Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

Abstract

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].