Convergence rates of solutions in apredator-preysystem withindirect pursuit-evasion interaction in domains of arbitrary dimension

In this paper, we deal with the following indirect pursuit-evasion model

$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+ u(\lambda-u+av), \quad x\in \Omega, t>0, \\ v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+ v(\mu-v-bu), \quad x\in \Omega, t>0, \\ { }{0 = \Delta w- w+v}, \quad x\in \Omega, t>0, \\ { }{0 = \Delta z- z+ u}, \quad x\in \Omega, t>0, \\ \end{array}\right. \ \ \ \ \ \ \ \ \ \ (\star) \end{align} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^N(N\geq1) $ with smooth boundary $ \partial\Omega $, where $ \chi, \xi, \lambda, \mu $ as well as $ a $ and $ b $ are positive parameters. This system is used to achieve some insight into possible dynamical properties of pursuit-evasion processes, in which the respective tactic movements are oriented along gradients of some indirectly produced stimuli, rather than following individuals directly. One main purpose of the present paper is to remove the restriction of $ N\leq3 $. Indeed, by using a iteration argument combined with suitable a priori estimates, we conclude that for any $ N\geq1 $, an associated initial-boundary value problem $ (\star) $ admits a unique global bounded classical solution. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when

$ \chi<\left\{\begin{array}{ll} 4\sqrt{\frac{a(1+ab)}{b(\lambda+a\mu)}}, \quad\; \; \mbox{if}\; \; \lambda>b\mu, \\ 4\sqrt{\frac{a}{b\lambda}}, \quad\; \; \mbox{if}\; \; \lambda\leq b\mu\ \end{array}\right. $

and $ \xi<4\sqrt{\frac{b(1+ab)}{a(\mu-b\lambda)_+}} $, the corresponding solution $ (u, v, w, z) $ of the system decays to $ (u_*, v_*, v_*, u_*) $ exponentially (or algebraically), where

$ u_* = \left\{\begin{array}{ll} \frac{\lambda+a\mu}{1+ab}, \quad\; \; \mbox{if}\; \; \lambda>b\mu, \\ \lambda, \quad\; \; \mbox{if}\; \; \lambda\leq b\mu\ \end{array}\right. $

and $ v_* = \frac{(\lambda-b\mu)_+}{1+ab} $. To the best of our knowledge, there is the first result on convergence rates of solutions of the system.