Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

Changtong Li; Dandan Cheng; Xiaozhou Feng; Mengyan Liu; Changtong Li; Dandan Cheng; Xiaozhou Feng; Mengyan Liu

doi:10.3934/math.2024702

AIMS Mathematics

2024, Volume 9, Issue 6: 14454-14472. doi: 10.3934/math.2024702

Previous Article Next Article

Research article Special Issues

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

School of Science, Xi'an Technological University, Xi'an 710032, China

Received: 03 January 2024 Revised: 10 April 2024 Accepted: 17 April 2024 Published: 22 April 2024
MSC : 35J55, 92D25

A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.
- nonlinear impulsive,
- global asymptotic stability,
- bifurcation,
- numerical simulation
Citation: Changtong Li, Dandan Cheng, Xiaozhou Feng, Mengyan Liu. Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response[J]. AIMS Mathematics, 2024, 9(6): 14454-14472. doi: 10.3934/math.2024702

Related Papers:

Abstract

A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.

References

[1]	P. Kalra, M. Kaur, Stability analysis of an eco-epidemiological SIN model with impulsive control strategy for integrated pest management considering stage-structure in predator, Int. J. Math. Model. Numer. Optim., 12 (2022), 43–68. https://doi.org/10.1504/IJMMNO.2022.119779 doi: 10.1504/IJMMNO.2022.119779
[2]	M. He, Z. Li, Stability of a fear effect predator-prey model with mutual interference or group defense, J. Biol. Dyn., 16 (2022), 480–498. https://doi.org/10.1080/17513758.2022.2091800 doi: 10.1080/17513758.2022.2091800
[3]	W. Gao, S. Tang, The effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity, Nonlinear Anal.: Hybrid Syst., 5 (2011), 540–553. https://doi.org/10.1016/j.nahs.2010.12.001 doi: 10.1016/j.nahs.2010.12.001
[4]	J. Liu, Q. Qi, B. Liu, S. Gao, Pest control switching models with instantaneous and non-instantaneous impulsive effects, Math. Comput. Simul., 205 (2023), 926–938. https://doi.org/10.1016/j.matcom.2022.10.027 doi: 10.1016/j.matcom.2022.10.027
[5]	Y. Xiao, F. Van Den, The dynamics of an eco-epidemic model with biological control, Ecol. Model., 168 (2003), 203–214. https://doi.org/10.1016/S0304-3800(03)00197-2 doi: 10.1016/S0304-3800(03)00197-2
[6]	X. Zhang, Y. Wang, Y. Liu, P. Dai, J. Dong, Z. Yu, Approaches biological control of pests of through landscape regulation: theory and practice, J. Ecol. Rural Environ., 31 (2015), 617–624. https://doi.org/10.11934/j.issn.1673-4831.2015.05.001 doi: 10.11934/j.issn.1673-4831.2015.05.001
[7]	B. Liu, Y. Zhang, L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Anal., 6 (2005), 227–243. https://doi.org/10.1016/j.nonrwa.2004.08.001 doi: 10.1016/j.nonrwa.2004.08.001
[8]	J. Wang, H. Cheng, X. Meng, B. G. S. A. Pradeep, Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse, Adv. Differ. Equ., 2017 (2017), 252. https://doi.org/10.1186/s13662-017-1300-5 doi: 10.1186/s13662-017-1300-5
[9]	H. Cheng, F. Wang, T. Zhang, Multi-state dependent impulsive control for Holling Ⅰ predator-prey model, Discrete Dyn. Nat. Soc., 2012 (2012), 181752. https://doi.org/10.1155/2012/181752 doi: 10.1155/2012/181752
[10]	D. Luo, Global bifurcation for a reaction-diffusion predator-prey model with Holling-Ⅱ functional response and prey-taxis, Chaos Soliton. Fract., 147 (2021), 110975. https://doi.org/10.1016/j.chaos.2021.110975 doi: 10.1016/j.chaos.2021.110975
[11]	M. Liu, C. Du, M. Deng, Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ stochastic predator-prey model with impulsive toxicant input in polluted environments, Nonlinear Anal.: Hybrid Syst., 27 (2018), 177–190. https://doi.org/10.1016/j.nahs.2017.08.001 doi: 10.1016/j.nahs.2017.08.001
[12]	L. Niu, X. Wang, Periodic solution for a delayed Leslie-Gower and Holling-Type Ⅲ predator-prey model incorporating prey cannibalism, Adv. Appl. Math., 9 (2020), 1170–1176. https://doi.org/10.12677/AAM.2020.98136 doi: 10.12677/AAM.2020.98136
[13]	Z. He, B. Li, Complex dynamic behavior of a discrete-time predator-prey system of Holling-Ⅲ type, Adv. Differ. Equ., 2014 (2014), 180. https://doi.org/10.1186/1687-1847-2014-180 doi: 10.1186/1687-1847-2014-180
[14]	M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ration-dependent functional responses and diffusion, Phys. D: Nonlinear Phenom., 196 (2004), 172–192. https://doi.org/10.1016/j.physd.2004.05.007 doi: 10.1016/j.physd.2004.05.007
[15]	X. Feng, C. Sun, Y. Wang, C. Li, Dynamics of a predator-prey model with nonlinear growth rate and B-D functional response, Nonlinear Anal., 70 (2023), 103766. https://doi.org/10.1016/j.nonrwa.2022.103766 doi: 10.1016/j.nonrwa.2022.103766
[16]	X. Feng, M. Liu, Y. Jiang, D. Li, Dynamics and stability of a fractional-order Tumor-Immune interaction model with B-D functional response and immunotherapy, Fractal Fract., 7 (2023), 200. https://doi.org/10.3390/fractalfract7020200 doi: 10.3390/fractalfract7020200
[17]	Z. Xiang, X. Song, The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response, Chaos Soliton. Fract., 39 (2009), 2282–2293. https://doi.org/10.1016/j.chaos.2007.06.124 doi: 10.1016/j.chaos.2007.06.124
[18]	Z. Zhao, L. Yang, L. Chen, Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type Ⅱ schemes, J. Appl. Math. Comput., 35 (2011), 119–134. https://doi.org/10.1007/s12190-009-0346-2 doi: 10.1007/s12190-009-0346-2
[19]	H. Guo, X. Song, An impulsive predator-prey system with modified Leslie-Gower and Holling type Ⅱ schemes, Chaos Soliton. Fract., 36 (2008), 1320–1331. https://doi.org/10.1016/j.chaos.2006.08.010 doi: 10.1016/j.chaos.2006.08.010
[20]	C. Li, X. Feng, Y. Wang, X. Wang, Complex dynamics of Beddington-DeAngelis-Type predator-prey model with nonlinear impulsive control, Complexity, 2020 (2020), 8829235. https://doi.org/10.1155/2020/8829235 doi: 10.1155/2020/8829235
[21]	M. A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ type 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
[22]	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., 7 (2006), 1104–1108. https://doi.org/10.1016/j.nonrwa.2005.10.003 doi: 10.1016/j.nonrwa.2005.10.003
[23]	I. U. Khan, S. Tang, The impulsive model with pest density and its change rate dependent feedback control, Discrete Dyn. Nat. Soc., 2020 (2020), 4561241. https://doi.org/10.1155/2020/4561241 doi: 10.1155/2020/4561241
[24]	C. Li, S. Tang, Analyzing a generalized pest-natural enemy model with nonlinear impulsive control, Open Math., 16 (2018), 1390–1411. https://doi.org/10.1515/math-2018-0114 doi: 10.1515/math-2018-0114

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)