Thanks to visit codestin.com
Credit goes to link.springer.com

Skip to main content
Springer Nature Link
Log in

Dynamics of a Delayed Predator–Prey System in Highland Pasture

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Studying the dynamical behavior of the time-delayed predator–prey model helps to predict the pattern of population change, thus maintaining ecological balance and promoting harmonious coexistence between humans and nature. The predator–prey model of a highland pasture with two-time delays was investigated in this work. Firstly, the shape of the critical manifold of the model and the existence of the equilibrium point are analyzed when the time delay is equal to zero. Using the geometric singular perturbation theory, the dynamical behavior in the first quadrant of the model is explored based on the distribution of the equilibrium points, including singular Hopf bifurcations, relaxation oscillations, homoclinic orbits, heteroclinic orbits, and canard explosion phenomena. Subsequently, we examined the local stability around the positive equilibrium point when the time delay is not equal to zero. We also derived parametric requirements for the Hopf bifurcation. Besides, the direction and stability of the bifurcation are analyzed by employing center manifold and normal form theory. Simultaneously, we performed numerical simulations to confirm our conclusions in every case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+
from £29.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

Data Availability

Data will be made available on request.

References

  1. Ji, J.P., Lin, G.H., Wang, L.: Effects of intraguild prey dispersal driven by intraguild predator-avoidance on species coexistence. Appl. Math. Model. 103, 51–67 (2022). https://doi.org/10.1016/j.apm.2021.10.026

    Article  MathSciNet  MATH  Google Scholar 

  2. Araujo, G., Moura, R.R.: Individual specialization and generalization in predator-prey dynamics: The determinant role of predation efficiency and prey reproductive rates. J. Theor. Biol. 537(21), 111026 (2022). https://doi.org/10.1016/j.jtbi.2022.111026

    Article  MATH  Google Scholar 

  3. Yousef, A.M., Algelany, A.M., Elsadany, A.A.: Codimension one and codimension two bifurcations in a discrete Kolmogorov type predator-prey model. J. Comput. Appl. Math. 428(15), 115171 (2023). https://doi.org/10.1016/j.cam.2023.115171

    Article  MathSciNet  MATH  Google Scholar 

  4. Djilali, S.: Threshold asymptotic dynamics for a spatial age-dependent cell-to-cell transmission model with nonlocal disperse. Discrete. Cont. Dyn-B. 28(7), 4108–4143 (2023). https://doi.org/10.3934/dcdsb.2023001

    Article  MathSciNet  MATH  Google Scholar 

  5. Lotka, A.J.: Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42(8), 1595–1599 (1920)

    Article  MATH  Google Scholar 

  6. Volterra, V.: Variazioni e fluttuazioni del numbero d’individui in specie animali conviventi. Mem. Acad. Lincei. 2, 31–113 (1926)

    MATH  Google Scholar 

  7. Ahn, I., Yoon, C.: Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis. J. Differ. Equ. 268(8), 4222–4255 (2020). https://doi.org/10.1016/j.jde.2019.10.019

    Article  MathSciNet  MATH  Google Scholar 

  8. Yan, X.P., Zhang, C.H.: Global stability of a delayed diffusive predator-prey model with prey harvesting of Michaelis-Menten type. Appl. Math. Lett. 114, 106904 (2021). https://doi.org/10.1016/j.aml.2020.106904

    Article  MathSciNet  MATH  Google Scholar 

  9. Roy, J., Banerjee, M.: Global stability of a predator-prey model with generalist predator. Appl. Math. Lett. 142, 108659 (2023). https://doi.org/10.1016/j.aml.2023.108659

    Article  MathSciNet  MATH  Google Scholar 

  10. Onana, M., Mewoli, B., Tewa, J.J.: Hopf bifurcation analysis in a delayed Leslie-Gower predator-prey model incorporating additional food for predators, refuge and threshold harvesting of preys. Nonlinear Dyn. 100, 3007–3028 (2020). https://doi.org/10.1007/s11071-020-05659-7

    Article  MATH  Google Scholar 

  11. Peng, M., Zhang, Z.D.: Bifurcation analysis and control of a delayed stage-structured predator-prey model with ratio-dependent Holling type III functional response. J. Vib. Control 26(13–14), 1232–1245 (2020). https://doi.org/10.1177/1077546319892144

    Article  MathSciNet  MATH  Google Scholar 

  12. Bentout, S., Djilali, S., Kumar, S.: Mathematical analysis of the influence of prey escaping from prey herd on three species fractional predator-prey interaction model. Phys. A 572, 125840 (2021). https://doi.org/10.1016/j.physa.2021.125840

    Article  MathSciNet  MATH  Google Scholar 

  13. Djilali, S., Cattani, C.: Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response. Chaos Solitons Fractals 151, 111258 (2021). https://doi.org/10.1016/j.chaos.2021.111258

