Skip to main content
SpringerLink
Log in
Book cover

Trends in Biomathematics: Modeling, Optimization and Computational Problems pp 167–188Cite as

A Plankton-Nutrient Model with Holling Type III Response Function

  • Chapter
  • First Online:

Abstract

A plankton model including the latest mathematical features introduced in a very recent specialistic contribution showing the emergence of the Holling type III response function is here formulated and developed in its deterministic and stochastic counterparts. The effects of additional food source and harvesting rate of zooplankton are analyzed. The results indicate that if the intensity of environmental fluctuation is kept under a certain threshold value, the control procedure proposed in the deterministic case is also valid in the presence of environmental disturbances.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. V.N. Afanas’ev, V.B. Kolmanowskii, V.R. Nosov, Mathematical Theory of Control Systems Design (Kluwer Academic, Dordrecht, 1996)

    Book  Google Scholar 

  2. M. Bandyopadhyay, J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability. Nonlinearity 18, 913–936 (2005)

    Article  MathSciNet  Google Scholar 

  3. E. Beretta, V.B. Kolmanowskii, L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simul. 45(3–4), 269–277 (1998)

    Article  MathSciNet  Google Scholar 

  4. F. Brauer, A.C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting. J. Math. Biol. 8, 55–71 (1979)

    Article  MathSciNet  Google Scholar 

  5. S. Chakraborty, J. Chattopadhyay, Nutrient-phytoplankton-zooplankton dynamics in the presence of additional food source — A mathematical study. J. Biol. Syst. 16(4), 547–564 (2008)

    Article  Google Scholar 

  6. K. Chakraborty, K. Das, Modeling and analysis of a two-zooplankton one-phytoplankton system in the presence of toxicity. Appl. Math. Model. 39(3–4), 1241–1265 (2015)

    Article  MathSciNet  Google Scholar 

  7. K. Chakraborty, S. Das, T.K. Kar, Optimal control of effort of a stage structured prey-predator fishery model with harvesting. Nonlinear Anal Real World Appl. 12(6), 3452–3467 (2011)

    Article  MathSciNet  Google Scholar 

  8. K. Chakraborty, M. Chakraborty, T.K. Kar, Optimal control of harvest and bifurcation of a prey-predator model with stage structure. Appl. Math. Comput. 217(21), 8778–8792 (2011)

    MathSciNet  MATH  Google Scholar 

  9. A. Chatterjee, S. Pal, Effect of dilution rate on the predictability of a realistic ecosystem model with instantaneous nutrient recycling. J. Biol. Syst. 19, 629 (2011)

    Article  MathSciNet  Google Scholar 

  10. A. Chatterjee, S. Pal, Role of constant nutrient input in a detritus based open marine plankton ecosystem model. Contemp. Math. Stat. 2, 71–91 (2013)

    Google Scholar 

  11. A. Chatterjee, S. Pal, S. Chatterjee, Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom. Appl. Math. Comput. 218, 3387–3398 (2011)

    MathSciNet  MATH  Google Scholar 

  12. C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. (Wiley Interscience, New York, 1990)

    MATH  Google Scholar 

  13. G. Dai, M. Tang, Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 58, 193–210 (1998)

    Article  MathSciNet  Google Scholar 

  14. T. Das, R.N. Mukherjee, K.S. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity. Appl. Math. Model. 33(5), 2282–2292 (2009)

    Article  MathSciNet  Google Scholar 

  15. M.R. Droop, Vitamin B12 in marine ecology. Nature 180, 1041–1042 (1957)

    Article  Google Scholar 

  16. A.M. Edwards, J.Brindley, Oscillatory behaviour in a three-component plankton population model. Dyn. Stab. Syst. 11(4), 347–370 (1996)

    Article  Google Scholar 

  17. A. Fan, P. Han, K. Wang, Global dynamics of a nutrient-plankton system in the water ecosystem. Appl. Math. Comput. 219, 8269–8276 (2013)

    MathSciNet  MATH  Google Scholar 

  18. E. González-Olivares, A. Rojas-Palma, Multiple limit cycles in a Gause type predator-prey model with holling Type III functional response and Allee effect on prey. Bull. Math. Biol. 73, 1378–1397 (2011)

    Article  MathSciNet  Google Scholar 

  19. E. González-Olivares, P.C. Tintinago-Ruiz, A. Rojas-Palma, A Leslie-Gower type predator-prey model with sigmoid functional response. Int. J. Comput. Math. 92, 1895–1909 (2015)

    Article  MathSciNet  Google Scholar 

  20. B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Application of Hopf Bifurcation (Cambridge University Press, Cambridge, 1981)

