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

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JournalBulletin of Mathematical Biology
DatePublished - Aug 2010
Issue number6
Volume72
Number of pages22
Pages (from-to)1361-1382
Original languageEnglish

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.

    Research areas

  • Growth diffusion, Marine ecosystem, Master equation, McKendrick, von Foerster equation, Predator, prey, Size, spectrum, Stochastic process, Systematic expansion, INDIVIDUAL-BASED MODEL, STRUCTURED FOOD WEBS, SIZE-SPECTRA, BODY-SIZE, ABUNDANCE, FISH, VARIABILITY, EQUATION, ECOLOGY

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