Mathematics – Probability
Scientific paper
2012-04-13
Mathematics
Probability
Scientific paper
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 $N \to \infty$. 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 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 reproducing each other. To illustrate the usefulness of our convergence result, we discuss certain applications in ecology and cell biology.
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