# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021240
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## Large time behavior in a predator-prey system with pursuit-evasion interaction

 School of Mathematics Renmin University of China Beijing, 100872, China

* Corresponding author: Yuanyuan Ke

Received  June 2021 Revised  August 2021 Early access October 2021

This work considers a pursuit-evasion model
 $$$\left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &w_t = \Delta w-w+v,\\ &z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1)$$$
with positive parameters
 $\chi$
,
 $\xi$
,
 $\mu$
,
 $\lambda$
,
 $a$
and
 $b$
in a bounded domain
 $\Omega\subset\mathbb{R}^N$
(
 $N$
is the dimension of the space) with smooth boundary. We prove that if
 $a<2$
and
 $\frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}>\max\{\chi,\xi\}$
, (1) possesses a global bounded classical solution with a positive constant
 $C_{\frac{N}{2}+1}$
corresponding to the maximal Sobolev regularity. Moreover, it is shown that if
 $b\mu<\lambda$
, the solution (
 $u,v,w,z$
) converges to a spatially homogeneous coexistence state with respect to the norm in
 $L^\infty(\Omega)$
in the large time limit under some exact smallness conditions on
 $\chi$
and
 $\xi$
. If
 $b\mu>\lambda$
, the solution converges to (
 $\mu,0,0,\mu$
) with respect to the norm in
 $L^\infty(\Omega)$
as
 $t\rightarrow \infty$
under some smallness assumption on
 $\chi$
with arbitrary
 $\xi$
.
Citation: Dayong Qi, Yuanyuan Ke. Large time behavior in a predator-prey system with pursuit-evasion interaction. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021240
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