×

A delay differential equations model of plankton allelopathy. (English) Zbl 0946.92031

Summary: We have studied the dynamics of planktonic growth with special consideration on time dependent fluctuations in density of the species. We propose a modified delay differential equation model of the growth of two species of plankton having competitive and allelopathic effects on each other. The model system shows a stable limit cycle oscillation when the allelopathic effect is of a stimulatory nature.

MSC:

92D40 Ecology
34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E.L. Rice, Allelopathy, 2nd ed., Academic Press, New York, 1984; E.L. Rice, Allelopathy, 2nd ed., Academic Press, New York, 1984
[2] Pratt, R., Influence of the size of the inoculum on the growth of Chlorella vulgaris in freshly prepared culture medium, Am. J. Bot., 27, 52 (1940)
[3] Pratt, R.; Fong, J., Studies on Chlorella vulgaris. II. Further evidence that Chlorella cells form a growth-inhibiting substance, Am. J. Bot., 27, 431 (1940)
[4] Rodhe, W., Enviromental requirements of fresh-water plankton algae, Symb. Bot. Ups., 10, 1 (1948)
[5] Rice, T. R., Biotic influences affecting population growth of planktonic algae, US Fish Wildl. Serv., Fish Bull., 54, 227 (1954)
[6] Pratt, R., Studies on Chlorella vulgaris. V. Some properties of the growth inhibitor formed by Chlorella cells, Am. J. Bot., 29, 142 (1942)
[7] Pratt, R., Studies on Chlorella vulgaris. IX. Influence on the growth of Chlorella of continuous removal of chlorellin from the solution, Am. J. Bot., 31, 418 (1944)
[8] Pratt, R., Studies on Chlorella vulgaris. XI. Relation between surface tension and accumalation of chlorellin, Am. J. Bot., 35, 634 (1948)
[9] Lefevre, M.; Nisbet, M., Sur la sécrétion par certaines espéces d’algues de substances inhibitrices d’autres espéces d’algues, C.R. Acad. Sci., 226, 107 (1948)
[10] Lefevre, M.; Nisbet, M.; Jakob, H., Action des substances excretées, en culture, par certaines espéces d’algues, sur le metabolisme d’autres espéces d’algues, Verh. Int. Ver. Limnol., 10, 259 (1949)
[11] Lefevre, M.; Jakob, H.; Nisbet, M., Sur la secretion, par certaines cyanophytes, de substances algostatiques dans les collections d’eau naturelles, C.R. Acad. Sci., 230, 2226 (1950)
[12] Lefevre, M.; Jakob, H.; Nisbet, M., Auto-et heteroantagonisme chez les algues d’eau douce, Ann. Stn. Cent. Hydrobiol. Appl., 4, 5 (1952)
[13] Jakob, H., Compatibilitiés et antagonismes entre algues du sol, C.R. Acad. Sci., 238, 928 (1954)
[14] Denffer, D., Über einen Wachstumhemmstoff in alternden Diatomeenkulturen, Biol. Zentralb., 67, 7 (1948)
[15] Fogg, G. E.; Westlake, D. F., The importance of extracellular products of algae in freshwater, Int. Ver. Theor. Angew. Limnol. Mitt., 12, 219 (1955)
[16] Fogg, G. E.; Nalewajko, C.; Watt, W. D., Extracellular products of phytoplankton photosynthesis, Proc. R. Soc. London, Ser. B, 162, 517 (1965)
[17] Jørgensen, E., Growth inhibiting substances formed by algae, Physiol. Plant., 9, 712 (1956)
[18] Proctor, V. W., Some controlling factors in the distribution of Haematococcus pluvialis, Eclogy, 38, 457 (1957)
[19] Proctor, V. W., Studies of algal antibiosis using Haematococcus and Chlamydomonas, Limnol. Oceanogr., 2, 125 (1957)
[20] McCraken, M. D.; Middaugh, R. E.; Middaugh, R. S., A chemical characterization of an algal inhibitor obtained from Chlamydomonas, Hydrobiologia, 70, 271 (1980)
[21] Pratt, D. M., Competition between Skeletonema costatum and Olisthodiscus luteus in Narragansett Bay and in culture, Limnol. Oceanogr., 11, 447 (1966)
[22] Nozawa, K., The effect of Peridinium toxin on other algae, Bull. Misaki Mar. Biol. Inst., Kyoto Univ., 12, 21 (1968)
[23] Martin, D. F.; Kutt, E. C.; Kim, Y. S., Use of a multiple diffusion chamber unit in culture studies. Application to Gomphosphaeria aponina, Environ. Lett., 7, 39 (1974)
[24] Berglund, H., Stimulation of growth of two marine algae by organic substances excreted by Enteromorpha linza in unialgal and axenic cultures, Physiol. Plant., 22, 1069 (1969)
[25] Maksimova, I. V.; Pimenova, M. N., Liberation of organic acids by green unicellular algae, Microbiology (Transl. of Mikrobiologiya), 38, 64 (1969)
[26] Fitzgerald, G. P., Some factors in competition or antagonism among bacteria, algae, and aquatic weeds, J. Phycol., 5, 351 (1969)
