Modeling and analysis of the effect of dissolved oxygen on the dynamics of Plankton system in shallow lakes

Considering the effect of dissolved oxygen concentration on the filtering rate of zooplankton, we propose and mathematically analyze an ordinary differential equation model involving the interactions among phytoplankton, zooplankton, and dissolved oxygen. For this three-component ecological model, we derive the stability conditions of boundary equilibria and the existence conditions of multiple positive equilibria. The local dynamical behaviors of the model are discussed by examining the singularity of a triple equilibrium on a manifold, revealing complex bifurcation phenomena. The theoretical results show that the system undergoes Saddle-node bifurcation, Hopf bifurcation, and Transcritical bifurcation, all of which are corroborated by numerical simulations. By selecting the zooplankton death rate and the maximum relative influence ratio of dissolved oxygen on the zooplankton filtering rate as two bifurcation parameters, we construct the corresponding bifurcation diagrams and provide biological interpretations for the model's dynamical behaviors under parameter variations. This study highlights that both the zooplankton death rate and dissolved oxygen concentration significantly influence the rich dynamical patterns of the plankton ecosystem, playing a crucial role in triggering algal bloom outbreaks.