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A Jump-Growth Model for Predator–Prey Dynamics: Derivation and Application to Marine Ecosystems

Bulletin of Mathematical Biology volume 72, pages 1361–1382 (2010)Cite this article

Abstract

This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker–Planck equation for the stochastic fluctuations. The McKendrick–von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation, and the McKendrick–von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.

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References

  • Aldous, D.J., 1999. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5(1), 3–48.

    Article  MATH  MathSciNet  Google Scholar 

  • Andersen, K.H., Beyer, J.E., 2006. Asymptotic size determines species abundance in the marine size spectrum. Am. Natur. 168, 54–61.

    Article  Google Scholar 

  • Andersen, K.H., Beyer, J.E., Lundberg, P., 2008. Trophic and individual efficiencies of size-structured communities. Proc. R. Soc., Ser. B 276, 109–114.

    Article  Google Scholar 

  • Anderson, C.N.K., Hsieh, C.-H., Sandin, S.A., Hewitt, R., Hollowed, A., Beddington, J., May, R.M., Sugihara, G., 2008. Why fishing magnifies fluctuations in abundance. Nature 452, 835–839.

    Article  Google Scholar 

  • Bardos, D.C., 2005. Probabilistic Gompertz model of irreversible growth. Bull. Math. Biol. 67, 529–545.

    Article  MathSciNet  Google Scholar 

  • Benoît, E., Rochet, M.-J., 2004. A continuous model of biomass size spectra governed by predation and the effects of fishing on them. J. Theor. Biol. 226, 9–21.

    Article  Google Scholar 

  • Blanchard, J.L., Jennings, S., Law, R., Castle, M.D., McCloghrie, P., Rochet, M.-J., Benoît, E., 2009. How does abundance scale with body size in coupled size-structured food webs? J. Animal Ecol. 78, 270–280.

    Article  Google Scholar 

  • Boudreau, P.R., Dickie, L.M., 1992. Biomass spectra of aquatic ecosystems in relation to fisheries yield. Can. J. Fisheries Aquat. Sci. 49, 1528–1538.

    Article  Google Scholar 

  • Brown, J.H., Gillooly, J.F., Allen, A.P., Savage, V.M., West, G.B., 2004. Toward a metabolic theory of ecology. Ecology 85, 1771–1789.

    Article  Google Scholar 

  • Camacho, J., Solé, R.V., 2001. Scaling in ecological size spectra. Europhys. Lett. 55, 774–780.

    Article  Google Scholar 

  • Cohen, J.E., Pimm, S.L., Yodzis, P., Saldana, J., 1993. Body sizes of animal predators and animal prey in food webs. J. Animal Ecol. 62, 67–78.

    Article  Google Scholar 

  • Finkelshtein, D., Kondratiev, Y., Kutoviy, O., 2009. Individual based model with competition in spatial ecology. SIAM J. Math. Anal. 41(1), 297–317. arXiv:0803.3565.

    Article  MATH  MathSciNet  Google Scholar 

  • Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434.

    Article  MathSciNet  Google Scholar 

  • Gillespie, D.T., 2000. The chemical Langevin equation. J. Chem. Phys. 113, 297–306.

    Article  Google Scholar 

  • Gurney, W.S.C., Veitch, A.R., 2007. The dynamics of size-at-age variability. Bull. Math. Biol. 69, 861–885.

    Article  MATH  MathSciNet  Google Scholar 

  • Heath, M.R., 1995. Size spectrum dynamics and the planktonic ecosystem of Loch Linnhe. ICES J. Marine Sci. 52, 627–642.

    Article  Google Scholar 

  • Hsieh, C.-H., Reiss, C.S., Hunter, J.R., Beddington, J.R., May, R.M., Sugihara, G., 2006. Fishing elevates variability in the abundance of exploited species. Nature 443, 859–862.

    Article  Google Scholar 

  • Jennings, S., Mackinson, S., 2003. Abundance-body mass relationships in size-structured food webs. Ecol. Lett. 6, 971–974.

    Article  Google Scholar 

  • Kerr, S.R., Dickie, L.M., 2001. The Biomass Spectrum: A Predator–Prey Theory of Aquatic Production. Columbia University Press, New York.

