Home Bifurcation analysis in a predator–prey model with strong Allee effect
Article
Licensed
Unlicensed Requires Authentication

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

  • Jingwen Zhu , Ranchao Wu EMAIL logo and Mengxin Chen
Published/Copyright: September 20, 2021
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 76 Issue 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.

References

[1] A. J. Lotka, “Undamped oscillations derived from the law of mass action,” J. Am. Chem. Soc., vol. 42, pp. 1595–1599, 1920. https://doi.org/10.1021/ja01453a010.Search in Google Scholar

[2] V. Volterra, “Fluctuation in the abundance of a species considered mathematically,” Nature, vol. 118, pp. 558–560, 1920. https://doi.org/10.1093/jof/18.5.558.Search in Google Scholar

[3] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57. New York, Marcel Dekker Incorporated, 1980.Search in Google Scholar

[4] S. Hsu and T. Huang, “Global stability for a class of predator-prey system,” SIAM J. Appl. Math., vol. 55, pp. 763–783, 1995. https://doi.org/10.1137/s0036139993253201.Search in Google Scholar

[5] P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, pp. 213–245, 1948. https://doi.org/10.2307/2332342.Search in Google Scholar

[6] P. H. Leslie and J. C. Gower, “The properties of a stochastic model for the predator-prey type of interaction between two species,” Biometrika, vol. 47, pp. 219–234, 1960. https://doi.org/10.1093/biomet/47.3-4.219.Search in Google Scholar

[7] J. B. Collings, “The effects of the functional response on the behavior of a mite predator-prey interaction model,” J. Math. Biol., vol. 36, pp. 149–168, 1997. https://doi.org/10.1007/s002850050095.Search in Google Scholar

[8] C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Mem. Entomol. Soc. Can., vol. 97, pp. 5–60, 1965. https://doi.org/10.4039/entm9745fv.Search in Google Scholar

[9] A. D. Bazykin, “Nonlinear Dynamics of Interacting Populations,” in World Sci. Ser. Nonlinear Sci. Ser. A, vol. 11, Singapore, World Scientific, 1998.10.1142/2284Search in Google Scholar

[10] D. P. Hu and H. J. Cao, “Stability and bifurcation analysis in a predator–prey system with Michaelis-Menten type predator harvesting,” Nonlinear Anal. R. World Appl., vol. 33, pp. 58–82, 2017. https://doi.org/10.1016/j.nonrwa.2016.05.010.Search in Google Scholar

[11] S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM J. Appl. Math., vol. 61, pp. 1445–1472, 2001.10.1137/S0036139999361896Search in Google Scholar

[12] C. Xiang, J. C. Huang, S. G. Ruan, and D. M. Xiao, “Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response,” J. Differ. Equations., vol. 268, pp. 4618–4662, 2020. https://doi.org/10.1016/j.jde.2019.10.036.Search in Google Scholar

[13] J. Huang, X. Xia, X. Zhang, and S. Ruan, “Bifurcation of codimension 3 in a predator-prey system of Leslie type with simplified Holling type IV functional response,” Int. J. Bifurc. Chaos., vol. 26, p. 1650034, 2016. https://doi.org/10.1142/s0218127416500346.Search in Google Scholar

[14] Y. Gong and J. Huang, “Bogdanov–Takens bifurcation in a Leslie–Gower predator-prey model with prey harvesting,” Acta Math. Appl. Sinica Eng. Ser., vol. 30, pp. 239–244, 2014. https://doi.org/10.1007/s10255-014-0279-x.Search in Google Scholar

[15] H. B. Shi, S. G. Ruan, Y. Su, and J. F. Zhang, “Spatiotemporal dynamics of a diffusive Leslie-Gower predator-prey model with ratio-dependent functional response,” Int. J. Bifurc. Chaos., vol. 25, p. 1530014, 2015. https://doi.org/10.1142/s0218127415300141.Search in Google Scholar

[16] M. Sen, M. Banerjee, and A. Morozov, “Bifurcation analysis of a ratio-dependent prey-predator model with the allee effect,” Ecol. Complex., vol. 11, pp. 12–27, 2012. https://doi.org/10.1016/j.ecocom.2012.01.002.Search in Google Scholar

