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} u_t=uₓₓ-χ (u/v v_x)_x+u(a-b u), & 0<x<L, \, t>0, 0=vₓₓ- μ v+ ν u, & 0<x<L, \, t>0 u_x(t,0)=u_x(t,L)=v_x(t,0)=v_x(t,L)=0, & t>0, {1} {cases} where χ, a, b, μ, ν are positive constants. Among others, we prove there are χ^*>0 and \χₖ^*\⊂ [χ^*,∞) (χ^*∈\χₖ^*\) such that the constant solution (a/b,ν//b) of (1) is locally stable when 0<χ<χ^* and is unstable when χ>χ^*, and under some generic condition, for each k≥ 1, a (local) branch of non-constant stationary solutions of (1) bifurcates from (a/b,ν//b) when χ passes through χₖ^*, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions \(u(·;χₙ),v(·;χₙ))\ of (1) with χ=χₙ(→ ∞) develops spikes at any x^* satisfying n→∞ u(x^*;χₙ)>a/b. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (a/b,ν//b) when χ passes through χ^* can be extended to χ=∞ and the stationary solutions on this global bifurcation extension are locally stable when χ 1 and develop spikes as χ→∞.