Transcritical Bifurcation and Flip Bifurcation of a New Discrete Ratio-Dependent Predator-Prey System

• Xianyi Li ^[1] ; Yuqing Liu ^[1]
  1. [1] Zhejiang University of Science and Technology

     Zhejiang University of Science and Technology

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • After a discrete two-species predator-prey system with ratio-dependent functional response is topologically and equivalently reduced, some new dynamical properties for the new discrete system are formulated. The one is for the existence and local stability for all equilibria of this new system. Although the corresponding results for the equilibrium E3 have been presented in a known literature, our results are more complete. The other is, what’s more important and difficult, to derive some sufficient conditions for the transcritical bifurcation and period-doubling bifurcation of this system at the equilibria E1, E2 and E3 to occur, which are completely new. Numerical simulations are performed to not only illustrate the theoretical results obtained but also find new dynamics—chaos occuring. Our results sufficiently display that this system is very sensitive to its parameters. Namely, the perturbations of different parameters in this system will produce different bifurcations.

• Referencias bibliográficas
  • 1. Zhuo, X., Zhang, X.: Stability for a new discrete ratio-dependent predator-prey system. Qualit. Theory Dyn. Syst. 17, 189–202 (2018)
  • 2. Rodrigo, C., Willy, S., Eduardo, S.: Bifurcations in a predator-prey model with general logistic growth and exponential fading memory....
  • 3. Sarker, M., Rana, S.: Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model. J. Egypt. Math. l Soc. 48, 1–16...
  • 4. Khan, A.Q.: Neimark-Sacker bifurcation of a two-dimensional discrete-time predator-prey model. Springer Plus 5, 1–10 (2016)
  • 5. Chen, Xiaoxing, Chen, Fengde: Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control....
  • 6. Xiang, C., Huang, J., Ruan, S., et al.: Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response....
  • 7. Luo, Y., Zhang, L., Teng, Z., et al.: Global stability for a nonautonomous reaction-diffusion predatorprey model with modified Leslie-Gower...
  • 8. Fan, Y., Li, W.: Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response. J. Math....
  • 9. Fan, M., Wang, K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Computer Model. 35,...
  • 10. Fazly, M., Hesaaraki, M.: Periodic solutions for discrete time predator-prey system with monotone functional responses. Comptes Rendus...
  • 11. Hu, D., Zhang, Z.: Four positive periodic solutions of a discrete time delayed predator-prey system with nonmonotonic functional response...
  • 12. Liu, X.: A note on the existence of periodic solution in discrete predator-prey models. Appl. Math. Model. 34, 2477–2483 (2010)
  • 13. Xia, Y., Cao, J., Lin, M.: Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses. Nonlinear...
  • 14. Yakubu, A.: The effects of planting and harvesting on endangered species in discrete competitive systems. Math. Biosci. 126, 1–20 (1995)
  • 15. Claudio, A., Aguirre, P., Flores, J., Heijster, P.: Bifurcation analysis of a predator-prey model with predator intraspecific interactions...
  • 16. Yang, W., Li, X.: Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses. Nonlinear...
  • 17. Ji, C., Jiang, D., Li, X.: Qualitative analysis of a stochastic ratio-dependent predator-prey system. J. Comput. Appl. Math. 235, 1326–1341...
  • 18. Li, W., Li, X.: Neimark-Sacker bifurcation of a semi-discrete hematopoiesis model. J. App. Anal. Comput. 8, 1679–1693 (2018)
  • 19. Wang, C., Li, X.: Stability and Neimark-Sacker bifurcation of a semi-discrete population model. J. Appl. Anal. Comput. 4, 419–435 (2014)
  • 20. Kuzenetsov, Y.: Elements of Applied Bifurcation Theory, 2nd edn. Springer- Verlag, New York (1998)
  • 21. Gallay, T.: A center-stable manifold theorem for differential equations in Banach spaces. Commun. Math. Phys. 152, 249–268 (1993)
  • 22. Jorba, A., Masdemont, J.: Dynamics in the center manifold of the collinear points of the restricted three body problem. Phys. D Nonlinear...
  • 23. Knobloch, E., Wiesenfeld, K.A.: Bifurcations in fluctuating systems: the center-manifold approach. J. Stat. Phys. 33, 611–637 (1983)
  • 24. Xin, B., Wu, Z.: Neimark-Sacker bifurcation analysis and 0–1 chaos test of an interactions model between industrial production and environmental...
  • 25. Berezansky, L., Braverman, E., Idels, L.: Mackey-Glass model of hematopoiesis with non-monotone feedback: stability, oscillation and control....
  • 26. Carr, J.: Application of Center Manifold Theorem. Springer-Verlag, New York (1981)
  • 27. Kuzenetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer-Verlag, New York (1998)
  • 28. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edn. London, New York Boca Raton (1999)
  • 29. Winggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York (2003)