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Mathematical Biosciences and Engineering

2024, Volume 21, Issue 1: 186-213. doi: 10.3934/mbe.2024009
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Dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment

  • School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

  • Academic Editor: Xiaodi Li
  • Received: 24 June 2023 Revised: 07 September 2023 Accepted: 13 September 2023 Published: 11 December 2023

  • In this paper, a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment is investigated. We obtain the sufficient and necessary conditions for persistence in mean and extinction of each species. The results show that the stochastic dynamics of the system are closely correlated with both time delays and environmental noises. Some numerical examples are introduced to illustrate the main results.

    Citation: Zeyan Yue, Sheng Wang. Dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 186-213. doi: 10.3934/mbe.2024009

    Related Papers:

  • In this paper, a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment is investigated. We obtain the sufficient and necessary conditions for persistence in mean and extinction of each species. The results show that the stochastic dynamics of the system are closely correlated with both time delays and environmental noises. Some numerical examples are introduced to illustrate the main results.


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    [1] M. Yavuz, N. Sene, Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate, Fractal Fract., 4 (2020), 35. https://doi.org/10.3390/fractalfract4030035 doi: 10.3390/fractalfract4030035
    [2] A. Chatterjee, S. Pal, A predator-prey model for the optimal control of fish harvesting through the imposition of a tax, Int. J. Optim. Control Theor. Appl., 13 (2023), 68–80. https://doi.org/10.11121/ijocta.2023.1218 doi: 10.11121/ijocta.2023.1218
    [3] D. Ghosh, P. K. Santra, G. S. Mahapatra, A three-component prey-predator system with interval number, Math. Model. Numer. Simul. Appl., 3 (2023), 1–16. https://doi.org/10.53391/mmnsa.1273908 doi: 10.53391/mmnsa.1273908
    [4] M. Liu, C. Bai, Optimal harvesting policy of a stochastic food chain population model, Appl. Math. Comput., 245 (2014), 265–270. https://doi.org/10.1016/j.amc.2014.07.103 doi: 10.1016/j.amc.2014.07.103
    [5] J. Yu, M. Liu, Stationary distribution and ergodicity of a stochastic food-chain model with Lévy jumps, Phys. A, 482 (2017), 14–28. https://doi.org/10.1016/j.physa.2017.04.067 doi: 10.1016/j.physa.2017.04.067
    [6] T. Zeng, Z. Teng, Z. Li, J. Hu, Stability in the mean of a stochastic three species food chain model with general Lévy jumps, Chaos Solitons Fractals, 106 (2018), 258–265. https://doi.org/10.1016/j.chaos.2017.10.025 doi: 10.1016/j.chaos.2017.10.025
    [7] H. Li, H. Li, F. Cong, Asymptotic behavior of a food chain model with stochastic perturbation, Phys. A, 531 (2019), 121749. https://doi.org/10.1016/j.physa.2019.121749 doi: 10.1016/j.physa.2019.121749
    [8] Q. Yang, X. Zhang, D. Jiang, Dynamical behaviors of a stochastic food chain system with ornstein-uhlenbeck process, J. Nonlinear Sci., 32 (2022), 34. https://doi.org/10.1007/s00332-022-09796-8 doi: 10.1007/s00332-022-09796-8
    [9] P. A. Naik, Z. Eskandari, M. Yavuz, J. Zu, Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect, J. Comput. Appl. Math., 413 (2022), 114401. https://doi.org/10.1016/j.cam.2022.114401 doi: 10.1016/j.cam.2022.114401
