said to be subcritical, critical or supercritical according as D < 0, D = 0 or D > 0. In the latter case extinction is of course still possible, and the extinction probabilities can be expressed in terms of the probability generating function d(s) = a + d i s i+1, s < 1. (2) (Note that D = 1 d (1 ) and this satisfies D 1.) It follows from Theorem 4 of [5] (see also [8]) that, when D > 0, (1 α i)s i = Ds/(d(s) s). Thus, writing b(s) = d(s) s, we see that α i s i 1 = 1 1 s D b(s). (3) It is interesting to note that α i tends to 0 as i tends to ; roughly speaking, the larger the initial population the less likely the population is to become extinct (in the supercritical case). However, as Pakes [8] notes, the convergence of α i to 0 can be very slow. For example, it is easy to show, letting s 1 in (3) and using L Hôpital s rule twice, that α i = d (1 )/(2D), and so this is finite if and only if the variance of the catastrophe size distribution is finite. 4. Explosions One interesting aspect of the present model is that the process may explode (that is, the population size may reach infinity in a finite time). Of course this can only occur in the supercritical case, for as we have already noted, the process hits 0 with probability 1 in the subcritical and critical cases. It is easy to exhibit explosive behaviour, for imagine that there is no catastrophe component in the model. We obtain the pure-birth process with birth rates q i,i+1 = f i a, and this is well known to be explosive if and only if g i <, where g i = 1/f i. If death transitions are included, that is, q i,i 1 = f i b, where a + b = 1, then the resulting birth-death process is explosive if and only if a > b (supercritical) and g i < (apply Theorem of [1]). It might therefore be conjectured that the general birth, death and catastrophe process is explosive if and only if it is supercritical and g i <. Our first result establishes that this is indeed the case. Theorem 1. The birth, death and catastrophe process with transition rates (1) is nonexplosive if and only if 1/f i = or id i a.

4 4 Ben Cairns and Phil Pollett Proof. Recall that D = a id i. We will use results contained in [4] to prove that Q is always regular (the process is non-explosive) if D 0, while if D > 0, then Q is regular if and only if 1/f i =. Theorem 1 of [4] applies to any conservative q-matrix Q = (q ij ) defined on the nonnegative integers that is upwardly skip free in that q i,i+1 > 0, for all i 1, and q ij = 0 if j > i + 1. It states that Q is regular if and only if ( R n = q n,n+1 n m 1 m=1 k=0 n=1 R n =, where R 0 = 1 and ) q nk R m 1, n 1. (4) This result was obtained using Reuter s [10] condition (Theorem (2) of [1]) to first establish that Q is regular if and only if q n,n+1 (u n+1 u n ) = u n + n m 1 m=1 k=0 q nk (u m u m 1 ), 0 u n 1, n 0, has only the trivial solution, and then proving that this is equivalent to n=1 R n = using a generalization of Reuter s lemma (Lemma of [1]). Set g i = 1/f i, so that (4) can be written where, for m < n + 1, Since v n (m) Then, (5) becomes R n = 1 q n,n+1 + v n (m 1) = 1 m 1 q nk = g n q n,n+1 a k=0 n m=1 R m 1 v (m 1) n, n 1, (5) ( k=n m 1 d k + g n k=1 d n k g n ) = 1 a k=n m+1 depends on m and n only though the difference n m, let us write v (m) n = v n m. R n = g n 1 n a + m=0 R m v n m, n 1, remembering that R 0 = 1. On introducing generating functions R(s) = n=0 R ns n, v(s) = v is i and g(s) = g is i, we find that R(s)(1 v(s)) = 1 + g(s)/a. It is then easy to prove that b(s) = a(1 s)(1 v(s)), which implies that and hence, if b(s) 0, R(s) b(s) 1 s d k. = a + g(s), (6) R(s) 1 s = a + g(s). (7) b(s)

