Nonlinear dynamics and chaotic and complex systems constitute some of the most fascinating developments of late twentieth century mathematics and physics. The implications have changed our understanding of important phenomena in almost every field of science, including biology and ecology. This article investigates complexity and chaos in the spatiotemporal dynamics of aquatic ecosystems. The dynamics of these biological communities exhibit an interplay between processes acting on a scale from hundreds of meters to kilometers, controlled by biology, and processes acting on a scale from dozens to hundreds of kilometers, dominated by the heterogeneity of hydrophysical fields. We focus on how biological processes affect spatiotemporal pattern formation. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal plankton dynamics, fractal properties of planktivorous fish school movements, and their interrelationships.

MSC codes

  1. 82C41
  2. 92B05
  3. 92D25
  4. 92D40


  1. chaos
  2. order
  3. scaling
  4. aquatic ecosystems
  5. predator-prey interaction
  6. modeling

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S. Levin, T. Powell, J. Steele, Patch dynamics, Lecture Notes in Biomathematics, Vol. 96, Springer‐Verlag, 1993xiv+307
M. R. Abbott and P. M. Zion (1985), Satellite observations of phytoplankton variability during an upwelling event, Continental Shelf Research, 4, pp. 661–680.
E. R. Abraham (1998), The generation of plankton patchiness by turbulent stirring, Nature, 391, pp. 577–580.
J. Adler (1966), Chemotaxis in bacteria, Science, 153, pp. 708–716.
J. Adler and B. Templeton (1967), The effect of environmental conditions on the motility of Escherichia coli, J. Gen. Microbiol., 46, pp. 175–184.
W. C. Allee (1931), Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago.
W. C. Allee et al. (1949), Principles of Animal Ecology, Saunders, Philadelphia.
J. C. Allen, W. M. Schaffer, and D. Rosko (1993), Chaos and extinction in ecological populations, Nature, 364, pp. 229–232.
W. Alt and G. Hoffmann, eds. (1990), Biological Motion, Lecture Notes in Biomath. 89, Springer‐Verlag, Berlin.
M. V. Angel and M. J. R. Fasham (1983), Eddies and biological processes, in Eddies in Marine Science, A. R. Robinson, ed., Springer‐Verlag, New York, pp. 492–524.
L. Armi and W. Zenk (1984), Large lenses of highly saline Mediterranean water, J. Phys. Oceanogr., 14, pp. 1560–1576.
L. Armi et al. (1988), The history and decay of a Mediterranean salt lens, Nature, 333, pp. 649–651.
A. Arnéodo, E. Bacry, and J. F. Muzy (1995), The thermodynamics of fractals revisited with wavelets, Phys. A, 213, pp. 232–275.
A. Arnéodo et al. (1996), Wavelet based fractal analysis of DNA sequences, Phys. D, 96, pp. 291–320.
F. A. Ascioti, E. Beltrami, T. O. Carroll, and C. Wirick (1993), Is there chaos in plankton dynamics?, J. Plankt. Res., 15, pp. 603–617.
E. Bacry, J. Muzy, A. Arneodo, Singularity spectrum of fractal signals from wavelet analysis: exact results, J. Statist. Phys., 70 (1993), 635–674
R. C. Bain, Jr. (1968), Predicting DO variations caused by algae, J. Sanitary Engineering Division, Proceedings of the American Society of Civil Engineers, pp. 867–881.
Grigory Barenblatt, Scaling, self‐similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, Vol. 14, Cambridge University Press, 1996xxii+386, With a foreword by Ya. B. Zeldovich
G. I. Barenblatt, N. L. Galerkina, and M. V. Luneva (1987), Evolution of a localized turbulent burst, Inzhenerno‐Fizicheskii Zhurnal, 53, pp. 733–740.
H. J. G. Baretta‐Bekker, E. K. Duursma, and B. R. Kuipers, eds. (1998), Encyclopedia of Marine Sciences, Springer‐Verlag, Berlin.
A. H. Barnard, P. M. Stegmann, and J. A. Yoder (1997), Seasonal surface variability in the South Atlantic Bight derived from CZCS and AVHRR imagery, Continental Shelf Research, 17, pp. 1181–1206.
J. A. Barth (1989), Stability of a coastal upwelling front. I. Model development and a stability theorem, J. Geophys. Res., 94, pp. 10844–10856.
M. J. Behrenfeldt and P. G. Falkowski (1997), A consumer’s guide to phytoplankton primary productivity models, Limnology and Oceanography, 42, pp. 1479–1491.
E. Beltrami (1989), A mathematical model of the brown tide, Estuaries, 12, pp. 13–17.
E. Beltrami (1996), Unusual algal blooms as excitable systems: The case of “brown‐tides,” Environmental Modeling and Assessment, 1, pp. 19–24.
E. Ben‐Jacob, H. Shmueli, O. Shochet, and A. Tenenbaum (1992), Adaptive self‐organization during growth of bacterial colonies, Phys. A, 87, pp. 378–424.
V. N. Biktashev et al. (1998), Excitation wave breaking in excitable media with linear shear flow, Phys. Rev. Lett., 81, pp. 2815–2818.
R. W. Blake (1983), Fish Locomotion, Cambridge University Press, Cambridge.
L. N. Bocharov (1990), Systems Analysis in Short‐Term Fishery Forecasts, Nauka, Leningrad.
M. C. Boerlijst, M. E. Lamers, and P. Hogeweg (1993), Evolutionary consequences of spiral waves in a host parasitoid system, Proc. Roy. Soc. Lond. B, 253, pp. 15–18.
M. J. Bowman, S. M. Chiswell, P. L. Lapennas, and R. A. Murtagh (1983), Coastal upwelling, cyclogenesis and squid fishing near Cape Farewell, New Zealand, in Coastal Oceanography, H. G. Gade, ed., NATO Conf. Ser. IV: Marine Sciences, Plenum Press, New York, pp. 279–310.
A. Bucklin (1991), Population genetic responses of the planktonic copepod Metridia pacifica to a coastal eddy in the California Current, J. Geophys. Res., 96, pp. 14799–14808.
