Fig. 10 | Scientific Reports

Fig. 10

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

Fig. 10

Continuation of the leading characteristic root and Hopf detection with delay. (a) Real part of the dominant eigenvalue \(\Re (\lambda )\) versus delay \(\tau\) showing a smooth crossing of the imaginary axis at the critical value \(\tau _c \approx 37.65\), marking the onset of a Hopf bifurcation. (b) Repeated computation confirming identical Hopf crossing behaviour. For \(\tau <\tau _c\) the system is stable; for \(\tau>\tau _c\), sustained oscillations emerge.

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