Existence of globally bounded solutions for a predator-prey system with indirect taxis-driven pursuit-evasion and parabolic-elliptic signal

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Abstract

In this paper, we consider a cross-diffusive model for pursuit-evasion processes which was described as

$\left\{\begin{array}{ll} u_{t}=\Delta u-\chi \nabla \cdot\left(u(u+1)^{r-1} \nabla w\right)+u\left(\lambda_{1}-\mu_{1} u^{r_{1}-1}+a v\right), & x \in \Omega, t>0, \\ v_{t}=\Delta v+\xi \nabla \cdot(v \nabla z)+v\left(\lambda_{2}-\mu_{2} v^{r_{2}-1}-b u\right), & x \in \Omega, t>0, \\ w_{t}=\Delta w-w+v, & x \in \Omega, t>0, \\ 0=\Delta z-z+u, & x \in \Omega, t>0 \end{array}\right.$

in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$ (N ≥ 1) under homogeneous Neumann boundary conditions. For nonnegative initial data with appropriate regularity, the unique globally bounded classical solution is obtained provided that min $\left\{\left(r_{1}-1\right)\left(r_{2}-1\right),\left(r_{1}-r\right)\left(r_{2}-1\right)\right\}>\frac{(N-2)_{+}}{N}$ , where $r, \lambda_{i}, \mu_{i}, \chi, \xi>0$ and r_i > 1(i = 1, 2). This generalizes the results of Li et al. [1], Zheng et al. [2], and Zheng and Zhang [3].

Keywords

pursuit-evasion / global existence / parabolic-elliptic / boundedness

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