Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects

Mengyun Xing; Mengxin He; Zhong Li; Mengyun Xing; Mengxin He; Zhong Li

doi:10.3934/mbe.2024034

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 1: 792-831. doi: 10.3934/mbe.2024034

Previous Article Next Article

Research article

Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects

1.
School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China
2.
College of Mathematics and Data Science, Minjiang University, Fuzhou 350108, China

Academic Editor: Xiangsheng Wang

Received: 14 October 2023 Revised: 04 December 2023 Accepted: 11 December 2023 Published: 20 December 2023

In this paper, we investigate the dynamic behavior of a modified Leslie-Gower predator-prey model with the Allee effect on both prey and predator. It is shown that the model has at most two positive equilibria, where one is always a hyperbolic saddle and the other is a weak focus with multiplicity of at least three by concrete example. In addition, we analyze the bifurcations of the system, including saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. The results show that the model has a cusp of codimension three and undergoes a Bogdanov-Takens bifurcation of codimension two. The system undergoes a degenerate Hopf bifurcation and has two limit cycles (the inner one is stable and the outer one is unstable). These enrich the dynamics of the modified Leslie-Gower predator-prey model with the double Allee effects.
- Leslie-Gower,
- Allee effect,
- limit cycle,
- Hopf bifurcation,
- Bogdanov-Takens bifurcation
Citation: Mengyun Xing, Mengxin He, Zhong Li. Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 792-831. doi: 10.3934/mbe.2024034

Related Papers:

Abstract

In this paper, we investigate the dynamic behavior of a modified Leslie-Gower predator-prey model with the Allee effect on both prey and predator. It is shown that the model has at most two positive equilibria, where one is always a hyperbolic saddle and the other is a weak focus with multiplicity of at least three by concrete example. In addition, we analyze the bifurcations of the system, including saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. The results show that the model has a cusp of codimension three and undergoes a Bogdanov-Takens bifurcation of codimension two. The system undergoes a degenerate Hopf bifurcation and has two limit cycles (the inner one is stable and the outer one is unstable). These enrich the dynamics of the modified Leslie-Gower predator-prey model with the double Allee effects.

