Asymptotic dynamics in continuous structured populations with mutations, competition and mutualism

This paper deals with a class of integro-differential equations arising in evolutionary biology to model the dynamics of specialist and generalist species related by mutualistic interactions. The effects of mutation events, proliferative phenomena and competition are taken into account. Specialist population is assumed to be structured by a continuous phenotypical trait related to the ability of individuals to ingest specific resources and a parameter ϵ is introduced to model the average size of mutations. A well-posedness result is here proposed and the asymptotic behavior of the density of specialist individuals in the space of the phenotypical traits is studied in the limit ϵ→0. In particular, under a suitable time rescaling, we prove the weak convergence of such a density to a sum of Diracʼs masses. A characterization of the set of concentration points is provided.