Selection-mutation dynamics with asymmetrical reproduction kernels

TL;DR

This paper investigates the long-term behavior of a sexually reproducing population with asymmetric inheritance, using mathematical models to understand trait distribution concentration and population viability under specific reproductive assumptions.

Contribution

It introduces a novel selection-mutation model with asymmetric trait inheritance and analyzes its asymptotic behavior, including concentration phenomena and non-extinction conditions.

Findings

Derived non-extinction conditions for the population.

Established BV estimates on total population.

Obtained Lipschitz estimates for Hamilton-Jacobi solutions.

Abstract

We study a family of selection-mutation models of a sexual population structured by a phenotypical trait. The main feature of these models is the asymmetric trait heredity or fecundity between the parents : we assume that each individual inherits mostly its trait from the female or that the trait acts on the female fecundity but does not affect male. Following previous works inspired from principles of adaptive dynamics, we rescale time and assume that mutations have limited effects on the phenotype. Our goal is to study the asymptotic behavior of the population distribution. We derive non-extinction conditions and BV estimates on the total population. We also obtain Lipschitz estimates on the solutions of Hamilton-Jacobi equations that arise from the study of the population distribution concentration at fittest traits. Concentration results are obtained in some special cases by using a Lyapunov functional.