A Plankton-Nutrient Model with Holling Type III Response Function

  • Anal Chatterjee
  • Samares Pal
  • Ezio VenturinoEmail author


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.



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”.


  1. 1.
    V.N. Afanas’ev, V.B. Kolmanowskii, V.R. Nosov, Mathematical Theory of Control Systems Design (Kluwer Academic, Dordrecht, 1996)CrossRefGoogle Scholar
  2. 2.
    M. Bandyopadhyay, J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability. Nonlinearity 18, 913–936 (2005)MathSciNetCrossRefGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    F. Brauer, A.C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting. J. Math. Biol. 8, 55–71 (1979)MathSciNetCrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 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)MathSciNetzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 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. 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)MathSciNetzbMATHGoogle Scholar
  12. 12.
    C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. (Wiley Interscience, New York, 1990)zbMATHGoogle Scholar
  13. 13.
    G. Dai, M. Tang, Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 58, 193–210 (1998)MathSciNetCrossRefGoogle Scholar
  14. 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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M.R. Droop, Vitamin B12 in marine ecology. Nature 180, 1041–1042 (1957)CrossRefGoogle Scholar
  16. 16.
    A.M. Edwards, J.Brindley, Oscillatory behaviour in a three-component plankton population model. Dyn. Stab. Syst. 11(4), 347–370 (1996)CrossRefGoogle Scholar
  17. 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)MathSciNetzbMATHGoogle Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 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)MathSciNetCrossRefGoogle Scholar
  20. 20.
    B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Application of Hopf Bifurcation (Cambridge University Press, Cambridge, 1981)zbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefGoogle Scholar
  22. 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)MathSciNetGoogle Scholar
  23. 23.
    M.Y. Li, J.S. Muldowney, Global Stability for the SEIR model in epidemiology. Math. BioSci. 125, 155–164 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M.Y. Li, H. Shu, Global dynamics of an in-host viral model with intracellular delay. Bull. Math. Biol. 72, 1492–1505 (2010)MathSciNetCrossRefGoogle Scholar
  25. 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. 26.
    J. Luo, Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication. Math. BioSci. 245, 126–136 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    A. Martin, S. Ruan, Predator-prey models with delay and prey harvesting. J. Math. Biol. 43, 247–267 (2001)MathSciNetCrossRefGoogle Scholar
  28. 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)MathSciNetCrossRefGoogle Scholar
  29. 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)MathSciNetzbMATHGoogle Scholar
  30. 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)MathSciNetCrossRefGoogle Scholar
  31. 31.
    S. Pal, A. Chatterjee, Coexistence of plankton model with essential multiple nutrient in chemostat. Int. J. Biomath. 6, 28–42 (2013)MathSciNetzbMATHGoogle Scholar
  32. 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)CrossRefGoogle Scholar
  33. 33.
    F. Rao, The complex dynamics of a stochastic toxic- phytoplankton- zooplankton model. Adv. Difference Equ. 2014, 22 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    S. Ruan S, Oscillations in Plankton models with nutrient recycling. J. Theor. Biol. 208, 15–26 (2001)Google Scholar
  35. 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)MathSciNetCrossRefGoogle Scholar
  36. 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)MathSciNetzbMATHGoogle Scholar
  37. 37.
    P.K. Tapaswi, A. Mukhopadhyay, Effects of environmental fluctuation on plankton allelopathy. J. Math. Biol. 39, 39–58 (1999)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KalyaniKalyaniIndia
  2. 2.Dipartimento di Matematica “Giuseppe Peano”Università di TorinoTorinoItaly

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