Analysis of a stochastic food chain model with nonlinear prey refuge and Allee effect driven by Ornstein-Uhlenbeck process

Shujie Yang; Binghui Zhao; Xiaohui Ai; Shujie Yang; Binghui Zhao; Xiaohui Ai

doi:10.3934/math.2025782

AIMS Mathematics

2025, Volume 10, Issue 8: 17494-17517. doi: 10.3934/math.2025782

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Research article

Analysis of a stochastic food chain model with nonlinear prey refuge and Allee effect driven by Ornstein-Uhlenbeck process

School of Science, Northeast Forestry University, China

Received: 18 April 2025 Revised: 10 July 2025 Accepted: 29 July 2025 Published: 01 August 2025
MSC : 60H10, 60H30, 92D25, 92D40

In this paper, we investigated a food chain model driven by the Ornstein-Uhlenbeck process, incorporating the Holling type Ⅱ functional response, nonlinear prey refuge, and the Allee effect in the top predator. First, the biological significance of the Ornstein-Uhlenbeck process was illustrated, and its rationality was explained. Subsequently, the existence and uniqueness of the global solution of the model were established, and its ultimate boundedness was analyzed. Then, by constructing a Lyapunov function and applying Itô's formula, the existence of the stationary distribution of the model was demonstrated. Furthermore, the conditions for the system extinction were provided. Finally, numerical simulations were conducted to verify the theoretical results and confirm the validity of the conclusions.
- Ornstein-Uhlenbeck process,
- nonlinear prey refuge,
- Allee effect,
- stationary distribution,
- extinction
Citation: Shujie Yang, Binghui Zhao, Xiaohui Ai. Analysis of a stochastic food chain model with nonlinear prey refuge and Allee effect driven by Ornstein-Uhlenbeck process[J]. AIMS Mathematics, 2025, 10(8): 17494-17517. doi: 10.3934/math.2025782

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Abstract

In this paper, we investigated a food chain model driven by the Ornstein-Uhlenbeck process, incorporating the Holling type Ⅱ functional response, nonlinear prey refuge, and the Allee effect in the top predator. First, the biological significance of the Ornstein-Uhlenbeck process was illustrated, and its rationality was explained. Subsequently, the existence and uniqueness of the global solution of the model were established, and its ultimate boundedness was analyzed. Then, by constructing a Lyapunov function and applying Itô's formula, the existence of the stationary distribution of the model was demonstrated. Furthermore, the conditions for the system extinction were provided. Finally, numerical simulations were conducted to verify the theoretical results and confirm the validity of the conclusions.

