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.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
V.N. Afanas’ev, V.B. Kolmanowskii, V.R. Nosov, Mathematical Theory of Control Systems Design (Kluwer Academic, Dordrecht, 1996)
M. Bandyopadhyay, J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability. Nonlinearity 18, 913–936 (2005)
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)
F. Brauer, A.C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting. J. Math. Biol. 8, 55–71 (1979)
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)
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)
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)
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)
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)
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)
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)
C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. (Wiley Interscience, New York, 1990)
G. Dai, M. Tang, Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 58, 193–210 (1998)
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)
M.R. Droop, Vitamin B12 in marine ecology. Nature 180, 1041–1042 (1957)
A.M. Edwards, J.Brindley, Oscillatory behaviour in a three-component plankton population model. Dyn. Stab. Syst. 11(4), 347–370 (1996)
A. Fan, P. Han, K. Wang, Global dynamics of a nutrient-plankton system in the water ecosystem. Appl. Math. Comput. 219, 8269–8276 (2013)
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)
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)
B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Application of Hopf Bifurcation (Cambridge University Press, Cambridge, 1981)
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)
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)
M.Y. Li, J.S. Muldowney, Global Stability for the SEIR model in epidemiology. Math. BioSci. 125, 155–164 (1995)
M.Y. Li, H. Shu, Global dynamics of an in-host viral model with intracellular delay. Bull. Math. Biol. 72, 1492–1505 (2010)
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)
J. Luo, Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication. Math. BioSci. 245, 126–136 (2013)
A. Martin, S. Ruan, Predator-prey models with delay and prey harvesting. J. Math. Biol. 43, 247–267 (2001)
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)
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)
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)
S. Pal, A. Chatterjee, Coexistence of plankton model with essential multiple nutrient in chemostat. Int. J. Biomath. 6, 28–42 (2013)
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)
F. Rao, The complex dynamics of a stochastic toxic- phytoplankton- zooplankton model. Adv. Difference Equ. 2014, 22 (2014)
S. Ruan S, Oscillations in Plankton models with nutrient recycling. J. Theor. Biol. 208, 15–26 (2001)
Y. Sekerci, S. Petrovskii, Mathematical modelling of spatiotemporal dynamics of oxygen in a plankton system. Math. Model. Nat. Phenom. 10(2), 96–114 (2015)
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)
P.K. Tapaswi, A. Mukhopadhyay, Effects of environmental fluctuation on plankton allelopathy. J. Math. Biol. 39, 39–58 (1999)
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
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
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
DOI: https://doi.org/10.1007/978-3-319-91092-5_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91091-8
Online ISBN: 978-3-319-91092-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)