title: Scale-invariant model of marine population dynamics.
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.
authors.affiliation: Grupo Interdisciplinar de Sistemas Complejos, Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, E-28911 Leganés, Madrid, Spain. jcapitan@math.uc3m.es
authors.forename: José A
authors.initials: JA
authors.lastname: Capitán
authors.forename: Gustav W
authors.initials: GW
authors.lastname: Delius
mesh_terms: D000465:Algorithms; D000818:Animals; D001522:Behavior, Animal; D017753:Ecosystem; D005399:Fishes; D008386:Marine Biology; D008954:Models, Biological; D015233:Models, Statistical; D010933:Plankton / Q000378:metabolism; D011157:Population Dynamics; D011235:Predatory Behavior; D012623:Seawater; D013269:Stochastic Processes