Global qualitative analysis of a phytoplankton–zooplankton model in the presence of toxicity

TLDR: In this paper, the authors investigate the dynamical behavior of a toxic-phytoplankton-zooplankton model with delay and establish the existence of local Hopfbifurcation.

Abstract: In this paper, we investigate the dynamical behavior of toxic-phytoplankton–zooplankton model with delay. Here, we describe a phytoplankton–zooplankton system that exhibits a Holling type II functional response in the presence of toxicity. Combined effort (E) is used to harvest the population. It is assumed that the phytoplankton is affected directly by an external toxic substance and the feeding of zooplankton on the affected phytoplankton is influenced indirectly by the toxic substance. Firstly we consider the elementary dynamical properties of the toxic-phytoplankton–zooplankton interacting model system in absence of time delay. Then we establish the existence of local Hopfbifurcation as the time delay crosses a threshold value and also prove the existence of stability switching phenomena. Theoretical results of this paper justified with numerical simulations.