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

Sepideh Mirrahimi (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: Key-Words: Fractional reaction-diffusion equation; Hamilton-Jacobi equation; viscosity solutions; AMS Class No: 35K57; nonlocal reaction term; asymptotic analysis; viscosity solutions AMS Class No: 35K57; 35B25; 47G20; 49L25; 92D15

Domaines: Equations aux dérivées partielles [math.AP]

Sepideh Mirrahimi. Singular limits for models of selection and mutations with heavy-tailed mutation distribution. Journal de Mathématiques Pures et Appliquées, 2019, ⟨10.1016/j.matpur.2019.10.001⟩. ⟨hal-01849418v3⟩