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This paper investigates the dynamics of biomass in a marine ecosystem. Astochastic process is defined in which organisms undergo jumps in body size asthey catch and eat smaller organisms. Using a systematic expansion of themaster equation, we derive a deterministic equation for the macroscopicdynamics, which we call the deterministic jump-growth equation, and a linearFokker-Planck equation for the stochastic fluctuations. The McKendrick--vonFoerster equation, used in previous studies, is shown to be a first-orderapproximation, appropriate in equilibrium systems where predators are muchlarger than their prey. The model has a power-law steady state consistent withthe approximate constancy of mass density in logarithmic intervals of body massoften observed in marine ecosystems. The behaviours of the stochastic process,the deterministic jump-growth equation and the McKendrick--von Foersterequation are compared using numerical methods. The numerical analysis shows twoclasses of attractors: steady states and travelling waves.

A jump-growth model for predator-prey dynamics: derivation and application to marine ecosystems