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 language||English (US)|
|Number of pages||22|
|Journal||Discrete and Continuous Dynamical Systems - Series S|
|State||Published - 2019|
Bibliographical noteFunding 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.
one of the referees pointed out reference  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.