Modelling the depletion of atmospheric fuzzy plankton-oxygen dynamics in a lake due to photosynthetic activity of ecosystem under Climate change

S Saravanakumar, B Sridevi, A Eswari, Lakshmanan Rajendran

Article ID: 556
Vol 1, Issue 2, 2018, Article identifier:

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This paper proposes a new approach to the depletion of plankton-oxygen dynamic systems that deal with non-spatial processes in fuzzy environment. In this paper, an approximate analytical method to solve the non-linear fuzzy differential equations in a plankton-oxygen dynamics is presented. This kinetic mechanism is based on the system of non-linear reaction diffusion equations. An approximate fuzzy analytical expression of concentration profiles of oxygen, phytoplankton and zooplankton has been derived using the Homotopy perturbation method for all hypothetical values of the parameters. Analytical results are compared with the numerical results and graphical representation of previous result, a satisfactory agreement.


Phytoplankton; Homotopy perturbation method; Simulation technique; Oxygen Depletion; fuzzy differential equations

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Fasham M (1978) The statistical and mathematical analysis of plankton patchiness. Oceanogr Mar Biol Ann Rev 16:43–79

Okubo A (1980) Diffusion and ecological problems: mathematical models. Springer, Berlin

E.E. Holmes, M.A. Lewis, J.E. Banks, and R.R. Veit (1994), Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75, pp. 17–29.

Bain Jr R C (1968) Predicting variations caused by algae; J. Sanitary Eng. Div. 867–881

Ritschard R L (1992) Marine algae as a CO2 sink; Water Air Soil Pollut. 64 289–303

Duinker J and Wefer G 1994 Das CO2 -Problem und die Rolle des Ozeans; Naturwissenschaften 81 237–242

Malin G (1997) Sulphur, climate and the microbial maze; Nature (London) 387 857–859

Charlson RJ, Lovelock JE, Andreae MO, Warren SG (1987) Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate. Nature 326:655–661

Williamson P, Gribbin J (1991) How plankton change the climate. N Sci 129:48–52

Raymont J E G (1980) Plankton and productivity in the ocean (Oxford: Pergamon)

Sommer U (1994) Planktonologie (Berlin: Springer)

Lotka A J (1925) Elements of physical biology (Baltimore: Williams and Wilkins)

Volterra V (1926) Variations and fluctuations of the numbers of individuals in animal species living together; J. Cons. Perm. Int. Ent. Mer. 3 3–51

Segel L A and Jackson J L (1972) Dissipative structure: an explanation and an ecological example; J. Theor. Biol. 37 545–559

Dubois D (1975) A model of patchiness for prey–predator plankton populations; Ecol. Model. 1 67–80

Levin S A and Segel L A (1976) Hypothesis for origin of planktonic patchiness; Nature (London) 259 659

Vinogradov M E and Menshutkin V V (1977) The modeling of open sea ecosystems; in The sea: Ideas and observations on progress in the study of the sea 6 (ed.) E D Goldberg (New York: John Wiley)

Mimura M and Murray J D (1978) On a diffusive prey–predator model which exhibits patchiness; J. Theor. Biol. 75 249– 262

Mayzaud P and Poulet S A (1978) The importance of the time factor in the response of zooplankton to varying concentrations of naturally occurring particulate matter; Limnol. Oceanogr. 23 1144–1154

Beltrami E (1989) A mathematical model of the brown tide; Estuaries 12 13–17

Beltrami E (1996) Unusual algal blooms as excitable systems: The case of brown–tides; Environ. Modelling Assessment 1 19–24

Truscott J E and Brindley J (1994a) Ocean plankton populations as excitable media; Bull. Math. Biol. 56 981–998

Truscott J E and Brindley J (1994b) Equilibria, stability and excitability in a general class of plankton population models; Philos. Trans. R. Soc. London Ser. A 347 703–718

Petrovskii SV, Morozov AY, Venturino E (2002) Allee effect makes possible patchy invasion in a predatorprey system. EcolLett 5:345–352

Petrovskii SV, Kawasaki K, Takasu F, Shigesada N (2001) Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species. Japan J IndApplMaths 18:459 – 481

M Jayaweera, and T Asaeda, Modeling of biomanipulation in shallow, eutrophic lakes: An application to Lake Bleiswijkse Zoo, the Netherlands: Ecological Modelling [ECOL. MODEL.], vol. 85, no. 2-3, pp. 113-127, 1996.

Busenberg, S., Kumar, S.K., Austin, P., Wake, G., (1990). The dynamics of a model of a plankton-nutrient interaction. Bull. Math. Biol. 52, 677–696.

Hallam, G., (1978). Structural sensitivity of grazing formulation in nutrient controlled plankton models. J. Math. Biol. 5, 261–280

Jørgensen, S. E. (1988). Fundamentals of Ecological Modelling. Elsevier, Amsterdam.

A.K. Misra, Mathematical Modeling and Analysis of Eutrophication of Water Bodies Caused by Nutrients, Nonlinear Anal. Model. Control, 12(4), pp. 511–524, 2007.

A.K. Misra, Mathematical modeling and analysis of the depletion of dissolved oxygen in water bodies, Proc. Nat. Acad. Sci. India A, 78(IV), pp. 331–340, 2008

J. B. Shukla, A. K. Misra, and P. Chandra, Modeling and analysis of the algal bloom in a lake caused by discharge of nutrients, App. Math. And comp. 196 (2008), pp. 782 - 790.

Sekerci, Y., Petrovskii, S., (2015). Mathematical modelling of plankton–oxygen dynamics under the climate change. Bull. Math. Biol. 77 (12), 2325–2353.

Sergei Petrovskii ,YadigarSekerci , EzioVenturinoc ,” Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change. Journal of Theoretical Biology 424 (2017) 91–109

J.H. He, Homotopy perturbation technique.Comput.Methods Appl. Mech. Eng. 178(3–4), 257–262(1999).

J.H. He, A coupling method of a homotopy technique and a perturbation technique for Non-linear problems. Int. J. Non Linear Mech. 35(1), 37–43 (2000).

J.H. He, Homotopy perturbation method for solving boundary value problems. Phys.

Lett. A 350(1–2),87–88 (2006). doi:10.1016/j.physleta.2005.10.005

A. Eswari, L. Rajendran, Analytical solution of steady state current at a microdisk

Biosensor, J. Electroanal. Chem., 641 (2010) 35-44.

S. Saravanakumar, B. Sridevi, A. Eswari, L. Rajendran, Analysis of empirical fuzzy rate equations in homogeneous reaction diffusion Thomas model, TAGA JOURNAL VOL. 14 ISSN: 1748-0345.



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