Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism

Tingting Ma; Xinzhu Meng; Tingting Ma; Xinzhu Meng

doi:10.3934/mbe.2022282

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 6: 6040-6071. doi: 10.3934/mbe.2022282

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Research article

Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism

Tingting Ma ,
Xinzhu Meng ^,

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Received: 26 January 2022 Revised: 24 March 2022 Accepted: 01 April 2022 Published: 12 April 2022

We investigate a new cross-diffusive prey-predator system which considers prey refuge and fear effect, where predator cannibalism is also considered. The prey and predator that partially depends on the prey are followed by Holling type-Ⅱ terms. We first establish sufficient conditions for persistence of the system, the global stability of constant steady states are also investigated. Then, we investigate the Hopf bifurcation of ordinary differential system, and Turing instability driven by self-diffusion and cross-diffusion. We have found that the $ d_{12} $ can suppress the formation of Turing instability, while the $ d_{21} $ promotes the appearance of the pattern formation. In addition, we also discuss the existence and nonexistence of nonconstant positive steady state by Leray-Schauder degree theory. Finally, we provide the following discretization reaction-diffusion equations and present some numerical simulations to illustrate analytical results, which show that the establishment of prey refuge can effectively protect the growth of prey.
- cross-diffusion,
- fear effect,
- predator cannibalism,
- Hopf bifurcation,
- Turing instability
Citation: Tingting Ma, Xinzhu Meng. Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 6040-6071. doi: 10.3934/mbe.2022282

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Abstract

We investigate a new cross-diffusive prey-predator system which considers prey refuge and fear effect, where predator cannibalism is also considered. The prey and predator that partially depends on the prey are followed by Holling type-Ⅱ terms. We first establish sufficient conditions for persistence of the system, the global stability of constant steady states are also investigated. Then, we investigate the Hopf bifurcation of ordinary differential system, and Turing instability driven by self-diffusion and cross-diffusion. We have found that the $ d_{12} $ can suppress the formation of Turing instability, while the $ d_{21} $ promotes the appearance of the pattern formation. In addition, we also discuss the existence and nonexistence of nonconstant positive steady state by Leray-Schauder degree theory. Finally, we provide the following discretization reaction-diffusion equations and present some numerical simulations to illustrate analytical results, which show that the establishment of prey refuge can effectively protect the growth of prey.

References

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