Stability and bifurcation analysis of a nutrient-phytoplankton model.

摘要

This Master thesis consists of six chapters, which are mainly concerned with the stability and bifurcation analysis of a Nutrient ( N), Phytoplankton (A) model.;In chapter 2, we introduce a two dimensional (N, A) model to describe the nutrient-phytoplankton interactions, and investigate the dynamical properties of this model. We show the existence of a boundary equilibrium point, and use geometerical and analytical methods to find conditions for the existence of none, one, or at most two positive equilibrium points. We then analyze the stability of each equilibrium point.;In chapter 3, we modify the previous model by introducing a time delay τ, and discuss its effect on the stability of each equilibrium point, by investigating the distribution of the roots in the corresponding characteristic equation.;In chapter 4, we discuss the bifurcations. By using the projection method, we prove the existence of a saddle-node bifurcation for the system without delay. And by using the center manifold theory and normal form method, we study the direction of Hopf bifurcation and the stability of the periodic solutions for both systems, and we prove the existence of Hopf-Zero bifurcation for the system with delay.;In chapter 1, some existing Nutrient-Phytoplankton models and the motivation for this work are presented.;In chapter 5, we provide numerical simulations to verify our theoretical predictions in the previous chapters, and biological interpretations based on these simulations.;In the last chapter, we summarize the results obtained in the previous chapters and provide suggestions to improve the model.