A scale-invariant model of marine population dynamics

TL;DR

This paper develops a scale-invariant size-structured model of marine populations, explaining the observed power-law size spectrum and analyzing the stability of the steady state under various ecological processes.

Contribution

It introduces a novel scale-invariant population model based on a Markov process, extending previous models and analyzing stability with respect to ecological rate functions.

Findings

The model predicts a power-law size spectrum consistent with observations.

Including maintenance and reproduction stabilizes the population dynamics.

Steady state stability depends on the balance between reproduction and overall density.

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, 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 astrong stabilising effect. Furthermore, the steady state is unstable against a change in the overall population density unless the reproduction rate exceeds a certain threshold.