Bifurcation and Chaos in a Phytoplankton–Zooplankton Model with Holling Type-II Response and Toxicity

Abstract

Phytoplankton and zooplankton’s interconnection coordinate in various dynamical processes that occur in ecological population and make it a fascinating subject matter to explore. Time delay is an additional factor that plays an imperative part in the dynamical frameworks. For a stable aquatic ecosystem, the growth of both zooplankton and phytoplankton should be in steady state but in previous eras, there have been a universal explosion in destructive plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So in this paper, we studied dynamical characteristics of a discrete-time phytoplankton–zooplankton model with Holling type-II predational functional response and toxic substance distribution. More specifically, we studied local stability with topological classification at equilibrium solutions, periodic points and possible bifurcation analysis of the model under consideration. It is proved that at trivial and semitrivial equilibrium solutions the phytoplankton–zooplankton model does not undergo flip bifurcation but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution. It is also proved that the interior equilibrium solution model does not undergo the fold bifurcation. We also studied chaos in the sense of feedback control strategy. Finally, extensive numerical simulations are provided to demonstrate theoretical results.