Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response

Xianyi Li; Xingming Shao; Xianyi Li; Xingming Shao

doi:10.3934/era.2023003

Electronic Research Archive

2023, Volume 31, Issue 1: 37-57. doi: 10.3934/era.2023003

Previous Article Next Article

Research article Special Issues

Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response

Xianyi Li ^,,
Xingming Shao

Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received: 10 September 2022 Revised: 09 October 2022 Accepted: 14 October 2022 Published: 20 October 2022

In this paper, we use a semi-discretization method to explore a predator-prey model with Michaelis-Menten functional response. Firstly, we investigate the local stability of fixed points. Then, by using the center manifold theorem and bifurcation theory, we demonstrate that the system experiences a flip bifurcation and a Neimark-Sacker bifurcation at a fixed point when one of the parameters goes through its critical value. To illustrate our results, numerical simulations, which include maximum Lyapunov exponents, fractal dimensions and phase portraits, are also presented.
- discrete predator-prey system,
- semi-discretization method,
- flip bifurcation,
- Neimark-Sacker bifurcation,
- center manifold theorem
Citation: Xianyi Li, Xingming Shao. Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response[J]. Electronic Research Archive, 2023, 31(1): 37-57. doi: 10.3934/era.2023003

Related Papers:

Abstract

In this paper, we use a semi-discretization method to explore a predator-prey model with Michaelis-Menten functional response. Firstly, we investigate the local stability of fixed points. Then, by using the center manifold theorem and bifurcation theory, we demonstrate that the system experiences a flip bifurcation and a Neimark-Sacker bifurcation at a fixed point when one of the parameters goes through its critical value. To illustrate our results, numerical simulations, which include maximum Lyapunov exponents, fractal dimensions and phase portraits, are also presented.

