Fast–slow dynamics in a cyanobacteria–phosphorus model

Understanding how cyanobacterial blooms emerge from biological processes requires models that account for both rapid physiological adjustments and slower nutrient dynamics. In this study, we analyze a cyanobacterial-phosphorus model using dynamic simulation, quasi-steady-state approximation (QSSA), single-parameter sensitivity analysis, and global uncertainty propagation. Comparisons between the full model and QSSA show that while QSSA can reproduce the rapid equilibrium of the internal phosphorus quota, it systematically overestimates dissolved phosphorus during periods of high external loading because it ignores short-term biomass-nutrient feedbacks. Sensitivity analyses reveal threshold-like transitions in biomass behavior for initial biomass and growth rate. The model abruptly shifts from low steady state biomass to one in which biomass grows rapidly. In a broader parameter set, the model indicates two qualitatively distinct long-term states. A low-biomass, light-limited state and a high-biomass, bloom-prone state reflect the existence of a bifurcated structure in the ecological dynamics. Uncertainty analysis confirms the coexistence of two distinct dynamical states including rapid collapse to a low-biomass state and sustained convergence toward a high-biomass attractor under combinations of growth capacity, light availability, and nutrient loading. Partial rank correlation analysis identifies growth rate and light attenuation as dominant controls on bloom magnitude, while nutrient storage parameters exert secondary influence. This study indicates that cyanobacterial model can experience ecological bifurcations, where small changes in environmental or physiological conditions shift the ecosystem between alternative stable states. Therefore, we show a vital characteristic of such models by combining fast–slow analysis, sensitivity analysis, and global uncertainty propagation which is crucial for studying eutrophication and designing effective nutrient management strategies.