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

TL;DR

This paper analyzes the asymptotic behavior of a nonlocal reaction-diffusion model with heavy-tailed mutations, showing phenotypic distribution concentrates as a moving Dirac mass, extending Hamilton-Jacobi methods to fat-tailed kernels.

Contribution

It extends Hamilton-Jacobi approach to models with heavy-tailed mutation kernels, demonstrating concentration phenomena and supersolution convergence in this context.

Findings

Phenotypic distribution concentrates as a Dirac mass over time.

The WKB transformation converges to a minimal viscosity supersolution.

The approach applies to fat-tailed mutation kernels in evolutionary models.

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. We prove that asymptotically the phenotypic distribution density concentrates as a Dirac mass which evolves in time. This work extends 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.