    [10] P. A. Naik, Z. Eskandari, H. E. Shahraki, Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model, Math. Model. Numer. Simul. Appl., 1 (2021), 95–101. https://doi.org/10.53391/mmnsa.2021.01.009 doi: 10.53391/mmnsa.2021.01.009
    [11] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993.
    [12] F. Shakeri, M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model., 48 (2008), 486–498. https://doi.org/10.1016/j.mcm.2007.09.016 doi: 10.1016/j.mcm.2007.09.016
    [13] K. Golpalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992.
    [14] F. A. Rihan, H. J. Alsakaji, Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species, Discrete. Contin. Dyn. Syst. Ser. S, 15 (2020), 245–263. https://doi.org/10.3934/dcdss.2020468 doi: 10.3934/dcdss.2020468
    [15] H. J. Alsakaji, S. Kundu, F. A. Rihan, Delay differential model of one-predator two-prey system with Monod-Haldane and holling type Ⅱ functional responses, Appl. Math. Comput., 397 (2021), 125919. https://doi.org/10.1016/j.amc.2020.125919 doi: 10.1016/j.amc.2020.125919
    [16] J. Roy, D. Barman, S. Alam, Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment, Biosystems, 197 (2020), 104176. https://doi.org/10.1016/j.biosystems.2020.104176 doi: 10.1016/j.biosystems.2020.104176
    [17] Q. Liu, D. Jiang, Influence of the fear factor on the dynamics of a stochastic predator-prey model, Appl. Math. Lett., 112 (2021), 106756. https://doi.org/10.1016/j.aml.2020.106756 doi: 10.1016/j.aml.2020.106756
    [18] N. Tuerxun, Z. Teng, Global dynamics in stochastic n-species food chain systems with white noise and Lévy jumps, Math. Methods Appl. Sci., 45 (2022), 5184–5214. https://doi.org/10.1002/mma.8101 doi: 10.1002/mma.8101
    [19] Q. Zhang, D. Jiang, Z. Liu, D. O'Regan, Asymptotic behavior of a three species eco-epidemiological model perturbed by white noise, J. Math. Anal. Appl., 433 (2016), 121–148. https://doi.org/10.1016/j.jmaa.2015.07.025 doi: 10.1016/j.jmaa.2015.07.025
    [20] Y. Zhao, L. You, D. Burkow, S. Yuan, Optimal harvesting strategy of a stochastic inshore-offshore hairtail fishery model driven by Lévy jumps in a polluted environment, Nonlinear Dyn., 95 (2019), 1529–1548. https://doi.org/10.1007/s11071-018-4642-y doi: 10.1007/s11071-018-4642-y
    [21] L. Liu, X. Meng, T. Zhang, Optimal control strategy for an impulsive stochastic competition system with time delays and jumps, Physica A, 477 (2017), 99–113. https://doi.org/10.1016/j.physa.2017.02.046 doi: 10.1016/j.physa.2017.02.046
    [22] J. R. Beddington, R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463–465. https://doi.org/10.1126/science.197.4302.463 doi: 10.1126/science.197.4302.463
    [23] X. Zou, W. Li, K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170–174. https://doi.org/10.1016/j.aml.2012.08.006 doi: 10.1016/j.aml.2012.08.006
    [24] X. Zou, K. Wang, Optimal harvesting for a stochastic regime-switching logistic diffusion system with jumps, Nonlinear Anal. Hybrid Syst., 13 (2014), 32–44. https://doi.org/10.1016/j.nahs.2014.01.001 doi: 10.1016/j.nahs.2014.01.001
    [25] H. Qiu, W. Deng, Optimal harvesting of a stochastic delay logistic model with Lévy jumps, J. Phys. A: Math. Theor., 49 (2016), 405601. https://doi.org/10.1088/1751-8113/49/40/405601 doi: 10.1088/1751-8113/49/40/405601
