# American Institute of Mathematical Sciences

2013, 10(3): 637-647. doi: 10.3934/mbe.2013.10.637

## Finite element approximation of a population spatial adaptation model

 1 Dpto. de Matemáticas, Universidad de Oviedo, c/ Calvo Sotelo, 33007-Oviedo, Spain, Spain

Received  May 2012 Revised  October 2012 Published  April 2013

In [18], 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 attraction to more favorable environmental regions was included. In this article, we study numerically a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model, in which a time non-local drift term is considered, we propose a numerical discretization in terms of a mass-preserving time semi-implicit finite element method. Finally, we provied the results of some biologically inspired numerical experiments showing qualitative differences between the original model of [18] and the model proposed in this article.
Citation: Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637
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