Stability and bifurcation for logistic Keller--Segel models on compact graphs

TL;DR

This paper analyzes the stability and bifurcation behavior of chemotaxis models on metric graphs, identifying critical parameters for stability, bifurcation points, and providing numerical bifurcation data for various graph structures.

Contribution

It introduces a threshold for chemotaxis sensitivity that determines stability regimes and characterizes bifurcation points leading to non-constant steady states on metric graphs.

Findings

Identified a critical chemotaxis sensitivity threshold $st$ for stability.

Established parameter intervals for global convergence to steady states.

Numerically computed bifurcation points for several graph types.

Abstract

This paper concerns asymptotic stability, instability, and bifurcation of constant steady state solutions of the parabolic-parabolic and parabolic-elliptic chemotaxis models on metric graphs. We determine a threshold value $\chi^*>0$ of the chemotaxis sensitivity parameter that separates the regimes of local asymptotic stability and instability, and, in addition, determine the parameter intervals that facilitate global asymptotic convergence of solutions with positive initial data to constant steady states. Moreover, we provide a sequence of bifurcation points for the chemotaxis sensitivity parameter that yields non-constant steady state solutions. In particular, we show that the first bifurcation point coincides with threshold value $\chi^*$ for a generic compact metric graph. Finally, we supply numerical computation of bifurcation points for several graphs.