Singular limits for models of selection and mutations with heavy-tailed mutation distribution

Abstract

In this article, we perform an asymptotic analysis of a nonlocal reaction-diffusion equation, with a fractional laplacian as the diffusion term and with a nonlocal reaction term. Such equation models the evolutionary dynamics of a phenotypically structured population.We perform a rescaling considering large time and small effect of mutations, but still with algebraic law. With such rescaling, we expect that the phenotypic density will concentrate as a Dirac mass which evolves in time. To study such concentration phenomenon, we extend an approach based on Hamilton-Jacobi equations with constraint, that has been developed to study models from evolutionary biology, to the case of fat-tailed mutation kernels. However, unlike previous works within this approach, the WKB transformation of the solution does not converge to a viscosity solution of a Hamilton-Jacobi equation but to a viscosity supersolution of such equation which is minimal in a certain class of supersolutions. Such property allows to derive the concentration of the population density as an evolving Dirac mass, under monotony conditions on the growth rate, similarly to the case with thin-tailed mutation kernels.