A stability analysis of the power-law steady state of marine size spectra
arXiv:1001.3826
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.
This final version will appear in J. Math. Biol. In v2 we added section 2.5 with a discussion of fluctuations due to demographic stochasticity and section 3.7 with a plot of the correlation function. In v3 we verified the analytic results for the correlation function through simulations. 24 pages, 7 figures