BIFURCATION AND CHAOS ANALYSIS OF A TWO-DIMENSIONAL DISCRETE-TIME PREDATOR–PREY MODEL

Tamer El-Azab; M. Y. Hamada; H. El-Metwally

doi:10.11948/20220285

2023 Volume 13 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(4): 1910-1930

Next Article Previous Article

Tamer El-Azab, M. Y. Hamada, H. El-Metwally. BIFURCATION AND CHAOS ANALYSIS OF A TWO-DIMENSIONAL DISCRETE-TIME PREDATOR–PREY MODEL[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1910-1930. doi: 10.11948/20220285

Citation:

Tamer El-Azab, M. Y. Hamada, H. El-Metwally. BIFURCATION AND CHAOS ANALYSIS OF A TWO-DIMENSIONAL DISCRETE-TIME PREDATOR–PREY MODEL[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1910-1930. doi: 10.11948/20220285

BIFURCATION AND CHAOS ANALYSIS OF A TWO-DIMENSIONAL DISCRETE-TIME PREDATOR–PREY MODEL

1.
Mathematics Department, Faculty of Engineering, German International University, Cairo, Egypt
2.
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Author Bio: Email: Tamerazab@mans.edu.eg(T. El-Azab); Email: Helmetwally@mans.edu.eg(H. El-Metwally)
Corresponding author: Email: moe_hamada87@hotmail.com(M. Y. Hamada)

Abstract

Abstract

The dynamical behavior of a discrete predator–prey system with a nonmonotonic functional response is investigated in this work. We study the local asymptotic stability of the positive equilibrium of the system by examining the characteristic equation of the linearized system corresponding to the model. By choosing the growth rate as a bifurcation parameter, the existence of Neimark-Sacker and period-doubling bifurcations at the positive equilibrium is established. Furthermore, the effects of perturbations on the system dynamics are investigated. Finally, examples are presented to illustrate our main results.
- Discrete predator–prey model /
- Neimark-Sacker bifurcation /
- flip bifurcation /
- center Manifold Theorem
MSC: 39A10

FullText(HTML)

