Bifurcation Analysis of Phytoplankton and Zooplankton Interaction System with Two Delays

Abstract

In this paper, the system of describing the interactions between poisonous phytoplankton and zooplankton is presented. It focuses on the effects of two delays on the dynamic behavior of the system. At first, the properties of solutions including positivity and boundedness are given. Next, the stability of equilibria and the existence of local Hopf bifurcation are established when delays change and cross some threshold values. Especially, the existence of global periodic solutions is discussed when the two delays are equal. Furthermore, the implicit algorithm is derived for deciding the properties of the branching periodic solutions by using center manifold theory. Some numerical simulations are performed for supporting the theoretical results. Finally, some conclusions are given.