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, S., Huang, C.D., Song, X.Y.: Detection of Hopf bifurcations induced by pregnancy and maturation delays in a spatial predator-prey model via crossing curves method. Chaos Solitons Fractals 175, 114012 (2023). https://doi.org/10.1016/j.chaos.2023.114012

    Article  MathSciNet  MATH  Google Scholar 

  15. Souna, F., Djilali, S., Lakmeche, A.: Spatiotemporal behavior in a predator-prey model with herd behavior and cross-diffusion and fear effect. Eur. Phys. J. Plus. 136(5), 474 (2021). https://doi.org/10.1140/epjp/s13360-021-01489-7

    Article  MATH  Google Scholar 

  16. Bentout, S., Djilali, S., Atangana, A.: Bifurcation analysis of an age-structured prey-predator model with infection developed in prey. Math. Method. Appl. Sci. 45(3), 1189–1208 (2022). https://doi.org/10.1002/mma.7846

    Article  MathSciNet  MATH  Google Scholar 

  17. Wei, W., Xu, W., Liu, J.K., Song, Y., Zhang, S.: Stochastic bifurcation and Break-out of dynamic balance of predator-prey system with Markov switching. Appl. Math. Model. 117, 563–576 (2023). https://doi.org/10.1016/j.apm.2022.12.034

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, H.J., Wei, Z.C.: Stability analysis of discrete-time switched systems with all unstable subsystems. Discrete. Cont. Dyn-S. (2024). https://doi.org/10.3934/dcdss.2024040

    Article  MATH  Google Scholar 

  19. Wang, C., Zhang, X.: Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III. J. Differ. Equ. 267(6), 3397–3441 (2019). https://doi.org/10.1016/j.jde.2019.04.008

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, S.M., Wang, C., Wu, K.L.: Relaxation oscillations of a slow-fast predator-prey model with a piecewise smooth functional response. Appl. Math. Lett. 113, 106852 (2021). https://doi.org/10.1016/j.aml.2020.106852

    Article  MathSciNet  MATH  Google Scholar 

  21. Chowdhury, P.R., Banerjee, M., Petrovskii, S.: Canards, relaxation oscillations, and pattern formation in a slow-fast ratio-dependent predator-prey system. Appl. Math. Model. 109, 519–535 (2022). https://doi.org/10.1016/j.apm.2022.04.022

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhao, L., Shen, J.H.: Relaxation oscillations in a slow-fast predator-prey model with weak Allee effect and Holling-IV functional response. Commun. Nonlinear Sci. Numer. Simul. 112, 106517 (2022). https://doi.org/10.1016/j.cnsns.2022.106517

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, J., Li, S.M., Wang, X.L.: Canard, homoclinic loop, and relaxation oscillations in a Lotka-Volterra system with Allee effect in predator population. Chaos 33(7), 073130 (2023). https://doi.org/10.1063/5.0152946

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, Q., Zhang, Y.Y., Xiao, Y.N.: Canard phenomena for a slow-fast predator-prey system with group defense of the prey. J. Math. Anal. Appl. 527(1), 127418 (2023). https://doi.org/10.1016/j.jmaa.2023.127418

    Article  MathSciNet  MATH  Google Scholar 

  25. Tiwari, V., Tripathi, J.P., Upadhyay, R.K., Wu, Y.P., Wang, J.S., Sun, G.Q.: Predator-prey interaction system with mutually interfering predator: role of feedback control. Appl. Math. Model. 87, 222–244 (2020). https://doi.org/10.1016/j.apm.2020.04.024

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, X.M., Liu, Z.H.: Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay. Appl. Math. Model. 91, 530–548 (2021). https://doi.org/10.1016/j.apm.2020.08.054

    Article  MathSciNet  MATH  Google Scholar 

  27. Yuan, Y., Fu, X.L.: Asymptotic behavior of an age-structured prey-predator system with distributed delay. J. Differ. Equ. 317, 121–152 (2022). https://doi.org/10.1016/j.jde.2022.01.062

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, S., Yuan, S.L., Jin, Z., Wang, H.: Bifurcation analysis in a diffusive predator-prey model with spatial memory of prey, Allee effect and maturation delay of predator. J. Differ. Equ. 357, 32–63 (2023). https://doi.org/10.1016/j.jde.2023.02.009

    Article  MathSciNet  MATH  Google Scholar 

  29. Bhunia, B., Ghorai, S., Kar, T.K., Biswas, S., Bhutia, L.T., Debnath, P.: A study of a spatiotemporal delayed predator-prey model with prey harvesting: Constant and periodic diffusion. Chaos Solitons Fractals 175, 113967 (2023). https://doi.org/10.1016/j.chaos.2023.113967

