A bifurcation analysis of a multi compartment plankton-zooplankton-nutrient interaction.

Abstract

This thesis concentrates on understanding the long term behaviour of a multi-compartment phytoplankton-zooplankton-nutrient interaction. A variable-yield model is considered, in which the rate of carbon uptake by phytoplankton necessary for its growth is governed by cell quota i.e. the ratio of external nutrient (nitrogen) and the internal nutrient (carbon). The internal and external nutrient of the phytoplankton are governed by separate equations. The work addresses the question 'How complex should a model be?', besides attempting to understand analytical and qualitative model behaviour. The simplest model considered consists of four ordinary differential equations relating to one pool or compartment, and is then extended to eight ordinary differential equations: (four equations for each pool) of the two compartments, and finally to twelve ordinary differential equations: (four equations in each of the three compartments). Chapter 1 introduces the basic mathematical model, and critiques its formulation based on various ecological studies on phytoplankton-zooplankton-nutrient interactions. Local stability analysis necessary to investigate stability of our model is discussed, together with an introductory explanation of bifurcation theory, which is used by modellers to tune and adjust the dynamical system. Thus by altering system parameters the behaviour of the system may change gradually or even abruptly. Abrupt changes occur at bifurcating values of the parameter. Chapter 1 also includes a manual for running the software XPP and also AUTO, sophisticated software to study bifurcation and hence important model behaviour. Chapter 2 provides a complete analysis of the behaviour of the one compartment model. It includes local stability analysis for all the solutions and a global analysis for the null solution. A detailed study of this simplest model includes a complete profile of bifurcation diagrams executed by the software AUTO, with information on the behaviour of the steady state and periodic solutions for comparison with an extension of the analytical results. Chapter 3 presents analytical as well as numerical studies of the two compartment model. Two cases are considered, one with an equal growth parameter of the phytoplankton and the other where the growth parameter in each compartment is different. Stability analysis for the first case is examined by both local stability analysis and bifurcation analysis, but other case can only be done numerically via a complete profile of bifurcation diagrams using the software AUTO. Chapter 4 presents stability analysis for the three compartment model via bifurcation diagrams generated numerically using AUTO. This chapter considers 3 cases: the equal growth parameter of phytoplankton, different growth parameters of phytoplankton and both different growth parameters and different diffusion parameters. Chapter 5 presents the various conclusions drawn from all of three models considered.