Bifurcations of a Two-Dimensional Discrete-Time Predrator-Prey Model

Sabina Hrustić; Samra Moranjkić; Zehra Nurkanović

doi:10.5644/SJM.21.02.08

Authors

Sabina Hrustić University of Tuzla, Departmen of Mathematics, U. Vejzagića 4, Tuzla, Bosnia and Herzegovina https://orcid.org/0000-0002-6379-2768
Samra Moranjkić University of Tuzla, Departmen of Mathematics, U. Vejzagića 4, Tuzla, Bosnia and Herzegovina https://orcid.org/0000-0003-2402-6312
Zehra Nurkanović University of Tuzla, Departmen of Mathematics, U. Vejzagića 4, Tuzla, Bosnia and Herzegovina https://orcid.org/0000-0002-0917-8433

DOI:

https://doi.org/10.5644/SJM.21.02.08

Keywords:

Difference equations, fixed point, Neimark-Sacker bifurcation

Abstract

In this paper, we study the dynamics and bifurcation of a two-dimensional discrete-time predator-prey model. The existence and local stability of the equilibrium points of the model are analyzed algebraically. It is shown that the model can undergo a transcritical bifurcation at equilibrium point on the $x$-axis and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium point. Some numerical simulations are presented to illustrate our theoretical results.

Statistics

Abstract: 5 / PDF: 1

References

E. Beretta, V. Capasso, and F. Rinaldi, Global stability results for a generalized Lotka-Volterra system with distributed delays, J. Math. Biol., 26 (1988), 661–688.

E. Bešo, S. Kalabušić, and E. Pilav, A Generalized Beddington Host–Parasitoid Model with an Arbitrary Parasitism Escape Function, International Journal of Bifurcation and Chaos, Vol. 34, No. 10 (2024).

F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Texts in Applied Mathematics, Springer, New York, 2011.

A. Cima, A. Gasull, and V. Manosa, Asymptotic Stability for Block Triangular Maps, Sarajevo Journal of Mathematics, Vol.18(31), no.1 (2022), 25–44.

M. Garić-Demirović, S. Hrustić, S. Moranjkić, M. Nurkanović, and Z. Nurkanović, The Existence of Li-Yorke Chaos in Certain Predator-Prey System of Difference Equations, Sarajevo Journal of Mathematics, Vol.18(31), no.1 (2022), 45–62.

M. Garić-Demirović, S. Hrustić, and S. Moranjkić, Global Dynamics of certain non-symetric second order difference equation with quadratic term, Sarajevo Journal of Mathematics, Vol 15(28), no. 2 (2019), 155–167.

M. Garić-Demirović, S. Moranjkić, M. Nurkanović, and Z. Nurkanović, Stability, Neimark–Sacker Bifurcation, and Approximation of the Invariant Curve of Certain Homogeneous Second-Order Fractional Difference Equation, Discrete Dynamics in Nature and Society, vol. 2020 (2020), 12.

J. Hale, H. Buttanri, and H. Kocak, Dynamics and Bifurcations, Texts in Applied Mathematics, Springer, New York, 2012.

J. Hofbauer and J.W. Thus, Multiple limit cycles for predator-prey models, Math. Biosci., 99 (1990), 71–75.

B. Hong and C. Zhang, Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect, Mathematics, 11(6):1399 (2023).

S. Hrustić, S. Moranjkić, and Z. Nurkanović, Local Stability, Period-Doubling and 1:2 Resonance Bifurcation for a Discretized Chemical Reaction System, MATCH Communications in Mathematical and in Computer Chemistry, 93 (2025), 349–378.

S. Hrustić, M.R.S. Kulenović, Z. Nurkanović, and E. Pilav, Birkhoff normal forms, KAM theory and symmetries for certain second order rational difference equation with quadratic term, Int. J. Differ. Equ., 10 (2015), 181–199.

K. C. Hung and S. H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex–concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differ. Equ., 251 (2011), 223–237.

Z. Y. Hu, Z. D. Teng, and L. Zhang, Stability and bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response, Nonlinear Anal. Real World Appl., 12 (2011), 2356–2377.

M. Kot, Elements of mathematical ecology, Cambridge University Press, Cambridge, 2001.

M.R.S. Kulenović and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC: Boca Raton, FL, USA; London, UK, 2002.

M.R.S. Kulenović, S. Moranjkić, and Z. Nurkanović, Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms, J. Nonlinear Sci. Appl., 10(7) (2017), 3477–3489.

M.R.S. Kulenović, S. Moranjkić, M. Nurkanović, and Z. Nurkanović, Global asymptotic stability and Naimark-Sacker bifurcation of certain mix monotone difference equation, Discrete Dyn. Nat. Soc., 2018 (2018), 1–22.

Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, NY, 2023.

P. Liu and S.N. Elaydi, Discrete competitive and cooperative models of Lotka–Volterra type, J. Comput. Anal. Appl., 3(1) (2001), 53–73.

X. Liu and X. Dongmei, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32(1) (2007), 80–94.

L. Perko, Differential Equations and Dynamical Systems, Springer, Berlin, Germany, 2000.

J. Sotomayor, Generic bifurcations of dynamical systems, Dynamical Systems, 1973 (1973), 549–560.

S. H. Streipert, G.S.K. Wolkowicz, and M. Bohner, Derivation and Analysis of a Discrete Predator–Prey Model, Bulletin of Mathematical Biology, 84:67 (2022).

P-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Corr. Math. Phy., 10:113–121, 1838.