Instability in a generalized Keller–Segel model

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.

Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model

“…Starting in [13], nonlinear instability was established from linear instability for dissipative systems [14], based on a bootstrap lemma. Then, in [8], nonlinear instability was proved with matrix theoretical tools under a sufficient condition that the chemotactic feedback is sufficiently strong. Next, the authors in [15] considered the classical Turing instability over a time scale before blow-up of solutions.…”

Nonnegativity of exact and numerical solutions of some chemotactic models

“…This reinforces the need to satisfy (45) if we want to find the smallest value of m(u 0 ) that yields (bounded) pattern formation. Third, an analysis of [35] (or its extension in [10]) shows that any constant solution u * of (43) is unstable if u * > 1 + µ 1 (Ω) where µ 1 (Ω) is the smallest nonzero eigenvalue of the Laplacian with Neumann boundary conditions on Ω. These results suggest that to avoid convergence to the biologically uninteresting constant stationary states, we should choose the initial iterate to have mean value m(u 0 ) > 1 + µ 0 .…”

“…, N , represented as components of a vector function v. We are interested in the situation where one of these chemicals is a chemoattractant for the species. This situation is modeled by the following system of equations, taken from [10], which we refer to as a generalized Keller-Segel system throughout this paper:…”