    [26] G. Denaro, D. Valenti, A. L. Cognata, B. Spagnolo, A. Bonanno, G. Basilone, et al., Spatio-temporal behaviour of the deep chlorophyll maximum in Mediterranean Sea: Development of a stochastic model for picophytoplankton dynamics, Ecol. Complex., 13 (2013), 21–34. https://doi.org/10.4414/pc-d.2013.00242 doi: 10.4414/pc-d.2013.00242
    [27] D. Valenti, A. Fiasconaro, B. Spagnolo, Stochastic resonance and noise delayed extinction in a model of two competing species, Phys. A, 331 (2004), 477–486. https://doi.org/10.1016/j.physa.2003.09.036 doi: 10.1016/j.physa.2003.09.036
    [28] R. Grimaudo, P. Lazzari, C. Solidoro, D. Valenti, Effects of solar irradiance noise on a complex marine trophic web, Sci. Rep., 12 (2022), 12163. https://doi.org/10.1038/s41598-022-12384-1 doi: 10.1038/s41598-022-12384-1
    [29] P. Lazzari, R. Grimaudo, C. Solidoro, D. Valenti, Stochastic 0-dimensional biogeochemical flux model: Effect of temperature fluctuations on the dynamics of the biogeochemical properties in a marine ecosystem, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 105994. https://doi.org/10.1016/j.cnsns.2021.105994 doi: 10.1016/j.cnsns.2021.105994
    [30] D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio, S. Mazzola, et al., Noise induced phenomena in the dynamics of two competing species, Math. Model. Nat. Phenom., 11 (2016), 158–174. https://doi.org/10.1051/mmnp/201611510 doi: 10.1051/mmnp/201611510
    [31] D. Valenti, G. Denaro, B. Spagnolo, S. Mazzola, G. Basilone, F. Conversano, et al., Stochastic models for phytoplankton dynamics in Mediterranean Sea, Ecol. Complex., 27 (2016), 84–103. https://doi.org/10.1016/j.ecocom.2015.06.001 doi: 10.1016/j.ecocom.2015.06.001
    [32] D. Valenti, G. Denaro, F. Conversano, C. Brunet, A. Bonanno, G. Basilone, et al., The role of noise on the steady state distributions of phytoplankton populations, J. Stat. Mech., 2016 (2016), 054044. https://doi.org/10.1088/1742-5468/2016/05/054044 doi: 10.1088/1742-5468/2016/05/054044
    [33] G. Denaro, D. Valenti, B. Spagnolo, A. Bonanno, G. Basilone, S. Mazzola, et al., Dynamics of two picophytoplankton groups in mediterranean sea: Analysis and prediction of the deep chlorophyll maximum by a stochastic reaction-diffusion-taxis model, PLoS One, 8 (2013), e66765. https://doi.org/10.1371/journal.pone.0066765 doi: 10.1371/journal.pone.0066765
    [34] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, 1985.
    [35] B. Spagnolo, D. Valenti, A. Fiasconaro, Noise in ecosystems: A short review, Math. Biosci. Eng., 1 (2004), 185–211. https://doi.org/10.3934/mbe.2004.1.185 doi: 10.3934/mbe.2004.1.185
    [36] 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
    [37] X. Li, A. Gray, D. Jiang, X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11–28. https://doi.org/10.1016/j.jmaa.2010.10.053 doi: 10.1016/j.jmaa.2010.10.053
    [38] Y. Cai, S. Cai, X. Mao, Stochastic delay foraging arena predator-prey system with Markov switching, Stoch. Anal. Appl., 38 (2020), 191–212. https://doi.org/10.1080/07362994.2019.1679645 doi: 10.1080/07362994.2019.1679645
    [39] A. La Cognata, D. Valenti, B. Spagnolo, A. A. Dubkov, Two competing species in super-diffusive dynamical regimes, Eur. Phys. J. B, 77 (2010), 273–279. https://doi.org/10.1140/epjb/e2010-00239-6 doi: 10.1140/epjb/e2010-00239-6
    [40] J. Bertoin, Lévy Processes, Cambridge University Press, 1996.
