Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity

doi:10.1007/s42967-023-00365-8

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (6): 2257-2268.doi: 10.1007/s42967-023-00365-8

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Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity

Miqin Chen¹, Wensheng Yang^1,2

1. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, Fujian, China;
2. FJKLMAA and Center for Applied Mathematics of Fujian Province (FJNU), Fuzhou 350117, Fujian, China

收稿日期:2023-05-08 修回日期:2023-11-21 发布日期:2025-12-24
通讯作者: Wensheng Yang, E-mail:ywensheng@126.com E-mail:ywensheng@126.com
基金资助:
This work was supported by the National Natural Science Foundation of China (11672074), and the Natural Science Foundation of Fujian Province, China (2022J01192).

Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity

Miqin Chen¹, Wensheng Yang^1,2

1. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, Fujian, China;
2. FJKLMAA and Center for Applied Mathematics of Fujian Province (FJNU), Fuzhou 350117, Fujian, China

Received:2023-05-08 Revised:2023-11-21 Published:2025-12-24
Contact: Wensheng Yang, E-mail:ywensheng@126.com E-mail:ywensheng@126.com
Supported by:
This work was supported by the National Natural Science Foundation of China (11672074), and the Natural Science Foundation of Fujian Province, China (2022J01192).

摘要/Abstract

摘要： In this paper, we investigate the global dynamics of a predator-prey model with a general growth rate function and carrying capacity. We prove that the origin is unstable using the blow-up method. Also, by constructing a new Lyapunov function and using LaSalle’s invariance principle, we obtain the global stability of the positive equilibrium state of the system. In addition, the system undergoes the Hopf bifurcation at the positive equilibrium point when the predator birth rate $ \delta $ is used as the bifurcation parameter. Finally, two examples are given to verify the feasibility of the theoretical results. One example is given to reconsider the global stability of the positive equilibrium of a Leslie-Gower predator-prey model with prey cannibalism, and the obtained results confirm the conjecture proposed by Lin et al. (Adv Differ Equ 2020, 153, 2020). The other example is given to verify the occurrence of the Hopf bifurcation of a Leslie-Gower predator-prey model with a square root response function, and obtain the Hopf bifurcation diagram by the numerical simulation.

关键词: General carrying capacity, General growth rate function, Global stability, Lyapunov function, Hopf bifurcation

Abstract: In this paper, we investigate the global dynamics of a predator-prey model with a general growth rate function and carrying capacity. We prove that the origin is unstable using the blow-up method. Also, by constructing a new Lyapunov function and using LaSalle’s invariance principle, we obtain the global stability of the positive equilibrium state of the system. In addition, the system undergoes the Hopf bifurcation at the positive equilibrium point when the predator birth rate $ \delta $ is used as the bifurcation parameter. Finally, two examples are given to verify the feasibility of the theoretical results. One example is given to reconsider the global stability of the positive equilibrium of a Leslie-Gower predator-prey model with prey cannibalism, and the obtained results confirm the conjecture proposed by Lin et al. (Adv Differ Equ 2020, 153, 2020). The other example is given to verify the occurrence of the Hopf bifurcation of a Leslie-Gower predator-prey model with a square root response function, and obtain the Hopf bifurcation diagram by the numerical simulation.

Key words: General carrying capacity, General growth rate function, Global stability, Lyapunov function, Hopf bifurcation

中图分类号:

Miqin Chen, Wensheng Yang. Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2257-2268.

参考文献

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Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity

Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity

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