Self-organized clusters in diffusive run-and-tumble processes

Patrick Flynn, Quinton Neville, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We analyze a simplistic model for run-and-tumble dynamics, motivated by observations of complex spatio-temporal patterns in colonies of myxobacteria. In our model, agents run with fixed speed either left or right, and agents turn with a density-dependent nonlinear turning rate, in addition to diffusive Brownian motion. We show how a very simple nonlinearity in the turning rate can mediate the formation of self-organized stationary clusters and fronts. Phenomenologically, we demonstrate the formation of barriers, where high concentrations of agents at the boundary of a cluster, moving towards the center of a cluster, prevent the agents caught in the cluster from escaping. Mathematically, we analyze stationary solutions in a four-dimensional ODE with a conserved quantity and a reversibility symmetry, using a combination of bifurcation methods, geometric arguments, and numerical continuation. We also present numerical results on the temporal stability of the solutions found here.

Original languageEnglish (US)
Pages (from-to)1187-1208
Number of pages22
JournalDiscrete and Continuous Dynamical Systems - Series S
Issue number4
StatePublished - 2019

Bibliographical note

Funding Information:
2010 Mathematics Subject Classification. Primary: 92C15, 35Q92; Secondary: 37N25. Key words and phrases. Run-and-tumble, homoclinic, heteroclinic, myxobacteria, stability. The authors were partially supported by NSF grant DMS-1612441. ∗ Corresponding author: Arnd Scheel.

Funding Information:
one of the referees pointed out reference [2] with closely related analysis. Most of this work was carried out during an REU project on “Complex Systems” at the University of Minnesota, funded through NSF grant DMS–1311740.


  • Heteroclinic
  • Homoclinic
  • Myxobacteria
  • Run-and-tumble
  • Stability

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