Instability in a generalized Keller-Segel model

TL;DR

This paper generalizes the Keller-Segel model to include multiple reacting chemical compounds, analyzes the stability of homogeneous states, and identifies conditions under which chemotactic feedback causes instability and pattern formation.

Contribution

It extends the classical Keller-Segel model to multiple chemicals and provides verifiable criteria for instability using matrix theory.

Findings

Strong chemotactic feedback leads to instability.

Homogeneous states can become unstable under certain conditions.

The instability mechanism applies to a broader class of models.

Abstract

We present a generalized Keller-Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller-Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.