References

[1]	A. J. Lotka, Elements of physical biology, Nature, 116 (1925), 461. https://doi.org/10.1038/116461b0 doi: 10.1038/116461b0
[2]	V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558–560. https://doi.org/10.1038/118558a0 doi: 10.1038/118558a0
[3]	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/10.1093/biomet/47.3-4.219 doi: 10.1093/biomet/47.3-4.219
[4]	C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293–320. https://doi.org/10.4039/ent91293-5 doi: 10.4039/ent91293-5
[5]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398. https://doi.org/10.4039/ent91385-7 doi: 10.4039/ent91385-7
[6]	G. T. Skalski, J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the holling type Ⅱ model, Ecology, 82 (2001), 3083–3092. https://doi.org/10.2307/2679836 doi: 10.2307/2679836
[7]	F. Courchamp, L. Berec, J. Gascoigne, Allee effects in ecology and conservation, Environ. Conserv., 36 (2009), 80–85. http://dx.doi.org/10.1017/S0376892909005384 doi: 10.1017/S0376892909005384
[8]	W. Allee, Animal aggregations, Nature, 128 (1931), 940–941. https://doi.org/10.1038/128940b0 doi: 10.1038/128940b0
[9]	F. Courchamp, T. Clutton-Brock, B. Grenfell, Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405–410. https://doi.org/10.1016/S0169-5347(99)01683-3 doi: 10.1016/S0169-5347(99)01683-3
[10]	P. A. Stephens, W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14 (1999), 401–405. https://doi.org/10.1016/S0169-5347(99)01684-5 doi: 10.1016/S0169-5347(99)01684-5
[11]	C. M. Taylor, A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895–908. https://doi.org/10.1111/j.1461-0248.2005.00787.x doi: 10.1111/j.1461-0248.2005.00787.x
[12]	S. A. A. Ahmed, W. A. I. Elmorsi, Hopf bifurcation and stability analysis for a delayed prey-predator model subject to a strong Allee effect in the prey species, Partial Differ. Equ. Appl. Math., 14 (2025), 101199. https://doi.org/10.1016/j.padiff.2025.101199 doi: 10.1016/j.padiff.2025.101199
[13]	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. Simulat., 188 (2021), 1–22. https://doi.org/10.1016/j.matcom.2021.03.035 doi: 10.1016/j.matcom.2021.03.035
[14]	C. Zhang, J. Lu, M. Liu, H. Zhang, Stable patterns with jump-discontinuity for a phytoplankton–zooplankton system with both Allee and fear effect, Physica D, 472 (2025), 134481. https://doi.org/10.1016/j.physd.2024.134481 doi: 10.1016/j.physd.2024.134481
[15]	S. Vinoth, R. Sivasamy, K. Sathiyanathan, B. Unyong, R. Vadivel, N. Gunasekaran, A novel discrete-time Leslie–Gower model with the impact of Allee effect in predator population, Complexity, 2022, 6931354. https://doi.org/10.1155/2022/6931354 doi: 10.1155/2022/6931354
[16]	S. K. Sasmal, J. Chattopadhyay, An eco-epidemiological system with infected prey and predator subject to the weak Allee effect, Math. Biosci., 246 (2013), 260–271. https://doi.org/10.1016/j.mbs.2013.10.005 doi: 10.1016/j.mbs.2013.10.005
[17]	S. L. Lima, L. M. Dill, Behavioral decisions made under the risk of predation: A review and prospectus, Can. J. Zool., 68 (1990), 619–640. https://doi.org/10.1139/z90-092 doi: 10.1139/z90-092
[18]	R. Tollrian, S. Duggen, L. C. Weiss, C. Laforsch, M. Kopp, Density-dependent adjustment of inducible defenses, Sci. Rep., 5 (2015), 12736. https://doi.org/10.1038/srep12736 doi: 10.1038/srep12736
[19]	H. Molla, S. Sarwardi, M. Sajid, Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge, J. Math. Comput. Sci., 25 (2021), 150–165. http://dx.doi.org/10.22436/jmcs.025.02.04 doi: 10.22436/jmcs.025.02.04
[20]	H. Molla, M. S. Rahman, S. Sarwardi, Dynamical study of a prey–predator model incorporating nonlinear prey refuge and additive Allee effect acting on prey species, Model. Earth Syst. Environ., 7 (2021), 749–765. https://doi.org/10.1007/s40808-020-01049-5 doi: 10.1007/s40808-020-01049-5
[21]	M. M. Haque, S. Sarwardi, Dynamics of a harvested prey–predator model with prey refuge dependent on both species, Int. J. Bifurcat. Chaos, 28 (2018), 1830040. https://doi.org/10.1142/s0218127418300409 doi: 10.1142/s0218127418300409
[22]	E. González-Olivares, B. González-Yañez, R. Becerra-Klix, R. Ramos-Jiliberto, Multiple stable states in a model based on predator-induced defenses, Ecol. Complex., 32 (2017), 111–120. https://doi.org/10.1016/j.ecocom.2017.10.004 doi: 10.1016/j.ecocom.2017.10.004
