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Spatio-temporal chaos in a chemotaxis model. (English) Zbl 1255.37026

In this paper, the dynamics of a one-dimensional Keller-Segel type model for chemotaxis incorporating a logistic cell growth term was explore. The capacity of the model to self-organise into multiple cellular aggregations was demonstrated, 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, it was found that 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, it was argued that the spatio-temporal irregularity observed here describes a form of spatio-temporal chaos. Brief discussion on the results in the context of previous applications of chemotaxis models, including tumour invasion, embryonic development and ecology was followed.

MSC:

37N25 Dynamical systems in biology
92C17 Cell movement (chemotaxis, etc.)
92C37 Cell biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)

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