Steady states and spatiotemporal dynamics of a diffusive predator-prey system with predator harvesting

Rongjie Yu; Hengguo Yu; Min Zhao; Rongjie Yu; Hengguo Yu; Min Zhao

doi:10.3934/math.20241170

AIMS Mathematics

2024, Volume 9, Issue 9: 24058-24088. doi: 10.3934/math.20241170

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Research article

Steady states and spatiotemporal dynamics of a diffusive predator-prey system with predator harvesting

1.
School of Mathematics and Physics, Wenzhou University, Wenzhou 325035, China
2.
Key Laboratory for Subtropical Oceans & Lakes Environment and Biological Resources Utilization Technology of Zhejiang, Wenzhou University, Wenzhou 325035, China
3.
School of Life and Environmental Science, Wenzhou University, Wenzhou 325035, China

Received: 30 June 2024 Revised: 25 July 2024 Accepted: 31 July 2024 Published: 13 August 2024
MSC : 35B32, 35J65, 92D25

From the perspective of ecological control, harvesting behavior plays a crucial role in the ecosystem natural cycle. This paper proposes a diffusive predator-prey system with predator harvesting to explore the impact of harvesting on predatory ecological relationships. First, the existence and boundedness of system solutions were investigated and the non-existence and existence of non-constant steady states were obtained. Second, the conditions for Turing instability were given to further investigate the Turing patterns. Based on these conditions, the amplitude equations at the threshold of instability were established using weakly nonlinear analysis. Finally, the existence, direction, and stability of Hopf bifurcation were proven. Furthermore, numerical simulations were used to confirm the correctness of the theoretical analysis and show that harvesting has a strong influence on the dynamical behaviors of the predator-prey systems. In summary, the results of this study contribute to promoting the research and development of predatory ecosystems.
- harvesting,
- non-constant steady states,
- Turing instability,
- Hopf bifurcation
Citation: Rongjie Yu, Hengguo Yu, Min Zhao. Steady states and spatiotemporal dynamics of a diffusive predator-prey system with predator harvesting[J]. AIMS Mathematics, 2024, 9(9): 24058-24088. doi: 10.3934/math.20241170

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Abstract

From the perspective of ecological control, harvesting behavior plays a crucial role in the ecosystem natural cycle. This paper proposes a diffusive predator-prey system with predator harvesting to explore the impact of harvesting on predatory ecological relationships. First, the existence and boundedness of system solutions were investigated and the non-existence and existence of non-constant steady states were obtained. Second, the conditions for Turing instability were given to further investigate the Turing patterns. Based on these conditions, the amplitude equations at the threshold of instability were established using weakly nonlinear analysis. Finally, the existence, direction, and stability of Hopf bifurcation were proven. Furthermore, numerical simulations were used to confirm the correctness of the theoretical analysis and show that harvesting has a strong influence on the dynamical behaviors of the predator-prey systems. In summary, the results of this study contribute to promoting the research and development of predatory ecosystems.

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