Bifurcation and control of a predator-prey system with unfixed functional responses

Lizhi Fei; Xingwu Chen

doi:10.3934/dcdsb.2021292

Article Contents

2022, Volume 27, Issue 10: 5701-5721. Doi: 10.3934/dcdsb.2021292

This issue Previous Article Boundedness of the complex Chen system Next Article Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels

Bifurcation and control of a predator-prey system with unfixed functional responses

Lizhi Fei^1, and
Xingwu Chen^2, ,

1.
College of Science, Nanchang Institute of Technology, Nanchang, Jiangxi 330000, China
2.
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

^* Corresponding author: Xingwu Chen
^* Corresponding author: Xingwu Chen

Received: May 2021

Revised: September 2021

Early access: December 2021

Published: October 2022

The second author is supported by NSFC grant 11871355

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Abstract

In this paper we investigate a discrete-time predator-prey system with not only some constant parameters but also unfixed functional responses including growth rate function of prey, conversion factor function and predation probability function. We prove that the maximal number of fixed points is $ 3 $ and give necessary and sufficient conditions of exactly $ j $($ j = 1,2,3 $) fixed points, respectively. For transcritical bifurcation and Neimark-Sacker bifurcation, we provide bifurcation conditions depending on these unfixed functional responses. In order to regulate the stability of this biological system, a hybrid control strategy is used to control the Neimark-Sacker bifurcation. Finally, we apply our main results to some examples and carry out numerical simulations for each example to verify the correctness of our theoretical analysis.

Keywords:

Mathematics Subject Classification: Primary: 37G10, 39A28; Secondary: 37N25.

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Figure 1. phase portraits of system (5.1) when parameter $ r_0 $ is set to different values

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Figure 2. transcritical bifurcation graphs of system (5.1) for $ r_0\in [0.5, 2] $

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Figure 3. phase portraits of system (5.2) when parameter $ r_0 $ is set to different values

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Figure 4. transcritical bifurcation graphs of system (5.2) for $ r_0\in [0.5, 4] $

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Figure 5. phase portraits and bifurcation graphs of system (5.2) for $ \gamma $ vary in the small neighborhood of $ \gamma = 7.961845698 $, the initial value is (0.25, 0.75)

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Figure 6. time-series and phase-plane graphs for system (5.3) with θ=0.99

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Figure 7. phase portraits and bifurcation graphs of system (5.4) when $ m $ vary in the small neighborhood of $ m = 6 $, the initial value is (0.999, 0.999)

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Figure 8. time-series and phase-plane graphs for system (5.5) with θ=0.99

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