    Google Scholar 

  • Law, R., Plank, M.J., James, A., Blanchard, J.L., 2009. Size-spectra dynamics from stochastic predation and growth of individuals. Ecology 90(3), 802–811.

    Article  Google Scholar 

  • Marquet, P.A., Quiñones, R.A., Abades, S., Labra, F., Tognelli, M., Arim, M., Rivadeneira, M., 2005. Scaling and power-laws in ecological systems. J. Exp. Biol. 208, 1749–1769.

    Article  Google Scholar 

  • Maury, O., Faugeras, B., Shin, Y.-J., Poggiale, C., Ari, T.B., Marsac, F., 2007. Modelling environmental effects on the size-structured energy flow through marine ecosystems. Part 1: The model. Prog. Oceanography 74, 479–499.

    Article  Google Scholar 

  • McKendrick, A.G., 1926. Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 40, 98–130.

    Google Scholar 

  • Okubo, A., Levin, S.A., 2001. Diffusion and Ecological Problems, 2nd edn. Springer, New York.

    Google Scholar 

  • Paloheimo, J.E., Dickie, L.M., 1966. Food and growth of fishes. III. Relations among food, body size, and growth efficiency. J. Fisheries Res. Board Can. 23, 1209–1248.

    Google Scholar 

  • Pfister, C.A., Stevens, F.R., 2002. The genesis of size variability in plants and animals. Ecology 83, 59–72.

    Article  Google Scholar 

  • Platt, T., Denman, K., 1978. The structure of pelagic marine ecosystems. J. Conseil Int. l’Explor. Mer 173, 60–65.

    Google Scholar 

  • Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Sheldon, R.W., Prakash, A., Sutcliffe Jr., W.H., 1972. The size distribution of particles in the ocean. Limnology Oceanography 17, 327–340.

    Article  Google Scholar 

  • Sheldon, R.W., Sutcliffe, W.H., Paranjape, M.A., 1977. Structure of pelagic food chain and relationship between plankton and fish production. J. Fisheries Res. Board Can. 34, 2344–2353.

    Google Scholar 

  • Shin, Y.-J., Cury, P., 2004. Using an individual-based model of fish assemblages to study the response of size spectra to changes in fishing. Can. J. Fisheries Aquat. Sci. 61, 414–431.

    Article  Google Scholar 

  • Silvert, W., Platt, T., 1978. Energy flux in the pelagic ecosystem: a time-dependent equation. Limnology Oceanography 23, 813–816.

    Article  Google Scholar 

  • Silvert, W., Platt, T., 1980. Dynamic energy-flow model of the particle size distribution in pelagic ecosystems. In: Kerfoot, W.C. (Ed.), Evolution and Ecology of Zooplankton Communities, pp. 754–763. University Press of New England, Hanover.

    Google Scholar 

  • Smoluchowski, M., 1916. Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys Z. 17, 557–585.

    Google Scholar 

  • van Kampen, N.G., 1992. Stochastic Processes in Physics and Chemistry. Elsevier Science, Amsterdam.

    Google Scholar 

  • von Foerster, H., 1959. Some remarks on changing populations. In: Stohlman, J.F. (Ed.), The Kinetics of Cellular Proliferation, pp. 382–407. Grune and Stratton, New York

    Google Scholar 

  • Ware, D.M., 1978. Bioenergetics of pelagic fish: theoretical change in swimming speed and ration with body size. J. Fisheries Res. Board Can. 35, 220–228.

    Google Scholar 

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Authors and Affiliations

  1. Departments of Biology and Mathematics, University of York, York, YO10 5DD, UK

    Samik Datta, Gustav W. Delius & Richard Law

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  1. Samik Datta
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  2. Gustav W. Delius
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  3. Richard Law
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Correspondence to Samik Datta.

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Datta, S., Delius, G.W. & Law, R. A Jump-Growth Model for Predator–Prey Dynamics: Derivation and Application to Marine Ecosystems. Bull. Math. Biol. 72, 1361–1382 (2010). https://doi.org/10.1007/s11538-009-9496-5

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  • DOI: https://doi.org/10.1007/s11538-009-9496-5

Keywords

  • Growth diffusion
  • Marine ecosystem
  • Master equation
  • McKendrick–von Foerster equation
  • Predator–prey
  • Size-spectrum
  • Stochastic process
  • Systematic expansion