Cross-Diffusion-Driven Positive Solutions in a Predator-Prey Model with Prey Refuge and Crowley-Martin Functional Response

JIANG Hongling

doi:10.1051/wujns/2026315415

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Cross-Diffusion-Driven Positive Solutions in a Predator-Prey Model with Prey Refuge and Crowley-Martin Functional Response

Mathematics | 更新时间：2026-10-29

- Cross-Diffusion-Driven Positive Solutions in a Predator-Prey Model with Prey Refuge and Crowley-Martin Functional Response
- Wuhan University Journal of Natural Sciences Vol. 31, Issue 5, Pages: 415-428(2026)
- 作者机构：
  
  宝鸡文理学院数学与信息科学学院, 陕西宝鸡 721013
- 作者简介：
  
  JIANG Hongling, male, Associate professor, Ph. D., research direction: partial differential equations. E-mail: jhonglings@163.com
- 基金信息：
- DOI：10.1051/wujns/2026315415
  CLC： O175.25
- Received：08 May 2026，
  
  Published：24 October 2026
- 稿件说明：
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姜洪领.交叉扩散驱动的具有避难所和Crowley-Martin功能反应的捕食-食饵模型的正解[J].武汉大学学报自然科学英文版,2026,31(05):415-428.

JIANG Hongling. Cross-Diffusion-Driven Positive Solutions in a Predator-Prey Model with Prey Refuge and Crowley-Martin Functional Response[J]. ,Wuhan Univ J of Nat Sci, 2026, 31(5): 415-428.
姜洪领.交叉扩散驱动的具有避难所和Crowley-Martin功能反应的捕食-食饵模型的正解[J].武汉大学学报自然科学英文版,2026,31(05):415-428. DOI： 10.1051/wujns/2026315415.

JIANG Hongling. Cross-Diffusion-Driven Positive Solutions in a Predator-Prey Model with Prey Refuge and Crowley-Martin Functional Response[J]. ,Wuhan Univ J of Nat Sci, 2026, 31(5): 415-428. DOI： 10.1051/wujns/2026315415.

摘要

本文研究了在Neumann边界条件下

一类具有食饵避难所和Crowley-Martin型功能反应函数的修正Leslie-Gower捕食-食饵模型。基于Routh-Hurwitz判据

推导了无扩散模型和扩散模型唯一正常数稳态解的存在性与稳定性条件。此外

研究了交叉扩散模型唯一正常数稳态解的Turing不稳定性。进而

利用Young不等式、Poincaré不等式和Leray-Schauder度理论

分别得到了扩散模型和交叉扩散模型非常数正稳态解的不存在性与存在性结果。最后

通过若干组数值模拟

展示了正常数稳态解的稳定性、Turing不稳定性、非常数稳态解的存在性以及食饵避难所的作用。结果表明

交叉扩散是驱动模型中非常数正解(包括稳态解)产生的重要因素

即交叉扩散可导致捕食者和被捕食者形成非均匀共存模式。

Abstract

This paper investigates a modified Leslie-Gower predator-prey model with prey refuge and a Crowley-Martin type functional response under Neumann boundary conditions. Based on the Routh-Hurwitz criterion

we derive the existence and stability conditions of the unique positive constant equilibrium for the diffusion-free model and the diffusion model. Moreover

we study Turing instability of the unique positive constant equilibrium for the cross-diffusion model. Then

using Young's inequality

Poincaré's inequality and Leray-Schauder degree theory

we obtain the nonexistence and existence of nonconstant positive equilibria for the diffusion model and the cross-diffusion model

respectively. Finally

several numerical simulations are provided to illustrate the stability of positive constant equilibria

Turing instability

the existence of nonconstant equilibria and the role of the prey refuge. The results indicate that cross-diffusion is an important factor driving the emergence of nonconstant positive solutions (including equilibria) in the model; that is

cross-diffusion can lead to a non-uniform coexistence pattern of predator and prey.

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references

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