Ensemble behaviour in population processes with applications to ecological systems

P.K. Pollett

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Abstract: We study Markovian models for population processes in continuous time, addressing questions concerning the behaviour of ensembles of individuals (equilibrium, quasi-equilibrium and time-dependent behaviour), and in particular what can be deduced from models for individual behaviour. It is self evident that ensemble behaviour is precisely the combined behaviour of individuals, so let me be more precise by way of three examples of populations, which will be used throughout to illustrate our major results. I will make the distinction between the two kinds of models (or processes) by referring to them simply as the "individual model" or the "ensemble model" (or process).

Population 1. Our first example is a population network, frequently called a metapopulation (see for example Gilpin and Hanski (1991)), where a fixed number n of individuals occupies geographically separated regions or patches. Patches may become empty, but can be recolonized through migration from other patches. From the point of view of the individual, it spends a period of time in a given patch and might then emigrate to another patch, spend a period there, and so forth. Assuming individuals do not affect each other's progress through the network, one could model the progress of the individual as a random walk on the patches, and thus evaluate quantities such as the probability pj(t) that the individual occupies patch j at time t. Our intuition tells us that, for the ensemble, the proportion of individuals in patch j should be approximately equal to pj(t). So strong is this intuition that scientists frequently model population proportions using individual-level models.

In Section 2 we give a careful examination of whether it is reasonable to approximate random proportions of individuals that share a characteristic using probabilities derived from individual-based models. We are able to make a very precise statement for a very general class of models.

Population 2. This is a variant of Population 1 where we allow death or external emigration from any patch. We will study two cases: (i) the open network, where there is external immigration to one or more patches, and (ii) the closed network, where there are n individuals to begin with, each eventually disappearing from the network through death or external emigration. As before, individuals are assumed not to affect one another's progress, but now individuals (perhaps arriving from outside the network) perform a random walk on the patches but then eventually leave. In contrast to Population 1, the total number of individuals is random. Yet, we would expect to be able to draw similar conclusions concerning ensemble proportions. Furthermore, as the population would be expected to settle down to a stable equilibrium, we might ask whether it is also reasonable to approximate the equilibrium proportion of individuals occupying patch j using the equilibrium probability that an individual is in patch j. This is certainly reasonable in the open case, but even closed metapopulations can exhibit "quasi" equilibrium behaviour over reasonable time scales before extinction occurs; see Pollett (1999).

We examine these questions in Sections 4 and 4, evaluating stationary quasi-stationary distributions for the open and closed ensembles, respectively, and describing their relationship with the corresponding distributions for the individual model.

Population 3. Our final example is a population of organisms, each having a life time that consists of several distinct stages (for example, the butterfly life cycle comprises egg, larva, pupa and adult). Again our intuition tells us that the proportion of the population in stage s should be close to the proportion of time ps that an individual spends in stage s of its life cycle. Results proved in Section 2 confirm this. The quasi-equilibrium behaviour of this population is examined in Section 4.

We begin by describing a general individual model and then construct the corresponding ensemble model. The individual model is Markovian with a specified set of transition rates Q. Since Q is arbitrary, the model is very flexible. In the ensemble model individuals are assigned to the various states, each then moving independently according to Q.

Keywords: Stochastic models; Markov chains; population processes

Acknowledgement: This worked was funded by the Australian Research Council.

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Last modified: 31 July 2007