J. E. Capella, L. B. Quetin, E. E. Hofmann, and R. M. Ross (1992), Models of the early life history of Euphausia superba. Part II. Lagrangian calculations, Deep‐Sea Res., 39, pp. 1201–1220.
R. J. Charlson, J. E. Lovelock, M. O. Andreae, and S. G. Warren (1987), Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate, Nature, 326, pp. 655–661.
A. B. Chhabra, C. Meneveau, R. V. Jensen, and K. R. Sreenivasan (1987), Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence, Phys. Rev. A, 40, pp. 5284–5294.
The Coastal Transition Zone Group (1988), The coastal transition zone program, EOS, 69, pp. 669–707.
L. H. N. Cooper (1961), Vertical and horizontal movements in the ocean, in Oceanography, AAAS, Washington, D.C., pp. 599–622.
D. H. Cushing (1975), Marine Ecology and Fisheries, Cambridge University Press, Cambridge.
K. L. Daly and W. O. Smith, Jr. (1993), Physical‐biological interactions influencing marine plankton production, Ann. Rev. Ecological Systems, 24, pp. 555–585.
Guy David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics, Vol. 1465, Springer‐Verlag, 1991x+107
F. Davidson, Chaotic wakes and other wave‐induced behavior in a system of reaction‐diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 1303–1313
P. de Kepper, V. Castets, E. Dulos, and J. Boissonade (1991), Turing‐type chemical patterns in the chlorite‐iodide‐malonic acid reaction, Phys. D, 49, pp. 161–169.
D. L. DeAngelis (1992), Dynamics of Nutrient Cycling and Food Webs, Chapman and Hall, London.
K. L. Denman (1976), Covariability of chlorophyll and temperature in the sea, Deep‐Sea Res., 23, pp. 539–550.
F. Doveri et al. (1993), Seasonality and chaos in a plankton‐fish model, Theoret. Population Biology, 43, pp. 159–183.
M. R. Droop (1983), 25 years of algal growth kinetics, Botanica Marina, 26, pp. 99–112.
D. Dubois (1975), A model of patchiness for prey‐predator plankton populations, Ecological Modelling, 1, pp. 67–80.
J. Duinker and G. Wefer (1994), Das CO2‐Problem und die Rolle des Ozeans, Naturwissenschaften, 81, pp. 237–242.
Steven Dunbar, Traveling waves in diffusive predator‐prey equations: periodic orbits and point‐to‐periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057–1078
W. Ebeling and L. Schimansky‐Geier (1980), Nonequilibrium phase transitions and nucleation in reacting systems, in Proceedings of the 6th International Conference on Thermodynamics, Merseburg, pp. 95–100.
W. Ebenhöh (1980), A model of the dynamics of plankton patchiness, Modeling, Identification and Control, 1, pp. 69–91.
G. T. Evans and S. Parslow (1985), A model of annual plankton cycles, Biological Oceanography, 3, pp. 327–347.
M. J. R. Fasham (1978), The statistical and mathematical analysis of plankton patchiness, Oceanography and Marine Biology Ann. Rev., 16, pp. 43–79.
Jens Feder, Fractals, Physics of Solids and Liquids, Plenum Press, 1988xxvi+283, With a foreword by Benoit B. Mandelbrot
K. N. Fedorov (1983), Physical Nature and Structure of the Ocean Fronts, Gidrometeoizdat, Leningrad.
K. N. Fedorov and A. I. Ginzburg (1988), The Subsurface Layer of the Ocean, Gidrometeoizdat, Leningrad.
K. N. Fedorov and A. I. Ginzburg (1989), Mushroom‐like currents (vortex dipoles): One of the most widespread forms of non‐stationary coherent motions in the ocean, in Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, Elsevier Oceanography Ser. 50, J. C. J. Nihoul and B. M. Jamart, eds., Elsevier, Amsterdam, pp. 1–15.
A. Fernö et al. (1998), The challenge of the herring in the Norwegian sea: Making optimal collective spatial decisions, SARSIA, 83, pp. 149–167.
P. C. Fiedler and H. J. Bernard (1987), Tuna aggregation and feeding near fronts observed in satellite imagery, Continental Shelf Research, 7, pp. 871–881.
R. Fild, Mária Burger, R. Fild, Mária Burger, R. Fild, Mária Burger, Kolebaniya i begushchie volny v khimicheskikh sistemakh, “Mir”, 1988, 720–0, Translated from the English by A. B. Rovinskii˘ and V. R. Fed’kina; Translation edited by A. M. Zhabotinskii˘
R. A. Fisher (1937), The wave of advance of advantageous genes, Annals of Eugenics, 7, pp. 355–369.
R. H. Fleming (1939), The control of diatom populations by grazing, Journal du Conseil Permanent International pour l’Exploration de la Mer, 14, pp. 210–227.
G. Flierl, D. Grünbaum, S. Levin, and D. Olson (1999), From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biology, 196, pp. 397–454.
T. D. Foster (1974), The hierarchy of convection, in Processus de Formation des eaux Oceaniques Profondes en Particulier en Mediterranee Occidentale, Colloques Intern. du CNRS 115, Paris, pp. 237–241.
P. J. S. Franks (1997), Spatial patterns in dense algal blooms, Limnology and Oceanography, 42, pp. 1297–1305.
A. Garfinkel, M. Spano, W. Ditto, and J. Weiss (1992), Controlling cardiac chaos, Science, 257, pp. 1230–1235.
G. Gerisch (1968), Cell aggregation and differentiation in Dictyostelium, Current Topics in Developmental Biology, Vol. 3, A. A. Moscona and A. Monroy, eds., Academic Press, New York.
G. Gerisch (1971), Periodische Signale steuern die Musterbildung in Zellverbänden, Naturwissenschaften, 58, pp. 430–438.
C. Godfray and M. Hassell (1997), Chaotic beetles, Science, 275, pp. 323–326.
R. M. Goodwin (1967), A growth cycle, in Socialism, Capitalism and Economic Growth, C. H. Feinstein, ed., Cambridge University Press, Cambridge, pp. 54–58.
