10.1007/s40096-022-00494-w

Theoretical and numerical bifurcation analysis of a predator–prey system with ratio-dependence

  • Z. Eskandari1,
  • Z. Avazzadeh*2,
  • R. Khoshsiar Ghaziani3,
  1. Department of Mathematics,Faculty of Science, Fasa University, Fasa, IR
  2. Department of Mathematical Sciences, University of South Africa, Florida, ZA
  3. Department of Mathematical Sciences, Shahrekord University, Shahrekord, IR

Published in Issue 2023-02-03

How to Cite

Eskandari, Z., Avazzadeh, Z., & Ghaziani, R. K. (2023). Theoretical and numerical bifurcation analysis of a predator–prey system with ratio-dependence. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00494-w
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Abstract The present paper investigates the critical normal form coefficients for the one-parameter and two-parameter bifurcations of a two-dimensional discrete-time ratio-dependence predator–prey model. The discrete-time ratio-dependence predator–prey model exhibits the period-doubling, Neimark-Sacker, and strong resonance bifurcations. Based on the critical coefficients, it can be determined which scenario corresponds to each bifurcation. This paper investigates the complex dynamics of the model numerically by using MatcotM which is a MATLAB package.

Keywords

  • The ratio-dependence model,
  • Normal form coefficient,
  • Bifurcation,
  • Chaos,
  • Numerical continuation.

References

  1. Panday et al. (2021) Dynamics of a stage-structured predator-prey model: cost and benefit of fear-induced group defense https://doi.org/10.1016/j.jtbi.2021.110846
  2. Ghori et al. (2022) Global dynamics and bifurcation analysis of a fractional-order SEIR epidemic model with saturation incidence rate 45(7) (pp. 3665-3688) https://doi.org/10.1002/mma.8010
  3. Sen et al. (2012) Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect (pp. 12-27) https://doi.org/10.1016/j.ecocom.2012.01.002
  4. Elabbasy et al. (2007) Bifurcation analysis, chaos and control in the Burgers mapping 4(3) (pp. 171-185)
  5. Frank et al. (2021) Population dynamic regulators in an empirical predator-prey system https://doi.org/10.1016/j.jtbi.2021.110814
  6. Lehtinen (2021) Ecological and evolutionary consequences of predator-prey role reversal: Allee effect and catastrophic predator extinction https://doi.org/10.1016/j.jtbi.2020.110542
  7. Tripathi et al. (2018) Interaction between prey and mutually interfering predator in prey reserve habitat: Pattern formation and the Turing Hopf bifurcation 355(15) (pp. 7466-7489) https://doi.org/10.1016/j.jfranklin.2018.07.029
  8. Jiao et al. (2016) Dynamics of a periodic switched predator-prey system with impulsive harvesting and hibernation of prey population 353(15) (pp. 3818-3834) https://doi.org/10.1016/j.jfranklin.2016.06.035
  9. Wang, J., Wang, M.: The dynamics of a predator-prey model with diffusion and indirect prey-taxis. J. Dyn. Differ. Equ. 1–20 (2019)
  10. Arancibia-Ibarra et al. (2021) Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response
  11. Naik et al. (2021) Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model 1(2) (pp. 95-101)
  12. Eskandari et al. (2021) Dynamics and bifurcations of a discrete-time prey-predator model with Allee effect on the prey population https://doi.org/10.1016/j.ecocom.2021.100962
  13. Elsadany et al. (2020) Qualitative properties and bifurcations of discrete-time Bazykin-Berezovskaya predator-prey model 13(06) https://doi.org/10.1142/S1793524520500400
  14. Lin et al. (2020) Dynamics and chaos control for a discrete-time Lotka-Volterra model (pp. 126760-126775) https://doi.org/10.1109/ACCESS.2020.3008522
  15. Eskandari and Alidousti (2021) Generalized flip and strong resonances bifurcations of a predator-prey model (pp. 275-287) https://doi.org/10.1007/s40435-020-00637-8
  16. Khan and Khalique (2020) Neimark-Sacker bifurcation and hybrid control in a discrete-time Lotka-Volterra model 43(9) (pp. 5887-5904) https://doi.org/10.1002/mma.6331
  17. Wang and Fečkan (2020) Dynamics of a discrete nonlinear prey-predator model 30(04) https://doi.org/10.1142/S0218127420500558
  18. Mishra and Tiwari (2021) Stability and bifurcation analysis of a prey-predator model 31(04) https://doi.org/10.1142/S0218127421500590
  19. Atabaigi (2017) Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response 7(2) (pp. 411-426)
  20. Khan (2017) Stability and Neimark-Sacker bifurcation of a ratio-dependence predator-prey model 40(11) (pp. 4109-4117) https://doi.org/10.1002/mma.4290
  21. Eskandari and Alidousti (2020) Stability and codimension 2 bifurcations of a discrete time SIR model 357(15) (pp. 10937-10959) https://doi.org/10.1016/j.jfranklin.2020.08.040
  22. Tassaddiq et al. (2020) A ratio-dependent nonlinear predator-prey model with certain dynamical results (pp. 195074-195088) https://doi.org/10.1109/ACCESS.2020.3030778
  23. Sarwardi et al. (2012) Ratio-dependent predator-prey model of interacting population with delay effect 69(3) (pp. 817-836) https://doi.org/10.1007/s11071-011-0307-9
  24. Fan and Wang (2002) Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system 35(9–10) (pp. 951-961) https://doi.org/10.1016/S0895-7177(02)00062-6
  25. Fazly and Hesaaraki (2007) Periodic solutions for a discrete time predator-prey system with monotone functional responses 345(4) (pp. 199-202) https://doi.org/10.1016/j.crma.2007.06.021
  26. Hu and Zhang (2008) Four positive periodic solutions of a discrete time delayed predator-prey system with nonmonotonic functional response and harvesting 56(12) (pp. 3015-3022) https://doi.org/10.1016/j.camwa.2008.09.009
  27. Xia et al. (2007) Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses 8(4) (pp. 1079-1095) https://doi.org/10.1016/j.nonrwa.2006.06.007
  28. Yang and Li (2009) Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses 10(2) (pp. 1068-1072) https://doi.org/10.1016/j.nonrwa.2007.11.022
  29. Naik et al. (2022) Multiple bifurcations of a discrete-time prey-predator model with mixed functional response 32(4) https://doi.org/10.1142/S021812742250050X
  30. Naik et al. (2022) Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect https://doi.org/10.1016/j.cam.2022.114401
  31. Kuznetsov (2013) Springer
  32. Kuznetsov and Meijer (2005) Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues 26(6) (pp. 1932-1954) https://doi.org/10.1137/030601508
  33. Kuznetsov and Meijer (2019) Cambridge USIRversity Press https://doi.org/10.1017/9781108585804
  34. Govaerts et al. (2007) Numerical methods for two-parameter local bifurcation analysis of maps 29(6) (pp. 2644-2667) https://doi.org/10.1137/060653858