Impact of planktivorous fish on delay-induced plankton fish ecosystem model

Abstract

Models of plankton-fish based ecosystems with delay have received a great deal of attention in the last few decades. This paper deals with a plankton-fish based ecosystem involving phytoplankton, zooplankton and planktivorous fish. Initial analysis of various possible equilibrium points and their steady states behavior demonstrates that mortality rate and harvesting rate of planktivorous fish and intrinsic growth rate of zooplankton play key roles to stabilize the system. Further, conditions for global stability and direction of Hopf-bifurcation is also discussed. A path of optimal harvesting policy is derived by introducing the Pontryagins maximum principle. Further, a discrete time delay term due to gestation in the functional response is introduced in the growth equation of planktivorous fish. In addition, estimate for the length of time delay to preserve the stability of the model system is exploded. Existence of Hopf bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter. Computer simulations are implemented to help the analytical results.