Complicate Dynamics of a Discrete Predator-prey Model with Competition Between Predators


  • Shaosan Xia
  • Xianyi Li



Discrete predator-prey system with competition, Semidiscretization method, Neimark-Sacker bifurcation.


We consider a discrete predator-prey model with competition between predators in this paper. By simplifying the corresponding continuous predator-prey model, and using the semidiscretization method to obtain a new discrete model, we discuss the existence and local stability of nonnegative fixed points of the new discrete model. What’s more important, by using the bifurcation theory, we derive the sufficient conditions for the occurrences of Neimark-Sacker bifurcation and the stability of closed orbit bifurcated. Finally, the numerical simulation are presented, which not only illustrate the existence of NeimarkSacker bifurcation but also reveal some new dynamic phenomena of this model.


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Md, S. & Rana, S. [2019] “Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model,” J.Egypt.Math.Soc. 2019, doi:10.1186/s42787-019-0055-4.

Khan, A. Q. [2016] “Neimark-Sacker bifurcation of a twodimensional discrete-time predator-prey model,” Adv.Differ.Equ. 2016, doi:10.1186/s40064-015-1618-y.

Rodrigo, C., Willy, S. & Eduardo, S. [2017] “Bifurcations in a predatorprey model with general logistic growth and exponential fading memory,” Appl.Math.Model. 45, 134–147.

Holling, C. [1959] “The functional response of predator to prey density and its role in mimicry and population regulation,” Mem.Entomol.Soc.Can. 91, 385-398.

Berryman, A. A., Gutierrez, A. P. & Arditi, R. [1995] “Credible, Parsimonious and useful predator–prey models - A reply to Abrams, Gleeson, and Sarnelle,” Entomological Society of America. 76, 1980–1985.

Saunders, M. C. [2006] “Complex Population Dynamics: A Theoretical/Empirical Synthesis,” Entomological Society of America. 35, 1139-1139.

Akcakaya, HR., et al. [1995] “Ratio-dependent predation: an abstraction that works,” Ecology. 76, 995–1004.

Cosner, C., et al. [1999] “Effects of spatial grouping on the functional response of predators,” Theor.Popul.Biol. 56, 65–75.

Gutierrez, AP. [1992] “Physiological basis of ratio-dependent predatorprey theory: the metabolic pool model as a paradigm,” Ecology. 73, 1552–1563.

May, R. [1973] “Stability and complexity in model ecosystems with a new introduction by the author,” Princeton University Press.

Pielou, EC. [1969] “An introduction to mathematical ecology,” Biometrical Journal. 13, doi:10.1002/bimj.19710130308.

Din, Q. [2017] “Complexity and chaos control in a discrete-time prey-predator model,” Commun.Nonliner.Sci.Numer. Simul. 49, 113-134.

Li, W. & Li, X. Y. [2018] “Neimark–Sacker bifurcation of a semi-discrete hematopoiesis model,” J.Appl.Anal.Comput. 8, 1679–1693.

Hu, Z. Y., Teng, Z. D. & Zhang, L. [2011] “Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic func-tional response,” Nonlinear. Anal. Real. World. Appl. 12, 2356–2377.

Wang, C. & Li, X. Y. [2015] “Further investigations into the stability and bifurcation of a discrete predator-prey model,” J.Math.Anal.Appl. 422, 920–939.

Wang, C. & Li, X. Y. [2014] “Stability and Neimark–Sacker bifurcation of a semi-discrete population model,” J.Appl.Anal.Comput. 4, 419–435.

Li, M. S., Zhou, X. L. & Xu, J. M. [2020] “Dynamic properties of a discrete population model with diffusion,” Adv.Differ.Equ. 2020, doi:10.1186/s13662-020-03033-w.

Gallay, T. [1993] “A center-stable manifold theorem for differential equations in Banach spaces,” Commun.Math.Phys. 152, 249–268.

Liu, W., Cai, D. & Shi, J. [2018] “Dynamic behaviors of a discretetime predator–prey bioeconomic system,” Adv.Differ.Equ. 2018, doi:10.1186/s13662-018-1592-0.

Jorba, A. & Masdemont, J. [1999] “Dynamics in the center manifold of the collinear points of the restricted three body problem,” Physica D : Nonlinear Phenomena. 132, 189–213.

Zhao, M., Xuan, Z. & Li, C. [2016] “Dynamics of a discrete-time predator-prey system,” Adv.Differ.Equ. 2016, doi:10.1186/s13662016-0903-6.

Winggins, S. [2003] “Introduction to Applied Nonlinear Dynamical Systems and Chaos,” Springer-Verlag, New York.







How to Cite

Complicate Dynamics of a Discrete Predator-prey Model with Competition Between Predators. (2022). Academic Journal of Science and Technology, 3(3), 119-123.

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