References

[1]	R. Arditi, L. R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theoret. Biol., 139 (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5 doi: 10.1016/S0022-5193(89)80211-5
[2]	L. B. Slobodkin, The role of minimalism in art and science, Am. Nat., 127 (1986), 257–265. https://doi.org/10.1086/284484 doi: 10.1086/284484
[3]	M. J. Coe, D. H. Cumming, J. Phillipson, Biomass and production of large African herbivores in relation to rainfall and primary production, Oecologia, 22 (1976), 341–354. https://doi.org/10.1007/BF00345312 doi: 10.1007/BF00345312
[4]	H. Liu, H. Cheng, Dynamic analysis of a prey-predator model with state-dependent control strategy and square root response function, Adv. Differ. Equations, 1 (2018). https://doi.org/10.1186/s13662-022-03729-1 doi: 10.1186/s13662-022-03729-1
[5]	F. Bian, W. Zhao, Y. Song, R. Yue, Dynamical analysis of a class of prey-predator model with Beddington-Deangelis functional response, stochastic perturbation, and impulsive toxicant input, Complexity, 3 (2017), 1–18. https://doi.org/10.1155/2017/3742197 doi: 10.1155/2017/3742197
[6]	P. A. Abrams, L. R. Ginzburg, The nature of predation: prey dependent, ratio dependent or neither?, Trends Ecol. Evol., 15 (2000), 337–341. https://doi.org/10.1016/S0169-5347(00)01908-X doi: 10.1016/S0169-5347(00)01908-X
[7]	H. R. Akcakaya, R. Arditi, L. R. Ginzburg, Ratio-dependent predation: an abstraction that works, Ecology, 76 (1995), 995–1004. https://doi.org/10.2307/1939362 doi: 10.2307/1939362
[8]	L. Wang, G. Feng, Stability and Hopf bifurcation for a ratio-dependent predator-prey system with stage structure and time delay, Adv. Differ. Equations, 255 (2015). https://doi.org/10.1186/s13662-015-0548-x. doi: 10.1186/s13662-015-0548-x
[9]	Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389–406. https://doi.org/10.1007/s002850050105 doi: 10.1007/s002850050105
[10]	R. Shi, L. Chen, The study of a ratio-dependent predator-prey model with stage structure in the prey, Nonlinear Dyn., 58 (2009), 443–451. https://doi.org/10.1007/s11071-009-9491-2 doi: 10.1007/s11071-009-9491-2
[11]	R. Xu, Z. Ma, Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure, Chaos, Solitons Fractals, 38 (2008), 669–684. https://doi.org/10.1016/j.chaos.2007.01.019 doi: 10.1016/j.chaos.2007.01.019
[12]	R. Xu, Q. Gan, Z. Ma, Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay, J. Comput. Appl. Math., 230 (2009), 187–203. https://doi.org/10.1016/j.cam.2008.11.009 doi: 10.1016/j.cam.2008.11.009
[13]	W. Li, J. Ji, L. Huang, Z. Guo, Global dynamics of a controlled discontinuous diffusive SIR epidemic system, Appl. Math. Lett., 121 (2021). https://doi.org/10.1016/j.aml.2021.107420 doi: 10.1016/j.aml.2021.107420
[14]	W. Li, J. Ji, L. Huang, Dynamic of a controlled discountinous computer worm system, P. Am. Math. Soc., 148 (2020), 4389–4403. https://doi.org/10.1090/proc/15095 doi: 10.1090/proc/15095
[15]	Q. Din, Complexity and chaos control in a discrete-time prey-predator model, Commun. Nonlinear Sci., 49 (2017), 113–134. https://doi.org/10.1016/j.cnsns.2017.01.025 doi: 10.1016/j.cnsns.2017.01.025
[16]	J. Huang, S. Liu, S. Ruan, D. Xiao, Bifurcations in a discrete predator-prey model with nonmonotonic functional response, J. Math. Anal. Appl., 464 (2018), 201–230. https://doi.org/10.1016/j.jmaa.2018.03.074 doi: 10.1016/j.jmaa.2018.03.074
[17]	A. Singh, P. Deolia, Dynamical analysis and chaos control in discrete-time prey-predator model, Commun. Nonlinear Sci., 90 (2020). https://doi.org/10.1016/j.cnsns.2020.105313. doi: 10.1016/j.cnsns.2020.105313
[18]	H. Singh, J. Dhar, H. Bhatti, Discrete-time bifurcation behavior of a prey-predator system with generalized predator, Adv. Differ. Equations, 206 (2015), 1–15. https://doi.org/10.1186/s13662-015-0546-z doi: 10.1186/s13662-015-0546-z
[19]	X. Jiang, C. Chen, X. Zhang, M. Chi, H. Yan, Bifurcation and chaos analysis for a discrete ecological developmental system, Nonlinear Dyn., 104 (2021), 4671–4680. https://doi.org/10.1007/s11071-021-06474-4 doi: 10.1007/s11071-021-06474-4
[20]	X. Jiang, X. Chen, Bifurcation and control for a predator-prey system with two delays, IEEE T. Circuits-Ⅱ., 68 (2021), 376–380. https://doi.org/10.1109/TCSII.2020.2987392 doi: 10.1109/TCSII.2020.2987392
[21]	W. Li, X. Li, Neimark-Sacker bifurcation of a semi-discrete hematopoiesis model, J. Appl. Anal. Comput., 8 (2018), 1679–1693. https://doi.org/10.11948/2018.1679 doi: 10.11948/2018.1679
[22]	C. Wang, X. Li, Stability and Neimark-Sacker bifurcation of a semi-discrete population model, J. Appl. Anal. Comput., 4 (2014), 419–435. https://doi.org/10.11948/2014024 doi: 10.11948/2014024
[23]	Y. Kuzenetsov, Elements of Apllied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2004. https://doi.org/10.1007/978-1-4757-3978-7nosfx=y
[24]	C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2$^{nd}$ edition, Boca Raton, New York, 1999. https://doi.org/10.1112/S0024609397343616
[25]	S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 2003. https://doi.org/10.1007/b97481
[26]	J. Carr, Application of Center Manifold Theory, Springer-Verlag, New York, 1981. https://doi.org/10.1007/978-1-4612-5929-9
[27]	J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-1140-2

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)