On the Fisher Infinitesimal Model Without Variability

Abstract

We study the long-time behavior of solutions to a kinetic equation inspired by a model of sexual populations structured in phenotypes. The model features a nonlinear integral reproduction operator derived from the Fisher infinitesimal operator and a trait-dependent selection term. The reproduction operator describes here the inheritance of the mean parental traits to the offspring without variability. We show that, under assumptions on the growth of the selection rate, Dirac masses are stable around phenotypes for which the difference between the selection rate and its minimum value is less than \(\frac{1}{2}\). Moreover, we prove the convergence in some Fourier-based distance of the centered and rescaled solution to a stationary profile under some conditions on the initial moments of the solution.