Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food supply

Ting Yu; Qinglong Wang; Shuqi Zhai; Ting Yu; Qinglong Wang; Shuqi Zhai

doi:10.3934/mbe.2023676

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 15094-15119. doi: 10.3934/mbe.2023676

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Research article

Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food supply

School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China

Academic Editor: Yang Kuang

Received: 21 April 2023 Revised: 28 May 2023 Accepted: 04 July 2023 Published: 17 July 2023

In this manuscript, a novel ratio-dependent predator-prey bioeconomic model with time delay and additional food supply is investigated. We first change the bioeconomic model into a normal version by virtue of the differential-algebraic system theory. The local steady-state of equilibria and Hopf bifurcation could be derived by varying time delay. Later, the formulas of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. Moreover, employing the Pontryagin's maximum principle and considering the instantaneous annual discount rate, the optimal harvesting problem of the model without time delay is analyzed. Finally, four numeric examples are carried out to verify the rationality of our analytical findings. Our analytical results show that Hopf bifurcation occurs in this model when the value of bifurcation parameter, the time delay of the maturation time of prey, crosses a critical value.
- predator-prey bioeconomic model,
- ratio-dependent,
- time delay,
- Hopf bifurcation,
- optimal harvesting
Citation: Ting Yu, Qinglong Wang, Shuqi Zhai. Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food supply[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 15094-15119. doi: 10.3934/mbe.2023676

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Abstract

In this manuscript, a novel ratio-dependent predator-prey bioeconomic model with time delay and additional food supply is investigated. We first change the bioeconomic model into a normal version by virtue of the differential-algebraic system theory. The local steady-state of equilibria and Hopf bifurcation could be derived by varying time delay. Later, the formulas of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. Moreover, employing the Pontryagin's maximum principle and considering the instantaneous annual discount rate, the optimal harvesting problem of the model without time delay is analyzed. Finally, four numeric examples are carried out to verify the rationality of our analytical findings. Our analytical results show that Hopf bifurcation occurs in this model when the value of bifurcation parameter, the time delay of the maturation time of prey, crosses a critical value.

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