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.
Highlights
Density-dependent models are well-known in population biology
The birth and death rates of individuals may depend on the density of the population, where the term density refers to the population size under a suitably chosen normalization
Density-dependent models are appealing because one can account for interactions among individuals by appropriately specifying the birth and death rates as functions of the population density
Summary
Density-dependent models are well-known in population biology. In these models, the birth and death rates of individuals may depend on the density of the population, where the term density refers to the population size under a suitably chosen normalization. Since the discovery of resources is tied to the movement of individuals, it is reasonable to assume that both migration and birth-death mechanisms operate at the same timescale For such spatial models, Oelschlager [31] has shown in a multi-type setting that the dynamics converges in the infinite population limit to a system of reaction-diffusion partial differential equations. Our Fleming-Viot convergence result shows that one can recover this sampling mechanism in the infinite population limit if the dynamics has an equilibrating density regulation mechanism that acts at a faster timescale than other events It is well-known that a Fleming-Viot process arises in the infinite population limit of an appropriately scaled version of the Wright-Fisher or the Moran model (see [17] and [13]).
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