    Article  MathSciNet  MATH  Google Scholar 

  30. Du, W.T., Xiao, M., Ding, J., Yao, Y., Wang, Z.X., Yang, X.S.: Fractional-order PD control at Hopf bifurcation in a delayed predator-prey system with trans-species infectious diseases. Math. Comput. Simul. 205, 414–438 (2023). https://doi.org/10.1016/j.matcom.2022.10.014

    Article  MathSciNet  MATH  Google Scholar 

  31. Jiao, J.F., Chen, C.: Bogdanov-Takens bifurcation analysis of a delayed predator-prey system with double Allee effect. Nonlinear Dyn. 104, 1697–1707 (2021). https://doi.org/10.1007/s11071-021-06338-x

    Article  MATH  Google Scholar 

  32. Pati, N.C., Ghosh, B.: Delayed carrying capacity induced subcritical and supercritical Hopf bifurcations in a predator-prey system. Math. Comput. Simul. 195, 171–196 (2022). https://doi.org/10.1016/j.matcom.2022.01.008

    Article  MathSciNet  MATH  Google Scholar 

  33. Jiao, X.B., Li, X.D., Yang, Y.P.: Dynamics and bifurcations of a Filippov Leslie-Gower predator-prey model with group defense and time delay. Chaos Solitons Fractals 162, 112436 (2022). https://doi.org/10.1016/j.chaos.2022.112436

    Article  MathSciNet  MATH  Google Scholar 

  34. Liang, Z.W., Meng, X.Y.: Stability and Hopf bifurcation of a multiple delayed predator-prey system with fear effect, prey refuge and Crowley-Martin function. Chaos Solitons Fractals 175, 113955 (2023). https://doi.org/10.1016/j.chaos.2023.113955

    Article  MathSciNet  MATH  Google Scholar 

  35. Sarif, N., Sarwardi, S.: Complex dynamical study of a delayed prey-predator model with fear in prey and square root harvesting of both species. Chaos 33(3), 033112 (2023). https://doi.org/10.1063/5.0135181

    Article  MathSciNet  MATH  Google Scholar 

  36. Yousef, A.M.: Stability and further analytical bifurcation behaviors of Moran-Ricker model with delayed density dependent birth rate regulation. J. Comput. Appl. Math. 355, 143–161 (2019). https://doi.org/10.1016/j.cam.2019.01.012

    Article  MathSciNet  MATH  Google Scholar 

  37. Eskandari, Z., Alidousti, J., Avazzadeh, Z.: Rich dynamics of discrete time-delayed Moran-Ricker model. Qual. Theor. Dyn. Syst. 22(3), 98 (2023). https://doi.org/10.1007/s12346-023-00774-3

    Article  MathSciNet  MATH  Google Scholar 

  38. Xu, C.J., Zhang, W., Aouiti, C., Liu, Z.X., Yao, L.Y.: Bifurcation insight for a fractional-order stage-structured predator-prey system incorporating mixed time delays. Math. Method. Appl. Sci. 46(8), 9103–9118 (2023). https://doi.org/10.1002/mma.9041

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, B., Yuan, Z.M., Eskandari, Z.: Dynamics and bifurcations of a discrete-time Moran-Ricker model with a time delay. Mathematics. 11(11), 2446 (2023). https://doi.org/10.3390/math11112446

    Article  MATH  Google Scholar 

  40. Ma, Y.Y., Dong, N., Liu, N., Xie, L.L.: Spatiotemporal and bifurcation characteristics of a nonlinear prey-predator model. Chaos Solitons Fractals 165, 112851 (2022). https://doi.org/10.1016/j.chaos.2022.112851

    Article  MathSciNet  MATH  Google Scholar 

  41. Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points–fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2), 286–314 (2001). https://doi.org/10.1137/S0036141099360919

    Article  MathSciNet  MATH  Google Scholar 

  42. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001). https://doi.org/10.1006/jdeq.2000.3929

    Article  MathSciNet  MATH  Google Scholar 

  43. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and applications of Hopf bifurcation. Cambridge University Press (1981)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant No.12172333 and the Natural Science Foundation of Zhejiang through grant No.LY20A020003.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Haolan Wang & Youhua Qian

Authors
  1. Haolan Wang
    View author publications

    Search author on:PubMed Google Scholar

  2. Youhua Qian
    View author publications

    Search author on:PubMed Google Scholar

Contributions

Haolan Wang: conceptualization, methodology, software, investigation, formal analysis, writing-original draft. Youhua Qian: conceptualization, methodology, software, supervision, funding acquisition.

Corresponding author

Correspondence to Youhua Qian.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H., Qian, Y. Dynamics of a Delayed Predator–Prey System in Highland Pasture. Qual. Theory Dyn. Syst. 24, 44 (2025). https://doi.org/10.1007/s12346-024-01207-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-024-01207-5

Keywords