The Role of Ceilings in Population Models Ben Cairns‡

Department of Mathematics

Supervisor: Phil Pollett Assoc. Supervisor: Hugh Possingham

‡ bjc@maths.uq.edu.au, http://www.maths.uq.edu.au/~bjc

Introduction

Population ceilings are features of many population models, in which they play important roles as representatives of the physical limits on the size of the population. Here we present an overview of the use of population ceilings in the mathematical modelling of populations. We will argue that fixed ceilings to populations are often misplaced in cases where the value of the ceiling does not have a clear physical interpretation beyond that of a maximum population size. A population ceiling may still be useful, however, if the aim is to approximate an unbounded population by one that is bounded, but such a value must be chosen with care.

The use of population ceilings

In many cases, hard limits on the size of a population are imposed by its environment. For example, in a classical metapopulation the number of suitable patches is also the ceiling for the number of occupied patches (the ‘size’ of the metapopulation). Other natural limits such as minimum home ranges or the availability of nesting sites may limit a population. In [1], for example, the population of breeding pairs of the California Spotted Owl, Strix occidentalis occidentalis, is limited by a ceiling that represents the total

number of suitable nesting sites. In many circumstances, however, soft-limiting dynamics may be preferable to a hard limit. Even unbounded models (those which do not impose a hard upper limit on the population) can be guaranteed to remain finite, and otherwise faithfully represent the dynamics of the population. If the goal is to approximate a population with a finite model, however, ceilings play an important role.

Approximating unbounded populations

One form of unbounded population model is the birth, death and catastrophe process, a continuous-time Markov chain model for the size of a population. Birth, death and catastrophe processes represent the dynamics of a population as rates at which the population makes (Markovian) transitions from one size to another. Figure 2 illustrates a birth, death and catastrophe process in which, for a population of i individuals,

births occur at a rate Bi, deaths occur at a rate Di, and catastrophe drops in

population from size i to size j occur at rates Cij. The model has a ceiling if

the boundary, N, is finite.

Birth death and catastrophe processes are typically represented by a transition rate matrix, Q, which uniquely determines the behaviour of the

population model. The elements, qij, of this matrix are given by

When a population is not subject to a hard upper bound, it may be difficult to analyse its behaviour. Expected times to extinction (or simply extinction times) are found by solving systems of linear equations, and in the case of unbounded models these systems of equations are infinite. If, in Figure 2, N is made finite, then the model is subject to an upper bound, which allows the approximation of the true extinction times by values that are much easier to obtain: the solution, T, to the finite system of equations

where M is the restriction of Q to states i, j 2 {1, … , N}.

This approach assumes that the ceiling N is reflecting. When reflecting, if the population reaches N, then it will remain there until it drops to a lower value. We may instead take N to be absorbing such that the model allows the population to jump from N up into some absorbing state A. This latter form of ceiling has an important advantage in approximating unbounded populations (Figure 1). The expected time to either extinction or to absorption at A is always less than or equal to the extinction time for the unbounded model, and so provides a conservative estimate of this quantity (which may be calculated according to the program laid out in §9.2 of [2]). Figure 3 illustrates that conversely, where the ceiling is reflecting, very bad approximate extinction times may be obtained for poor choices of N.

Australian Research Council

Centre of Excellence for Mathematics

and Statistics of Complex Systems

Figure 2. A birth, death and catastrophe process with a ceiling if N is finite. The ceiling is absorbing if BN > 0 and otherwise reflecting. Both forms of ceiling are useful in approximating population processes.

Di Bi

Ci, i-1Ci, j

CN, j

D1

…i – 1 i N A…0

C•, 0

DN BN

Figure 1. True and approximate extinction times for the model with linear birth and death rates and binomial catastrophes considered in [3]. The expected time to extinction predicted by a model with a ceiling (red) differs from that of an unbounded model with the same structure (blue). A large proportion of the expected time to extinction for the bounded population occurs after the population hits the ceiling, since the expected time to either hit the ceiling or go extinct (green) is much smaller than the expected time only to extinction.

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References

[1] M.C. Anderson & D. Mahato (1995). Demographic models and reserve designs for the California Spotted Owl. Ecological Applications 5, 639—647.

[2] W.J. Anderson (1991). Continuous-time Markov Chains: An Applications-Oriented Approach. Springer-Verlag, New York.

[3] P.J. Brockwell (1986). The extinction time of a general birth and death process with catastrophes. Journal of Applied Probability 23, 851—858.

Figure 3. Comparison of approximate extinction times for various choices of the reflecting ceiling N for a population model with linear birth and death rates, and binomial catastrophes only when the population is above the critical point of 100 individuals. Poor choices of N may produce either over- or underestimates of the extinction times, unlike approximations using an absorbing N, which reliably underestimate the expected time to extinction.

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N = 120

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N = 150

N =