Mathematical Properties of a Class of Integro-differential Models from Population Genetics

Abstract

We propose a mathematical analysis of an integro-differential model arising in population genetics. The model describes the dynamics of fitness distribution in an asexual population under the effect of mutation and selection. These two processes are represented by two nonlocal terms. First, we prove the existence and uniqueness of the solution, and we derive asymptotic estimates of the distribution as the fitness tends to $\pm \infty$. Based on these asymptotic estimates, we show that the cumulant generating function of the distribution is well defined and satisfies a linear nonlocal transport equation that we solve explicitly. This explicit formula allows us to characterize the dependence of the long time behavior of the distribution with respect to the mutation kernel. On the one hand, if the kernel contains some beneficial mutations, the distribution diverges, which is reminiscent of the results of Alfaro and Carles [SIAM J. Appl. Math., 74 (2014), pp. 1919-1934] who analyzed a mutator-replicator equat...