Some deterministic structured population models which are limit of stochastic individual based models

TL;DR

This paper establishes a rigorous framework connecting stochastic individual-based population models to their deterministic limits, proving a law of large numbers and applying it to various examples.

Contribution

It provides a general framework for the limit of stochastic population models to deterministic models, fulfilling a program proposed by Diekmann et al. (2001).

Findings

Established a law of large numbers for the models.

Provided a general framework for stochastic to deterministic limits.

Applied the results to multiple population models.

Abstract

The aim of this paper is to tackle part of the program set by Diekmann et al. in their seminal paper Diekmann et al. (2001). We quote "It remains to investigate whether, and in what sense, the nonlinear determin-istic model formulation is the limit of a stochastic model for initial population size tending to infinity" We set a precise and general framework for a stochastic individual based model : it is a piecewise deterministic Markov process defined on the set of finite measures. We then establish a law of large numbers under conditions easy to verify. Finally we show how this applies to old and new examples.