Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting

Consider that some zooplankton can be harvested for food and some phytoplankton can liberate toxin; a toxin producing phytoplankton–zooplankton model with delay and selective harvesting is proposed and investigated. We discuss the stability of equilibria and perform the analysis of Hopf bifurcation. More precisely, the global asymptotical stability of equilibria is investigated by the Lyapunov method and Dulac theorem. In addition, the computing formulas of stability and direction of the Hopf bifurcating periodic solutions are also given. Furthermore, we prove that there exists at least one positive periodic solution as a time delay varies in some regions by using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc. 350:4799–4838, 1998) for functional differential equations. Finally, the impact of harvesting is discussed along with numerical results to provide some support to the analytical findings.