Stationary solutions of a volume-filling chemotaxis model with logistic growth and their stability

Author name used in this publication: Zhi-An Wang2011-2012 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a d-dimensional box Td=(0,π)d  (d=1,2,3). It is proved that given any general perturbation of magnitude δ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln⁡(1/δ). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns

A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis

We consider two models which were both designed to describe the movement of eukaryotic cells responding to chemical signals. Besides a common standard parabolic equation for the diffusion of a chemoattractant, like chemokines or growth factors, the two models differ for the equations describing the movement of cells. The first model is based on a quasilinear hyperbolic system with damping, the other one on a degenerate parabolic equation. The two models have the same stationary solutions, which may contain some regions with vacuum. We first explain in details how to discretize the quasilinear hyperbolic system through an upwinding technique, which uses an adapted reconstruction, which is able to deal with the transitions to vacuum. Then we concentrate on the analysis of asymptotic preserving properties of the scheme towards a discretization of the parabolic equation, obtained in the large time and large damping limit, in order to present a numerical comparison between the asymptotic behavior of these two models. Finally we perform an accurate numerical comparison of the two models in the time asymptotic regime, which shows that the respective solutions have a quite different behavior for large times.Comment: One sentence modified at the end of Section 4, p. 1