5 Extinction times for a catastrophe process 5 We know, from Section V.12 of Harris [6], that 1/b(s) has a power series expansion with positive coefficients and with radius of convergence σ, where σ is the smallest zero of b on (0, 1]. Furthermore, σ = 1 or σ < 1 according as D 0 or D < 0, and b(s) > 0 for all s [0, σ). Thus, if D < 0, then letting s σ in (7) gives R(σ) =, implying that n=0 R n =. If D 0, then because D = b (1 ), letting s 1 in (6) gives D n=0 R n = a + n=1 g n (> 0). Therefore, if D = 0, then then n=0 R n = if and only if n=0 R n =, while if D > 0, g i =. This completes the proof. 5. Expected extinction times We shall restrict our attention to the subcritical case (D < 0), where extinction occurs with probability 1, and make some remarks about the other cases at the end of this section. For a general Markov chain with transition rates Q = (q ij, i, j S), whose state space consists of an irreducible class {1, 2,... } and a single absorbing state 0 that is reached with probability 1, the expected absorption time τ i, starting in state i, is the minimal non-negative solution to the system of equations q ij z j + 1 = 0, i 1, (8) j 0 with z 0 = 0. In the present case, (8) becomes i 1 f i az i+1 f i z i + f i d i j z j + 1 = 0, i 1, with the empty sum being interpretted as 0 when i = 1. This can be written i 1 az i+1 z i + d i j z j + g i = 0, i 1. (9) On multiplying by s i and then summing over i, we find that the generating function h(s) = z is i 1 of any solution (z i, i 1) to (9) must satisfy b(s)h(s) az 1 + g(s) = 0. (We delay addressing the question of whether the solution is non-negative.) We have already noted that 1/b(s) has a power series expansion with positive coefficients and with radius of convergence σ, and that b(s) > 0 for all s [0, σ). (In the present subcritical case, σ < 1.) Let us write e(s) = 1/b(s) = j=0 e js j, s < σ, where e j > 0, noting that a = b(0) = 1/e 0. Letting κ = az 1, we obtain ( ) i h(s) = z 1 + κe i g j e i j s i, and hence, for i 2, i 1 z i = κe i 1 g j e i 1 j. (10)

6 6 Ben Cairns and Phil Pollett Now, (e i ) is an increasing sequence. To see this, observe that if we had g i = 0 for all i, then we would have z i = κe i 1 and then, from (9), i 1 ae i e i 1 + d i j e j 1 = 0, i 1, (11) with the empty sum being interpretted as 0 when i = 1. Hence, (e i ) satisfies i 1 a(e i e i 1 ) = d i j (e i 1 e j 1 ) + e i 1 j=i d j, i 1. Since e 0 = 1/a > 0 and d j > 0 for at least one j 1, a straightforward inductive argument shows that (e i ) is increasing. Therefore, referring to (10), (z i ) is the difference of two non-negative increasing sequences. Thus, in order to ensure that (z i ) itself is non-negative, we require κ sup i 1 h i, where h i = 1 e i i g j e i j = i ( ) ei j g j. (12) e i The minimal solution is then obtained on setting κ = sup i 1 h i. Since (e i ) is increasing, we have 0 < e i j /e i 1 for all i j 0. Moreover, because σ is the radius of convergence of j=0 e js j, we have σ = lim i e i 1 /e i, whenever this limit exists, implying that e i j /e i σ j for each j. Hence, formally, h i g(σ). Once we prove that this limit exists and equals sup i 1 h i, we may set κ = g(σ) to obtain the minimal non-negative solution to (9). To achieve this, we will draw further on branching process theory. Since e(s) has a power series expansion with positive coefficients, then so does π(s) = aπ 1 s 0 du, s < σ. b(u) Indeed, writing π(s) = π is i, it is easy to see that π i /π 1 = ae i 1 /i, i 1. However, the coefficients (π i ) form a stationary measure on {1, 2,... } for the Markov Branching Process with offspring distribution (q i, i 0), where q 0 = a, q 1 = 0 and q i = d i 1 for i 2 (refer to the corollary of Theorem V.12.2 of [6]). Theorem 1(e) of [14] then gives iπ i σ i aπ 1 /(1 d (σ)), as i, whenever D 0 (note that d (σ) < 1 since b (σ) < 0 when D 0). Consequently, e i σ i+1 1/(1 d (σ)). Thus, by the monotonicity of this limit, e i 1 /e i σ and hence e i j /e i σ j for j = 1,..., i. Furthermore, e i j /e i σ j as i. Applying the Dominated Convergence Theorem to (12) shows that if g(σ) <, then h i g(σ) and hence sup i 1 h i = g(σ) because h i g(σ). On the other hand, Fatou s Lemma always gives lim inf i h i g(σ), so that if g(σ) =, then sup i 1 h i =. We have therefore proved the following result.

