Individual and patch behaviour in structured metapopulation models

TL;DR

This paper analyzes structured metapopulation models, showing that individuals experience near-deterministic environments and behave almost independently, with their dynamics approximated by Markov jump processes influenced by the population's average drift.

Contribution

It demonstrates that in large population models, individual and patch behaviors can be approximated by nearly deterministic and independent Markov processes, extending understanding of metapopulation dynamics.

Findings

Individuals experience an almost deterministic environment.

Small groups behave as independent Markov jump processes.

The model applies to patches, hosts, and parasites.

Abstract

Density dependent Markov population processes with countably many types can often be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, provided that the total population size is large. They also exhibit diffusive stochastic fluctuations on a smaller scale about this deterministic path. Here, it is shown that the individuals in such processes experience an almost deterministic environment. Small groups of individuals behave almost independently of one another, evolving as Markov jump processes, whose transition rates are prescribed functions of time. In the context of metapopulation models, we show that `individuals' can represent either patches or the individuals that migrate among the patches; in host--parasite systems, they can represent both hosts and parasites.