Stability and Bifurcation Analysis of an Allelopathic Phytoplankton–Zooplankton Model

This work proposes a new mathematical model incorporating allelopathy into the dynamic interaction between phytoplankton and zooplankton populations. The proposed nonlinear system is formulated to investigate how allelopathic effects influence the long-term behavior of the aquatic ecosystem. We first establish the model’s basic mathematical properties, including the existence, positivity, and boundedness of solutions, to ensure biological relevance. The equilibrium points of the system are then determined, and their local stability is also investigated. Furthermore, conditions for Hopf bifurcation are derived to examine the emergence of periodic oscillations in population dynamics. Numerical simulations are carried out to support the analytical results and illustrate the influence of model parameters on system behavior. The findings indicate that allelopathic interactions can significantly affect the stability and coexistence of phytoplankton and zooplankton populations, providing valuable insights into aquatic ecosystem dynamics.