Patchiness in a minimal nutrient-phytoplankton model.

We present a minimal two-component model that can exhibit various types of spatial patterns including patchiness. The model, comprising nutrients and phytoplankton, includes the effect of nutrient uptake by phytoplankton as a Holling type II functional response, and also includes the effect of zooplankton grazing on phytoplankton as a Holling type II non-dynamical term. The mean-field model without the diffusion and advection terms shows both bistability and limit-cycle oscillations as a few parameters such as the input rate of nutrients and the maximum feeding rate of zooplankton are changed. If the parameter values are chosen from the limit-cycle oscillation region, the corresponding reaction-advection-diffusion equations show spatial pattern formations by the combined effects of advection and diffusion by turbulent stirring and mixing, and biological interactions. As the nutrient input is increased, the system behaviour changes from the extinction of the entire phytoplankton to the formation of filamentous patterns, patchiness patterns and homogeneous distributions. These observations suggest that the spatial pattern of phytoplankton can function as an indicator to evaluate the eutrophication level in aquatic ecosystems.