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

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 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 slow migration mechanism and the fast density regulation mechanism, the population dynamics converges to a Fleming-Viot process as the scaling parameter N approaches ∞. 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 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 mutation and 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. A part of this work was done when I was a graduate student at University of Wisconsin, Madison. I wish to sincerely thank my thesis adviser, Prof. Thomas G. Kurtz, for his continuing support and encouragement. This work was completed while I was holding a postdoctoral appointment under Prof. Sylvie Méléard and Prof. Vincent Bansaye at École Polytechnique, Paris. The hospitality and support provided by them during my stay is gratefully acknowledged. This work was supported by the professoral chair Jean Marjoulet, the project MANEGE (Modèles Aléatoires en Écologie, Génétique et Évolution) of ANR (French national research agency) and the Chair Modelisation Mathematique et Biodiversite VEOLIA-École Polytechnique-MNHN-F.X.