Impact of time delay in a plankton–fish system with nonlinear harvesting and external toxicity

Abstract

In this study, we explore the dynamics of a plankton–fish system in which external toxic substances have adverse impacts on the plankton populations as well as the fish communities. Fish population is assumed to grow in a logistic fashion due to food sources other than zooplankton, and is being harvested at a nonlinear rate. Mathematically, we analyze the system for the feasibility and stability of the ecologically meaningful equilibria. We also obtain criteria for the optimal harvesting policy of the fishery resources. We modify the considered model to encapsulate the effect of negative feedback time delay involved in the growth of the fish population. By considering this time delay as a bifurcation parameter, we derive conditions for the existence of Hopf bifurcation; direction and stability of the bifurcating periodic solutions are also discussed. Extensive numerical simulations are done for the systems with and without time delay. The environmental toxins are found to affect the equilibrium abundances of plankton and fish population in the aquatic system. In the absence of time delay, we observe that the growths in fish population due to uptake of zooplankton and other food sources have potential to suppress the chaotic disorder and bring forth stability in the ecosystem. On the other hand, the excessive growth of phytoplankton due to overloading of nutrients and the increasing carrying capacity of the ecosystem for the fish population are found to exhibit destabilizing roles on the system’s dynamics. Finally, we observe that the larger magnitude of negative feedback time delay in the growth of fish population drives the system to a chaotic zone.