A stability analysis of the power-law steady state of marine size spectra

TL;DR

This study analyzes the stability of power-law steady states in marine ecosystems using different dynamical models, deriving eigenvalue spectra and identifying conditions that promote stability.

Contribution

It derives the eigenvalue spectrum for the linearized operator of three models and reveals stability conditions for the marine size spectra steady state.

Findings

The steady state of the McKendrick-von Foerster equation without diffusion is always unstable.

Eigenvalue spectra of the diffusion-augmented equation approximate those of the jump-growth model.

Stability increases with lower predator-prey mass ratios, larger diet breadths, and higher feeding efficiencies.

Abstract

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a "jump-growth" equation, a first order approximation which is the widely used McKendrick-von Foerster equation, and a second order approximation which is the McKendrick-von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick-von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick-von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator : prey mass ratio, a large diet breadth and a high feeding efficiency.