Bifurcations of a discrete-time Phytoplankton-Zooplankton model

Abstract

This work mainly investigates the bifurcation problems and topological classifications of a discrete phytoplankton-zooplankton model, whose continuous version was proposed by Truscott and Brindley in 1994. The model is derived by using the semidiscretization technique. Within this study, trivial and semi-trivial fixed points are identified, as well as an interior fixed point that emerges based on specific parametric conditions.Subsequently, an exploration of their topological classifications is undertaken, employing linear stability theory in the vicinity of the trivial, semi-trivial, and interior fixed points.Leveraging the center manifold theorem and bifurcation theory, it is feasible to derive conditions under which the flip and Neimark-Sacker bifurcations are expected to transpire. To validate these findings and draw conclusions, numerical simulations are conducted which produce proof of a Neimark-Sacker bifurcation. Through these comprehensive analyses and simulations, the main aim of this research is to effectively augment the understanding of the dynamics of the model and affirm the validity of our results.