# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021292
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## Bifurcation and control of a predator-prey system with unfixed functional responses

 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

Received  May 2021 Revised  September 2021 Early access December 2021

Fund Project: The second author is supported by NSFC grant 11871355

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.

Citation: Lizhi Fei, Xingwu Chen. Bifurcation and control of a predator-prey system with unfixed functional responses. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021292
##### References:

show all references

##### References:
phase portraits of system (5.1) when parameter $r_0$ is set to different values
transcritical bifurcation graphs of system (5.1) for $r_0\in [0.5, 2]$
phase portraits of system (5.2) when parameter $r_0$ is set to different values
transcritical bifurcation graphs of system (5.2) for $r_0\in [0.5, 4]$
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)
time-series and phase-plane graphs for system (5.3) with θ=0.99
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)
time-series and phase-plane graphs for system (5.5) with θ=0.99
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