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Asymptotic Stability, Bifurcation Analysis and Chaos Control in a Discrete Evolutionary Ricker Population Model with Immigration

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Advances in Discrete Dynamical Systems, Difference Equations and Applications (ICDEA 2021)

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Abstract

This research deals with the derivation and dynamical analysis of a discrete-time evolutionary Ricker population model. The model is built using Evolutionary Game Theory and takes into consideration the effect of immigration. The positive fixed point’s existence and local asymptotic stability are examined. Further, it is shown that the evolutionary model experiences Neimark–Sacker bifurcation (NSB) and period doubling bifurcation (PDB) in a small neighborhood of the positive fixed point under certain conditions. To make the chaotic behavior predictable and stable, three different chaos control strategies are applied. Detailed numerical simulations are carried out to not only verify our theoretical results but also exhibit the rich dynamics of the derived system.

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References

  1. Auerbach, D., Grebogi, C., Ott, E., Yorke, J.A.: Controlling chaos in high dimensional systems. Phys. Rev. Lett. 69(24), 3479 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Assas, L., Elaydi, S., Kwessi, E., Livadiotis, G., Ribble, D.: Hierarchical competition models with Allee effects. J. Biol. Dyn. 9(sup1), 32–44 (2015). https://doi.org/10.1080/17513758.2014.923118

    Article  MathSciNet  MATH  Google Scholar 

  3. Assas, L., Dennis, B., Elaydi, S., Kwessi, E., Livadiotis, G.: Hierarchical competition models with the Allee effect II: the case of immigration. J. Biol. Dyn. 9(1), 288–316 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cushing, J.M.: A darwinian ricker equation. In: Baigent, S., Bohner, M., Elaydi, S. (eds.) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol. 341. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60107-2_10

  5. Cushing, J.M.: A strong ergodic theorem for some nonlinear matrix models. Nat. Resour. Model. 3(3), 331–357 (1989)

    Google Scholar 

  6. Cushing, J.M.: An evolutionary Beverton-Holt model. In: AlSharawi, Z., Cushing, J.M., Elaydi, S. (eds.) Theory and Applications of Difference equations and Discrete Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol. 102, pp. 127–141 (2014)

    Google Scholar 

  7. Cushing, J.M., Levarge, S., Chitnis, N., Henson, S.M.: Some discrete competition models and the competitive exclusion principle. J. Diff. Equ. Appl. 10(13–15), 1139–1151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cushing, J., Stefanco, K.: A darwinian dynamics model for the evolution of post-reproduction survival. J. Biol. Syst. 29(02), 433–450 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cushing, J., et al.: The evolutionary dynamics of a population model with strong Allee effect. Math. Biosci. Eng. 12, 4 (2015)

    Google Scholar 

  10. Choua, Y.H., Chowb, Y., Huc, X., Jang, S.R.J.: A Ricker type predator prey system with hunting cooperation in discrete time. Math. Comput. Simul. 190, 570–586 (2021)

    Google Scholar 

  11. Darwin, C.: On the Origin of Species. Harvard University Press, Cambridge (2001)

    Google Scholar 

  12. Din, Q.: Complexity and chaos control in a discrete time prey-predator model. Commun. Nonlinear Sci. Numer. Simul. 49, 113–34 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005)

    MATH  Google Scholar 

  14. Elaydi, S.: Discrete Chaos, Applications in Science and Engineering, 2nd edn. Chapman and Hall/CRC, London (2008)

    MATH  Google Scholar 

  15. Elaydi, S., Kwessi, E., Livadiotis, G.: Hierarchical competition models with the Allee effect III: multi-species. J. Biol. Dyn. 12(1), 271–287 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Elaydi, S.: Global dynamics of discrete dynamical systems and difference equations. In: Elaydi S., Potzsche C., Sasu A. (eds.) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics & Statistics, vol. 287. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20016-9_3

  17. Feia, L., Chena, X., Hanb, B.: Bifurcation analysis and hybrid control of a discrete-time predator-prey model. J. Diff. Equ. Appl. 27(1), 102–117 (2021)

    Article  MathSciNet  Google Scholar 

  18. Liu, X.L., Xiao, D.M.: Complex dynamic behaviors of a discrete-time predator-prey system. Chaos, Solitons Fractals 32, 80–94 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, B., He, Z.: Bifurcations and chaos in a two-dimensional discrete Hindmarsh Rose model. Nonlinear Dyn. 76(697–715), 20 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Livadiotis, G., Elaydi, S.: General Allee effect in two-species population biology. J. Biol. Dyn. 6, 959–973 (2012)

    Article  MATH  Google Scholar 

  21. Livadiotis, G., Assas, L., Elaydi, S., Kwessid, E., Ribblef, D.: Competition models with Allee effects. J. Diff. Eq. 20(8), 1127–1151 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mokni, K., Elaydi, S., Ch-Chaoui, M., Eladdadi, A.: Discrete evolutionary population models: a new approach. J. Biol. Dyn. 14(1), 454–478 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Murry, J.D.: Mathematical Biology. Springer, New York (1989)

    Google Scholar 

  24. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Khan, A.Q., Ma, J., Dongmei, X.: Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect. J. Biol. Dyn. 11(1), 121–146 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khan, A.Q., Abdullah, E., Ibrahim, T.F.: Supercritical Neimark Sacker bifurcation and hybrid control in a discrete time glycolytic oscillator model. Math. Problems Eng. 2020, Article ID 7834076, 15

    Google Scholar 

  27. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)

    Google Scholar 

  28. Romeiras, F.J., Grebogi, C., Ott, E., Dayawansa, W.: Controlling chaotic dynamical systems. Physica D 58(1–4), 165–192 (1992)

    Google Scholar 

  29. Tan, W., Gao, J., Fan, W.: Bifurcation analysis and chaos control in a discrete epidemic system. Discr. Dyn. Nat. Soc. 2015, Article ID 974868 (2015)

    Google Scholar 

  30. Vincent, T.L., Brown, J.S.: Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics. Cambridge University Press, Cambridge, UK (2005)

    Book  MATH  Google Scholar 

  31. Zhang, L., Zou, L.: Bifurcations and control in a discrete predator prey model with strong Allee effect. Int. J. Bifur. Chaos 28(5) (2018)

    Google Scholar 

  32. Zhu, J., Wu, R., Chen, M.: Bifurcation analysis in a predator prey model with strong Allee effect. Zeitschrift fur Naturforschung A 76(12), 000010151520210178 (2021). https://doi.org/10.1515/zna-2021-0178

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Acknowledgements

The authors would like to thank Professor Saber Elaydi for his comments on an early draft of this manuscript. K.M. thanks her main supervisor, Professor Saber Elaydi, for the excellent advice and support. The authors also thank the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. MRI Laboratory, Faculté Polydisciplinaire Khouribga, Sultan My Slimane University, BP 145 Khouribga principale, 25000, Khouribga, Morocco

    Karima Mokni & Mohamed Ch-Chaoui

Authors
  1. Karima Mokni
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  2. Mohamed Ch-Chaoui
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Corresponding author

Correspondence to Karima Mokni .

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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. University of Rhode Island, Kingston, RI, USA

    Mustafa R. S. Kulenović

  3. University of Sarajevo, Sarajevo, Bosnia and Herzegovina

    Senada Kalabušić

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Mokni, K., Ch-Chaoui, M. (2023). Asymptotic Stability, Bifurcation Analysis and Chaos Control in a Discrete Evolutionary Ricker Population Model with Immigration. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_17

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