Skip to main content
SpringerLink
Log in

Analysis of a Stochastic Phytoplankton–Zooplankton Model under Non-degenerate and Degenerate Diffusions

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

Phytoplankton is an important indicator organism to evaluate the quality of water environment, which may reflect the nutritional level of the sea area. Conversely, environmental conditions can directly affect the community structure of phytoplankton. A stochastic phytoplankton–zooplankton model considering non-degenerate and degenerate diffusions is formulated in this paper. What’s more, in both systems, we obtain sufficient conditions of extinction and ergodicity. Our results demonstrate that the weaker white noise can ensure the permanence of zooplankton and the stronger white noise can lead to the disappearance of zooplankton. Moreover, the threshold value of extinction and persistence can serve as a theoretical basis for controlling phytoplankton and zooplankton. Numerical examples are performed on the analysis results of the two cases to confirm our theoretical results. In addition, we also provide a real-life case study, the validity of the model is verified based on experimental data, and it is shown that fluctuation of external environment and the consumption of phytoplankton by zooplankton will affect the growth and number of phytoplankton. Effectively controlling the quantity of phytoplankton can delay the occurrence of water bloom or red tide.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Bao, K., Rong, L., Zhang, Q.: Analysis of a stochastic SIRS model with interval parameters. Discrete Contin. Dyn. Syst. B 24(9), 4827–4849 (2019)

    MathSciNet  MATH  Google Scholar 

  • Ben Arous, G., Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Relat. Fields 90, 377–402 (1991)

    Article  Google Scholar 

  • Coutinho, M., Brito, A., Pereira, P., et al.: A phytoplankton tool for water quality assessment in semi-enclosed coastal lagoons: open vs closed regimes. Estuar. Coastal Shelf Sci. 110, 134–146 (2012)

    Article  Google Scholar 

  • Du, N., Nguyen, D., Yin, G.: Conditions for permanence and ergodicity of certain stochastic predator–prey models. J. Appl. Probab. 53(01), 187–202 (2016)

    Article  MathSciNet  Google Scholar 

  • Fogg, G.: The ecological significance of extracellular products of phytoplankton photosynthesis. Bot. Mar. 26(1), 3–14 (1983)

    Article  Google Scholar 

  • Han, B., Jiang, D., Hayat, T., et al.: Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation. Chaos Solitons Fractals (2020). https://doi.org/10.1016/j.chaos.2020.110238

    Article  MathSciNet  Google Scholar 

  • Has’miniskii, R.: Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations. Theory Probab. Appl. 5(2), 196–214 (2006)

    MathSciNet  Google Scholar 

  • Havens, K., Elia, A., Taticchi, M., et al.: Zooplankton–phytoplankton relationships in shallow subtropical versus temperate lakes Apopka (Florida, USA) and trasimeno (Umbria, Italy). Hydrobiologia 628(1), 165–175 (2009)

    Article  Google Scholar 

  • He, S., Tang, S., Cai, Y., et al.: A stochastic epidemic model coupled with seasonal air pollution: analysis and data fitting. Stoch. Env. Res. Risk Assess. 34(1), 2245–2257 (2020)

    Article  Google Scholar 

  • Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 433, 525–546 (2001)

    Article  MathSciNet  Google Scholar 

  • Ikeda, N., Watanabe, S.: A comparison theorem for solutions of stochastic differential equations and applications. J. Math. 14, 619–633 (1977)

    MathSciNet  MATH  Google Scholar 

  • Jassby, D., Platt,T.: Mathematical formulation of the relationship between photosynthesis and light for phytoplankton. Limnol. Oceanogr. (1976)

  • Khas’minskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Heidelberg (2012)

    Book  Google Scholar 

  • Liu, Q., Jiang, D., Hayat, T., et al.: Dynamical behavior of a stochastic model of gene expression with distributed delay and degenerate diffusion. Stoch. Anal. Appl. 36, 584–599 (2018)

    Article  MathSciNet  Google Scholar 

  • Luo, J.: Phytoplankton–zooplankton dynamics in periodic environments taking into account eutrophication. Math. Biosci. 245(2), 126–136 (2013)

    Article  MathSciNet  Google Scholar 

  • Majumder, A., Adak, D., Bairagi, N.: Phytoplankton-zooplankton interaction under environmental stochasticity: survival, extinction and stability. Appl. Math. Model. 89(2), 1382–1404 (2021)

