Stochastic dynamics of nutrient-phytoplankton-by-product systems under environmental variability: Gaussian noise and feedback regulation

In this study, we develop and analyze a stochastic nutrient–phytoplankton–by-product model that captures the coupled effects of resource limitation, self-inhibition, and random environmental variability in aquatic ecosystems. Starting from a deterministic formulation, we establish the existence of positive equilibria and derive local stability conditions using the characteristic polynomial and Routh–Hurwitz criteria. Bifurcation analysis identifies parameter thresholds where steady states lose stability, leading to oscillatory or unstable nutrient–phytoplankton cycles. Extending the model into a stochastic framework with multiplicative Gaussian noise, we prove the existence of a unique positive global solution, demonstrate boundedness, and derive persistence and extinction criteria through Lyapunov and generator-based methods. Numerical simulations implemented using Higham’s higher-order stochastic scheme confirm the analytical results, showing that environmental fluctuations can induce transient blooms, damped oscillations, or stochastic collapses depending on parameter values. The results highlight how noise intensity and feedback regulation jointly shape long-term ecosystem stability, offering quantitative insight into how variability in nutrient supply and inhibitory by-products influences phytoplankton under fluctuating environmental conditions.