[27] Monahan, T. J.; Trainor, F. R., Stimulatory properties of filtrate from the green alga Hormotila blennista. I. Description, J. Phycol., 6, 263 (1970)
[28] Harris, D. O., Growth inhibitor produced by the green algae (Volvocaceae), Arch. Mikrobiol., 76, 47 (1971)
[29] Harris, D. O., A model system for the study of algal growth inhibitors, Arch. Protistenkd., 113, 230 (1971)
[30] Harris, D. O., Inhibition of oxygen evolution in Volvox globator by culture filtrates from Pandorina morum, Microbios, 3, 73 (1971)
[31] Harris, D. O.; Caldwell, C. D., Possible mode of action of a photosynthetic inhibitor produced by Pandorina morum, Arch. Microbiol., 95, 193 (1974)
[32] Harris, D. O.; Parekh, M. C., Further observations on an algicide produced by Pandorina morum, a colonial green flagellate, Microbios, 9, 259 (1974)
[33] K.M. Khailov, Ecological Metabolism in the Sea (in Russian), Naukova Dumka, Kiev, 1971; K.M. Khailov, Ecological Metabolism in the Sea (in Russian), Naukova Dumka, Kiev, 1971
[34] K.M. Khailov (Ed.), Biochemical Trophodynamics in Marin Coastal Ecosystem (in Russian), Naukova Dumka, Kiev, 1974; K.M. Khailov (Ed.), Biochemical Trophodynamics in Marin Coastal Ecosystem (in Russian), Naukova Dumka, Kiev, 1974
[35] Murphy, T. P.; Lean, D. R.S.; Nalewajko, C., Blue green algae: Their excretion of Feselective chelators enables them to dominate other algae, Science, 192, 900 (1976)
[36] Huntsman, S. A.; Barber, R. T., Modification of phtoplankton growth by excreted compounds in low-density populations, J. Phycol., 11, 10 (1975)
[37] Keating, K. I., Allelopathic influence on blue-green bloom sequence in a eutrophic lake, Science, 196, 885 (1977)
[38] Keating, K. I., Blue-green algal inhibition of diatom growth: Trasnsition from mesotrophic to eutropic community structure, Science, 199, 971 (1978)
[39] Wolfe, J. M.; Rice, E. L., Allelopathic interaction between algae, J. Chem. Ecol., 5, 533 (1979)
[40] Sharp, J. H.; Underhill, P. A.; Hughes, D. J., Interaction (allelopathy) between marine diatoms: Thalassiosira pseudonana and Phaeodactylum tricornutum, J. Phycol., 15, 353 (1979)
[41] Mason, C. P.; Gleason, F. K., An antibiotic from Scytonema hofmanni (Cyanophyta), J. Phycol. (suppl.), 17, 8 (1981)
[42] J.A. Hellebust, in: W.D.P. Stewart (Ed.), Extracellular Products in Algal Physiology and Biochemistry, University of California, Berkeley, 1974, p. 838; J.A. Hellebust, in: W.D.P. Stewart (Ed.), Extracellular Products in Algal Physiology and Biochemistry, University of California, Berkeley, 1974, p. 838
[43] Porter, J. W.; Targitt, N. M., Sponge (Planktoris) kill a coral through water borne allelochemic, Biol. Bull., 175, 230 (1988)
[44] Gupta, N. N.; Houdeshell, J., A differential-difference equations model of dynamic aquatic ecosystem, Int. J. Syst. Sci., 7, 481 (1976)
[45] J. Maynard-Smith, Models in Ecology, Cambridge University, Cambridge, 1974, p. 146; J. Maynard-Smith, Models in Ecology, Cambridge University, Cambridge, 1974, p. 146
[46] Chattopadhyay, J., Effect of toxic substances on a two-species competitive system, Ecol. Modelling, 84, 287 (1996)
[47] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kulwer Academic, Dordrecht, 1992; K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kulwer Academic, Dordrecht, 1992 · Zbl 0752.34039
[48] Freedman, H. I.; Sree Hari Rao, V., The trade-off between mutual interference and time lags in predator-prey-systems, Bull. Math. Biol., 45, 991 (1983) · Zbl 0535.92024
[49] Thingstad, T. F.; Langeland, T. I., Dynamics of chemostat culture: The effect of a delay in cell response, J. Theoret. Biol., 48, 149 (1974)
[50] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993; Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993 · Zbl 0777.34002
[51] J. Hale, Theory of Functional Differential Equations, Springer, New York, 1977; J. Hale, Theory of Functional Differential Equations, Springer, New York, 1977 · Zbl 0352.34001
[52] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University, Cambridge, 1981; B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University, Cambridge, 1981
[53] Beltrami, E.; Carroll, T. O., Modeling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32, 857 (1994) · Zbl 0825.92122
[54] Akehurst, S. C., Observations on pond life, with special reference to the possible causation of swarming of phytoplankton, R. Microsc. Soc. J., 51, 237 (1931)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.