  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.
