Chemotaxis models with signal-dependent sensitivity and a logistic-type source, II: Persistence and stabilization
This paper is Part II of a series on global existence and asymptotic behavior of positive solutions to \begin{equation*} \begin{cases} \displaystyle u_t=\Delta u-\chi_0\nabla\cdot\left(\frac{u^m}{(1+v)^\beta}\nabla v\right)+au-bu^{1+\alpha}, & x\in\Omega, \cr \displaystyle 0=\Delta v-\mu v+\nu u^\gamma, & x\in\Omega, \cr \displaystyle \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded and smooth domain. The parameters $\alpha,\gamma,m,\mu,\nu$ are positive, $\chi_0$ is real, and $a,b,\beta$ are nonnegative. In Part I, we established boundedness and global existence. Here, we study persistence and stabilization, quantifying how $\beta$ and $\chi_0$ influence long-time dynamics. First, we prove uniform persistence if $m\ge 1$. Next, for $a,b>0$, the unique positive equilibrium is $(u^*,v^*) = \left((\tfrac{a}{b})^{1/\alpha},(\tfrac{\nu}{\mu})(\tfrac{a}{b})^{\gamma/\alpha}\right)$. We identify a threshold $\chi^*(u^*)$: $(u^*,v^*)$ is linearly stable if $\chi_0<\chi^*(u^*)$, with local exponential decay, unstable if $\chi_0>\chi^*(u^*)$. We also give conditions ensuring every bounded solution converges exponentially to $(u^*,v^*)$. For $a=b=0$, we study stability of the constant equilibria under mass constraint, obtaining a linear stability threshold and global stabilization. We extend the Lyapunov method from $m=1$ to $m>1$ and the rectangle/ODE method from $\beta=0$ to $\beta>0$. For $m\ge 1$, signal saturation (large $\beta$) or repulsion ($\chi_0<0$) prevents aggregation and promotes relaxation. In Part III, we study bifurcation and pattern formation when $\chi_0$ passes through critical thresholds.