A Hamilton-Jacobi approach to evolution of dispersal

The evolution of dispersal is a classical question in evolutionary biology, and it has been studied wide range mathematical models. A selection-mutation model, which the population structured by space phenotypic trait, with trait acting directly on (diffusion) rate, was formulated Perthame Souganidis [Math. Model. Nat. Phenom. 11:154–166, 2016] to study random toward evolutionarily stable strategy. For rare mutation limit, shown that equilibrium concentrates single associated smallest rate. In this paper, we consider corresponding equation characterize asymptotic behaviors time-dependent solutions under mild convexity assumptions underlying Hamiltonian function.