C. H. Greene et al. (1992), The migration behavior, fine structure, and bioluminescent activity of krill sound–scattering layer, Limnology and Oceanography, 37, pp. 650–658.
K.‐U. Grusa, Mathematical analysis of nonlinear dynamic processes, Pitman Research Notes in Mathematics Series, Vol. 176, Longman Scientific & Technical, 1988xiv+450, An introduction to processes governed by partial differential equations
Daniel Grünbaum, Akira Okubo, Modelling social animal aggregations, Lecture Notes in Biomath., Vol. 100, Springer, Berlin, 1994, 296–325
S. Gueron, S. A. Levin, and D. I. Rubenstein (1996), The dynamics of herds: From individuals to aggregations, J. Theoret. Biology, 182, pp. 85–98.
J. A. Gulland, ed. (1977), Fish Population Dynamics, Wiley, London.
Hermann Haken, Synergetics—an introduction, Springer‐Verlag, 1977xii+325, Nonequilibrium phase transitions and self‐organization in physics, chemistry and biology
G. M. Hallegraeff (1988), Plankton: A Microscopic World, E. J. Brill, Leiden.
A. Hastings (1993), Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations, Ecology, 74, pp. 1362–1372.
L. R. Haury et al. (1986), Biological consequences of a recurrent eddy off Point Conception, California, J. Geophys. Res., 91, pp. 12937–12956.
T. L. Hayward and A. W. Mantyla (1990), Physical, chemical and biological structure of a coastal eddy near Cape Mendocino, J. Marine Res., 48, pp. 825–850.
V. Hensen (1892), Ergebnisse der in dem Atlantischen Ocean von Mitte Juli bis bis Anfang November 1889 ausgeführten Plankton‐Expedition der Humboldt‐Stiftung, Kiel und Leipzig, 1892.
T. Höfer, J. A. Sherratt, and P. K. Maini (1995), Cellular pattern formation during Dictyostelium aggregation, Phys. D, 85, pp. 425–444.
E. E. Hofmann, J. E. Capella, R. M. Ross, and L. B. Quetin (1992), Models of the early life history of Euphausia superba. Part I. Time and temperature dependence during the descent‐ascent cycle, Deep‐Sea Res., 39, pp. 1177–1200.
John Holland, Adaptation in natural and artificial systems, University of Michigan Press, 1975ix+183, An introductory analysis with applications to biology, control, and artificial intelligence
C. S. Holling (1959), Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91, pp. 385–398.
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.
R. R. Hood, M. R. Abbott, A. Huyer, and P. M. Kosro (1990), Surface patterns in temperature, flow, phytoplankton biomass and species composition in the coastal transition zone off northern California, June to August 1988, J. Geophys. Res., 95, pp. 18081–18094.
J. Huisman and F. J. Weissing (1999), Biodiversity of plankton by species oscillations and chaos, Nature, 402, pp. 407–410.
A. Huth and C. Wissel (1994), The simulation of fish schools in comparison with experimental data, Ecological Modelling, 75/76, pp. 135–145.
S. Ikegami, I. Imai, J. Kato, and H. Ohtake (1995), Chemotaxis toward inorganic phosphate in the red tide alga Chattonella antiqua, J. Plankt. Res., 17, pp. 1587–1591.
G. R. Ivanitsky, A. B. Medvinsky, and M. A. Tsyganov (1991), From disorder to order as applied to the movement of microorganisms, Sov. Phys. Usp., 34, pp. 289–316.
G. R. Ivanitsky, A. B. Medvinsky, and M. A. Tsyganov (1994), From the dynamics of population autowaves generated by living cells to neuroinformatics, Physics‐Uspekhi, 37, pp. 961–989.
V. S. Ivlev (1945), Biologicheskaya produktivnost’ vodoemov, Uspekhi Sovremennoi Biologii, 19, pp. 98–120.
I. R. Jenkinson and B. A. Biddanda (1995), Bulk‐phase viscoelastic properties of seawater: Relationship with plankton components, J. Plankt. Res., 17, pp. 2251–2274.
S. E. Jørgensen (1994), Fundamentals of Ecological Modelling, Elsevier, Amsterdam.
Daniel Joseph, Stability of fluid motions. I, Springer‐Verlag, 1976xiii+282, Springer Tracts in Natural Philosophy, Vol. 27
J. P. Kahane and P. G. Lemarié‐Rieusset (1995), Fourier Series and Wavelets, Gordon and Breach, London.
V. M. Kamenkovich, M. N. Koshlyakov, and A. S. Monin (1987), Synoptic Eddies in the Ocean, 2nd ed., Gidrometeoizdat, Leningrad.
K. Kawasaki, A. Mochizuki, and N. Shigesada (1995), A mathematical model of pattern formation in a bacterial colony, Control & Measurement, 34, pp. 811–816 (in Japanese).
K. Kawasaki et al. (1995), Modeling spatio‐temporal patterns generated by Bacillus subtilis, J. Theoret. Biology, 188, pp. 177–185.
E. F. Keller and L. A. Segel (1970), Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biology, 26, pp. 399–415.
E. F. Keller and L. A. Segel (1971a), Model for chemotaxis, J. Theoret. Biology, 30, pp. 225–234.
E. F. Keller and L. A. Segel (1971b), Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biology, 30, pp. 235–248.
H. Kierstead and L. B. Slobodkin (1953), The size of water masses containing plankton blooms, J. Marine Res., 12, pp. 141–147.
A. Kolmogorov, I. Petrovskii, and N. Piskunov (1937), Étude de l’equation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin d’Université de Moscou, Serie Internationale, Section A, 1, pp. 1–25.
N. Kopell and L. N. Howard (1973), Plane wave solutions to reaction‐diffusion equations., Stud. Appl. Math., 52, pp. 291–328.
A. G. Kostianoy and I. M. Belkin (1989), A survey of observations on intrathermocline eddies in the world ocean, in Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, Elsevier Oceanography Ser. 50, J. C. J. Nihoul and B. M. Jamart, eds., Elsevier, Amsterdam, pp. 821–841.
