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This paper investigates the stability of the power-law steady state oftenobserved 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 usedMcKendrick-von Foerster equation, and a second order approximation which is theMcKendrick-von Foerster equation with a diffusion term. All of these yield apower-law steady state. We derive, for the first time, the eigenvalue spectrumfor the linearised evolution operator, under certain constraints on theparameters. This provides new knowledge of the stability properties of thepower-law steady state. It is shown analytically that the steady state of theMcKendrick-von Foerster equation without the diffusion term is always unstable.Furthermore, numerical plots show that eigenvalue spectra of the McKendrick-vonFoerster equation with diffusion give a good approximation to those of thejump-growth equation. The steady state is more likely to be stable with a lowpreferred predator : prey mass ratio, a large diet breadth and a high feedingefficiency.

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