Bifurcation analysis in a discrete predator–prey model with herd behaviour and group defense

Jie Xia; Xianyi Li; Jie Xia; Xianyi Li

doi:10.3934/era.2023229

Electronic Research Archive

2023, Volume 31, Issue 8: 4484-4506. doi: 10.3934/era.2023229

Previous Article Next Article

Research article Special Issues

Bifurcation analysis in a discrete predator–prey model with herd behaviour and group defense

Jie Xia ,
Xianyi Li ^,

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

Received: 28 March 2023 Revised: 31 May 2023 Accepted: 01 June 2023 Published: 15 June 2023

In this paper, we utilize the semi-discretization method to construct a discrete model from a continuous predator-prey model with herd behaviour and group defense. Specifically, some new results for the transcritical bifurcation, the period-doubling bifurcation, and the Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). Our results not only formulate simpler forms for the existence conditions of these bifurcations, but also clearly present the conditions for the direction and stability of the bifurcated closed orbits. Numerical simulations are also given to illustrate the existence of the derived Neimark-Sacker bifurcation.
Citation: Jie Xia, Xianyi Li. Bifurcation analysis in a discrete predator–prey model with herd behaviour and group defense[J]. Electronic Research Archive, 2023, 31(8): 4484-4506. doi: 10.3934/era.2023229

Related Papers:

Abstract

In this paper, we utilize the semi-discretization method to construct a discrete model from a continuous predator-prey model with herd behaviour and group defense. Specifically, some new results for the transcritical bifurcation, the period-doubling bifurcation, and the Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). Our results not only formulate simpler forms for the existence conditions of these bifurcations, but also clearly present the conditions for the direction and stability of the bifurcated closed orbits. Numerical simulations are also given to illustrate the existence of the derived Neimark-Sacker bifurcation.

References

[1]	R. Arditi, L. R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. 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, 2018 (2018), 63. https://doi.org/10.1186/s13662-018-1507-0 doi: 10.1186/s13662-018-1507-0
[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, 2017 (2017), 3742197. https://doi.org/10.1155/2017/3742197 doi: 10.1155/2017/3742197
[6]	Y. Lv, Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering two different prey behaviours' transition, Nonlinear Dyn., 107 (2022), 1357–1381. https://doi.org/10.1007/s11071-021-07058-y doi: 10.1007/s11071-021-07058-y
[7]	R. A. De Assis, R. Pazim, M. C. Malavazi, P. P. da C. Petry, L. M. E. de Assis, E. Venturino, A mathematical model to describe the herd behaviour considering group defense, Appl. Math. Nonlinear Sci., 5 (2020), 11–24. https://doi.org/10.2478/amns.2020.1.00002 doi: 10.2478/amns.2020.1.00002
[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, 2015 (2015), 255. 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]	Q. Din, Complexity and chaos control in a discrete-time prey-predator model, Commun. Nonlinear Sci. Numer. Simul., 49 (2017), 113–134. https://doi.org/10.1016/j.cnsns.2017.01.025 doi: 10.1016/j.cnsns.2017.01.025
[14]	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
[15]	A. Singh, P. Deolia, Dynamical analysis and chaos control in discrete-time prey-predator model, Commun. Nonlinear Sci. Numer. Simul., 90 (2020), 105313. https://doi.org/10.1016/j.cnsns.2020.105313 doi: 10.1016/j.cnsns.2020.105313
[16]	H. Singh, J. Dhar, H. S. Bhatti, Discrete-time bifurcation behavior of a prey-predator system with generalized predator, Adv. Differ. Equations, 2015 (2015), 206. https://doi.org/10.1186/s13662-015-0546-z doi: 10.1186/s13662-015-0546-z
[17]	Z. Ba, X. Li, Period-doubling bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with Allee effect and cannibalism, Electron. Res. Arch., 31 (2023), 1405–1438. https://doi.org/10.3934/era.2023072 doi: 10.3934/era.2023072
[18]	W. Yao, X. Li, Bifurcation difference induced by different discrete methods in a discrete predator-prey model, J. Nonlinear Model. Anal., 4 (2022), 64–79. https://doi.org/10.12150/jnma.2022.64 doi: 10.12150/jnma.2022.64
[19]	J. Dong, X. Li, Bifurcation of a discrete predator-prey model with increasing functional response and constant-yield prey harvesting, Electron. Res. Arch., 30 (2022), 3930–3948. https://doi.org/10.3934/era.2022200 doi: 10.3934/era.2022200
[20]	X. Li, X. Shao, Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response, Electron. Res. Arch., 31 (2023), 37–57. https://doi.org/10.3934/era.2023003 doi: 10.3934/era.2023003
[21]	Z. Pan, X. Li, Stability and Neimark–Sacker bifurcation for a discrete Nicholson's blowflies model with proportional delay, J. Differ. Equations Appl., 27 (2021), 250–260. https://doi.org/10.1080/10236198.2021.1887159 doi: 10.1080/10236198.2021.1887159
[22]	Y. A. Kuzenetsov, Elements of Apllied Bifurcation Theory, 2$^{nd}$ edition, Springer-Verlag, New York, 1998. https://doi.org/10.1007/b98848
[23]	C. Robinson, Dynamical Systems: Stability, Symbolic and Chaos, 2$^{nd}$ edition, Boca Raton, London, New York, 1999.
[24]	S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003. https://doi.org/10.1007/b97481
[25]	J. Carr, Application of Center Manifold Theory, Springer-Verlag, NewYork, 1982. https://doi.org/10.1007/978-1-4612-5929-9
[26]	J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, NewYork, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[27]	V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal. Real World Appl., 12 (2011), 2319–2338. https://doi.org/10.1016/j.nonrwa.2011.02.002 doi: 10.1016/j.nonrwa.2011.02.002
[28]	A. Buscarino, L. Fortuna, M. Frasca, G. Sciuto, A Concise Guide to Chaotic Electronic Circuits, Springer International Publishing, 2014. https://doi.org/10.1007/978-3-319-05900-6

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)