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Bifurcation analysis in a predator–prey model with strong Allee effect

  • Jingwen Zhu , Ranchao Wu EMAIL logo und Mengxin Chen
Veröffentlicht/Copyright: 20. September 2021
Zeitschrift für Naturforschung A
Aus der Zeitschrift Zeitschrift für Naturforschung A Band 76 Heft 12

Abstract

In this paper, strong Allee effects on the bifurcation of the predator–prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The existence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor’s theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occurrence of Bogdanov–Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexistence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation. Numerical simulations and phase portraits are also given to verify our theoretical analysis.

Keywords: Bogdanov–Takens bifurcation; Hopf bifurcation; predator–prey model; saddle-node bifurcation; strong Allee effect
Corresponding author: Ranchao Wu, School of Mathematics and Center for Pure Mathematics, Anhui University, Hefei 230601, China, E-mail:

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11971032

Award Identifier / Grant number: 62073114

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was supported by the National Natural Science Foundation of China (Nos. 11971032, 62073114).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2021-06-24
Revised: 2021-08-31
Accepted: 2021-09-03
Published Online: 2021-09-20
Published in Print: 2021-12-20

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents
  4. Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma
  5. Bifurcation analysis in a predator–prey model with strong Allee effect
  6. Hydrodynamics
  7. Mixed initial-boundary value problems describing motions of Maxwell fluids with linear dependence of viscosity on the pressure
  8. Quantum Theory
  9. Spin coherent states, Bell states, spin Hamilton operators, entanglement, Husimi distribution, uncertainty relation and Bell inequality
  10. Solid State Physics & Materials Science
  11. Tailoring the optical properties of tin oxide thin films via gamma irradiation
  12. Thermodynamics & Statistical Physics
  13. Impact of partially thermal electrons on the propagation characteristics of extraordinary mode in relativistic regime
  14. The first-order phase transition of melting for molecular crystals by Frost–Kalkwarf vapor- and sublimation-pressure equations
  15. Mathieu-state reordering in periodic thermodynamics
  16. Corrigendum
  17. Corrigendum to: Study of ferrofluid flow and heat transfer between cone and disk
https://doi.org/10.1515/zna-2021-0178
Schlagwörter für diesen Artikel
Bogdanov–Takens bifurcation; Hopf bifurcation; predator–prey model; saddle-node bifurcation; strong Allee effect

Artikel in diesem Heft

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents
  4. Nonlinear excitations and dynamic features of dust ion-acoustic waves in a magnetized electron–positron–ion plasma
  5. Bifurcation analysis in a predator–prey model with strong Allee effect
  6. Hydrodynamics
  7. Mixed initial-boundary value problems describing motions of Maxwell fluids with linear dependence of viscosity on the pressure
  8. Quantum Theory
  9. Spin coherent states, Bell states, spin Hamilton operators, entanglement, Husimi distribution, uncertainty relation and Bell inequality
  10. Solid State Physics & Materials Science
  11. Tailoring the optical properties of tin oxide thin films via gamma irradiation
  12. Thermodynamics & Statistical Physics
  13. Impact of partially thermal electrons on the propagation characteristics of extraordinary mode in relativistic regime
  14. The first-order phase transition of melting for molecular crystals by Frost–Kalkwarf vapor- and sublimation-pressure equations
  15. Mathieu-state reordering in periodic thermodynamics
  16. Corrigendum
  17. Corrigendum to: Study of ferrofluid flow and heat transfer between cone and disk
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