• 论文 •

### BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS

1. 1. School of Sciences, Southwest Petroleum University, Chengdu, 610500, China;
2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China;
3. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
• 收稿日期:2020-09-25 修回日期:2021-03-24 发布日期:2022-06-24
• 通讯作者: Shuyan QIU,E-mail:shuyanqiu0701@126.com E-mail:shuyanqiu0701@126.com
• 基金资助:
The first author is supported by the Young Scholars Development Fund of SWPU (202199010087) and the Scientific Research Starting Project of SWPU (2021QHZ016). The second author is supported by the National Natural Science Foundation of China (11771062 and 11971082), the Fundamental Research Funds for the Central Universities (2019CDJCYJ001), and Chongqing Key Laboratory of Analytic Mathematics and Applications.

### BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS

Shuyan QIU1, Chunlai MU2, Hong YI3

1. 1. School of Sciences, Southwest Petroleum University, Chengdu, 610500, China;
2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China;
3. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
• Received:2020-09-25 Revised:2021-03-24 Published:2022-06-24
• Contact: Shuyan QIU,E-mail:shuyanqiu0701@126.com E-mail:shuyanqiu0701@126.com
• Supported by:
The first author is supported by the Young Scholars Development Fund of SWPU (202199010087) and the Scientific Research Starting Project of SWPU (2021QHZ016). The second author is supported by the National Natural Science Foundation of China (11771062 and 11971082), the Fundamental Research Funds for the Central Universities (2019CDJCYJ001), and Chongqing Key Laboratory of Analytic Mathematics and Applications.

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 $t\rightarrow \infty$.

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 $t\rightarrow \infty$.

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