Stability and bifurcation analysis of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response

Shuangte Wang; Hengguo Yu; Shuangte Wang; Hengguo Yu

doi:10.3934/mbe.2021391

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 7877-7918. doi: 10.3934/mbe.2021391

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Stability and bifurcation analysis of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response

Shuangte Wang ^{1,2
,
,},
Hengguo Yu ^{1,3
,
,}

1.
School of Mathematics and Physics, Wenzhou University, Wenzhou 325035, China
2.
Liushi No.3 Middle School, Wenzhou 325604, China
3.
School of Life and Environmental Science, Wenzhou University, Wenzhou 325035, China

Academic Editor: Cristiana J. Silva

Received: 17 June 2021 Accepted: 26 August 2021 Published: 10 September 2021

In the paper, stability and bifurcation behaviors of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response are studied theoretically and numerically. Mathematical theory works mainly give some critical threshold conditions to guarantee the existence and stability of all possible equilibrium points, and the occurrence of Hopf bifurcation and Bogdanov-Takens bifurcation. Numerical simulation works mainly display that the Bazykin's predator-prey ecosystem has complex dynamic behaviors, which also directly proves that the theoretical results are effective and feasible. Furthermore, it is easy to see from numerical simulation results that some key parameters can seriously affect the dynamic behavior evolution process of the Bazykin's predator-prey ecosystem. Moreover, limit cycle is proposed in view of the supercritical Hopf bifurcation. Finally, it is expected that these results will contribute to the dynamical behaviors of predator-prey ecosystem.
- Bazykin's predator-prey ecosystem,
- stability,
- Hopf bifurcation,
- Bogdanov-Takens bifurcation,
- limit cycle
Citation: Shuangte Wang, Hengguo Yu. Stability and bifurcation analysis of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7877-7918. doi: 10.3934/mbe.2021391

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Abstract

In the paper, stability and bifurcation behaviors of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response are studied theoretically and numerically. Mathematical theory works mainly give some critical threshold conditions to guarantee the existence and stability of all possible equilibrium points, and the occurrence of Hopf bifurcation and Bogdanov-Takens bifurcation. Numerical simulation works mainly display that the Bazykin's predator-prey ecosystem has complex dynamic behaviors, which also directly proves that the theoretical results are effective and feasible. Furthermore, it is easy to see from numerical simulation results that some key parameters can seriously affect the dynamic behavior evolution process of the Bazykin's predator-prey ecosystem. Moreover, limit cycle is proposed in view of the supercritical Hopf bifurcation. Finally, it is expected that these results will contribute to the dynamical behaviors of predator-prey ecosystem.

References

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