Dynamical Analysis of a Phytoplankton–Zooplankton System with Harvesting Term and Holling III Functional Response

Abstract

A toxin-producing phytoplankton and zooplankton system is investigated. Considering that zooplankton can be harvested for food in some bodies of water, the harvesting term is introduced to zooplankton population. Firstly, from the ordinary differential equation (ODE) system, we obtain the global asymptotic stability of equilibrium and optimal capture problem. Secondly, based on the ODE system, the diffusion term is introduced and the global asymptotic stability of the steady state solution is obtained. As a result, the diffusion cannot affect the global asymptotic stability of equilibrium, and Turing instability cannot occur. Once again, a delayed differential equation (DDE) system is put forward. The global asymptotic stability of boundary equilibrium and the existence of local Hopf bifurcation at positive equilibrium are discussed. Furthermore, it is proved that there exists at least one positive periodic solution as delay varies in some region by using the global Hopf result of Wu for functional differential equations. Lastly, some numerical simulations are carried out for supporting the theoretical analyses and the positive impacts of harvesting effort, and the release rate of toxin is given. The unstable interval of the positive equilibrium becomes smaller and smaller with the increase of harvesting effort or the release rate of toxin.