Abstract
The cyclic nature of blooms is a very interesting characteristic of marine plankton ecosystem. Release of toxins by some phytoplanktons has an important role on planktonic interactions and hence on regulating the blooms. A mathematical model describing the phytoplankton-zooplankton interaction with these characteristics is studied. The time needed for liberation of toxins by phytoplanktons is considered. To account for the spatial heterogeneity of an aquatic environment, diffusivity of different plankton populations is also incorporated into the system. Stability and bifurcation behaviour of different steady states are analysed.
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Duinker, J. and Wefer, G.: Das CO2-Problem und die Rolle des Ozeans, Naturwissenschahten 81 (1994), 237–242.
Provasoli, L.: Recent progress: An overview, in Taylor and Seliger (eds.), Toxic Dinoflagellate Blooms, Elsevier/North-Holland, 1978, p. 1.
Nishijima, T. and Hata, Y.: The dynamics of vitamin B12 and its relation to the outbreak of Chattonella red tides in Harima Nada, the Seto inland sea, in T. Okaichi (ed.), Red Tides: Biology, Environmental Science and Toxicology, Elsevier, Amsterdam, 1989, p. 257.
Truscott, J.E. and Brindley, J.: Ocean Plankton Populations as Excitable Media, Bull. Math. Biol. 56 (1994), 981–998.
Almeida Machado, P.: Dinoflagellate blooms on the Brazilian South Atlantic coast, in Taylor and Seligers (eds.), Toxic Dinoflagellate Blooms, Elsevier, 1978, p. 29.
Work, T.M. et al.: Domonic acid in toxication of brown pelicans and cormorants in Santa Cruz, California, in T.J. Smayda and Y. Shimuza (eds.), Toxic Phytoplankton Blooms in the Sea, Vol. 3, Elsevier, 1993, pp. 643–649.
Steidinger, K.A. et al.: Pfiesteria Piscicida a New Toxic Dinoflagellate Genus and Species of the Order Dinamoebales, J. Phycol. 32 (1996), 157–164.
Nielsen, T.G. et al.: Effects of a Chrysochromulina polylepis Subsurface Bloom on the Plankton Community, Mar. Ecol. Prog. Ser. 62 (1990), 21–35.
Aure, J. and Rey, F.: Oceanographic Conditions in the Sandsfjord System, Western Norway, After a Bloom of the Prymnesiophyte Prymnesium parvum Carter in August 1990, Sarsia 76 (1992), 247–254.
Hallegraeff, G.M.: A Review of Harmful Algae Blooms and the Apparent Global Increase, Phycologia 32 (1993), 79–99.
Keating, K.I.: Algal metabolite influence on bloom sequence in eutrophic freshwater ponds. E.P.A. Ecological Monograph Series (EPA. 600/3-76-081. Washington, DC), 1976, p. l48.
Kirk, K. and Gilbert, J.: Variations in Herbivore Response to Chemical Defences: Zooplankton Foraging on Toxic Cyanobacteria, Ecology 73 (1992), 2208.
Fay, P.: The Blue-Greens, Arnold, London, 1983.
Buskey, E.J. and Stockwell, D.A.: Effects of a persistent ‘brown tide’ on zooplankton population in the Laguna Madre of Southern Texas, in T.J. Smayda and Y. Shimuzu (eds.), Toxic Phytoplankton Blooms in the Sea. Elsevier, Amsterdam, 1993, pp. 659–666.
Estep, K.W. et al.: Predation by Copepods upon Natural Populations of Phaeocystis pouchetii as a Function of the Physiological State of the Prey, Mar. Ecol. Prog. Ser. 67 (1990), 235–249.
Hansen, F.C.: Tropic Interaction Between Zooplankton and Phaeocystis cf. Globosa., Helgol Meeresunters 49, 283–293.
Huntley, M.E. et al.: Chemically Mediated Rejection of Dinoflagellate Prey by the Copepods Calanus pacificus and Paracalanus parvus: Mechanism, Occurance and Significance, Mar. Ecol. Prog. Ser. 28 (1986), 105–120.
Buskey, E.J. and Hyatt, C.J.: Effects of the Texas (USA) Brown Tide Alga on Planktonic Grazers, Mar. Ecol. Prog. Ser. 126 (1995), 285–292.
Nejstgaard, J.C. and Solberg, P.T.: Repression of Copepod Feeding and Fecundity by the Toxic Haptophyte Prymnesium patelliferum, Sarsia 81 (1996), 339–344.
Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer, Dordrecht, 1992.
Cushing, J.M.: Integrodifferential Equations and Delay Models in Population Dynamics, Lect. Notes in Biomath., Vol. 20, Springer-Verlag, Berlin, 1977.
Kuang, Y.: Delay-Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
MacDonald, N.: Biological Delay Systems: Linear Stability Theory, Cambridge University Press, New York, 1989.
Chattopadhyay, J. et al.: A Delay Differential Equation Model on Harmful Algal Blooms in the Presence of Toxic Substances, IMA. J. Math. Appl. Med. Biol. 19 (2002), 137–161.
Steele, J.H.: Stability of plankton ecosystem, in M.B. Usher and M.H. Williams (eds.), Ecological Stability, Chapman and Hall, London, 1974.
Sjoberge, S.: Are Pelagic Systems Inherently Unstable? A Model Study, Ecol. Model. 3 (1977), 17–37.
Okubo, A.: Diffusion and Ecological Problems: Modern Perspective, Interdisciplinary Applied Mathematics., Vol. 14, Springer, Berlin, 2001.
Murray, J.D.: Mathematical Biology, Springer, Berlin, 2002.
Edwards, A.M. and Brindley, J.: Zooplankton Mortality and the Dynamical Behaviour of Plankton Population Model, Bull. Math. Biol. 61 (1999), 303.
Beltrami, E. and Carroll, T.O.: Modelling the Role of Viral Diseases in Recurrent Phytoplankton Blooms, J. Math. Biol. 32 (1994), 857–863.
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Mukhopadhyay, B., Bhattacharyya, R. A Delay-Diffusion Model of Marine Plankton Ecosystem Exhibiting Cyclic Nature of Blooms. J Biol Phys 31, 3–22 (2005). https://doi.org/10.1007/s10867-005-2306-x
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DOI: https://doi.org/10.1007/s10867-005-2306-x