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

TL;DR

This paper develops a jump-growth model for predator-prey biomass dynamics in marine ecosystems, deriving deterministic and stochastic equations, and compares their behaviors with classical models through numerical analysis.

Contribution

It introduces a novel jump-growth model derived from a stochastic process, extending previous predator-prey models with a more accurate representation of biomass dynamics.

Findings

The model predicts power-law steady states consistent with observed marine biomass distributions.

Numerical analysis reveals two attractor classes: steady states and travelling waves.

The McKendrick--von Foerster equation is shown to be a first-order approximation in equilibrium systems.

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.