Pattern Formation and Cross-Diffusion for a Chemotaxis Model

Abstract

Chemotaxis is the feature movement of cell or an organism along a chemical concentration gradient. The mathematical analysis of chemotaxis modelss how a plenitude of spatial patterns such as the chemotaxis models applied to skin pigmentation patterns, that lead to aggregations of one type of pigment cells into a striped spatial pattern. The analysis of pattern formation can be traced to a seminal paper by Turing [1], who established that an action-diffusion system can generate stable nonuniform patterns in space if the components of the system interact with each other.Our motivation is the numerical simulations of the pattern formation for a volume-filling chemotaxis model. In [2], the effect of volume-filling is expressed through a nonlinear squeezing probability. We investigate pattern formation using Turing's principle and the standard argument used by Murray [3,4]. Next, we introduce an implicit nite volume scheme; it is presented on a general mesh satisfying the orthogonality condition [5,6].The originality of this scheme is the upstream approach to discretize thecross-diffusion term. Finally, we present some numerical results showing the spatial patterns for the chemotaxis model.