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

In the current paper, we study stability, bifurcation, and spikes of positive stationary solutions of the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, cases uₜ=uₗₗ- (uv vₓ) ₓ+u (a-b u), ₙ), v (;ₙ) ) \} of (1) with =ₙ () develops spikes at any x^* satisfying ₍ u (x^*;ₙ) >ab. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (ab, ab) when passes through ^* can be extended to = and the stationary solutions on this global bifurcation extension are locally stable when 1 and develop spikes as.