# American Institute of Mathematical Sciences

July  2020, 25(7): 2373-2390. doi: 10.3934/dcdsb.2020014

## Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation

 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

* Corresponding author: sanling@usst.edu.cn

Received  April 2019 Revised  July 2019 Published  July 2020 Early access  April 2020

Fund Project: The work was supported by China NSF Grant 11671260, Shanghai Leading Academic Discipline Project Grant XTKX2012, Hujiang Foundation Grant B14005

In this paper, a stochastic chemostat model with two distributed delays and nonlinear perturbation is proposed. We first transform the stochastic model into an equivalent high-dimensional system. Then we prove the existence and uniqueness of global positive solution of the model. Based on Khasminskii's theory, we study the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Then we also establish sufficient conditions for the extinction of the plankton. Finally, numerical simulations are carried out to illustrate the theoretical results and to conclude our study, which shows that environmental noise experienced by limiting nutrient completely determines the persistence and extinction of the plankton.

Citation: Xingwang Yu, Sanling Yuan. Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2373-2390. doi: 10.3934/dcdsb.2020014
##### References:
 [1] E. Beretta, G. I. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99-111.  doi: 10.1007/BF00171521. [2] E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays: boundedness, local and global asymptotic stability, Differential Equations and Dynamical Systems, 2 (1994), 19-40. [3] G. I. Bischi, Effects of time lags on transient characteristics of a nutrient cycling model, Math. Biosci., 109 (1992), 151-175.  doi: 10.1016/0025-5564(92)90043-V. [4] Y. L. Cai, Y. Kang, M. Banerjee and W. M. Wang, A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 14 (2016), 893-910.  doi: 10.4310/CMS.2016.v14.n4.a1. [5] F. Campillo, M. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model., 222 (2011), 2676-2689.  doi: 10.1016/j.ecolmodel.2011.04.027. [6] J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.  doi: 10.2307/1934845. [7] R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type Ⅲ, J. Differ. Equ., 249 (2010), 2316-2356.  doi: 10.1016/j.jde.2010.06.021. [8] H. I. Freedman and Y. T. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol., 31 (1993), 513-527.  doi: 10.1007/BF00173890. [9] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [10] R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. [11] X.-Z. He and S. G. Ruan, Global stability in chemostat-type plankton models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253-271.  doi: 10.1007/s002850050128. [12] X.-Z. He, S. G. Ruan and H. X. Xia, Global stability in chemostat-type equations with distributed delays, SIAM J. Math. Anal., 29 (1998), 681-696.  doi: 10.1137/S0036141096311101. [13] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equ., 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [14] V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961. [15] D. X. Jia, T. H. Zhang and S. L. Yuan, Pattern dynamics of a diffusive toxin producing phytoplankton-zooplankton model with three-dimensional patch, Int. J. Bifurcat. Chaos, 29 (2019), 1930011. doi: 10.1142/S0218127419300118. [16] J. Jiang, A. L. Shen, H. Wang and S. L. Yuan, Regulation of phosphate uptake kinetics in the bloom-forming dinoflagellates Prorocentrum donghaiense with emphasis on two-stage dynamic process, J. Theor. Biol., 463 (2019), 12-21.  doi: 10.1016/j.jtbi.2018.12.011. [17] D. Li, S. Q. Liu and J. A. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differ. Equ., 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [18] M. Liu and M. L. Deng, Permanence and extinction of a stochastic hybrid model for tumor growth, Appl. Math. Lett., 94 (2019), 66-72.  doi: 10.1016/j.aml.2019.02.016. [19] Q. Liu and D. Q. Jiang, Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay, Discrete Contin. Dyn. Syst., 22 (2017), 2479-2500.  doi: 10.3934/dcdsb.2017127. [20] Q. Liu and D. Q. Jiang, Stationary distribution and extinction of a stochastic predator-prey model with distributed delay, Appl. Math. Lett., 78 (2018), 79-87.  doi: 10.1016/j.aml.2017.11.008. [21] Q. Liu, D. Q. Jiang, T. Hayat and A. Alsaedi, Long-time behavior of a stochastic logistic equation withdistributed delay and nonlinear perturbation, Physica A, 508 (2018), 289-304.  