V. I. Krinsky, A. B. Medvinsky, and A. V. Panfilov (1986), Evolution of Autowave Vortices: Waves in the Heart, 1st ed., Znanie, Moscow.
Yu. Kuznetsov, S. Muratori, S. Rinaldi, Bifurcations and chaos in a periodic predator‐prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 117–128
R. M. Laurs, P. C. Fiedler, and D. R. Montgomery (1984), Albacore tuna catch distributions relative to environmental features observed from satellites, Deep‐Sea Res., 31, pp. 1085–1099.
S. Levin, T. Powell, J. Steele, Patch dynamics, Lecture Notes in Biomathematics, Vol. 96, Springer‐Verlag, 1993xiv+307
S. A. Levin and L. A. Segel (1976), Hypothesis for origin of planktonic patchiness, Nature, 259, p. 659.
S. Levin, T. Powell, J. Steele, Patch dynamics, Lecture Notes in Biomathematics, Vol. 96, Springer‐Verlag, 1993xiv+307
M. A. Lewis and P. Kareiva (1993), Allee dynamics and the spread of invading organisms, Theoret. Population Biology, 43, pp. 141–158.
B. L. Li (2000), Fractal geometry applications in description and analysis of patch patterns and patch dynamics, Ecological Modelling, 132, pp. 33–50.
B. L. Li and C. Loehle (1995), Wavelet analysis of multiscale permeabilities in the subsurface, Geophys. Res. Lett., 22, pp. 3123–3126;
Correction, Geophys. Res. Lett., 23, pp. 1059–1061.
L. Lipsitz and A. Goldberger (1992), Loss of complexity and aging: Potential applications of fractals and chaos theory to senescence, J. Amer. Medical Association, 267, pp. 1806–1809.
Hans‐Walter Lorenz, Nonlinear dynamical economics and chaotic motion, Springer‐Verlag, 1993xvi+319
A. J. Lotka (1925), Elements of Physical Biology, Williams and Wilkins, Baltimore, MD.
D. Ludwig, D. D. Jones, and C. S. Holling (1978), Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Animal Ecology, 47, pp. 315–332.
R. Luther (1906), Räumliche Ausbreitung chemischer Reaktionen, Zeitschrift für Elektrochemie, 12, pp. 596–600.
D. L. Mackas and C. M. Boyd (1979), Spectral analysis of zooplankton spatial heterogeneity, Science, 204, pp. 62–64.
D. L. Mackas, L. Washburn, and S. L. Smith (1991), Zooplankton community pattern associated with a California Current cold filament, J. Geophys. Res., 96, pp. 14781–14797.
B. R. MacKenzie, T. J. Miller, S. Cyr, and W. C. Leggett (1994), Evidence for a dome‐shaped relationship between turbulence and larval fish ingestion rates, Limnology and Oceanography, 39, pp. 1790–1799.
H. Malchow (1993), Spatio‐temporal pattern formation in nonlinear nonequilibrium plankton dynamics, Proc. Roy. Soc. Lond. B, 251, pp. 103–109.
H. Malchow (1994), Nonequilibrium structures in plankton dynamics, Ecological Modelling, 75/76, pp. 123–134.
H. Malchow (1995), Flow‐ and locomotion‐induced pattern formation in nonlinear population dynamics, Ecological Modelling, 82, pp. 257–264.
H. Malchow (1996), Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system, J. Marine Systems, 7, pp. 193–202.
H. Malchow (1998), Flux‐induced instabilities in ionic and population‐dynamical interaction systems, Zeitschrift für Physikalische Chemie, 204, pp. 35–107.
H. Malchow (2000a), Motional instabilities in prey‐predator systems, J. Theoret. Biology, 204, pp. 639–647.
H. Malchow (2000b), Nonequilibrium spatio‐temporal patterns in models of nonlinear plankton dynamics, Freshwater Biology, 45, pp. 239–251.
H. Malchow, S. V. Petrovskii, and A. B. Medvinsky (2002), Numerical study of plankton‐fish dynamics in a spatially structured and noisy environment, Ecological Modelling, 149, pp. 247–255.
H. Malchow and L. Schimansky‐Geier (1985), Noise and Diffusion in Bistable Nonequilibrium Systems, Teubner‐Texte zur Physik 5, Teubner‐Verlag, Leipzig.
H. Malchow and N. Shigesada (1994), Nonequilibrium plankton community structures in an ecohydrodynamic model system, Nonlinear Processes in Geophys., 1, pp. 3–11.
Horst Malchow, Birgit Radtke, Malaak Kallache, Alexander Medvinsky, Dmitry Tikhonov, Sergei Petrovskii, Spatio‐temporal pattern formation in coupled models of plankton dynamics and fish school motion, Nonlinear Anal. Real World Appl., 1 (2000), 53–67, Spatial heterogeneity in ecological models (Alcalá de Henares, 1998)
G. Malin (1997), Sulphur, climate and the microbial maze, Nature, 387, pp. 857–859.
Benoit Mandelbrot, Fractals: form, chance, and dimension, W. H. Freeman and Co., 1977xvi+365, Translated from the French
Benoit Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., 1982v+460, Schriftenreihe für den Referenten. [Series for the Referee]
T. Marks‐Tarlow (1999), The self as a dynamical system, Nonlinear Dynamics, Psychology, and Life Sciences, 3, pp. 311–345.
C. Marrase, E. Saiz, and J. M. Redondo, eds. (1997), Lectures on plankton and turbulence, Scientia Marina, 61 (Supl. 1), pp. 1–238.
M. Matsushita and H. Fujikawa (1990), Diffusion‐limited growth in bacterial colony formation, Phys. A, 168, pp. 498–506.
R. May (1974), Biological populations with nonoverlapping generations: Stable points, stable cycles and chaos, Science, 186, pp. 645–667.
P. Mayzaud and S. A. Poulet (1978), The importance of the time factor in the response of zooplankton to varying concentrations of naturally occurring particulate matter, Limnology and Oceanography, 23, pp. 1144–1154.
