Aspects of a zooplankton, phytoplankton, phosphorus system

Abstract

Abstract In this paper a three-dimensional dynamical system which models the three-species system made up of phytoplankton, zooplankton and organic phosphorus nutrient in a lake environment is studied. The system is part of a more general limnological model for eutrophic lakes and impoundments which has been developed by Battelle Northwest Laboratories. It is shown that this system, henceforth referred to as Z-P-P, has a phase portrait comprised of a plane portrait embedded in the three-dimensional space R 3 as an “attractor”. ∗ Under a small perturbation of the nutrient equation it is shown that the system is essentially a classical Volterra-Lotka system embedded in a three-dimensional phase space R 3 . The system derived from Z-P-P by the addition of a term to the nutrient equation which represents the organic phosphorus contribution of dying phytoplankton is also considered. The equilibria of this system are studied and what can be deduced of the phase portrait is compared with that of the above systems. It is found that these phase portraits are qualitatively indifferent to the form of the growth rate functions for zooplankton and phytoplankton provided they are monotone increasing. Some discussion about the stability of these systems is included. Throughout this paper results are interpreted in limnological terms.