Delayed dynamics and detoxification in nutrient-phytoplankto-by-product systems: mechanisms driving bloom stability and oscillations.

Phytoplankton blooms emerge from the interplay between nutrient availability, biomass growth, and inhibitory by-products such as toxins or exudates. Here, we develop a mechanistic nutrient-phytoplankton-by-product model that couples Beddington-DeAngelis nutrient uptake, by-product-mediated inhibition, and nutrient-dependent detoxification. Analytical results demonstrate that the system remains biologically feasible and bounded, and that a threshold condition governs bloom initiation. Linear stability and bifurcation analyses reveal how detoxification delays can trigger oscillatory bloom behaviour. Across ecologically realistic parameter regimes, the system tends to a stable coexistence state-either directly or through damped oscillations-rather than exhibiting repeated bloom-crash cycles. Global sensitivity analysis (PRCC and Sobol indices) highlights by-product production, inhibition strength, detoxification rate, toxin-linked mortality, and saturation effects as dominant regulators of stability and damping time. Introducing an explicit ecological delay exposes a critical threshold at which a Hopf bifurcation arises, converting the stable equilibrium into sustained oscillations. Numerical simulations confirm the transversality condition and indicate a supercritical onset. Collectively, these results provide a quantitative diagnostic for distinguishing transient from sustained bloom oscillations and identify measurable ecological processes-particularly detoxification and delayed feedback-that govern transitions between stable and oscillatory regimes.