Hopf bifurcation in a periodic toxin producing phytoplankton model

Abstract

Harmful algal blooms (HABs) caused by toxin-producing phytoplankton (TPP) have become increasingly common worldwide. Understanding the complex interactions between TPP and other organisms in the ecosystem is crucial. This study focuses on the Hopf bifurcation analysis of plankton interactions between TPP and zooplankton, with uptake function and a periodic toxin production. The maximum toxin liberation rate is considered as a bifurcation parameter. The aim is to determine how the toxin liberation rate affects the system. One of the proposed models assumes constant toxin production by TPP, resulting in an autonomous system of ordinary differential equations. To incorporate natural day and night, tidal, or seasonal cycles, the model is extended to a periodic system. The study examines the existence of steady states and trivial periodic solutions and analyses the stability of both models. Moreover, using the concept of uniform persistence, we derive sufficient conditions for the coexistence of the periodic system based on the model parameters. Due to instability of equilibria, we observe Hopf bifurcations in the constant toxin-producing model, providing insights into the system's dynamic behaviour. Numerical simulations are performed to validate the analytical findings of the proposed models and their implications.