Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model

In a recent study (K.J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxismodel, Physica D, 240 (4), 363-375, 2011) a model for chemotaxis incorporatinglogistic growth was investigated for its pattern formation properties. In particular,a variety of complex spatio-temporal patterning was found, including stationary,periodic and chaotic. Complicated dynamics appear to arise through asequence of ``merging and emerging'' events: the merging of two neighbouringaggregates or the emergence of a new aggregate in an open space. In thispaper we focus on a time-discrete dynamical system motivated by thesedynamics, which we call the merging-emerging system (MES). We introduce this newclass of set-valued dynamical systems and analyse its capacity to generate similar``pattern formation'' dynamics. The MES shows remarkably close correspondencewith patterning in the logistic chemotaxis model, strengthening our assertionthat the characteristic length scales of merging and emerging are responsiblefor the observed dynamics. Furthermore, the MES describes a novel class ofpattern-forming discrete dynamical systems worthy of study in its own right.