Complex dynamic of plankton–fish interaction with quadratic harvesting and time delay

Abstract

The present paper deals with a food chain model consisting of three species phytoplankton, zooplankton and fish. We have divided the present paper into two parts. In the first part, we have assumed that the fish population is harvested using a non-linear harvesting function. Considering this rate of harvesting ‘E’ as a control parameter, we have estimated different ranges of harvesting parameter for maintaining the sustainability in the plankton ecosystem. Moreover, the bifurcation analysis of the system is carried out using normal form theorem by taking ‘E’ as bifurcation parameter. In the second part, a digestion delay corresponding to zooPlankton–fish interaction is introduced for more realistic consideration of the real world problem. Taking harvesting parameter in the stability range, the effect of time delay on the given system is investigated. This research demonstrate that for a certain range of delay, system enters into the excited state with the existence of stability switches which seems new findings for the Plankton–fish system. Explicit results are derived for stability and direction of the bifurcating periodic solution by using normal form theory and center manifold arguments. To validate our analytical findings numerical simulations are also executed.