Dynamical Analysis In A Kind Of Plankton System With Multiple Delays

In this paper, a mathematical plankton model with two discrete time delays is considered. This plankton model describes two harmful phytoplankton and a zooplankton with competition and predator-prey.Firstly, we consider the stability of the equilibriums. We first analyze the model with one delay. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Meanwhile, the phenomenon of stability switches is found under certain conditions. Then, we consider the model with multiple delays. We also prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases.Secondly, the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using the theory of normal form and center manifold.Finally, numerical simulations are given to support the theoretical results. In case of one delay, we get a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases, and the phenomenon of stability switches is found. In case of multiple delays, we get Hopf bifurcation is supercritical and the bifurcating periodic solutions are orbitally asymptotically stable.