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BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS
Shuyan QIU, Chunlai MU, Hong YI

Acta mathematica scientia,Series B. 2022, 42 (3 ):
1035-1057.
DOI: 10.1007/s10473-022-0313-7
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 $t\rightarrow \infty$.

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