Global stability and Hopf-bifurcation in a zooplankton–phytoplankton model

Abstract

We deal with a predator–prey model, representing a resource (phytoplankton) and two predators (zooplankton) system with toxin-producing delay. The response function is assumed here to be concave in nature. Firstly, the stability criterion of the model is analyzed both from a local and a global point of view. Our results imply that the toxin’s intrinsic characteristics, such as toxic liberation rate and toxin-producing delay, will not change the stability of the system irreversibly. Secondly, Hopf bifurcation of both systems with delay and without delay can occur via system parameters pertaining to the toxin. Our results indicate that the toxin produced by phytoplankton may be used as a bio-control agent for the Harmful Algal Bloom problems. Furthermore, the explicit algorithm for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results.