References

[1]	H. Agiza, E. ELabbasy, H. EL-Metwally and A. Elsadany, Chaotic dynamics of a discrete prey–predator model with holling type Ⅱ, Nonlinear Analysis: Real World Applications, 2009, 10(1), 116–129. doi: 10.1016/j.nonrwa.2007.08.029 CrossRef Google Scholar
[2]	E. S. Allman and J. A. Rhodes, Mathematical models in biology: an introduction, Cambridge University Press, 2004. Google Scholar
[3]	A. Almutairi, H. El-Metwally, M. Sohaly and I. Elbaz, Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology, Advances in Difference Equations, 2021, 2021(1), 1–32. doi: 10.1186/s13662-020-03162-2 CrossRef Google Scholar
[4]	A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 1992, 73(5), 1530–1535. doi: 10.2307/1940005 CrossRef Google Scholar
[5]	J. Carr, Applications of centre manifold theory, Springer Science & Business Media, 2012, 35. Google Scholar
[6]	J. D. Crawford, Introduction to bifurcation theory, Reviews of Modern Physics, 1991, 63(4), 991. doi: 10.1103/RevModPhys.63.991 CrossRef Google Scholar
[7]	Q. Din, Complexity and chaos control in a discrete-time prey-predator model, Communications in Nonlinear Science and Numerical Simulation, 2017, 49, 113–134. doi: 10.1016/j.cnsns.2017.01.025 CrossRef Google Scholar
[8]	L. Edelstein-Keshet, Mathematical models in biology, SIAM, 2005. Google Scholar
[9]	H. El-Metwally, M. Sohaly and I. Elbaz, Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson's blowflies application, Nonlinear Dynamics, 2021, 105(2), 1713–1722. doi: 10.1007/s11071-021-06696-6 CrossRef Google Scholar
[10]	S. N. Elaydi, Discrete chaos: with applications in science and engineering, Chapman and Hall/CRC, 2007. Google Scholar
[11]	A. -E. A. Elsadany, H. El-Metwally, E. Elabbasy and H. Agiza, Chaos and bifurcation of a nonlinear discrete prey-predator system, Computational Ecology and Software, 2012, 2(3), 169. Google Scholar
[12]	Q. Fang and X. Li, Complex dynamics of a discrete predator–prey system with a strong allee effect on the prey and a ratio-dependent functional response, Advances in Difference Equations, 2018, 2018(1), 1–16. doi: 10.1186/s13662-017-1452-3 CrossRef Google Scholar
[13]	H. Freedman, A model of predator-prey dynamics as modified by the action of a parasite, Mathematical biosciences, 1990, 99(2), 143–155. doi: 10.1016/0025-5564(90)90001-F CrossRef Google Scholar
[14]	S. Gao and L. Chen, The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses, Chaos, Solitons & Fractals, 2005, 24(4), 1013–1023. Google Scholar
[15]	W. Govaerts, Y. A. Kuznetsov, R. K. Ghaziani and H. Meijer, Cl matcontm: A toolbox for continuation and bifurcation of cycles of maps, Universiteit Gent, Belgium, and Utrecht University, The Netherlands, 2008. Google Scholar
[16]	J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer Science & Business Media, 2013, 42. Google Scholar
[17]	M. Y. Hamada, T. El-Azab and H. El-Metwally, Allee effect in a ricker type predator-prey model, Journal of Mathematics and Computer Science, 2022, 29(03), 239–251. doi: 10.22436/jmcs.029.03.03 CrossRef Google Scholar
[18]	M. Y. Hamada, T. El-Azab and H. El-Metwally, Bifurcation analysis of a two-dimensional discrete-time predator–prey model, Mathematical Methods in the Applied Sciences, 2023, 46(4), 4815–4833. doi: 10.1002/mma.8807 CrossRef Google Scholar
[19]	M. Y. Hamada, T. El-Azab and H. El-Metwally, Bifurcations and dynamics of a discrete predator–prey model of ricker type, Journal of Applied Mathematics and Computing, 2022, 69(1), 113–135. Google Scholar
[20]	H. El-Metwally, AQ. Khan and M. Y. Hamada, Allee effect in a Ricker type discrete-time predator–prey model with Holling type-Ⅱ functional response, Journal of Biological Systems, 2023, 31(2), 1–20. Google Scholar
[21]	F. C. Hoppensteadt, Mathematical methods of population biology, Cambridge University Press, 1982, 4. Google Scholar
[22]	G. Iooss, Bifurcation of maps and applications, Elsevier, 1979. Google Scholar
[23]	F. Kangalgil and S. Kartal, Stability and bifurcation analysis in a host–parasitoid model with hassell growth function, Advances in Difference Equations, 2018, 2018(1), 1–15. doi: 10.1186/s13662-017-1452-3 CrossRef Google Scholar
[24]	A. Q. Khan, Bifurcations of a two-dimensional discrete-time predator–prey model, Advances in Difference Equations, 2019, 2019(1), 1–23. doi: 10.1186/s13662-018-1939-6 CrossRef Google Scholar
[25]	A. Q. Khan and H. El-Metwally, Global dynamics, boundedness, and semicycle analysis of a difference equation, Discrete Dynamics in Nature and Society, 2021, 2021, 1–10. Google Scholar
[26]	Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 2004, 112. Google Scholar
[27]	Y. A. Kuznetsov and H. G. E. Meijer, Numerical Bifurcation Analysis of Maps: From Theory to Software, Cambridge University Press, 2019. Google Scholar
[28]	X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator–prey system, Chaos, Solitons & Fractals, 2007, 32(1), 80–94. Google Scholar
[29]	J. D. Murray, Mathematical biology: I. an introduction. interdisciplinary applied mathematics, Mathematical Biology, Springer, 2002. Google Scholar
[30]	J. M. Smith and M. Slatkin, The stability of predator-prey systems, Ecology, 1973, 54(2), 384–391. doi: 10.2307/1934346 CrossRef Google Scholar
[31]	S. Wiggins and M. Golubitsky, Introduction to applied nonlinear dynamical systems and chaos, Springer, 1990, 2. Google Scholar
[32]	W. Zhang, Discrete dynamical systems, bifurcations and chaos in economics, elsevier, 2006. Google Scholar
[33]	J. Zhao and Y. Yan, Stability and bifurcation analysis of a discrete predator–prey system with modified holling–tanner functional response, Advances in Difference Equations, 2018, 2018(1), 1–18. doi: 10.1186/s13662-017-1452-3 CrossRef Google Scholar