Analysis of a stochastic Leslie-Gower three-species food chain system with Holling-II functional response and Ornstein-Uhlenbeck process

Ruyue Hu; Chi Han; Yifan Wu; Xiaohui Ai; Ruyue Hu; Chi Han; Yifan Wu; Xiaohui Ai

doi:10.3934/math.2024920

AIMS Mathematics

2024, Volume 9, Issue 7: 18910-18928. doi: 10.3934/math.2024920

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Research article

Analysis of a stochastic Leslie-Gower three-species food chain system with Holling-II functional response and Ornstein-Uhlenbeck process

School of Science, Northeast Forestry University, China

Received: 28 March 2024 Revised: 12 May 2024 Accepted: 23 May 2024 Published: 05 June 2024
MSC : 60H10, 60H30, 92D25, 92D40

This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: The existence and uniqueness of the global solution of the model are given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito's formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion.
- Leslie-Gower model,
- Holling-II functional response,
- Ornstein-Uhlenbeck process,
- stationary distribution,
- extinction
Citation: Ruyue Hu, Chi Han, Yifan Wu, Xiaohui Ai. Analysis of a stochastic Leslie-Gower three-species food chain system with Holling-II functional response and Ornstein-Uhlenbeck process[J]. AIMS Mathematics, 2024, 9(7): 18910-18928. doi: 10.3934/math.2024920

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Abstract

This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: The existence and uniqueness of the global solution of the model are given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito's formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion.

