Chemotaxis, the directed motion of cells or organisms up chemical gradients, is a ubiquitous signal-response mechanism in nature with examples ranging from the movement of bacteria to the formation of blood capillary networks. In this paper, we investigate the existence and stability of steady state solutions to a system of partial differential equations used to model a class of chemotactic phenomena. This system is of Keller-Segel type with an additional volume-filling term. In particular, we are interested in how the initial cell density determines the possible asymptotic states of the system and the related properties of coarsening and meta-stability. We present existence results for continua of spatially heterogeneous steady state solutions where the bifurcation parameter is a measure of initial concentrations. These results are augmented with an analysis of the stability of both uniform and non-uniform solutions. We present numerical evidence that the spectrum of the linearisation associated with the steady states has a specific form and conjecture that coarsening in this model results from the underlying continuum of problems associated with different initial data.
Davidson, F. A., & Savory, A. I. M. (2014). Existence and meta-stability of steady states of a model for chemotaxis. Nonlinear Analysis: Real World Applications, 17, 101-113. https://doi.org/10.1016/j.nonrwa.2013.10.009