关键词: Chemotaxis, predator-prey, boundedness, stabilization

Abstract: This work explores the predator-prey chemotaxis system with two chemicals \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta u+\chi\nabla\cdot(u\nabla v)+\mu_1u(1-u-a_1w),\quad &x\in \Omega, t>0,\\ v_t=\Delta v-\alpha_1 v+\beta_1w,\quad &x\in \Omega, t>0,\\ w_t=\Delta w-\xi\nabla \cdot(w\nabla z)+\mu_2 w(1+a_2u-w),\quad &x\in\Omega, t>0,\\ z_t=\Delta z-\alpha_2 z+\beta_2u,\quad &x\in \Omega, t>0,\\ \end{array} \right. \end{eqnarray*} in an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^n$ under homogeneous Neumann boundary conditions. The parameters in the system are positive.
We first prove that if $n\leq3$, the corresponding initial-boundary value problem admits a unique global bounded classical solution, under the assumption that $\chi, \xi$, $\mu_i, a_i, \alpha_i$ and $\beta_i(i=1,2)$ satisfy some suitable conditions. Subsequently, we also analyse the asymptotic behavior of solutions to the above system and show that
$\bullet$ when $a_1<1$ and both $\frac{\mu_1}{\chi^2}$ and $\frac{\mu_2}{\xi^2}$ are sufficiently large, the global solution $(u, v, w, z)$ of this system exponentially converges to $(\frac{1-a_1}{1+a_1a_2}, \frac{\beta_1(1+a_2)}{\alpha_1(1+a_1a_2)}, \frac{1+a_2}{1+a_1a_2}, \frac{\beta_2(1-a_1)}{\alpha_2(1+a_1a_2)})$ as $t\rightarrow \infty$;
$\bullet$ when $a_1>1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently large, the global bounded classical solution $(u, v, w, z)$ of this system exponentially converges to $(0, \frac{\alpha_1}{\beta_1}, 1, 0)$ as $t\rightarrow \infty$;
$\bullet$ when $a_1=1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently large, the global bounded classical solution $(u, v, w, z)$ of this system polynomially converges to $(0, \frac{\alpha_1}{\beta_1}, 1, 0)$ as $t\rightarrow \infty$.

Key words: Chemotaxis, predator-prey, boundedness, stabilization