doi: 10.1016/j.physa.2018.05.054. [22] Q. Luo and X. R. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [23] N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27. Springer-Verlag, Berlin-New York, 1978. [24] X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester, 1997. [25] R. M. Nisbet and W. S. C. Gurney, Model of material cycling in a closed ecosystem, Nature, 264 (1976), 633-634.  doi: 10.1038/264633a0. [26] S. G. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, J. Math. Biol., 31 (1993), 633-654.  doi: 10.1007/BF00161202. [27] S. G. Ruan, The effect of delays on stability and persistence in plankton models, Nonlinear Analysis, 24 (1995), 575-585.  doi: 10.1016/0362-546X(95)93092-I. [28] S. G. Ruan and X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math., 58 (1998), 170-192.  doi: 10.1137/S0036139996299248. [29] K. Y. Song, W. B. Ma, S. B. Guo and H. Yan, A class of dynamic models describing microbial flocculant with nutrient competition and metabolic products in wastewater treatment, Adv. Differ. Equ., 2018 (2018), Paper No. 33, 14 pp. doi: 10.1186/s13662-018-1473-6. [30] W. M. Wang, Y. L. Cai, J. L. Li and Z. J. Gui, Periodic behavior in a FIV model with seasonality as well as environment fluctuations, J. Frankl. Inst., 354 (2017), 7410-7428.  doi: 10.1016/j.jfranklin.2017.08.034. [31] L. Wang and D. Q. Jiang, A note on the stationary distribution of the stochastic chemostat model with general response functions, Appl. Math. Lett., 73 (2017), 22-28.  doi: 10.1016/j.aml.2017.04.029. [32] R. H. Whittaker, Communities and Ecosystems, Macmillan, New York, 1975. [33] G. S. K. Wolkowicz, H. X. Xia and S. G. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math., 57 (1997), 1281-1310.  doi: 10.1137/S0036139995289842. [34] G. S. K. Wolkowicz, H. X. Xia and J. H. Wu, Global dynamics of a chemostat competition model with distributed delay, J. Math. Biol., 38 (1999), 285-316.  doi: 10.1007/s002850050150. [35] D. M. Wu, H. Wang and S. L. Yuan, Stochastic sensitivity analysis of noise-induced transitions in a predator-prey model with environmental toxins, Math. Biosci. Eng., 16 (2019), 2141-2153.  doi: 10.3934/mbe.2019104. [36] D. Y. Xu, Y. M. Huang and Z. G. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.  doi: 10.3934/dcds.2009.24.1005. [37] C. Q. Xu and S. L. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett., 48 (2015), 62-68.  doi: 10.1016/j.aml.2015.03.012. [38] C. Q. Xu and S. L. Yuan, Competition in the chemostat: A stochastic multi-species model and its asymptotic behavior, Math. Biosci., 280 (2016), 1-9.  doi: 10.1016/j.mbs.2016.07.008. [39] C. Q. Xu, S. L. Yuan and T. H. Zhang, Average break-even concentration in a simple chemostat model with telegraph noise, Nonlinear Anal. Hybrid. Syst., 29 (2018), 373-382.  doi: 10.1016/j.nahs.2018.03.007. [40] X. W. Yu, S. L. Yuan and T. H. Zhang, The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms, Nonlinear Dyn., 91 (2018), 1653-1668.  doi: 10.1007/s11071-017-3971-6. [41] X. W. Yu, S. L. Yuan and T. H. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 359-374.  doi: 10.1016/j.cnsns.2017.11.028. [42] X. W. Yu, S. L. Yuan and T. H. Zhang, Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling, Nonlinear Anal. Hybrid. Syst., 34 (2019), 209-225.  doi: 10.1016/j.nahs.2019.06.005. [43] X. W. Yu, S. L. Yuan and T. H. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249-264.  doi: 10.1016/j.amc.2018.11.005. [44] S. Q. Zhang, X. Z. Meng, T. Feng and T. H. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Anal. Hybrid. Syst., 26 (2017), 19-37.  doi: 10.1016/j.nahs.2017.04.003. [45] S. N. Zhao, S. L. Yuan and H. Wang, Threshold behavior in a stochastic algal growth model with stoichiometric constraints and seasonal variation, J. Differ. Equ. doi: 10.1016/j.jde.2019.11.004. [46] Y. Zhao, L. You, D. Burkow and S. L. 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.  doi: 10.1007/s11071-018-4642-y. [47] Y. Zhao, S. L. Yuan and 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.  doi: 10.1007/s11538-015-0086-4. [48] Y. Zhao, S. L. Yuan and T. H. Zhang, Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 266-276.  doi: 10.1016/j.cnsns.2016.08.013. [49] Y. Zhao, S. L. Yuan and T. H. Zhang, The stationary distribution and ergodicity of astochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 131-142.  doi: 10.1016/j.cnsns.2016.01.013.

