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

Yuxuan Liang; Chuchu Du; Kaiqiang Li; Jiashan Zheng; Yuxuan Liang; Chuchu Du; Kaiqiang Li; Jiashan Zheng

doi:10.3934/era.2025191

Electronic Research Archive

2025, Volume 33, Issue 7: 4222-4240. doi: 10.3934/era.2025191

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Research article

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

School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, China

Received: 18 April 2025 Revised: 26 June 2025 Accepted: 07 July 2025 Published: 16 July 2025

In this paper, we consider a cross-diffusive model for pursuit-evasion processes which was described as
$ \begin{equation*} \left\{\begin{aligned} & u_{t} = \Delta u-\chi\nabla\cdot(u(u+1)^{r-1}\nabla w)+u(\lambda_{1}-\mu_{1}u^{r_{1}-1}+av), &x\in\Omega, t>0, \\ & v_{t} = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda_{2}-\mu_{2}v^{r_{2}-1}-bu), &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{aligned}\right. \end{equation*} $
in a smooth bounded domain $ \Omega \subset \mathbb{R}^N $ ($ N \geq 1 $) under homogeneous Neumann boundary conditions. For nonnegative initial data with appropriate regularity, the unique globally bounded classical solution is obtained provided that $ \min \{(r_{1}-1)(r_{2}-1), (r_{1}-r)(r_{2}-1) \} > \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].
- pursuit-evasion,
- global existence,
- parabolic-elliptic,
- boundedness
Citation: Yuxuan Liang, Chuchu Du, Kaiqiang Li, Jiashan Zheng. Existence of globally bounded solutions for a predator-prey system with indirect taxis-driven pursuit-evasion and parabolic-elliptic signal[J]. Electronic Research Archive, 2025, 33(7): 4222-4240. doi: 10.3934/era.2025191

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Abstract

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

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

in a smooth bounded domain $ \Omega \subset \mathbb{R}^N $ ($ N \geq 1 $) under homogeneous Neumann boundary conditions. For nonnegative initial data with appropriate regularity, the unique globally bounded classical solution is obtained provided that $ \min \{(r_{1}-1)(r_{2}-1), (r_{1}-r)(r_{2}-1) \} > \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].

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