Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures

Abstract

In this paper we consider a selection-mutation model with an advection term formulated on the space of finite signed measures on \begin{document}$ \mathbb{R}^d $\end{document} . The selection-mutation kernel is described by a family of measures which allows the study of continuous and discrete kernels under the same setting. We rescale the selection-mutation kernel to obtain a diffusively rescaled selection-mutation model. We prove that if the rescaled selection-mutation kernel converges to a pure selection kernel then the solution of the diffusively rescaled model converges to a solution of an advection-diffusion equation.