References

[1]	P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245. https://doi.org/10.2307/2332342 doi: 10.2307/2332342
[2]	P. H. Leslie, J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219–234. https://doi.org/10.2307/2333294 doi: 10.2307/2333294
[3]	A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697–699. https://doi.org/10.1016/s0893-9659(01)80029-x doi: 10.1016/s0893-9659(01)80029-x
[4]	S. B. Hsu, T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763–783. https://doi.org/10.1137/s0036139993253201 doi: 10.1137/s0036139993253201
[5]	T. Lindstrom, Qualitative analysis of a predator-prey system with limit cycles, J. Math. Biol., 31 (1993), 541–561. https://doi.org/10.1007/bf00161198 doi: 10.1007/bf00161198
[6]	I. Hanski, L. Hansson, H. Henttonen, Specialist predators, generalist predators and the microtine rodent cycle, J. Anim. Ecol., 60 (1991), 353–367. https://doi.org/10.2307/5465 doi: 10.2307/5465
[7]	M. A. Aziz-Alaoui, M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069–1075. https://doi.org/10.1016/s0893-9659(03)90096-6 doi: 10.1016/s0893-9659(03)90096-6
[8]	A. F. Nindjin, M. A. Aziz-Alaoui, M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104–1118. https://doi.org/10.1016/j.nonrwa.2005.10.003 doi: 10.1016/j.nonrwa.2005.10.003
[9]	C. Xiang, J. Huang, H. Wang, Linking bifurcation analysis of Holling-Tanner model with generalist predator to a changing environment, Stud. Appl. Math., 149 (2022), 124–163. https://doi.org/10.1111/sapm.12492 doi: 10.1111/sapm.12492
[10]	Y. L. Zhu, K. Wang, Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes, J. Math. Anal. Appl., 384 (2011), 400–408. https://doi.org/10.1016/j.jmaa.2011.05.081 doi: 10.1016/j.jmaa.2011.05.081
[11]	M. K. Singh, B. K. Singh, Poonam, C. Cattani, Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model, Math. Biosci. Eng., 20 (2023), 9625–9644. https://doi.org/10.3934/mbe.2023422 doi: 10.3934/mbe.2023422
[12]	J. Gine, C. Valls, Nonlinear oscillations in the modified Leslie-Gower model, Nonlinear Anal. Real World Appl., 51 (2020), 103010. https://doi.org/10.1016/j.nonrwa.2019.103010 doi: 10.1016/j.nonrwa.2019.103010
[13]	M. M. Chen, Y Takeuchi, J. F. Zhang, Dynamic complexity of a modified Leslie-Gower predator-prey system with fear effect, Commun. Nonlinear Sci. Numer. Simulat., 119 (2023), 107109. https://doi.org/10.1016/j.cnsns.2023.107109 doi: 10.1016/j.cnsns.2023.107109
[14]	W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, 1931. https://doi.org/10.5962/bhl.title.7313
[15]	C. Arancibia-Ibarra, The basins of attraction in a modified May-Holling-Tanner predator-prey model with Allee affect, Nonlinear Anal., 185 (2018), 15–28. https://doi.org/10.1016/j.na.2019.03.004 doi: 10.1016/j.na.2019.03.004
[16]	C. Arancibia-Ibarra, J. Flores, Dynamics of a Leslie-Gower predator-prey model with Holling type II functional response, Allee effect and a generalist predator, Math. Comput. Simulation, 188 (2021), 1–22. https://doi.org/10.1016/j.matcom.2021.03.035 doi: 10.1016/j.matcom.2021.03.035
[17]	W. Q. Yin, Z. Li, F. D. Chen, M. X. He, Modeling Allee effect in the Leslie-Gower predator-prey system incorporating a prey pefuge, Int. J Bifurcat. Chaos, 6 (2022), 2250086. https://doi.org/10.1142/s0218127422500869 doi: 10.1142/s0218127422500869
[18]	Y. Z. Liu, Z. Li, M. X. He, Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect, Math. Biosci. Eng., 20 (2023), 8632–8665. https://doi.org/10.3934/mbe.2023379 doi: 10.3934/mbe.2023379
[19]	N. Martinez-Jeraldo, P. Aguirre, Allee effect acting on the prey species in a Leslie-Gower predation model, Nonlinear Anal. Real World Appl., 45 (2019), 895–917. https://doi.org/10.1016/j.nonrwa.2018.08.009 doi: 10.1016/j.nonrwa.2018.08.009
[20]	P. J. Pal, P. K. Mandal, Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect, Math. Comput. Simulation, 97 (2014), 123–146. https://doi.org/10.1016/j.matcom.2013.08.007 doi: 10.1016/j.matcom.2013.08.007
[21]	Y. J. Li, M. X. He, Z. Li, Dynamics of a ratio-dependent Leslie-Gower predator-prey model with Allee effect and fear effect, Math. Comput. Simulation, 201 (2022), 417–439. https://doi.org/10.1016/j.matcom.2022.05.017 doi: 10.1016/j.matcom.2022.05.017
[22]	Z. C. Shang, Y. H. Qiao, Multiple bifurcations in a predator-prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting, Math. Comput. Simulation, 205 (2023), 745–764. https://doi.org/10.1016/j.matcom.2022.10.028 doi: 10.1016/j.matcom.2022.10.028
[23]	P. Feng, Y. Kang, Dynamics of a modified Leslie-Gower model with double Allee effects, Nonlinear Dyn., 80 (2015), 1051–1062. https://doi.org/10.1007/s11071-015-1927-2 doi: 10.1007/s11071-015-1927-2
[24]	M. T. Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22. https://doi.org/10.1016/j.jtbi.2017.02.002 doi: 10.1016/j.jtbi.2017.02.002
[25]	S. Zhou, Y. Liu, G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23–31. https://doi.org/10.1016/j.tpb.2004.06.007 doi: 10.1016/j.tpb.2004.06.007
[26]	Z. Zhang, T. Ding, W. Huang, Z. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, New Jersey, 1992. https://doi.org/10.1090/mmono/101
[27]	J. C. Huang, Y. J. Gong, J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Int. J Bifurcat. Chaos, 23 (2013), 1350164. https://doi.org/10.1142/s0218127413501642 doi: 10.1142/s0218127413501642
[28]	L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. https://doi.org/10.1007/978-1-4684-0392-3
[29]	M. Lu, J. C. Huang, S. G. Ruan, P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differ. Equation, 267 (2019), 1859–1898. https://doi.org/10.1016/j.jde.2019.03.005 doi: 10.1016/j.jde.2019.03.005
[30]	Y. F. Dai, Y. L. Zhao, B. Sang, Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response, Nonlinear Anal. Real World Appl., 50 (2019), 218–239. https://doi.org/10.1016/j.nonrwa.2019.04.003 doi: 10.1016/j.nonrwa.2019.04.003

Reader Comments

Your name:*

Email:*
© 2024 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)