A. B. Medvinsky, V. Yu. Shakhbazian, M. A. Tsyganov, and G. R. Ivanitsky (1991), Formation of demarcation zones when bacterial population waves are drawn together, FEMS Microbiol. Lett., 84, pp. 279–284.
A. B. Medvinsky, D. A. Tikhonov, J. Enderlein, and H. Malchow (2000), Fish and plankton interplay determines both plankton spatio‐temporal pattern formation and fish school walks: A theoretical study, Nonlinear Dynamics, Psychology, and Life Sciences, 4, pp. 135–152.
A. B. Medvinsky et al. (1993a), Instability of waves formed by motile bacteria, FEMS Microbiol. Lett., 112, pp. 287–290.
A. B. Medvinsky et al. (1993b), Formation of stationary demarcation zones between population autowaves propagating towards each other, Phys. D, 64, pp. 267–280.
A. B. Medvinsky et al. (1994), Bacterial population autowave patterns: Spontaneous symmetry bursting, Phys. D, 79, pp. 299–305.
, Self‐organization of complex structures, Proceedings of the International Conference held in Berlin, September 24–28, 1995, Gordon and Breach Science Publishers, 1997, 0–0, xxiv+596, From individual to collective dynamics
A. B. Medvinsky et al. (2001), Patchy environment as a factor of complex plankton dynamics, Phys. Rev. E, 64, 021915.
J. H. Merkin, V. Petrov, S. K. Scott, and K. Showalter (1996), Wave‐induced chemical chaos, Phys. Rev. Lett., 76, pp. 546–549.
Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, 1990xii+215, Ondelettes. [Wavelets]
L. Michaelis and M. Menten (1913), Die Kinetik der Invertinwirkung, Biochemische Zeitschrift, 49, pp. 333–369.
M. Mimura, J. Murray, On a diffusive prey‐predator model which exhibits patchiness, J. Theoret. Biol., 75 (1978), 249–262
M. Mimura, H. Sakaguchi, and M. Matsushita (2000), Reaction‐diffusion modelling of bacterial colony patterns, Phys. A, 282, pp. 283–303.
O. A. Misund et al. (1998), Distribution, migration and abundance of Norwegian spring spawning herring in relation to the temperature and zooplankton biomass in the Norwegian Sea as recorded by coordinated surveys in spring and summer 1996, SARSIA, 83, pp. 117–127.
A. S. Monin, V. M. Kamenkovich, and V. G. Kort (1974), Variability of the World Ocean, Gidrometeoizdat, Leningrad.
A. S. Monin and R. V. Ozmidov (1981), Ocean Turbulence, Gidrometeoizdat, Leningrad.
A. S. Monin and V. P. Krasitskii (1985), Phenomena on the Ocean Surface, Gidrometeoizdat, Leningrad.
J. Monod and F. Jacob (1961), General conclusions: Teleonomic mechanisms in cellular metabolism, growth and differentiation, Cold Spring Harbor Symposia on Quantitative Biology. 26, pp. 389–401.
J. D. Murray (1977), Lectures on Nonlinear Differential‐Equation Models in Biology, 1st ed., Clarendon Press, Oxford.
J. Murray, Mathematical biology, Biomathematics, Vol. 19, Springer‐Verlag, 1989xiv+767
J. F. Muzy, E. Bacry, and A. Arnéodo (1993), Multifractal formalism for fractal signals: The structure‐function approach versus the wavelet‐transform modulus‐maxima method, Phys. Rev. E, 47, pp. 875–884.
K. Nakata and R. Ishikawa (1975), Fluctuation of local phytoplankton abundance in coastal waters, Japanese J. Ecology, 25, pp. 201–205.
Ali Nayfeh, Balakumar Balachandran, Applied nonlinear dynamics, Wiley Series in Nonlinear Science, John Wiley & Sons Inc., 1995xvi+685, Analytical, computational, and experimental methods; A Wiley‐Interscience Publication
P. C. Newell (1983), Attraction and adhesion in the slime mold Dictyostelium, in Fungal Differentiation. A Contemporary Synthesis, Mycology Ser. 43, J. E. Smith, ed., Marcel Dekker, New York, pp. 43–71.
G. Nicolis, I. Prigogine, Self‐organization in nonequilibrium systems, Wiley‐Interscience [John Wiley & Sons], 1977xii+491, From dissipative structures to order through fluctuations
J. C. J. Nihoul, ed. (1980), Marine Turbulence, Elsevier Oceanography Ser. 28, Elsevier, Amsterdam.
S. Levin, T. Powell, J. Steele, Patch dynamics, Lecture Notes in Biomathematics, Vol. 96, Springer‐Verlag, 1993xiv+307
A. Nitzan, P. Ortoleva, and J. Ross (1974), Nucleation in systems with multiple stationary states, Faraday Symposia of The Chemical Society, 9, pp. 241–253.
H.‐S. Niwa (1996), Newtonian dynamical approach to fish schooling, J. Theoret. Biology, 181, pp. 47–63.
E. G. Njoku, T. P. Barnett, R. M. Laurs, and A. C. Vastano (1985), Advances in satellite sea surface temperature measurements and oceanographic applications, J. Geophys. Res., 90, pp. 11573–11586.
J. J. O’Brien and J. S. Wroblewski (1973), On advection in phytoplankton models, J. Theoret. Biology, 38, pp. 197–202.
H. T. Odum (1956), Primary production in flowing waters, Limnology and Oceanography, 1, pp. 102–117.
A. Okubo (1971), Oceanic diffusion diagrams, Deep‐Sea Res., 18, pp. 789–802.
Akira Okubo, Diffusion and ecological problems: mathematical models, Biomathematics, Vol. 10, Springer‐Verlag, 1980xiii+254, An extended version of the Japanese edition, Ecology and diffusion; Translated by G. N. Parker
A. Okubo (1986), Dynamical aspects of animal grouping: Swarms, schools, flocks and herds, Adv. Biophys., 22, pp. 1–94.
R. V. Ozmidov (1966), Scales in Ocean Turbulence, Oceanology, 6, pp. 393–398.
R. V. Ozmidov (1968), Horizontal Turbulence and Turbulent Exchange in the Ocean, Nauka, Moscow.
R. V. Ozmidov (1986), Intermittence of hydrophysical factors and its impact on the spatial structure of hydrochemical and hydrobiological ocean fields, in Study of the Black Sea Pelagic Ecosystems, M. E. Vinogradov and R. V. Ozmidov, eds., SIO, Moscow.
