# American Institute of Mathematical Sciences

March  2021, 26(3): 1469-1497. doi: 10.3934/dcdsb.2020169

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

 1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA 2 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Pabellón I - Ciudad Universitaria, Buenos Aires - Argentina

* Corresponding author: azmy.ackleh@louisiana.edu

Received  September 2019 Revised  February 2020 Published  May 2020

In this paper we consider a selection-mutation model with an advection term formulated on the space of finite signed measures on $\mathbb{R}^d$. 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.

Citation: Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169
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##### References:
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