Journal of Mathematical Biology

, Volume 63, Issue 4, pp 779–799 | Cite as

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

  • Samik DattaEmail author
  • Gustav W. Delius
  • Richard Law
  • Michael J. Plank


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. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.

Mathematics Subject Classification (2000)

Primary 92D40 Secondary 92D25 


Marine ecosystem Stability Size-spectrum McKendrick–von Foerster equation Predator–prey Growth diffusion Eigenvalues 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Samik Datta
    • 1
    Email author
  • Gustav W. Delius
    • 2
  • Richard Law
    • 3
  • Michael J. Plank
    • 4
  1. 1.Departments of Biology and MathematicsUniversity of YorkHeslingtonUK
  2. 2.Department of MathematicsUniversity of YorkHeslingtonUK
  3. 3.Department of BiologyUniversity of YorkHeslingtonUK
  4. 4.Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

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