期刊
JOURNAL OF MATHEMATICAL BIOLOGY
卷 63, 期 4, 页码 779-799出版社
SPRINGER
DOI: 10.1007/s00285-010-0387-z
关键词
Marine ecosystem; Stability; Size-spectrum; McKendrick-von Foerster equation; Predator-prey; Growth diffusion; Eigenvalues
资金
- Natural Environment Research Council UK
- Centre for Environment Fisheries and Aquaculture Science UK
- Royal Society of New Zealand [08-UOC-034]
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.
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