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  December 2019

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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020014
References:
[1]

E. BerettaG. I. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99-111.  doi: 10.1007/BF00171521.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

Y. L. CaiY. KangM. 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.  Google Scholar

[5]

F. CampilloM. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model., 222 (2011), 2676-2689.  doi: 10.1016/j.ecolmodel.2011.04.027.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.  Google Scholar

[10]

R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar

[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.  Google Scholar

[12]

X.-Z. HeS. 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.  Google Scholar

[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.  Google Scholar

[14] V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961.   Google Scholar
[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.  Google Scholar

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J. JiangA. L. ShenH. 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.  Google Scholar

[17]

D. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Q. LiuD. Q. JiangT. 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.  Google Scholar

[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.  Google Scholar

[23]

N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[24]

X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

W. M. WangY. L. CaiJ. 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.  Google Scholar

[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.  Google Scholar

[32]

R. H. Whittaker, Communities and Ecosystems, Macmillan, New York, 1975. Google Scholar

[33]

G. S. K. WolkowiczH. 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.  Google Scholar

[34]

G. S. K. WolkowiczH. 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.  Google Scholar

[35]

D. M. WuH. 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.  Google Scholar

[36]

D. Y. XuY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

C. Q. XuS. 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.  Google Scholar

[40]

X. W. YuS. 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.  Google Scholar

[41]

X. W. YuS. 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.  Google Scholar

[42]

X. W. YuS. 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.  Google Scholar

[43]

X. W. YuS. 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.  Google Scholar

[44]

S. Q. ZhangX. Z. MengT. 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.  Google Scholar

[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.  Google Scholar

[46]

Y. ZhaoL. YouD. 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.  Google Scholar

[47]

Y. ZhaoS. 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.  Google Scholar

[48]

Y. ZhaoS. 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.  Google Scholar

[49]

Y. ZhaoS. 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.  Google Scholar

show all references

References:
[1]

E. BerettaG. I. Bischi and F. Solimano, Stability in chemostat equations with delayed nutrient recycling, J. Math. Biol., 28 (1990), 99-111.  doi: 10.1007/BF00171521.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

Y. L. CaiY. KangM. 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.  Google Scholar

[5]

F. CampilloM. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model., 222 (2011), 2676-2689.  doi: 10.1016/j.ecolmodel.2011.04.027.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.  Google Scholar

[10]

R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar

[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.  Google Scholar

[12]

X.-Z. HeS. 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.  Google Scholar

[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.  Google Scholar

[14] V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, 1961.   Google Scholar
[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.  Google Scholar

[16]

J. JiangA. L. ShenH. 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.  Google Scholar

[17]

D. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Q. LiuD. Q. JiangT. 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.  Google Scholar

[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.  Google Scholar

[23]

N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27. Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[24]

X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

W. M. WangY. L. CaiJ. 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.  Google Scholar

[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.  Google Scholar

[32]

R. H. Whittaker, Communities and Ecosystems, Macmillan, New York, 1975. Google Scholar

[33]

G. S. K. WolkowiczH. 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.  Google Scholar

[34]

G. S. K. WolkowiczH. 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.  Google Scholar

[35]

D. M. WuH. 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.  Google Scholar

[36]

D. Y. XuY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

C. Q. XuS. 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.  Google Scholar

[40]

X. W. YuS. 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.  Google Scholar

[41]

X. W. YuS. 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.  Google Scholar

[42]

X. W. YuS. 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.  Google Scholar

[43]

X. W. YuS. 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.  Google Scholar

[44]

S. Q. ZhangX. Z. MengT. 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.  Google Scholar

[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.  Google Scholar

[46]

Y. ZhaoL. YouD. 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.  Google Scholar

[47]

Y. ZhaoS. 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.  Google Scholar

[48]

Y. ZhaoS. 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.  Google Scholar

[49]

Y. ZhaoS. 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.  Google Scholar

Figure 1.  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 $
Figure 2.  The solutions of stochastic system (8) and its corresponding deterministic system. Here, all parameter values are taken as in Fig. 1 except $ \sigma_2. $
Figure 3.  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
Figure 4.  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
Table 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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