R. V. Ozmidov (1998), Phytoplankton patches in the ocean under various regimes of the ocean turbulence, Oceanology, 38, pp. 7–15.
C. Pahl‐Wostl (1993), The influence of a hierarchy in time scales on the dynamics of, and the coexistence within, ensembles of predator‐prey pairs, Theoret. Population Biology, 43, pp. 159–183.
M. Pascual (1993), Diffusion‐induced chaos in a spatial predator‐prey system, Proc. Roy. Soc. Lond. B, 251, pp. 1–7.
T. Pedley, J. Kessler, Hydrodynamic phenomena in suspensions of swimming microorganisms, Annual Reviews, Palo Alto, CA, 1992, 313–358
J. Pedlosky (1987), Geophysical Fluid Dynamics, Springer‐Verlag, Berlin.
J. Pedlosky (1996), Oceanic Circulation Theory, Springer‐Verlag, Berlin.
Heinz‐Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and fractals, Springer‐Verlag, 1992xvi+984, New frontiers of science; With a foreword by Mitchell J. Feigenbaum Appendix A by Yuval Fisher Appendix B by Carl J. G. Evertsz and Benoit B. Mandelbrot
S. Petrovskii, Approximate determination of the magnitude of the critical size in the problem of the evolution of an impact, Inzh.‐Fiz. Zh., 66 (1994), 398–404
S. V. Petrovskii (1999), Plankton front waves accelerated by marine turbulence, J. Marine Systems, 21, pp. 179–188.
S. V. Petrovskii, M. E. Vinogradov, and A. Yu. Morozov (1998), Spatial‐temporal dynamics of a localized populational “burst” in a distributed prey‐predator system, Oceanology, 38, pp. 881–890.
S. Petrovskii, H. Malchow, A minimal model of pattern formation in a prey‐predator system, Math. Comput. Modelling, 29 (1999), 49–63
Sergei Petrovskii, Horst Malchow, Critical phenomena in plankton communities: KISS model revisited, Nonlinear Anal. Real World Appl., 1 (2000), 37–51, Spatial heterogeneity in ecological models (Alcalá de Henares, 1998)
S. V. Petrovskii and H. Malchow (2001a), Wave of chaos: New mechanism of pattern formation in spatiotemporal population dynamics, Theoret. Population Biology, 59, pp. 157–174.
Sergei Petrovskii, Horst Malchow, Spatio‐temporal chaos in an ecological community as a response to unfavourable environmental changes, Adv. Complex Syst., 4 (2001), 227–249
Sergei Petrovskii, Kohkichi Kawasaki, Fugo Takasu, Nanako Shigesada, Diffusive waves, dynamical stabilization and spatio‐temporal chaos in a community of three competitive species, Japan J. Indust. Appl. Math., 18 (2001), 459–481, Recent topics in mathematics moving toward science and engineering
O. M. Phillips (1977), The Dynamics of the Upper Ocean, 2nd ed., Cambridge University Press, Cambridge.
J. R. Platt (1961), Bioconvection patterns in cultures of free‐swimming organisms, Science, 133, pp. 1766–1767.
T. Platt (1972), Local phytoplankton abundance and turbulence, Deep‐Sea Res., 19, pp. 183–187.
S. Pond and G. L. Pickard (1978), Introductory Dynamic Oceanography, Pergamon Press, Oxford.
E. E. Popova, M. J. R. Fasham, A. V. Osipov, and V. A. Ryabchenko (1997), Chaotic behaviour of an ocean ecosystem model under seasonal forcing, J. Plankt. Res., 19, pp. 1495–1515.
R. Porep (1970), Der Physiologe und Planktonforscher Victor Hensen (1835–1924). Sein Leben und Werk, Kieler Beiträge zur Geschichte der Medizin und Pharmazie, R. Herrlinger, F. Kudlien, and G. E. Dann, Heft 9, Karl Wachholtz Verlag, Neumünster.
T. M. Powell et al. (1975), Spatial scales of current speed and phytoplankton biomass fluctuations in Lake Tahoe, Science, 189, pp. 1088–1090.
D. V. Radakov (1973), Schooling in the Ecology of Fish, Wiley, New York.
E. Ranta, V. Kaitala, and P. Lundberg (1997), The spatial dimension in population fluctuations, Science, 278, pp. 1621–1623.
J. E. G. Raymont (1980), Plankton and Productivity in the Oceans, 2nd ed., Pergamon Press, Oxford.
A. N. Reshetilov, A. B. Medvinsky, T. P. Eliseeva, V. Yu. Shakhbazian, M. A. Tsyganov, A. M. Boronin, and G. R. Ivanitsky (1992), pH track of expanding bacterial populations, FEMS Microbiology Lett., 94, pp. 59–62.
H. Reuter and B. Breckling (1994), Selforganization of fish schools: An object‐oriented model, Ecological Modelling, 75/76, pp. 147–159.
G. A. Riley (1946), Factors controlling phytoplankton populations on Georges Bank, J. Marine Res., 6, pp. 54–73.
G. A. Riley (1963), Theory of food‐chain relations in the ocean, in The Sea, Vol. 2, M. N. Hill, ed., Wiley, New York, pp. 438–463.
S. Rinaldi and S. Muratori (1993), Conditioned chaos in seasonally perturbed predator‐prey models, Ecological Modelling, 69, pp. 79–97.
The Ring Group (1981), Gulf Stream cold core rings: Their physics, chemistry and biology, Science, 212, pp. 1091–1100.
