Abstract
This paper is devoted to the study of a plankton–fish ecosystem model. The model represents the interaction between phytoplankton, zooplankton, and fish with Holling II functional response consisting of carrying capacity and constant intrinsic growth rate of phytoplankton. It is observed that if the carrying capacity of phytoplankton population crosses a certain critical value, the system enters into Hopf bifurcation. We have introduced discrete time delay due to gestation in the functional response term involved with the growth equation of planktivorous fish. We have studied the effect of time delay on the stability behavior. In addition, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Existence of Hopf bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter. It is observed that constant intrinsic growth rate of phytoplankton and mortality rate of planktivorous fish play an important role in ch...
Highlights
It is well known that plankton plays an integral role in marine ecosystem
The authors have discussed that at high fish densities, zooplankton is controlled by fish predation and algal biomass is light or nutrient limited, whereas at low fish densities, zooplankton is food limited and phytoplankton density is controlled by zooplankton grazing
As studied by Biktashev, Brindley, and Horwood (2003), the food source for fish larvae depends on the zooplankton dynamics, which in turn is coupled to larval populations through the zooplankton mortality term
Summary
It is well known that plankton plays an integral role in marine ecosystem. Phytoplankton and zooplankton are the main two types of plankton. Rehim and Imran (2012) have induced a discrete time delay to both the consume response function and distribution of toxic substance terms in phytoplankton–zooplankton model They have found out a range of gestation delays which initially impart stability, induce instability and lead to periodic behavior. Gazi and Bandyopadhyay (2006) have introduced discrete time delay due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. They have studied the effect of time delay on the stability behavior and obtained an estimate for the length of time delay to preserve the stability of the model system. We choose Holling type II functional form to describe the grazing phenomena with K1 and K2 as half saturation constant
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