7 Extinction times for a catastrophe process 7 Theorem 2. For the subcritical birth, death and catastrophe process with transition rates (1), let (e i, i 0) be the coefficients of the power series expansion of e(s) = 1/(d(s) s), s < σ, where d(s) is given by (2), and σ (< 1) is the smallest solution of d(s) = s in (0, 1]. Then, the expected extinction time τ i, starting in state i, is finite if and only if κ := σi /f i <, in which case τ 0 = 0 and i 1 τ i = κe i 1 e i 1 j /f j, i 1. (13) We conclude this section with some remarks on the critical and supercritical cases, both of which have σ = 1. Theorem 2 is certainly valid in the critical case provided d (1 ) < (which corresponds to the variance of the catastrophe size distribution being finite): the expected extinction times are all finite if and only if κ := 1/f i <, in which case (13) holds. This is true because, as i, e i 1 /i 2/d (1) > 0 (Theorem 1(c) of [14]) and hence e i j /e i 1. (The infinite variance case is mathematically delicate, and we will not pursue it further here.) In the supercritical case, where there is a probability α i of extinction, starting in i, which is strictly less than 1 (and therefore the expected extinction times are infinite), it is possible to evaluate expected extinction times conditional on extinction occurring. This can be done by interpreting the result of Theorem 2 for the (subcritical) birth, death and catastrophe process with transitions rates q ij = q ij α j /α i, and following the progamme laid out in [13]. 6. Examples First let us examine the linear case studied by Brockwell [2]. This has f i = ρi, where ρ > 0. So, g(s) = log(1 s)/ρ, s < 1, implying that g(σ) is finite whenever σ < 1. Setting κ = log(1 σ)/ρ, we get from (13) ( τ i = 1 ( ) ) 1 i 2 e j e i 1 log, i 1. ρ 1 σ i j 1 j=0 This is equivalent to τ i s i 1 = 1 ρb(s) log ( ) 1 s, s < σ, 1 σ which is equation (3.1) of [2]. Further examples of the subcritical birth, death and catastrophe process for which the expected extinction times are finite include the case (i) where events occur at a constant rate f i = ρ > 0, independent of the population size, giving κρ = σ/(1 σ), and, (ii) where f i = ρi(i + 1), giving ( ) ( ) 1 σ 1 κρ = 1 log. σ 1 σ

8 8 Ben Cairns and Phil Pollett If f i = ρβ i 1, where ρ, β > 0, then the expected extinction times are finite only if σ < β, in which case κρ = βσ/(β σ). Explicit results can be obtained in the case where the catastrophe size follows a geometric law. Suppose that d i = b(1 q)q i 1, i 1, where b (> 0) satisfies a + b = 1, and 0 q < 1. The simple birth-death process with linear birth and linear death rates is recovered on setting q = 0. It is easy to see that D = a b/(1 q), and so D < 0 or D 0 according as c > 1 or c 1, where c = q + b/a. We also have b(s) = (b + qa)s2 (1 + qa)s + a 1 qs = a(1 s)(1 cs), 1 qs and hence if D < 0, then σ = 1/c (< 1). The coefficients of the power series 1/b(s) = j=0 e js j are easily evaluated using partial fractions. If D < 0 (or indeed if D > 0), then e j = 1 q (c q)cj, j 0. a(1 c) We may evaluate τ i by substituting these expressions into (13). If, for example, f i = ρβ i 1, where ρ, β > 0, then if β = 1, τ i = 1 + (1 q)(i 1) ρ(b a(1 q)), i 1, while if β 1, the expected extinction times are finite only if β > a/(b + qa), in which case where γ = 1/β. τ i = 1 q (γ q)γi 1 ρ(b a(γ q))(1 γ), i 1, Acknowledgements We would like to thank the referee for helpful suggestions. The support of the Australian Research Council is gratefully acknowledged (Grant No. A ). The work of the first author is supported by a PhD scholarship from the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. References [1] Anderson, W. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer-Verlag, New York. [2] Brockwell, P. (1985). The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Probab. 17, [3] Brockwell, P., Gani, J. and Resnick, S. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Probab. 14,

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