    Article  MathSciNet  Google Scholar 

  • Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing, Chichester (1997)

    MATH  Google Scholar 

  • Mu, X., Jiang, D., Hayat, T., et al.: Dynamical behavior of a stochastic Nicholson’s blowflies model with distributed delay and degenerate diffusion. Nonlinear Dyn. 103, 2081–2096 (2021)

    Article  Google Scholar 

  • Pichór, K., Rudnicki, R.: Stability of Markov semigroups and applications to parabolic systems. J. Math. Anal. Appl. 215, 56–74 (1997)

    Article  MathSciNet  Google Scholar 

  • Reynolds, C.: The Ecology of Phytoplankton: References. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  • Ripa, J., Lundberg, P.: The route to extinction in variable environments. Oikos 90(1), 89–96 (2010)

    Article  Google Scholar 

  • Rudnicki, R.: Asymptotic Properties of the Fokker–Planck Equation, pp. 517–521. Springer Berlin Heidelberg, Berlin (1995)

    MATH  Google Scholar 

  • Rudnicki, R., Pichór, K.: Influence of stochastic perturbation on prey–predator systems. Math. Biosci. 206(1), 108–119 (2007)

    Article  MathSciNet  Google Scholar 

  • Rudnicki, R., Pichór, K., Tyran-Kamińska, M.: Markov Semigroups and Their Applications. Dynamics of Dissipation. Springer Berlin Heidelberg, Berlin (2002)

    MATH  Google Scholar 

  • Saha, T., Bandyopadhyay, M.: Dynamical analysis of toxin producing phytoplankton–zooplankton interactions. Nonlinear Anal. Real World Appl. 10(1), 314–332 (2009)

    Article  MathSciNet  Google Scholar 

  • Salmaso, N., Morabito, G., Buzzi, F., et al.: Phytoplankton as an indicator of the water quality of the deep lakes South of the Alps. Hydrobiologia 563(1), 167–187 (2006)

    Article  Google Scholar 

  • Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. Oikos 62, 271–282 (1991)

    Article  Google Scholar 

  • Shekhar, T., Kiran, B., Puttaiah, E., et al.: Phytoplankton as index of water quality with reference to industrial pollution. J. Environ. Biol. 29(2), 233–236 (2008)

    Google Scholar 

  • Talling, J.: Phytoplankton–zooplankton seasonal timing and the ‘clear-water phase’ in some English lakes. Freshw. Biol. 08(03), 1–18 (2003)

    Google Scholar 

  • Zhang, X., Yang, Q.: Threshold behavior in a stochastic SVIR model with general incidence rates. Appl. Math. Lett. (2021). https://doi.org/10.1016/j.aml.2021.107403

    Article  MathSciNet  MATH  Google Scholar 

  • Zu, L., Jiang, D., O’Regan, D., et al.: Ergodic property of a Lotka–Volterra predator–prey model with white noise higher order perturbation under regime switching. Appl. Math. Comput. 330, 93–102 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research is supported by the Natural Science Foundation of China (No. 11871473), Shandong Provincial Natural Science Foundation (Nos. ZR2019MA010, ZR2019MA006) and the Fundamental Research Funds for the Central Universities (No. 19CX02055A).

Author information

Authors and Affiliations

  1. School of Petroleum Engineering, Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum (East China) Ministry of Education, Qingdao, 266580, People’s Republic of China

    Xiaojie Mu & Daqing Jiang

  2. College of Science, China University of Petroleum (East China), Qingdao, 266580, People’s Republic of China

    Daqing Jiang

  3. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, 121589, Saudi Arabia

    Daqing Jiang & Ahmed Alsaedi

Authors
  1. Xiaojie Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daqing Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmed Alsaedi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

DJ designed the research and methodology. XM wrote the original draft. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Daqing Jiang.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Changpin Li.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, X., Jiang, D. & Alsaedi, A. Analysis of a Stochastic Phytoplankton–Zooplankton Model under Non-degenerate and Degenerate Diffusions. J Nonlinear Sci 32, 35 (2022). https://doi.org/10.1007/s00332-022-09787-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-022-09787-9

Keywords

Mathematics Subject Classification

Search

Navigation