    [41] A. L. Cognata, D. Valenti, A. A. Dubkov, B. Spagnolo, Dynamics of two competing species in the presence of Lévy noise sources, Phys. Rev. E, 82 (2010), 011121. https://doi.org/10.1103/PhysRevE.82.011121 doi: 10.1103/PhysRevE.82.011121
    [42] J. Bao, X. Mao, G. Yin, C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601–6616. https://doi.org/10.1016/j.na.2011.06.043 doi: 10.1016/j.na.2011.06.043
    [43] J. Bao, C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363–375. https://doi.org/10.1016/j.jmaa.2012.02.043 doi: 10.1016/j.jmaa.2012.02.043
    [44] M. Liu, K. Wang, Dynamics of a Leslie-Gower Holling-type Ⅱ predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204–213. https://doi.org/10.1016/j.na.2013.02.018 doi: 10.1016/j.na.2013.02.018
    [45] M. Liu, K. Wang, Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410 (2014), 750–763. https://doi.org/10.1016/j.jmaa.2013.07.078 doi: 10.1016/j.jmaa.2013.07.078
    [46] M. Liu, M. Deng, B. Du, Analysis of a stochastic logistic model with diffusion, Appl. Math. Comput., 266 (2015), 169–182. https://doi.org/10.1016/j.amc.2015.05.050 doi: 10.1016/j.amc.2015.05.050
    [47] X. Zhang, W. Li, M. Liu, K. Wang, Dynamics of a stochastic Holling Ⅱ one-predator two-prey system with jumps, Phys. A, 421 (2015), 571–582. https://doi.org/10.1016/j.physa.2014.11.060 doi: 10.1016/j.physa.2014.11.060
    [48] Y. Sabbar, Asymptotic extinction and persistence of a perturbed epidemic model with different intervention measures and standard Lévy jumps, Bull. Biomath., 1 (2023), 58–77. https://doi.org/10.59292/bulletinbiomath.2023004 doi: 10.59292/bulletinbiomath.2023004
    [49] Y. Zhao, S. Yuan, Q. Zhang, The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment, Appl. Math. Model., 40 (2016), 7583–7600. https://doi.org/10.1016/j.apm.2016.01.056 doi: 10.1016/j.apm.2016.01.056
    [50] B. Liu, L. Chen, Y. Zhang, The effects of impulsive toxicant input on a population in a polluted environment, J. Biol. Syst., 11 (2003), 265–274. https://doi.org/10.1142/S0218339003000907 doi: 10.1142/S0218339003000907
    [51] X. Yang, Z. Jin, Y. Xue, Week average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input, Chaos Solitons Fractals, 31 (2007), 726–735. https://doi.org/10.1016/j.chaos.2005.10.042 doi: 10.1016/j.chaos.2005.10.042
    [52] S. Wang, L. Wang, T. Wei, Optimal harvesting for a stochastic predator-prey model with S-type distributed time delays, Methodol. Comput. Appl. Probab., 20 (2018), 37–68. https://doi.org/10.1007/s11009-016-9519-2 doi: 10.1007/s11009-016-9519-2
    [53] M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969–2012. https://doi.org/10.1007/s11538-010-9569-5 doi: 10.1007/s11538-010-9569-5
    [54] S. Wang, L. Wang, T. Wei, Optimal harvesting for a stochastic logistic model with S-type distributed time delay, J. Differ. Equations Appl., 23 (2017), 618–632. https://doi.org/10.1080/10236198.2016.1269761 doi: 10.1080/10236198.2016.1269761
    [55] Q. Liu, Q. Chen, Dynamics of stochastic delay Lotka-Volterra systems with impulsive toxicant input and Lévy noise in polluted environments, Appl. Math. Comput., 256 (2015), 52–67. https://doi.org/10.1016/j.amc.2015.01.009 doi: 10.1016/j.amc.2015.01.009
    [56] S. Wang, G. Hu, T. Wei, On a three-species stochastic hybrid Lotka-Volterra system with distributed delay and Lévy noise, Filomat, 36 (2022), 4737–4750. https://doi.org/10.2298/FIL2214737W doi: 10.2298/FIL2214737W
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