R. L. Ritschard (1992), Marine algae as a CO2 sink, Water, Air and Soil Pollution, 64, pp. 289–303.
A. R. Robinson, ed. (1983), Eddies in Marine Science, Springer‐Verlag, New York.
V. B. Rodionov and A. G. Kostianoy (1998), Ocean Fronts of the North‐European Basin Seas, GEOS, Moscow.
W. L. Romey (1996), Individual differences make a difference in the trajectories of simulated schools of fish, Ecological Modelling, 92, pp. 65–77.
A. B. Rovinsky and M. Menzinger (1992), Chemical instability induced by a differential flow, Phys. Rev. Lett., 69, pp. 1193–1196.
M. Ruth and B. Hannon (1997), Modeling dynamic economic systems, 1st ed., Springer‐Verlag, New York.
V. A. Ryabchenko, M. J. R. Fasham, B. A. Kagan, and E. E. Popova (1997), What causes short term oscillations in ecosystem models of the ocean mixed layer?, J. Marine Systems, 13, pp. 33–50.
N. J. Savill and P. Hogeweg (1997), Modelling morphogenesis: From single cells to crawling slugs, J. Theoret. Biology, 184, pp. 229–235.
M. Scheffer (1989), Alternative stable states in eutrophic, shallow freshwater systems: A minimal model, Hydrobiol. Bull., 23, pp. 73–83.
M. Scheffer (1991a), Fish and nutrients interplay determines algal biomass: A minimal model, OIKOS, 62, pp. 271–282.
M. Scheffer (1991b), Should we expect strange attractors behind plankton dynamics—and if so, should we bother?, J. Plankt. Res., 13, pp. 1291–1305.
M. Scheffer (1998), Ecology of Shallow Lakes, Population and Community Biology Ser. 22, Chapman and Hall, London.
M. Scheffer et al. (1995), Super‐individuals a simple solution for modelling large populations on an individual basis, Ecological Modelling, 80, pp. 161–170.
M. Scheffer, S. Rinaldi, Yu. A. Kuznetsov, and E. H. van Nes (1997), Seasonal dynamics of Daphnia and algae explained as a periodically forced predator‐prey system, OIKOS, 80, pp. 519–532.
F. Schlögl (1972), Chemical reaction models for nonequilibrium phase transitions, Zeitschrift für Physik, 253, pp. 147–161.
Manfred Schroeder, Fractals, chaos, power laws, W. H. Freeman and Company, 1991xviii+429, Minutes from an infinite paradise
L. A. Segel (1977), A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J. Appl. Math., 32, pp. 653–665.
L. A. Segel and J. L. Jackson (1972), Dissipative structure: An explanation and an ecological example, J. Theoret. Biology, 37, pp. 545–559.
L. A. Segel and B. Stoeckly (1972), Instability of a layer of chemotactic cells, attractant and degrading enzyme, J. Theoret. Biology, 37, pp. 561–585.
J. A. Shapiro and C. Hsu (1989), Escherichia coli K‐12 cell‐cell interactions seen by time‐lapse video, J. Bacteriology, 171, pp. 5963–5974.
J. A. Shapiro and D. Trubatch (1991), Sequential events in bacterial colony morphogenesis, Phys. D, 49, pp. 214–223.
J. A. Sherratt (2001), Periodic travelling waves in cyclic predator‐prey systems, Ecology Lett., 4, pp. 30–37.
J. A. Sherratt, M. A. Lewis, and A. C. Fowler (1995), Ecological chaos in the wake of invasion, Proc. Nat. Acad. Sci. USA, 92, pp. 2524–2528.
J. A. Sherratt, B. T. Eagan, and M. A. Lewis (1997), Oscillations and chaos behind predator–prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. Roy. Soc. Lond. B, 352, pp. 21–38.
N. Shigesada and K. Kawasaki (1997), Biological Invasions: Theory and Practice, Oxford University Press, Oxford.
F. Siegert and C. J. Weijer (1991), Analysis of optical density wave propagation and cell movement in the cellular slime mold Dictyostelium discoideum, Phys. D, 49, pp. 224–232.
J. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196–218
U. Sommer (1994), Planktologie, Springer‐Verlag, Berlin.
U. Sommer (1996), Algen, Quallen, Wasserfloh: Die Welt des Planktons, Springer‐Verlag, Berlin.
J. H. Steele (1974), The Structure of Marine Ecosystems, Blackwell Scientific, Oxford.
J. H. Steele, ed. (1977), Fisheries Mathematics, Academic Press, London.
J. H. Steele, ed. (1978), Spatial Pattern in Plankton Communities, NATO Conference Series: IV, Marine Sciences, Vol. 3, Plenum Press, New York.
J. H. Steele and E. W. Henderson (1981), A simple plankton model, American Naturalist, 117, pp. 676–691.
J. H. Steele and E. W. Henderson (1992a), The role of predation in plankton models, J. Plankt. Res., 14, pp. 157–172.
J. H. Steele and E. W. Henderson (1992b), A simple model for plankton patchiness, J. Plankt. Res., 14, pp. 1397–1403.
, Self‐organization of complex structures, Proceedings of the International Conference held in Berlin, September 24–28, 1995, Gordon and Breach Science Publishers, 1997, 0–0, xxiv+596, From individual to collective dynamics
E. Steffen and H. Malchow (1996b), Multiple equilibria, periodicity, and quasiperiodicity in a model plankton community Senckenbergiana maritima, 27, pp. 137–143.
E. Steffen, H. Malchow, and A. B. Medvinsky (1997), Effects of seasonal perturbations on a model plankton community, Environmental Modeling and Assessment, 2, pp. 43–48.
O. Steinbock, H. Hashimoto, and S. C. Müller (1991), Quantitative analysis of periodic chemotaxis in aggregation patterns of Dictyostelium discoideum, Phys. D, 49, pp. 233–239.
