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

Abstract

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, \begin{cases} u_t=u_{xx}-χ(\frac{u}{v} v_x)_x+u(a-b u), & 0<x<L, \, t>0,\cr 0=v_{xx}- μv+ νu, & 0<x<L, \, t>0 \cr u_x(t,0)=u_x(t,L)=v_x(t,0)=v_x(t,L)=0, & t>0, \tag{1} \end{cases} where $\chi$, $a$, $b$, $\mu$, $\nu$ are positive constants. Among others, we prove there are ${\chi }^{\ast }>0$ and $\{{\chi }_k^{\ast }\}\subset [{\chi }^{\ast },\infty )$ (${\chi }^{\ast }\in \{{\chi }_k^{\ast }\}$) such that the constant solution $(\frac{a}{b},\frac{\nu }{\mu }\frac{a}{b})$ of (1) is locally stable when $0<\chi <{\chi }^{\ast }$ and is unstable when $\chi >{\chi }^{\ast }$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\frac{\nu }{\mu }\frac{a}{b})$ when $\chi$ passes through ${\chi }_k^{\ast }$, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions $\{(u(\cdot ;{\chi }_n),v(\cdot ;{\chi }_n))\}$ of (1) with $\chi ={\chi }_n(\to \infty )$ develops spikes at any $x^{\ast }$ satisfying ${\mathrm{liminf}}_{n\to \infty }u(x^{\ast };{\chi }_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\frac{\nu }{\mu }\frac{a}{b})$ when $\chi$ passes through ${\chi }^{\ast }$ can be extended to $\chi =\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $\chi \gg 1$ and develop spikes as $\chi \to \infty$.

47 pages