[17] Q. Q. Fang and X. Y. Li, “Complex dynamics of a discrete predator-prey system with a strong Allee effect on the prey and a ratio-dependent functional response,” Adv. Differ. Equ., vol. 320, pp. 1–16, 2018. https://doi.org/10.1186/s13662-018-1781-x.Search in Google Scholar

[18] L. F. Cheng and H. J. Cao, “Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee Effect,” Commun. Nonlinear Sci. Numer. Simulat., vol. 38, pp. 288–302, 2016. https://doi.org/10.1016/j.cnsns.2016.02.038.Search in Google Scholar

[19] M. A. McCarthy, “The Allee effect, finding mates and theoretical models,” Ecol. Model., vol. 103, pp. 99–102, 1997. https://doi.org/10.1016/s0304-3800(97)00104-x.Search in Google Scholar

[20] B. Dennis, “Allee effects: population growth, critical density, and the chance of extinction,” Nat. Resour. Model., vol. 3, pp. 481–538, 1989. https://doi.org/10.1111/j.1939-7445.1989.tb00119.x.Search in Google Scholar

[21] M. K. Andrew, D. Brian, M. L. Andrew, and M. D. John, “The evidence for Allee effects,” Popul. Ecol., vol. 52, pp. 341–354, 2009.10.1007/s10144-009-0152-6Search in Google Scholar

[22] W. C. Allee, Animal Aggregations: A Study in General Sociology, Chicago, The University of Chicago Press, 1931.10.5962/bhl.title.7313Search in Google Scholar

[23] Z. L. Zhu, M. X. He, Z. Li, and F. D. Chen, “Stability and bifurcation in a logistic model with allee effect and feedback control,” Int. J. Bifurc. Chaos, vol. 30, p. 2050231, 2020. https://doi.org/10.1142/s0218127420502314.Search in Google Scholar

[24] D. Y. Wu, H. Y. Zhao, and Y. Yuan, “Complex dynamics of a diffusive predator-prey model with strong Allee effect and threshold harvesting,” J. Math. Anal. Appl., vol. 469, pp. 982–1014, 2019. https://doi.org/10.1016/j.jmaa.2018.09.047.Search in Google Scholar

[25] W. Ni and M. Wang, “Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey,” J. Differ. Equ., vol. 261, pp. 4244–4274, 2016. https://doi.org/10.1016/j.jde.2016.06.022.Search in Google Scholar

[26] K. Vishwakarma and M. Sen, “Influence of Allee effect in prey and hunting cooperation in predator with Holling type-III functional response,” J. Appl. Math. Comput., 2021. https://doi.org/10.1007/s12190-021-01520-1.Search in Google Scholar

[27] Z. F. Zhang, T. R. Ding, W. Z. Huang, and Z. X. Dong, Qualitative Theory of Differential Equation, Beijing, Science Press, 1992.Search in Google Scholar

[28] L. Perko, Differential Equations and Dynamical Systems, in Texts in Applied Mathematics, vol. 7, 2nd ed. New York, Springer-Verlag, 2001.10.1007/978-1-4613-0003-8Search in Google Scholar

[29] J. Sotomayor, “Generic bifurcations of dynamical systems,” in Dynamical Systems, M. M. Peixoto, Ed., New York, Academic Press, 1973.Search in Google Scholar

[30] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed. New York, Springer-Verlag, 1990.10.1007/978-1-4757-4067-7Search in Google Scholar

[31] M. Lu and J. C. Huang, “Global analysis in Bazykins model with Holling II functional response and predator competition,” J. Differ. Equations., vol. 280, pp. 99–138, 2021. https://doi.org/10.1016/j.jde.2021.01.025.Search in Google Scholar
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

Articles in the same Issue

  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
Keywords for this article
Bogdanov–Takens bifurcation; Hopf bifurcation; predator–prey model; saddle-node bifurcation; strong Allee effect

Articles in the same Issue

  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
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 3.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/zna-2021-0178/html
Scroll to top button