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On a cell-growth model for plankton. (English) Zbl 1062.92065

From the paper: We study a model for cell growth in plankton based on a modified Fokker-Planck equation. The cells are assumed to be undergoing both growth and fission, and mortality is incorporated into the model. Let \(n(x,t)\) denote the density function of the number of cells of size \(x\) at time \(t\). Thus, for \(0\leq a<b\), the quantity \(\int^t_an(x,t)\,dx\) is the number of cells of size between \(a\) and \(b\) at time \(t\). The cell growth process can be modelled by a modified Fokker-Planck equation of the form \[ \begin{split}(\partial/\partial t)n(x,t)=(\partial^2/ \partial x^2) \bigl(D(x,t)n(x,t)\bigr)-(\partial/\partial x)\bigl(g(x,t)n(x,t) \bigr)\\ + \alpha^2B(\alpha x,t)n(\alpha x,t)-\bigl(B(x,t)+\mu(x,t)\bigr)n(x,t), \end{split} \] where \(D\) \((m^2/s)\) is the dispersion coefficient, \(g\) \((m/s)\) is the rate of growth and \(\mu\) \((1/s)\) is the rate of death. The function \(B(1/s)\) is the rate at which cells divide into \(\alpha\) equally sized daughter cells. Here \(\alpha>1\) is regarded as a constant, and the functions \(D,g,\mu\) and \(B\) are all non-negative. The partial differential equation is supplemented by the boundary conditions \[ \lim_{x\to\infty} n(x,t)=0,\;\lim_{x\to\infty}(\partial/ \partial x)n(x,t)=0;\;(\partial/ \partial x)\bigl(D(x,t)n(x,t)\bigr)-g(x,t)n(x,t) |_{x=0}=0, \] where the first two conditions place decay conditions on \(n\) as \(x\to\infty\) for any fixed time, and the last equation is a ‘no flux’ condition on the boundary \(x=0\). The model above is deterministic though the dependent variable is in fact a probability distribution evolving in time. Thus it is partially ‘stochastic’ in character.
The frequency distribution of diatoms (microscopic unicellular alga with silicified cell-walls, found as plankton) is shown to evolve in time as a steady-size distribution with constant shape, scaled by time. This distribution is preserved when the division occurs at a fixed size into two daughter cells of half-size. In cases where the parameters for growth, division frequency, dispersion and mortality are constants, the frequency distributions can be found explicitly and thus provide a benchmark for computations in more complex cases.

MSC:

92D40 Ecology
35K20 Initial-boundary value problems for second-order parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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