The Effect of Time Delay on the Dynamics of a Plankton-Nutrient System with Refuge

The interaction between aquatic organisms and plants is crucial to maintaining the ecological balance of the marine environment. This study aims to investigate the effects of providing refuge on phytoplankton and the delay in releasing hazardous substances by phytoplankton. To accomplish this, a nutrient-phytoplankton-zooplankton system was developed and analyzed to examine its equilibria and the conditions under which they exist. The model’s stability was also analyzed, both in the absence and presence of time delays. The inclusion of a time delay caused a Hopf bifurcation at the interior equilibrium of the model, resulting in a transition from stability to instability. The study found that as the time delay value increases, the system’s stable dynamics transition into an unstable state. However, because phytoplankton refuge is part of the system, the dynamics temporarily become stable, even though phytoplankton release the toxins over a long time. However, this stable behavior is disrupted when the time required for the toxin liberation process by phytoplankton is further extended.