A model of phytoplankton growth on multiple nutrients based on the Michaelis-Menten-Monod uptake, Droopʹs growth and Liebigʹs law

A model of phytoplankton population growing on more than one potentially limiting nutrient is formulated and investigated. The model is based on the Michaelis-Menten-Monod uptake function for each nutrient, the Droopʹs function for growth of phytoplankton and Liebigʹs law for growth on different nutrients. The model is analyzed in a simple set up of phytoplankton culture reactor. Conditions are specified for which steady phytoplankton existence state is stable. Since growth depends on internal nutrient content, the limiting nutrient may be recognized as the one having the smallest content in phytoplankton relative to the subsistence quota. According to the model, in steady state during equal limitation by several nutrients, the Redfield ratio is equal to the ratio of subsistence quotas and to the ratio of uptake rates. Contrary to wide spread use, the ratio of nutrients in water is not the Redfield ratio but a function of the growth rate. In oligotrophic waters, however, nutrients are in another ratio that may be used as an analog to the Redfield ratio in phytoplankton. The model may be used as a submodel of larger ecosystem models.