### Abstract

The problem of the vertical distribution of phytoplankton is considered in the presence of gravitational settling, turbulent mixing, population growth due to cell division and a constant rate of loss due to predation and natural death. Nutrients are assumed to be plentiful so that the production rate depends only on the light available for photosynthesis. The non-linear saturation of plankton growth is modeled by allowing the attenuation rate of light to be a linear function of the plankton density. The turbulent diffusivity is assumed constant which corresponds to a mixed layer depth very much greater than the depth of light penetration (euphotic depth). It is shown that an exact analytical solution of this non-linear problem is possible for an idealized model in which the functional dependence of production on light intensity is assumed to be a step function. Non-zero solutions are shown to exist only if the parameters characterizing the system are above a certain critical curve in a two dimensional parameter space. Numerical simulations using functional forms of the production curve that resemble the measured photosynthetic response of plankton, show, that the qualitative behavior of the system is similar to that of the idealized model presented. Comparisons are made with other analytical approaches to the problem.

Original language | English (US) |
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Pages (from-to) | 333-346 |

Number of pages | 14 |

Journal | Journal of Mathematical Biology |

Volume | 46 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2003 |

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### Keywords

- Nonlinear
- Plankton
- Self-shading
- Turbulence

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics

### Cite this

*Journal of Mathematical Biology*,

*46*(4), 333-346. https://doi.org/10.1007/s00285-002-0184-4