Relationships between stochastic and deterministic population models

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Abstract

The infinitesimal parameters of a variety of Markov population models can be written as q k (N) ,k+l=Nℝl(N-1k) where k,l ∈ ℤdand N is a parameter which is of the same order of magnitude as the population size. Under appropriate conditions, a family of Markov processes {XN} with these parameters satisfies \( \mathop{{\lim }}\limits_{{N \to \infty }} {N^{{ - 1}}}{X_N}(t) = X(t) \) , in probability where X(t) is a solution of the differential equation \( \mathop{{X = F(X) \equiv \sum\limits_l {l{\mathbb{R}_l}} (X)}}\limits^o \) .

Several approaches to studying the error N-1XN(t) - X(t) have been considered by a number of authors, including limit theorems with error bounds for the sequence √N(N-1XN(t) - X(t)).

These results can be generalized in a variety of ways. We will consider examples that are not Markovian (i.e. epidemic models in which the infectious period is not exponentially distributed) and examples that take into account the spatial distribution of the population. In particular we will obtain the solution of Fisher’s Equation ut = auxx + αu(1-u) as the deterministic limit of Markov process models for two competing plant species.