[23]	H. Wu, Dynamical analysis of two types of food chain ecological models, Master's Thesis, Anqing Normal University, 2023.
[24]	D. Gravel, F. Massol, M. A. Leibold, Stability and complexity in model meta-ecosystems, Nat. Commun., 7 (2016), 12457. https://doi.org/10.1038/ncomms12457 doi: 10.1038/ncomms12457
[25]	B. Yang, Y. Cai, K. Wang, W. Wang, Optimal harvesting policy of logistic population model in a randomly fluctuating environment, Physica A, 526 (2019), 120817. https://doi.org/10.1016/j.physa.2019.04.053 doi: 10.1016/j.physa.2019.04.053
[26]	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
[27]	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
[28]	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
[29]	B. Tian, L. Yang, X. Chen, Y. Zhang, A generalized stochastic competitive system with Ornstein–Uhlenbeck process, Int. J. Biomath., 14 (2021), 2150001. https://doi.org/10.1142/s1793524521500017 doi: 10.1142/s1793524521500017
[30]	D. Zhou, M. Liu, Z. Liu, Persistence and extinction of a stochastic predator–prey model with modified Leslie–Gower and Holling-type Ⅱ schemes, Adv. Differ. Equ., 2020 (2020), 179. https://doi.org/10.1186/s13662-020-02642-9 doi: 10.1186/s13662-020-02642-9
[31]	Y. Gao, S. Yao, Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ predator-prey stochastic model in polluted environments with impulsive toxicant input, Math. Biosci. Eng., 18 (2021), 4894–4918. https://doi.org/10.3934/mbe.2021249 doi: 10.3934/mbe.2021249
[32]	X. Mao, Stochastic differential equations and applications, 2 Eds., Oxford: Woodhead Publishing, 2008.
[33]	Y. Zeng, P. Yu, Multistable states in a predator–prey model with generalized Holling type Ⅲ functional response and a strong Allee effect, Commun. Nonlinear Sci. Numer. Simul., 131 (2024), 107846. https://doi.org/10.1016/j.cnsns.2024.107846 doi: 10.1016/j.cnsns.2024.107846
[34]	S. Mandal, S. Samanta, P. K. Tiwari, R. K. Upadhyay, Bifurcation analysis and exploration of noise-induced transitions of a food chain model with Allee effect, Math. Comput. Simulat., 228 (2025), 313–338. https://doi.org/10.1016/j.matcom.2024.09.015 doi: 10.1016/j.matcom.2024.09.015
[35]	A. A. Thirthar, P. Panja, S. J. Majeed, K. S. Nisar, Dynamic interactions in a two-species model of the mammalian predator–prey system: The influence of Allee effects, prey refuge, water resources, and moonlights, Partial Differ. Equ. Appl. Math., 11 (2024), 100865. https://doi.org/10.1016/j.padiff.2024.100865 doi: 10.1016/j.padiff.2024.100865
[36]	X. Wang, X. Wang, W. Yang, Long time behavior of a rumor model with Ornstein-Uhlenbeck process, Q. Appl. Math., 2024. https://doi.org/10.1090/qam/1701
[37]	Y. Tian, J. Zhu, J. Zheng, K. Sun, Modeling and analysis of a prey-predator system with prey habitat selection in an environment subject to stochastic disturbances, Electron. Res. Arch., 33 (2025), 744–767. https://doi.org/10.3934/era.2025034 doi: 10.3934/era.2025034
[38]	Y. Wu, G. Lv, Stationary distribution and probability density for a stochastic crime model, Math. Methods Appl. Sci., 48 (2025), 9440–9455. https://doi.org/10.1002/mma.10809 doi: 10.1002/mma.10809
[39]	X. Wang, X. Wang, W. Yang, A rumor propagation model integrated with psychological factors, Int. J. Biomath., 2024, 2450149. https://doi.org/10.1142/S1793524524501493 doi: 10.1142/S1793524524501493
[40]	M. Qurban, A. Khaliq, K. S. Nisar, N. A. Shah, Dynamics and control of a plant-herbivore model incorporating Allee's effect, Heliyon, 10 (2024), e30754. https://doi.org/10.1016/j.heliyon.2024.e30754 doi: 10.1016/j.heliyon.2024.e30754
[41]	R. Khasminskii, Stochastic stability of differential equations, Berlin, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-23280-0
[42]	R. S. Lipster, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217–228. https://doi.org/10.1080/17442508008833146 doi: 10.1080/17442508008833146
[43]	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
[44]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
[45]	S. E. Jørgensen, Handbook of environmental data and ecological parameters: Environmental sciences and applications, Pergamon, 1979. https://doi.org/10.1016/C2013-0-05856-2

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