Self-organized Clusters in Diffusive Run-and-Tumble Processes

TL;DR

This paper studies a simple run-and-tumble model with nonlinear turning rates that leads to self-organized clustering and barrier formation, providing mathematical and numerical analysis of stationary solutions and their stability.

Contribution

It introduces a minimal nonlinear model for run-and-tumble dynamics that explains cluster formation and barrier effects, supported by bifurcation analysis and numerical simulations.

Findings

Nonlinear turning rates induce stable clusters and barriers.

Stationary solutions analyzed via bifurcation and geometric methods.

Numerical results confirm stability and structure of clusters.

Abstract

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.