    MATH  Google Scholar 

  21. Z. Hu, Z. Teng, L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. Nonlinear Anal Real World Appl. 12(4), 2356–2377 (2011)

    Article  MathSciNet  Google Scholar 

  22. S.R.J. Jang, E.J. Allen, Deterministic and stochastic nutrient-phytoplankton-zooplankton models with periodic toxin producing phytoplankton. Appl. Math. Comput. 271, 52–67 (2015)

    MathSciNet  Google Scholar 

  23. M.Y. Li, J.S. Muldowney, Global Stability for the SEIR model in epidemiology. Math. BioSci. 125, 155–164 (1995)

    Article  MathSciNet  Google Scholar 

  24. M.Y. Li, H. Shu, Global dynamics of an in-host viral model with intracellular delay. Bull. Math. Biol. 72, 1492–1505 (2010)

    Article  MathSciNet  Google Scholar 

  25. Y. Li, D. Xie, J. A. Cui, The effect of continuous and pulse input nutrient on a lake model. J. Appl. Math. 2014, Article ID 462946 (2014)

    Google Scholar 

  26. J. Luo, Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication. Math. BioSci. 245, 126–136 (2013)

    Article  MathSciNet  Google Scholar 

  27. A. Martin, S. Ruan, Predator-prey models with delay and prey harvesting. J. Math. Biol. 43, 247–267 (2001)

    Article  MathSciNet  Google Scholar 

  28. A.Y. Morozov, Emergence of Holling type III zooplankton functional response: bringing together field evidence and mathematical modelling. J. Theor. Biol. 265, 45–54 (2010)

    Article  MathSciNet  Google Scholar 

  29. B. Mukhopadhyay, R. Bhattacharyya, On a three-tier ecological food chain model with deterministic and random harvesting: a mathematical study. Nonlinear Anal Model. Control 16(1), 77–88 (2011)

    MathSciNet  MATH  Google Scholar 

  30. M.R. Myerscough, B.F. Gray, W.L. Hogarth, J. Norbury, An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking. J. Math. Biol. 30, 389–411 (1992)

    Article  MathSciNet  Google Scholar 

  31. S. Pal, A. Chatterjee, Coexistence of plankton model with essential multiple nutrient in chemostat. Int. J. Biomath. 6, 28–42 (2013)

    MathSciNet  MATH  Google Scholar 

  32. S. Pal, S. Chatterjee, J. Chattopadhyay, Role of toxin and nutrient for the occurrence and termination of plankton bloom – Results drawn from field observations and a mathematical model. Biosystems 90, 87–100 (2007)

    Article  Google Scholar 

  33. F. Rao, The complex dynamics of a stochastic toxic- phytoplankton- zooplankton model. Adv. Difference Equ. 2014, 22 (2014)

    Article  MathSciNet  Google Scholar 

  34. S. Ruan S, Oscillations in Plankton models with nutrient recycling. J. Theor. Biol. 208, 15–26 (2001)

    Google Scholar 

  35. Y. Sekerci, S. Petrovskii, Mathematical modelling of spatiotemporal dynamics of oxygen in a plankton system. Math. Model. Nat. Phenom. 10(2), 96–114 (2015)

    Article  MathSciNet  Google Scholar 

  36. A. Sen, D. Mukherjee, B.C. Giri, P. Das, Stability of limit cycle in a prey-predator system with pollutant. Appl. Math. Sci. 5(21), 1025–1036 (2011)

    MathSciNet  MATH  Google Scholar 

  37. P.K. Tapaswi, A. Mukhopadhyay, Effects of environmental fluctuation on plankton allelopathy. J. Math. Biol. 39, 39–58 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research of Samares Pal is supported by UGC, New Delhi, India Ref. No. MRP-MAJ-MATH-2013-609. The research of Ezio Venturino has been partially supported by the project “Metodi numerici nelle scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano”.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kalyani, Kalyani, 741235, India

    Anal Chatterjee & Samares Pal

  2. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Torino, Italy

    Ezio Venturino

Authors
  1. Anal Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samares Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ezio Venturino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ezio Venturino .

Editor information

Editors and Affiliations

  1. Federal University of Rio de Janeiro, President, BIOMAT Consortium – International Institute for Interdisciplinary Sciences, Rio de Janeiro, Brazil, Rio de Janeiro, Brazil

    Rubem P. Mondaini

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chatterjee, A., Pal, S., Venturino, E. (2018). A Plankton-Nutrient Model with Holling Type III Response Function. In: Mondaini, R. (eds) Trends in Biomathematics: Modeling, Optimization and Computational Problems. Springer, Cham. https://doi.org/10.1007/978-3-319-91092-5_12

Download citation

Publish with us

Policies and ethics

Search

Navigation