# Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

## TL;DR

This paper analyzes the stability and bifurcation structure of steady states in a bacterial colony model with density-dependent motility, revealing conditions for uniform stability and pattern formation.

## Contribution

It provides a new analytical framework for understanding how bacterial growth rates influence pattern formation and stability in density-suppressed motility models.

## Key findings

- Large growth rates lead to stable uniform states.
- Lower growth rates induce pattern formation.
- Analytical results are supported by numerical simulations.

## Abstract

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.