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=Δu-χ_0\nabla\cdot\left(\frac{u^m}{(1+v)^β}\nabla v\right)+au-bu^{1+α}, & x\inΩ, \cr \displaystyle 0=Δv-μv+νu^γ, & x\inΩ, \cr \displaystyle \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{cases} \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ 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 $(u^{\ast },v^{\ast })=((\frac{a}{b}{)}^{1\mathrm{/}\alpha },(\frac{\nu }{\mu })(\frac{a}{b}{)}^{\gamma \mathrm{/}\alpha })$. We identify a threshold ${\chi }^{\ast }(u^{\ast })$: $(u^{\ast },v^{\ast })$ is linearly stable if ${\chi }_0<{\chi }^{\ast }(u^{\ast })$, with local exponential decay, unstable if ${\chi }_0>{\chi }^{\ast }(u^{\ast })$. We also give conditions ensuring every bounded solution converges exponentially to $(u^{\ast },v^{\ast })$. 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.