show all references

##### References:
 [1] E. Beretta, G. I. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99-111.  doi: 10.1007/BF00171521. [2] E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays: boundedness, local and global asymptotic stability, Differential Equations and Dynamical Systems, 2 (1994), 19-40. [3] G. I. Bischi, Effects of time lags on transient characteristics of a nutrient cycling model, Math. Biosci., 109 (1992), 151-175.  doi: 10.1016/0025-5564(92)90043-V. [4] Y. L. Cai, Y. Kang, M. Banerjee and W. M. Wang, A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 14 (2016), 893-910.  doi: 10.4310/CMS.2016.v14.n4.a1. [5] F. Campillo, M. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model., 222 (2011), 2676-2689.  doi: 10.1016/j.ecolmodel.2011.04.027. [6] J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.  doi: 10.2307/1934845. [7] R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type Ⅲ, J. Differ. Equ., 249 (2010), 2316-2356.  doi: 10.1016/j.jde.2010.06.021. [8] H. I. Freedman and Y. T. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol., 31 (1993), 513-527.  doi: 10.1007/BF00173890. [9] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [10] R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. [11] X.-Z. He and S. G. Ruan, Global stability in chemostat-type plankton models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253-271.  doi: 10.1007/s002850050128. [12] X.-Z. He, S. G. Ruan and H. X. Xia, Global stability in chemostat-type equations with distributed delays, SIAM J. Math. Anal., 29 (1998), 681-696.  doi: 10.1137/S0036141096311101. [13] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equ., 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [14] V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961. [15] D. X. Jia, T. H. Zhang and S. L. Yuan, Pattern dynamics of a diffusive toxin producing phytoplankton-zooplankton model with three-dimensional patch, Int. J. Bifurcat. Chaos, 29 (2019), 1930011. doi: 10.1142/S0218127419300118. [16] J. Jiang, A. L. Shen, H. Wang and S. L. Yuan, Regulation of phosphate uptake kinetics in the bloom-forming dinoflagellates Prorocentrum donghaiense with emphasis on two-stage dynamic process, J. Theor. Biol., 463 (2019), 12-21.  doi: 10.1016/j.jtbi.2018.12.011. [17] D. Li, S. Q. Liu and J. A. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differ. Equ., 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [18] M. Liu and M. L. Deng, Permanence and extinction of a stochastic hybrid model for tumor growth, Appl. Math. Lett., 94 (2019), 66-72.  doi: 10.1016/j.aml.2019.02.016. [19] Q. Liu and D. Q. Jiang, Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay, Discrete Contin. Dyn. Syst., 22 (2017), 2479-2500.  doi: 10.3934/dcdsb.2017127. [20] Q. Liu and D. Q. Jiang, Stationary distribution and extinction of a stochastic predator-prey model with distributed delay, Appl. Math. Lett., 78 (2018), 79-87.  doi: 10.1016/j.aml.2017.11.008. [21] Q. Liu, D. Q. Jiang, T. Hayat and A. Alsaedi, Long-time behavior of a stochastic logistic equation withdistributed delay and nonlinear perturbation, Physica A, 508 (2018), 289-304.  doi: 10.1016/j.physa.2018.05.054. [22] Q. Luo and X. R. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [23] N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27. Springer-Verlag, Berlin-New York, 1978. [24] X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester, 1997. [25] R. M. Nisbet and W. S. C. Gurney, Model of material cycling in a closed ecosystem, Nature, 264 (1976), 633-634.  doi: 10.1038/264633a0. [26] S. G. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, J. Math. Biol., 31 (1993), 633-654.  doi: 10.1007/BF00161202. [27] S. G. Ruan, The effect of delays on stability and persistence in plankton models, Nonlinear Analysis, 24 (1995), 575-585.  doi: 10.1016/0362-546X(95)93092-I. [28] S. G. Ruan and X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math., 58 (1998), 170-192.  doi: 10.1137/S0036139996299248. [29] K. Y. Song, W. B. Ma, S. B. Guo and H. Yan, A class of dynamic models describing microbial flocculant with nutrient competition and metabolic products in wastewater treatment, Adv. Differ. Equ., 2018 (2018), Paper No. 33, 14 pp. doi: 10.1186/s13662-018-1473-6. [30] W. M. Wang, Y. L. Cai, J. L. Li and Z. J. Gui, Periodic behavior in a FIV model with seasonality as well as environment fluctuations, J. Frankl. Inst., 354 (2017), 7410-7428.  doi: 10.1016/j.jfranklin.2017.08.034. [31] L. Wang and D. Q. Jiang, A note on the stationary distribution of the stochastic chemostat model with general response functions, Appl. Math. Lett., 73 (2017), 22-28.  