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

TY - JOUR

T1 - A stability analysis of the power-law steady state of marine size spectra

AU - Datta, Samik

AU - Delius, Gustav Walter

AU - Law, Richard

AU - Plank, Michael J.

PY - 2011/10/1

Y1 - 2011/10/1

N2 - 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. Under certain constraints on the parameters a mathematical analysis yields an eigenvalue spectrum for the linearised evolution operator, from which the stability properties of the steady state can be inferred. 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.

AB - 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. Under certain constraints on the parameters a mathematical analysis yields an eigenvalue spectrum for the linearised evolution operator, from which the stability properties of the steady state can be inferred. 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.

KW - Mathematical Biology

UR - http://www.scopus.com/inward/record.url?scp=80052708674&partnerID=8YFLogxK

U2 - 10.1007/s00285-010-0387-z

DO - 10.1007/s00285-010-0387-z

M3 - Article

SN - 0303-6812

VL - 63

SP - 779

EP - 799

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

IS - 4

ER -