A bifurcation analysis of a simple phytoplankton and zooplankton model

Abstract The paper presents a detailed study of a four component population dynamical model consisting of the biomasses of carbon of phytoplankton x 1 , nitrogen of phytoplankton x 2 , carbon of zooplankton x 3 and total free nutrient x 4 . In this model, both the influence of the internal (carbon) and external (nitrogen) nutrient on the growth of phytoplankton is included. This is usually termed a variable yield model in contrast to the constant yield model analyzed by J. Monod [Recherches sur la croissance des cultures bacteriennes, Herman, Paris, 1942.  [21] ]. The existence and stability of various steady states are discussed analytically and numerically using a computer simulation (AUTO) designed by Doedel [E.J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer. 30 (1981) 265–284; E. Doedel, X. Wang, T. Fairgrieve, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, in: Applied Mathematics Reports, California Institute of Technology, 1994]. Various bifurcation diagrams so obtained support the analytical results and identified complex behavior which cannot be obtained analytically. The purpose of this paper is as a preliminary step to determine the change in the richness of behavior as complexity, such as spatial compartments, is added, which in limiting form would be a continuous spatial dependence (as a partial differential equation model).