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

Halil Ibrahim Kurt, Wenxian Shen, Shuwen Xue

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 $χ$, $a$, $b$, $μ$, $ν$ are positive constants. Among others, we prove there are $χ^*>0$ and $\{χ_k^*\}\subset [χ^*,\infty)$ ($χ^*\in\{χ_k^*\}$) such that the constant solution $(\frac{a}{b},\fracνμ\frac{a}{b})$ of (1) is locally stable when $0<χ<χ^*$ and is unstable when $χ>χ^*$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\fracνμ\frac{a}{b})$ 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 $\{(u(\cdot;χ_n),v(\cdot;χ_n))\}$ of (1) with $χ=χ_n(\to \infty)$ develops spikes at any $x^*$ satisfying $\liminf_{n\to\infty} u(x^*;χ_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ^*$ can be extended to $χ=\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $χ\gg 1$ and develop spikes as $χ\to\infty$.

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 , , , , are positive constants. Among others, we prove there are and () such that the constant solution of (1) is locally stable when and is unstable when , and under some generic condition, for each , a (local) branch of non-constant stationary solutions of (1) bifurcates from when passes through , and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions of (1) with develops spikes at any satisfying . Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from when passes through can be extended to and the stationary solutions on this global bifurcation extension are locally stable when and develop spikes as .
Paper Structure (10 sections, 6 theorems, 164 equations, 23 figures)

This paper contains 10 sections, 6 theorems, 164 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Local stability and instability of the positive constant solution
  3. Local bifurcation and stability of bifurcating solutions
  4. Global bifurcation and properties of non-constant stationary solutions
  5. Spiky stationary solutions
  6. Numerical analysis
  7. Numerical scheme
  8. Numerical analysis for the case $a=b=\mu=\nu=L=1$
  9. Numerical simulations for the case $a=b=\mu=\nu=1$ and $L=6$
  10. Numerical simulations for the case $a=2$, $b=\nu=1$, $\mu=3$, $L=6$

Key Result

Theorem 2.1

There is $\chi^*=\chi^*(a,\mu)$ depending only on $a$ and $\mu$ such that when $0<\chi<\chi^*$, the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of main-eq0 is locally asymptotically stable with respect to perturbations in $C^+([0,L])$ and when $\chi>\chi^*$, $(\frac{a}{b},\frac{\nu}

Figures (23)

  • Figure 1: (a) super-critical pitchfork bifurcation (b) sub-critical pitchfork bifurcation
  • Figure 2: (a) limit profile, (b) evolution of $u(t,x;u_0)$ with $\chi=5$, $a=b=\mu=\nu=L=1$ and initial function $u_0=1+0.5\cos(\pi x)$
  • Figure 3: (a) limit profile, (b) evolution of $u(t,x;u_0)$ with $\chi=11.96$, $a=b=\mu=\nu=L=1$ and initial function $u_0=1+0.5\cos(\pi x)$
  • Figure 4: (a) limit profile, (b) evolution of $u(t,x;u_0)$ with $\chi=11.98$, $a=b=\mu=\nu=L=1$ and initial function $u_0=1+0.5\cos(\pi x)$
  • Figure 5: (a) limit profile, (b) evolution of $u(t,x;u_0)$ with $\chi=20$, $a=b=\mu=\nu=L=1$ and initial function $u_0=1+0.5\cos(\pi x)$
  • ...and 18 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.1
  • proof : Proof of Theorem \ref{['local-bif-thm']}
  • Theorem 4.1: Global bifurcation
  • Remark 4.1
  • ...and 12 more