Global phase diagrams of run-and-tumble dynamics: equidistribution, waves, and blowup

TL;DR

This paper develops a phase diagram for one-dimensional run-and-tumble dynamics, identifying conditions for decay, wave formation, or blowup, with applications to biological phenomena like bacterial ripples and fruiting bodies.

Contribution

It introduces a coarse phase diagram that classifies the long-term behavior of run-and-tumble systems based on tumbling dynamics and parameters, highlighting critical transitions.

Findings

Identifies phase boundaries for different dynamical behaviors.

Shows how slight parameter changes can switch phases.

Connects mathematical phases to biological phenomena.

Abstract

For spatially one-dimensional run-and-tumble dynamics with mass conservation we develop a coarse phase diagram, that discriminates between global decay to equidistributed constant states, existence of spatially non-trivial waves, and finite time blowup of solutions. Motivated by counter-migrating ripples of high and low population density and fruiting body formation in myxobacteria, we identify phase boundaries as particular critical tumbling dynamics that allow for switching between these spatio-temporal phases upon slight changes in mass densities or parameter values.