Sabine Stöcker, Models for tuna school formation, Math. Biosci., 156 (1999), 167–190, Epidemiology, cellular automata, and evolution (Sofia, 1997)
H. Stommel (1948), Trajectories of small bodies sinking slowly through convection cells, J. Marine Res., 8, pp. 24–29.
H. I. Sur, E. Ozsoy, Y. P. Ilyin, and U. Unluata (1996), Coastal/deep ocean interactions in the Black Sea and their ecological/environmental impacts, J. Marine Systems, 7, pp. 293–320.
H. U. Sverdrup (1938), On the process of upwelling, J. Marine Res., 1, pp. 115–164.
D. A. Tikhonov, J. Enderlein, H. Malchow, and A. B. Medvinsky (2001), Chaos and fractals in fish school motion, Chaos, Solitons and Fractals, 12, pp. 277–288.
I. A. Tikhonova et al. (2000), Structure formation in aquatic communities. The dependence of fish school movement and plankton spatial distributions on the phytoplankton growth rate, Biofizika, 45, pp. 352–359.
U. Timm and A. Okubo (1994), Gyrotaxis: A plume model for self‐focusing micro‐organisms, Bull. Math. Biol., 56, pp. 187–206.
J. E. Truscott (1995), Environmental forcing of simple plankton models, J. Plankt. Res., 17, pp. 2207–2232.
J. E. Truscott and J. Brindley (1994a), Ocean plankton populations as excitable media, Bull. Math. Biol., 56, pp. 981–998.
J. E. Truscott and J. Brindley (1994b), Equilibria, stability and excitability in a general class of plankton population models, Phil. Trans. Roy. Soc. Lond. A, 347, pp. 703–718.
A. Turing, Morphogenesis, Collected Works of A. M. Turing, North‐Holland Publishing Co., 1992xxvi+131, With a preface by P. N. Furbank; Edited by P. T. Saunders
B. N. Vasiev, P. Hogeweg, and A. V. Panfilov (1994), Simulation of Dictyostelium discoideum aggregation via reaction‐diffusion model, Phys. Rev. Lett., 73, pp. 3173–3176.
M. E. Vinogradov and V. V. Menshutkin (1977), The modelling of open–sea ecosystems, in The Sea: Ideas and Observations on Progress in the Study of the Seas, Vol. 6, E. D. Goldberg, ed., John Wiley, New York, pp. 891–921.
V. Volterra (1926), Fluctuations in the abundance of a species considered mathematically, Nature, 118, pp. 558–560.
V. B. Vozjinskaya (1964), Swimming algae of the West Pacific, Oceanology, 4, pp. 876–883.
J. J. Walsh, D. A. Dieterle, M. B. Meyers, and F. E. Muller‐Karger (1989), Nitrogen exchange at the continental margin: A numerical study of the Gulf of Mexico, Progr. Oceanogr., 23, pp. 245–301.
L. H. Weber, S. Z. El‐Sayed, and I. Hampton (1986), The variance spectra of phytoplankton, krill and water temperature in the Antarctic ocean south of Africa, Deep‐Sea Res., 33, pp. 1327–1343.
Wolfgang Weidlich, Günter Haag, Concepts and models of a quantitative sociology, Springer Series in Synergetics, Vol. 14, Springer‐Verlag, 1983xii+217, The dynamics of interacting populations
B. West and B. Deering (1995), The Lure of Modern Science: Fractal Thinking, World Scientific, River’s Edge, NJ.
P. H. Wiebe et al. (1976), Gulf Stream cold core rings: Large‐scale interaction sites for open ocean plankton communities, Deep‐Sea Res., 23, pp. 695–710.
P. Williamson and J. Gribbin (1991), How plankton change the climate, New Scientist, March 16, 1991, pp. 48–52.
D. S. Wilson (1992), Complex interactions in metacommunities, with implications for biodiversity and higher levels of selection, Ecology, 73, pp. 1984–2000.
H. Winet and T. Jahn (1972), On the origin of bioconvection fluid instabilities in Tetrahymena culture systems, Biorheology, 9, pp. 87–104.
Arthur Winfree, The geometry of biological time, Biomathematics, Vol. 8, Springer‐Verlag, 1980xiv+530
Arthur Winfree, When time breaks down, Princeton University Press, 1987xiv+340, The three‐dimensional dynamics of electrochemical waves and cardiac arrhythmias
C. Wissel (1989), Theoretische Ökologie, Springer‐Verlag, Berlin.
T. A. Witten and L. M. Sander (1981), Diffusion‐limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett., 47, pp. 1400–1403.
J. S. Wroblewski and J. J. O’Brien (1976), A spatial model of phytoplankton patchiness, Marine Biology, 35, pp. 161–175.
J. S. Wroblewski, J. J. O’Brien, and T. Platt (1975), On the physical and biological scales of phytoplankton patchiness in the ocean, Mémoires Société Royale des Sciences de Liège, 7, pp. 43–57.
H. Wu, B. L. Li, T. A. Springer, and W. H. Neill (2000), Modelling animal movement as a persistent random walk in two dimensions: Expected magnitude of net displacement, Ecological Modelling, 132, pp. 115–124.
T. Wyatt (1971), Production dynamics of Oikopleura dioica in the Southern North Sea, and the role of fish larvae which prey on them, Thalassia Jugoslavica, 7, pp. 435–444.
T. Wyatt (1973), The biology of Oikopleura dioica and Fritillaria borealis in the Southern Bight, Marine Biology, 22, pp. 137–158.
J. A. Yoder, S. G. Ackleson, R. T. Barber, P. Flament, and W. M. Balch (1994), A line in the sea, Nature, 371, pp. 689–692.
S. Levin, Frontiers in mathematical biology, Lecture Notes in Biomathematics, Vol. 100, Springer‐Verlag, 1994x+633

Information & Authors


Published In

cover image SIAM Review
SIAM Review
Pages: 311 - 370
ISSN (online): 1095-7200


Published online: 4 August 2006

MSC codes

  1. 82C41
  2. 92B05
  3. 92D25
  4. 92D40


  1. chaos
  2. order
  3. scaling
  4. aquatic ecosystems
  5. predator-prey interaction
  6. modeling



Alexander B. Medvinsky

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