Spatio-temporal chaos in a chemotaxis model

In this paper we explore the dynamics of a one-dimensional Keller–Segel type model for chemotaxis incorporating a logistic cell growth term. We demonstrate the capacity of the model to self-organise into multiple cellular aggregations which, according to position in parameter space, either form a stationary pattern or undergo a sustained spatio-temporal sequence of merging (two aggregations coalesce) and emerging (a new aggregation appears). This spatio-temporal patterning can be further subdivided into either a time-periodic or time-irregular fashion. Numerical explorations into the latter indicate a positive Lyapunov exponent (sensitive dependence to initial conditions) together with a rich bifurcation structure. In particular, we find stationary patterns that bifurcate onto a path of periodic patterns which, prior to the onset of spatio-temporal irregularity, undergo a “periodic-doubling” sequence. Based on these results and comparisons with other systems, we argue that the spatio-temporal irregularity observed here describes a form of spatio-temporal chaos. We discuss briefly our results in the context of previous applications of chemotaxis models, including tumour invasion, embryonic development and ecology.