## Abstract

A striking feature of the marine ecosystem is the regularity in its size spectrum: the abundance of organisms as a function of their weight approximately follows a power law over almost ten orders of magnitude. We interpret this as evidence that the population dynamics in the ocean is approximately scale-invariant. We use this invariance in the construction and solution of a size-structured dynamical population model. Starting from a Markov model encoding the basic processes of predation, reproduction, maintenance respiration, and intrinsic mortality, we derive a partial integro-differential equation describing the dependence of abundance on weight and time. Our model represents an extension of the jump-growth model and hence also of earlier models based on the McKendrick-von Foerster equation. The model is scale-invariant provided the rate functions of the stochastic processes have certain scaling properties. We determine the steady-state power-law solution, whose exponent is determined by the relative scaling between the rates of the density-dependent processes (predation) and the rates of the density-independent processes (reproduction, maintenance, and mortality). We study the stability of the steady-state against small perturbations and find that inclusion of maintenance respiration and reproduction in the model has a strong stabilizing effect. Furthermore, the steady state is unstable against a change in the overall population density unless the reproduction rate exceeds a certain threshold.

Original language | English |
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Article number | 061901 |

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Physical Review E |

Volume | 81 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jun 2010 |

## Keywords

- STRUCTURED FOOD WEBS
- BIOMASS-SIZE SPECTRA
- BODY-SIZE
- PELAGIC ECOSYSTEM
- FISH
- ABUNDANCE
- TEMPERATURE
- PREDATION
- MASS