Effects of Nutrient Input and Zooplankton Immunity to Phytoplankton-Derived Toxins on Shallow Take Algal Blooms and Dynamics

Algal blooms in shallow lakes have attracting increasing attention owing to their high frequency and severe ecological harm. Previous studies, which primarily focused on single influencing factors such as nutrient input or toxins released by phytoplankton, have demonstrated that excessive nutrient input can increase the incidence of algal blooms, while high toxin levels exert an inhibitory effect. This study proposes a general analytical framework for exploring zooplankton grazing responses to phytoplankton-derived toxins and nutrient uptake, with a focus on zooplankton immunity to such toxins. A novel nutrient-phytoplankton-zooplankton model is adopted to investigate the combined impacts of nutrient input and phytoplankton-derived toxins on algal blooms, considering zooplankton immunity. We conduct analyses on the existence, stability, and bifurcations of the model to explore its complex dynamic behaviors, including Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2 and a degenerate case with a nilpotent cusp of codimension at least 3. Specific forms of nutrient uptake and zooplankton grazing functions are employed to validate the relevant theoretical results and perform numerical simulations. Beyond existing findings, this study reveals that oscillation periods tend to lengthen under lower toxin levels or higher nutrient input when oscillations occur.