Bifurcation analysis of a predator-prey model with cross-diffusion and two delays

Hongyan Sun; Jianzhi Cao; Pengmiao Hao; Li Ma; Hongyan Sun; Jianzhi Cao; Pengmiao Hao; Li Ma

doi:10.3934/era.2025347

Electronic Research Archive

2025, Volume 33, Issue 12: 7866-7901. doi: 10.3934/era.2025347

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Bifurcation analysis of a predator-prey model with cross-diffusion and two delays

1.
College of Data Science and Software Engineering, Baoding University, Baoding 071000, China
2.
Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China

Received: 30 October 2025 Revised: 14 December 2025 Accepted: 16 December 2025 Published: 25 December 2025

In this paper, a predator-prey model with cross-diffusion and two delays is investigated. First, the conditions for local stability and Turing instability of positive steady-state solution are studied separately when the system was without and with diffusion. Second, the existence and the stability of Hopf bifurcation were investigated by computing stability switching curves in the parameter plane with two delays. Moreover, explicit formulas for determining the stability and the direction of the bifurcation periodic solutions were derived using the normal form theory and the center manifold theorem. Finally, the theoretical results were verified by numerical simulations.
- predator-prey system,
- two delays,
- cross-diffusion,
- stability switching curves,
- Hopf bifurcation
Citation: Hongyan Sun, Jianzhi Cao, Pengmiao Hao, Li Ma. Bifurcation analysis of a predator-prey model with cross-diffusion and two delays[J]. Electronic Research Archive, 2025, 33(12): 7866-7901. doi: 10.3934/era.2025347

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Abstract

In this paper, a predator-prey model with cross-diffusion and two delays is investigated. First, the conditions for local stability and Turing instability of positive steady-state solution are studied separately when the system was without and with diffusion. Second, the existence and the stability of Hopf bifurcation were investigated by computing stability switching curves in the parameter plane with two delays. Moreover, explicit formulas for determining the stability and the direction of the bifurcation periodic solutions were derived using the normal form theory and the center manifold theorem. Finally, the theoretical results were verified by numerical simulations.

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