Chemotaxis models with signal-dependent sensitivity and a logistic-type source, II: Persistence and stabilization

Abstract

This paper is Part II of a series on boundedness, global existence, and asymptotic behavior of positive classical solutions to the chemotaxis model $$\{\begin{array}{ll}{u}_t=\Delta u-{\chi }_0\nabla \cdot \left(\frac{u^m}{(1+v{)}^{\beta }}\nabla v\right)+au-b{u}^{1+\alpha },& x\in \Omega ,\\ 0=\Delta v-\mu v+\nu u^{\gamma },& x\in \Omega ,\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,& x\in \partial \Omega ,\end{array}\text{\hspace{1em}(CM)}$$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain, $\alpha ,\gamma ,m,\mu ,\nu >0$, ${\chi }_0\in \mathbb{R}$, and $a,b,\beta \ge 0$. Part I established biologically relevant parameter regimes ensuring boundedness and global existence of positive classical solutions. Here we study persistence and stabilization of globally defined bounded positive solutions, quantifying how $\beta$ in the sensitivity $\chi (v)={\chi }_0(1+v{)}^{-\beta }$ influences long-time dynamics. The factor $(1+v{)}^{-\beta }$ models signal-dependent desensitization, so large $\beta$ should promote stabilization, though it considerably complicates the analysis. We take $\mu =\nu =1$ with either $a=b=1$ (logistic source) or $a=b=0$ (minimal model). We show that when $m\ge 1$, every globally defined bounded positive solution stays uniformly away from zero in space. We determine exact $\beta$-dependent critical sensitivity thresholds for local stability of constant solutions and establish $\beta$-dependent thresholds for their global stability; both tend to $\infty$ as $\beta \to \infty$. Thus signal saturation (large $\beta$) or repulsion (${\chi }_0<0$) can prevent aggregation and promote relaxation to spatially homogeneous states. For $\beta >0$ and $m,\alpha ,\gamma$ not all 1, we develop new techniques, including a Lyapunov function, pointwise estimates for $\nabla v$, and a nontrivial extension of the rectangle/ODE method from $\beta =0$ to $\beta >0$.

arXiv comment39 pages