Application of Analytical Design of Aggregated Regulators Method to Toxin-Producing-Phytoplankton Models

Predator-prey models based on the LotkaVolterra system of differential equations are widely used to describe phytoplankton-zooplankton interactions. The toxin producing phytoplankton model introduces a zooplankton response to harmful substances produced by some species of phytoplankton, making it possible to account for the case when prey is responsible for decrease in predator population size. The article describes two new models with different control objectives which are both based on the toxin-producing phytoplankton model. Holling type II function is used for phytoplankton-zooplankton interactions. Control objectives include achieving a given phytoplankton population size and achieving proportionality (i.e. balance) between phytoplankton and zooplankton population sizes. Control signals for both models are derived analytically by applying the method of analytical design of aggregated regulators. Analytical expressions for stationary points of both models are given, as well as conditions for the systems to evolve to a stable state. Toxic phytoplankton and zooplankton coexistence criteria are postulated based on the proposed models. Results are validated with numerical simulations.