Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor

Jing Zhang; Shengmao Fu; Jing Zhang; Shengmao Fu

doi:10.3934/math.20241558

AIMS Mathematics

2024, Volume 9, Issue 11: 32514-32551. doi: 10.3934/math.20241558

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Research article

Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor

Jing Zhang ¹,
Shengmao Fu ^{1,2
,
,}

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China
2.
School of Mathematics and Statistics, Kashi University, Kashi, 844006, China

Received: 20 August 2024 Revised: 30 October 2024 Accepted: 07 November 2024 Published: 18 November 2024
MSC : 37G15, 35B32, 35B36, 35K57, 92D25

In this paper, the dynamic behavior of a diffusive Rosenzweig-MacArthur (R-M) predator-prey model with hyperbolic tangent functional response and fear effect was investigated. For the local system, we gave a detailed classification of equilibria and performed bifurcation analysis. It was shown by numerical simulation that both the capture rate and fear factor have a stabilizing effect. Furthermore, the existence of limit cycles was discussed when the prey was in low fear or carrying capacity was sufficiently large. For the reaction-diffusion system, we considered the local stability of a unique positive equilibrium, Turing instability of both positive equilibrium and homogeneous periodic orbits under weak fear effect or strong carrying capacity, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions under small fear cost and large diffusion coefficients, as well as the existence of positive nonconstant steady states. However, in the absence of fear effect, Turing instability of both positive equilibrium and homogeneous periodic orbits did not occur. Meanwhile, numerical examples were given to illustrate the corresponding analytic results.
- Rosenzweig-MacArthur model,
- hyperbolic tangent functional response,
- fear effect,
- Hopf bifurcation,
- Turing instability,
- steady state
Citation: Jing Zhang, Shengmao Fu. Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor[J]. AIMS Mathematics, 2024, 9(11): 32514-32551. doi: 10.3934/math.20241558

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Abstract

In this paper, the dynamic behavior of a diffusive Rosenzweig-MacArthur (R-M) predator-prey model with hyperbolic tangent functional response and fear effect was investigated. For the local system, we gave a detailed classification of equilibria and performed bifurcation analysis. It was shown by numerical simulation that both the capture rate and fear factor have a stabilizing effect. Furthermore, the existence of limit cycles was discussed when the prey was in low fear or carrying capacity was sufficiently large. For the reaction-diffusion system, we considered the local stability of a unique positive equilibrium, Turing instability of both positive equilibrium and homogeneous periodic orbits under weak fear effect or strong carrying capacity, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions under small fear cost and large diffusion coefficients, as well as the existence of positive nonconstant steady states. However, in the absence of fear effect, Turing instability of both positive equilibrium and homogeneous periodic orbits did not occur. Meanwhile, numerical examples were given to illustrate the corresponding analytic results.

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