Bifurcation analysis of a discrete Phytoplankton–Zooplankton model with linear predational response function and toxic substance distribution

Abstract

Phytoplanktons are drifting plants in an aquatic system. They provide food for marine animals and are compared to terrestrial plants in that having chlorophyll and carrying out photosynthesis. Zooplanktons are drifting animals found inside the aquatic bodies. For stable aquatic ecosystem, the growth of both Zooplankton and Phytoplankton should be in steady state but in previous eras, there has 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, a discretized two-dimensional Phytoplankton–Zooplankton model is investigated. The results for the existence and uniqueness, and conditions for local stability with topological classifications of the equilibrium solutions are determined. It is also exhibited that at trivial and semitrivial equilibrium solutions, discrete model does not undergo flip bifurcation, but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution under certain conditions. Further, state feedback method is deployed to control the chaos in the under consideration system. The extensive numerical simulations are provided to demonstrate theoretical results.