Spreading speed of locally regulated population models in macroscopically heterogeneous environments

Abstract

We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\epsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient\ldots). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\epsilon \to 0$ and then $K\to \infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to \infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\epsilon \to 0$ and $K\to \infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\epsilon$ are of the same order.

41 pages, 1 figure. Title changed. Assumptions on the model relaxed, and other modifications

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