Pattern Formation of a Keller-Segel Model with the Source Termup(1-u)

Shengmao Fu,Fenli Cao

doi:10.1155/2013/454513

Abstract

Nonlinear dynamics near an unstable constant equilibrium in a Keller-Segel model with the source termup(1-u)is considered. It is proved that nonlinear dynamics of a general perturbation is determined by the finite number of linear growing modes over a time scale ofln(1/δ), whereδis a strength of the initial perturbation.

Highlights

Mimura and Tsujikawa in [1] proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis, a dimensionless prototype of which readsUt = ∇ (Du∇U − χU∇V) + g (U), (1)Vt = DV∇2V + αU − βV, where U(x, t) is the cell density, V(x, t) is the concentration of chemotactic substance, Du > 0 is the amoeboid motility, χ > 0 is the chemotactic sensitivity, DV > 0 is the diffusion rate of cyclic adenosine monophosphate, α > 0 is the rate of cAMP secretion per unit density of amoebae, and β > 0 is the rate of degradation of cAMP in environment
For model (1), with a Logistic source term g(u) = u(1−u), Tello and Winkler [2] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al [3] numerically showed several spatiotemporal patterns in a rectangle
Kuto et al [4] considered some qualitative behaviors of stationary solutions from global and local viewpoints

Summary

Introduction

Mimura and Tsujikawa in [1] proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis, a dimensionless prototype of which reads. Painter and Hillen [6] demonstrated the capacity of (1) to selforganize into multiple cellular aggregations, which, according to position in parameter space, either form a stationary pattern or undergo a sustained spatiotemporal sequence of merging (two aggregations coalesce) and emerging (a new aggregation appears) Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. By using nonnegativity of solutions, Nakaguchi and Efendiev [9] managed significantly to improve dimension estimates with respect to the chemotactic parameter It is well-known that the asymptotic behavior of solutions relating to patterns can be described by the dynamical systems of equations and that the degrees of freedom of such processes, which characterize the richness of emerging patterns, correspond to the dimensions of their attractors. Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns

Local Stability of Positive Constant Equilibrium Solution

Bootstrap Lemma

Main Result

Findings

Conclusion