The Fleming-Viot limit of an interacting spatial population with fast density regulation

Abstract

We consider population models in which the individuals reproduce, die and also migrate in space. The population size scales according to some parameter $N$, which can have different interpretations depending on the context. Each individual is assigned a mass of 1/N and the total mass in the system is called \emph{population density}. The dynamics has an intrinsic density regulation mechanism that drives the population density towards an equilibrium. We show that under a timescale separation between the \emph{slow} migration mechanism and the \emph{fast} density regulation mechanism, the population dynamics converges to a Fleming-Viot process as the scaling parameter $N$ approaches $\infty$. We first prove this result for a basic model in which the birth and death rates can only depend on the population density. In this case we obtain a \emph{neutral} Fleming-Viot process. We then extend this model by including position-dependence in the birth and death rates, as well as, offspring dispersal and immigration mechanisms. We show how these extensions add \emph{mutation} and \emph{selection} to the limiting Fleming-Viot process. All the results are proved in a multi-type setting, where there are $q$ types of individuals interacting with each other. We illustrate the usefulness of our convergence result by discussing applications in population genetics and cell biology.