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

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.