References

[1]	P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245. https://doi.org/10.1093/biomet/35.3-4.213 doi: 10.1093/biomet/35.3-4.213
[2]	P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16–31. https://doi.org/10.1093/biomet/45.1-2.16 doi: 10.1093/biomet/45.1-2.16
[3]	Z. Song, B. Zhen, J. Xu, Species coexistence and chaotic behavior induced by multiple delays in a food chain system, Ecol. Complex., 19 (2014), 9–17. https://doi.org/10.1016/j.ecocom.2014.01.004 doi: 10.1016/j.ecocom.2014.01.004
[4]	M. A. Aziz-Alaoui, Study of a leslie-gower-type tritrophic population model, Chaos Soliton. Fract., 14 (2002), 1275–1293. https://doi.org/10.1016/S0960-0779(02)00079-6 doi: 10.1016/S0960-0779(02)00079-6
[5]	A. F. Nindjin, M. A. Aziz-Alaoui, Persistence and global stability in a delayed leslie-gower type three species food chain, J. Math. Anal. Appl., 340 (2008), 340–357. https://doi.org/10.1016/j.jmaa.2007.07.078 doi: 10.1016/j.jmaa.2007.07.078
[6]	J. Lv, K. Wang, Asymptotic properties of a stochastic predator-prey system with holling ii functional response, Commun. Nonlinear Sci., 16 (2011), 4037–4048. https://doi.org/10.1016/j.cnsns.2011.01.015 doi: 10.1016/j.cnsns.2011.01.015
[7]	Y. Li, H. Gao, Existence, uniqueness and global asymptotic stability of positive solutions of a predator-prey system with holling II functional response with random perturbation, Nonlinear Anal. Theor., 68 (2008), 1694–1705. https://doi.org/10.1016/j.na.2007.01.008 doi: 10.1016/j.na.2007.01.008
[8]	D. Jiang, N. Shi, X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588–597. https://doi.org/10.1016/j.jmaa.2007.08.014 doi: 10.1016/j.jmaa.2007.08.014
[9]	C. Ji, D. Jiang, X. Li, Qualitative analysis of a stochastic ratio-dependent predator-prey system, J. Comput. Appl. Math., 235 (2011), 1326–1341. https://doi.org/10.1016/j.cam.2010.08.021 doi: 10.1016/j.cam.2010.08.021
[10]	C. Ji, D. Jiang, N. Shi, Analysis of a predator-prey model with modiffed leslie-gower and holling-type ii schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482–498. https://doi.org/10.1016/j.jmaa.2009.05.039 doi: 10.1016/j.jmaa.2009.05.039
[11]	X. Huang, Z. Song, Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling, Math. Biosci. Eng., 21 (2024), 765–777. https://doi.org/10.3934/mbe.2024032 doi: 10.3934/mbe.2024032
[12]	X. Mao, G. Marion, E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95–110. https://doi.org/10.1016/S0304-4149(01)00126-0 doi: 10.1016/S0304-4149(01)00126-0
[13]	X. Zhang, R. Yuan, A stochastic chemostat model with mean-reverting ornstein-uhlenbeck process and monod-haldane response function, Appl. Math. Comput., 394 (2021), 125833. https://doi.org/10.1016/j.amc.2020.125833 doi: 10.1016/j.amc.2020.125833
[14]	Y. Cai, J. Jiao, Z. Gui, Y. Liu, W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210–226. https://doi.org/10.1016/j.amc.2018.02.009 doi: 10.1016/j.amc.2018.02.009
[15]	E. Allen, Environmental variability and mean-reverting processes, Discrete Cont. Dyn-B, 21 (2016), 2073–2089. https://doi.org/10.3934/dcdsb.2016037 doi: 10.3934/dcdsb.2016037
[16]	Q. Liu, D. Jiang, Analysis of a Stochastic Within-Host model of dengue infection with immune response and Ornstein-Uhlenbeck process, J. Nonlinear Sci., 34 (2024), 28. https://doi.org/10.1007/s00332-023-10004-4 doi: 10.1007/s00332-023-10004-4
[17]	X. Mao, Stochastic differential equations and applications, Elsevier, 2007.
[18]	R. K. Upadhyay, J. Chattopadhyay, Chaos to order: Role of toxin producing phytoplankton in aquatic systems, Nonlinear Anal. Model., 10 (2005), 383–396. https://doi.org/10.15388/NA.2005.10.4.15117 doi: 10.15388/NA.2005.10.4.15117
[19]	J. E. Truscott, J. Brindley, Ocean plankton populations as excitable media, B. Math. Biol., 56 (1994), 981–998. https://doi.org/10.1007/BF02458277 doi: 10.1007/BF02458277
[20]	R. M. May., Stability and complexity in model ecosystems, Princeton university press, 2019. http://doi.org/10.2307/j.ctvs32rq4
[21]	R. Khasminskii, Stochastic stability of differential equations, Springer Science & Business Media, 2011.
[22]	Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84. https://doi.org/10.1016/j.jmaa.2006.12.032 doi: 10.1016/j.jmaa.2006.12.032
[23]	X. Chen, B. Tian, X. Xu, H. Zhang, D. Li, A stochastic predator-prey system with modified LG-Holling type II functional response, Math. Comput. Simulat., 203 (2023), 449–485. https://doi.org/10.1016/j.matcom.2022.06.016 doi: 10.1016/j.matcom.2022.06.016
[24]	S. E. Jorgensen, Handbook of environmental data and ecological parameters, Oxford: Pergamon Press, 1979.
[25]	V. Rai, R. Sreenivasan, Period-doubling bifurcations leading to chaos in a model food-chain, Ecol. Model., 69 (1993), 63–77. https://doi.org/10.1016/0304-3800(93)90049-X doi: 10.1016/0304-3800(93)90049-X
[26]	R. K. Upadhyay, Chaotic dynamics in a three species aquatic population model with holling type II functional response, Nonlinear Anal. Model., 13 (2008), 103–115. https://doi.org/10.15388/NA.2008.13.1.14592 doi: 10.15388/NA.2008.13.1.14592
[27]	R. K. Upadhyay, R. K. Naji, N. Kumari, Dynamical complexity in some ecological models: Effect of toxin production by phytoplankton, Nonlinear Anal. Model., 12 (2007), 123–138. https://doi.org/10.15388/NA.2007.12.1.14726 doi: 10.15388/NA.2007.12.1.14726
[28]	Y. Zhao, S. Yuan, J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, B. Math. Biol., 77 (2015), 1285–1326. https://doi.org/10.1007/s11538-015-0086-4 doi: 10.1007/s11538-015-0086-4

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