Pattern formation in a Keller-Segel chemotaxis model with logistic growth

TL;DR

This study analyzes pattern formation in a Keller-Segel chemotaxis model with logistic growth, identifying bifurcation points, stability of solutions, and resulting complex cellular aggregation patterns through theoretical and numerical methods.

Contribution

It introduces a bifurcation analysis framework for chemotaxis models with logistic growth, revealing stability criteria and pattern formation mechanisms.

Findings

Homogeneous steady state loses stability as chemoattraction increases.

Existence of nonhomogeneous steady states bifurcating from the homogeneous state.

Numerical simulations show diverse stable patterns like spikes and stripes.

Abstract

In this paper we investigate pattern formation in Keller--Segel chemotaxis models over a multi--dimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate $\chi$ increases. Then using Crandall--Rabinowitz local theory with $\chi$ being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.