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We consider population models in which the individuals reproduce, die andalso migrate in space. The population size scales according to some parameter$N$, which can have different interpretations depending on the context. Eachindividual 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 regulationmechanism that drives the population density towards an equilibrium. We showthat under a timescale separation between the \emph{slow} migration mechanismand the \emph{fast} density regulation mechanism, the population dynamicsconverges 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 anddeath 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 includingposition-dependence in the birth and death rates, as well as, offspringdispersal and immigration mechanisms. We show how these extensions add\emph{mutation} and \emph{selection} to the limiting Fleming-Viot process. Allthe results are proved in a multi-type setting, where there are $q$ types ofindividuals interacting with each other. We illustrate the usefulness of ourconvergence result by discussing applications in population genetics and cellbiology.

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