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

## TL;DR

This paper explores how phytoplankton blooms are influenced by nutrient availability, growth, and inhibitory by-products, revealing how detoxification delays can lead to oscillatory behavior.

## Contribution

The study introduces a new mechanistic model linking nutrient uptake, by-product inhibition, and detoxification delays to explain bloom dynamics.

## Key findings

- Detoxification delays can trigger oscillatory bloom behavior through Hopf bifurcations.
- Stable coexistence is more common than repeated bloom-crash cycles in realistic ecological conditions.
- By-product production and detoxification rate are key factors regulating bloom stability.

## Abstract

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.