Stability and bifurcation analysis in predator-prey system involving Holling type-II functional response

Jocirei D. Ferreira; Wilmer L. Molina; Jhon J. Perez; Aida P. González; Jocirei D. Ferreira; Wilmer L. Molina; Jhon J. Perez; Aida P. González

doi:10.3934/mbe.2025094

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 10: 2559-2594. doi: 10.3934/mbe.2025094

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Stability and bifurcation analysis in predator-prey system involving Holling type-II functional response

1.
Institute of Exact and Earth Science, Federal University of Mato Grosso, Barra do Garças-MT, Brazil
2.
Department of Mathematics, University of Cauca, Popayán, Colombia

Received: 14 February 2025 Revised: 07 June 2025 Accepted: 04 July 2025 Published: 04 August 2025

In this article, we focused on the study of codimension-one Hopf bifurcations and the associated Lyapunov stability coefficients in the context of general two-dimensional reaction-diffusion systems defined on a finite fixed-length segment. Algebraic expressions for the first Lyapunov coefficients are provided for the infinite-dimensional system subject to Neumann boundary conditions. As an application, a diffusive predator-prey system modeling competing populations with a Holling type-II functional response for the predator was analyzed and studied under Neumann boundary conditions. Our main goal is to perform a detailed, local stability analysis of the proposed model, showing the existence of multiple spatially homogeneous and non-homogeneous periodic orbits, arising from the occurrence of a codimension-one Hopf bifurcation.
- reaction-diffusion system,
- Lyapunov coefficients,
- predator-prey system,
- spatially homogeneous bifurcation,
- spatially non-homogeneous bifurcation,
- finite-length segment
Citation: Jocirei D. Ferreira, Wilmer L. Molina, Jhon J. Perez, Aida P. González. Stability and bifurcation analysis in predator-prey system involving Holling type-II functional response[J]. Mathematical Biosciences and Engineering, 2025, 22(10): 2559-2594. doi: 10.3934/mbe.2025094

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Abstract

In this article, we focused on the study of codimension-one Hopf bifurcations and the associated Lyapunov stability coefficients in the context of general two-dimensional reaction-diffusion systems defined on a finite fixed-length segment. Algebraic expressions for the first Lyapunov coefficients are provided for the infinite-dimensional system subject to Neumann boundary conditions. As an application, a diffusive predator-prey system modeling competing populations with a Holling type-II functional response for the predator was analyzed and studied under Neumann boundary conditions. Our main goal is to perform a detailed, local stability analysis of the proposed model, showing the existence of multiple spatially homogeneous and non-homogeneous periodic orbits, arising from the occurrence of a codimension-one Hopf bifurcation.

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