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, u_t=u_xx-χ(u/v v_x)_x+u(a-b u), 00,0=v_xx- μv+ νu, 00 u_x(t,0)=u_x(t,L)=v_x(t,0)=v_x(t,L)=0, t>0, 1 where χ, a, b, μ, ν are positive constants. Among others, we prove there are χ^*>0 and {χ_k^*}⊂ [χ^*,∞) (χ^*∈{χ_k^*}) such that the constant solution (a/b,ν/μa/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/μa/b) when χ passes through χ_k^*, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions {(u(·;χ_n),v(·;χ_n))} of (1) with χ=χ_n(→∞) develops spikes at any x^* satisfying lim inf_n→∞ u(x^*;χ_n)>a/b. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (a/b,ν/μa/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→∞.