Boundedness and stabilization of solutions to a chemotaxis May–Nowak model

Abstract

The chemotaxis May–Nowak model

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=D_u \Delta u-\nabla \cdot (uf\left( u\right) \nabla v)-g\left( u\right) w+r-u, &{}x\in \Omega ,\quad t>0,\\ v_t=D_v \Delta v+g\left( u\right) w-v, &{}x\in \Omega ,\quad t>0,\\ w_t=D_w \Delta w+v-w, &{}x\in \Omega ,\quad t>0, \end{array} \right. \end{aligned}$$

is considered in a bounded domain $\Omega \subset \mathbb {R}^n (n\ge 1)$ with homogeneous Neumann boundary conditions and the parameters $D_u,D_v,D_w,r>0$. The chemotactic sensitivity function and the conversion function are given by $f\left( s\right) =K_{f}\left( 1+s\right) ^{-\alpha }$ and $g(s) = K_{g} s^{\beta }$ for all $s > 0$, respectively, where $K_{f}\in \mathbb {R}$, $K_{g}, \alpha , \beta >0$. The global boundedness of solution is shown if the following case holds:

$$\begin{aligned} \begin{aligned} \alpha > \max \left\{ \frac{n\beta }{4}, \frac{\beta }{2}, \frac{n(n+2)}{6n+8}\beta +\frac{1}{2} \right\} . \end{aligned} \end{aligned}$$

Moreover, for the large time behavior of the global smooth bounded solution, the basic reproduction number $R_{0}:=K_{g}r^{\beta }$ has an important effect (Lai and Zou in Bull Math Biol 76:2806-2833, 2014; Wang et al. in Nonlinear Anal RWA 33:253-283, 2017), and system has the infection-free steady state if $0<R_0<1$ and the coexistence equilibrium steady state if $R_0>1$ (Korobeinikov in Bull Math Biol 66:879-883, 2004). By constructing an appropriate energy function, under the conditions that $K_f$ and $K_g$ are appropriately mild, it is shown that: $\bullet$ If $R_0\in (0,1)$, then any global bounded solution converges to $\left( r, 0, 0\right)$ as $t\rightarrow \infty$; $\bullet$ If $R_0\in (1,\infty )$, $\beta =1$, then any global bounded solution converges to $\bigg ( \frac{1}{K_g}, r-\frac{1}{K_g}, r-\frac{1}{K_g} \bigg )$ as $t\rightarrow \infty$.