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

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.