Effect of delay on growth function of zooplankton in plankton ecosystem model and its consequence on the formation of plankton bloom

Abstract

Models of plankton-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with a plankton-based ecosystem involving nutrient, phytoplankton and zooplankton. The model system is studied analytically and the threshold conditions for the existence and stability of various steady states are worked out.

Next, we have introduced discrete time delay due to gestation in the functional response term involved with the growth equation of zooplankton. With delay differential equation model system we have studied the effect of time delay on the stability behavior. Next, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Moreover, the existence of Hopf bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter. Finally, it is observed that constant nutrient input and dilution rate of nutrient play important role to change the steady state behavior in presence of time lag in gestation of the zooplankton. Computer simulations have been carried out to illustrate various analytical results.