Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem

In [19], Sighesada, Kawasaki and Teramoto presented a system of partial differential equations for modeling spatial segregation of interacting species. Apart from competitive Lotka-Volterra (reaction) and population pressure (cross-diffusion) terms, a convective term modeling the populations atraction to more favorable environmental regions is included. In this article, we introduce a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model we briefly discuss its well-possedness and propose a numerical discretization in terms of a mass-preserving time semi-implicit finite differences scheme. Finally, we provied the results of two biologically inspired numerical experiments showing qualitative differences between the original model of [19] and the model proposed in this article.