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

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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.

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Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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Funding

Open access funding provided by Manipal Academy of Higher Education, Manipal. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Mathematics, Poornima University, Jaipur, 303905, Rajasthan, India

    Randhir Singh Baghel

  2. Department of Physics, Poornima University, Jaipur, 303905, Rajasthan, India

    Shrikant Verma

  3. Department of Mechatronics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, 576104, India

    Narendra Khatri

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  1. Randhir Singh Baghel
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  2. Shrikant Verma
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  3. Narendra Khatri
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Contributions

R.S.B. and S.V. conceptualized and designed the study. R.S.B. conducted the experimental work and data collection. S.V. performed the data analysis and interpretation. N.K. contributed to reviewing, editing, and redrafting the manuscript and handled the correspondence. All authors reviewed and approved the final version of the manuscript.

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Narendra Khatri.

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Baghel, R.S., Verma, S. & Khatri, N. Delayed dynamics and detoxification in nutrient-phytoplankto-by-product systems: mechanisms driving bloom stability and oscillations.
Sci Rep (2025). https://doi.org/10.1038/s41598-025-32146-z

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  • DOI: https://doi.org/10.1038/s41598-025-32146-z

Keywords

  • Phytoplankton-nutrient dynamics
  • Beddington-DeAngelis uptake
  • By-product interference (allelopathy)
  • Stability and Hopf bifurcation
  • Global sensitivity analysis (Sobol, PRCC)
  • Delay

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