A large-population limit for a Markovian model of group-structured populations

TL;DR

This paper demonstrates that a Markovian model of group-structured populations converges to a deterministic PDE model in the large-population limit, providing a rigorous justification for using PDEs to approximate complex stochastic dynamics.

Contribution

It establishes a rigorous mathematical link between stochastic Markovian models and deterministic PDEs for group-structured populations.

Findings

Sample paths converge to a deterministic solution

The PDE model accurately approximates the stochastic model in large populations

Provides a theoretical foundation for using PDEs in evolutionary dynamics

Abstract

A Markovian model of group-structured (two-level) population dynamics features births, deaths, and migrations of individuals, and fission and extinction of groups. These models are useful for studying group selection and other evolutionary processes that occur when individuals live in distinct groups. We show that the sample paths of a properly scaled sequence of these models converge in an appropriate Skorohod space to a deterministic trajectory that is a unique solution to a quasilinear evolution equation. The PDE model can therefore be justified as an approximation to the Markovian one.