doi: 10.1016/j.aml.2017.04.029. [32] R. H. Whittaker, Communities and Ecosystems, Macmillan, New York, 1975. [33] G. S. K. Wolkowicz, H. X. Xia and S. G. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math., 57 (1997), 1281-1310.  doi: 10.1137/S0036139995289842. [34] G. S. K. Wolkowicz, H. X. Xia and J. H. Wu, Global dynamics of a chemostat competition model with distributed delay, J. Math. Biol., 38 (1999), 285-316.  doi: 10.1007/s002850050150. [35] D. M. Wu, H. Wang and S. L. Yuan, Stochastic sensitivity analysis of noise-induced transitions in a predator-prey model with environmental toxins, Math. Biosci. Eng., 16 (2019), 2141-2153.  doi: 10.3934/mbe.2019104. [36] D. Y. Xu, Y. M. Huang and Z. G. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.  doi: 10.3934/dcds.2009.24.1005. [37] C. Q. Xu and S. L. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett., 48 (2015), 62-68.  doi: 10.1016/j.aml.2015.03.012. [38] C. Q. Xu and S. L. Yuan, Competition in the chemostat: A stochastic multi-species model and its asymptotic behavior, Math. Biosci., 280 (2016), 1-9.  doi: 10.1016/j.mbs.2016.07.008. [39] C. Q. Xu, S. L. Yuan and T. H. Zhang, Average break-even concentration in a simple chemostat model with telegraph noise, Nonlinear Anal. Hybrid. Syst., 29 (2018), 373-382.  doi: 10.1016/j.nahs.2018.03.007. [40] X. W. Yu, S. L. Yuan and T. H. Zhang, The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms, Nonlinear Dyn., 91 (2018), 1653-1668.  doi: 10.1007/s11071-017-3971-6. [41] X. W. Yu, S. L. Yuan and T. H. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 359-374.  doi: 10.1016/j.cnsns.2017.11.028. [42] X. W. Yu, S. L. Yuan and T. H. Zhang, Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling, Nonlinear Anal. Hybrid. Syst., 34 (2019), 209-225.  doi: 10.1016/j.nahs.2019.06.005. [43] X. W. Yu, S. L. Yuan and T. H. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249-264.  doi: 10.1016/j.amc.2018.11.005. [44] S. Q. Zhang, X. Z. Meng, T. Feng and T. H. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Anal. Hybrid. Syst., 26 (2017), 19-37.  doi: 10.1016/j.nahs.2017.04.003. [45] S. N. Zhao, S. L. Yuan and H. Wang, Threshold behavior in a stochastic algal growth model with stoichiometric constraints and seasonal variation, J. Differ. Equ. doi: 10.1016/j.jde.2019.11.004. [46] Y. Zhao, L. You, D. Burkow and S. L. 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.  doi: 10.1007/s11071-018-4642-y. [47] Y. Zhao, S. L. Yuan and 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.  doi: 10.1007/s11538-015-0086-4. [48] Y. Zhao, S. L. Yuan and T. H. Zhang, Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 266-276.  doi: 10.1016/j.cnsns.2016.08.013. [49] Y. Zhao, S. L. Yuan and T. H. Zhang, The stationary distribution and ergodicity of astochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 131-142.  doi: 10.1016/j.cnsns.2016.01.013.
The left is the stochastic trajectories of system (8); the right is the density functions of $N(t)$ and $P(t)$, respectively. Here, $r_1 = r_2 = 1$, $D = 0.3$, $N^0 = 2$, $a = 0.7$, $L = 0.3$, $b = 0.3$, $\gamma = 0.2$, $\alpha_1 = 0.1$, $\alpha_2 = 0.15$, $c = 0.58$, $\sigma_1 = 0.01$, $\sigma_2 = 0.08$
The solutions of stochastic system (8) and its corresponding deterministic system. Here, all parameter values are taken as in Fig. 1 except $\sigma_2.$
The solutions of stochastic system (8) when $\sigma_1 = 0$. Here, (a) $c = 0.5$, $\sigma_2 = 2.8$; (b) $c = 0.59$, $\sigma_2 = 2.8$. Other parameter values are taken as in Fig. 1
The solutions of stochastic system (8). Here, (a) $c = 0.48$, $\sigma_1 = 1.5$, $\sigma_2 = 2.8$, $\alpha_1 = 0.1$, $T_{K_1} = 10$, $T_{K_2} = 100$ (blue), $10$ (red), $1$ (green); (b) $c = 0.58$, $\sigma_1 = 0.01$, $\sigma_2 = 0.15$, $\alpha_2 = 0.1$, $T_{K_2} = 10$, $T_{K_1} = 1000$ (blue), $100$ (red), $1$ (green). Other parameters take the same values as in Fig. 1
Biological explanation of parameters
 Parameters Description $N^0$ the input concentration of the limiting nutrient $a$ the maximum uptake rate of nutrient $c$ the maximum specific growth rate of plankton $b$ the fraction of the nutrient recycled from the dead plankton $\gamma$ the death rate of plankton $D$ the washout rate of the limiting nutrient
 Parameters Description $N^0$ the input concentration of the limiting nutrient $a$ the maximum uptake rate of nutrient $c$ the maximum specific growth rate of plankton $b$ the fraction of the nutrient recycled from the dead plankton $\gamma$ the death rate of plankton $D$ the washout rate of the limiting nutrient
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