Abstract
We analyze three non-local models describing the evolutionary dynamics of a continuous phenotypic trait undergoing the joint action of mutations and selection. We establish the existence and uniqueness of the solutions to the Cauchy problem, and give a description of the long-time behaviour of the solution. In the first work we study the replicator-mutator equation in the unbounded domain and generalize to cases of selective values confining the known results in the harmonic case. Namely, the existence of a unique global regular solution, converging towards a universal profile; for this, we use spectral decomposition techniques of Schrödinger operators. In the second work, we discuss a model whose fitness value is density-dependent. In order to show the well-posedness of the equation, we combine two approaches. The first is based on the study of the cumulant generating functions, satisfying a non-local transport equation and making it possible to implicitly obtain the average trait. The second uses a change of variable (Avron-Herbst formula), allowing the solution to be written in terms of the average trait and the solution of the heat equation with the same initial data. Finally, we study a model whose mutation rate is proportional to the average value of the trait. We establish a bijective link between this last model and the second, thus making it possible to describe the dynamics of the solution in detail. In particular, we show the exponential growth of the average trait.