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

Thomas Hillen*, Jeffery Zielinski, Kevin J Painter

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


In a recent study (K.J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D. 240 (4), 363-375, 2011) a model for chemotaxis incorporating logistic 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 a sequence of "merging and emerging" events: the merging of two neighbouring aggregates or the emergence of a new aggregate in an open space. In this paper we focus on a time-discrete dynamical system motivated by these dynamics, which we call the merging-emerging system (MES). We introduce this new class of set-valued dynamical systems and analyse its capacity to generate similar "pattern formation" dynamics. The MES shows remarkably close correspondence with patterning in the logistic chemotaxis model, strengthening our assertion that the characteristic length scales of merging and emerging are responsible for the observed dynamics. Furthermore, the MES describes a novel class of pattern-forming discrete dynamical systems worthy of study in its own right.

Original languageEnglish
Pages (from-to)2513-2536
Number of pages24
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number10
Publication statusPublished - Dec 2013
Event9th AIMS Conference on Dynamical Systems, Differential Equations and Applications - Orlando, Florida, United States
Duration: 1 Jul 20125 Jul 2012


  • Chemotaxis patterns
  • merging
  • emerging
  • discrete dynamical systems
  • set-valued dynamical systems


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