Modelling and analysis of a phytoplankton zooplankton system with continuous and discrete time.

Abstract

In this study, a phytoplankton–zooplankton system has been modelled using a system of differential equations with piecewise constant arguments, which represents a new approach to modelling phytoplankton– zooplankton interaction. To analyse the dynamic behaviour of the model, we consider the solution of the system in a certain subinterval, which yields a system of difference equations. Some theoretical results on the boundedness character and local stability properties for the discrete dynamical system are obtained. In addition, we explain the biological dynamics of the bloom in the plankton model through Neimark–Sacker bifurcation and obtain the threshold values for different parameters that govern the periodic nature of the bloom. We conclude that, while other studies explained that the bloom depended on only one parameter, this study explains that the bloom depended on three different parameters, namely θ (rate of toxin production per phytoplankton), β (zooplankton growth efficiency) and K (environmental carrying capacity of phytoplankton).