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

In , Sighesada et al. 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 attraction 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 provide the results of two biologically inspired numerical experiments showing qualitative differences between the original model of and the model proposed in this article.