Scale-Invariant Model of Marine Population Dynamics Articles uri icon



publication date

  • June 2010

start page

  • 61901


  • 6


  • 81

International Standard Serial Number (ISSN)

  • 1539-3755

Electronic International Standard Serial Number (EISSN)

  • 1550-2376


  • 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.