Seasonal dynamics and stoichiometry of the planktonic community in the NW Mediterranean Sea: a 3D modeling approach

Abstract

The 3D hydrodynamic Model for Applications at Regional Scale (MARS3D) was coupled with a biogeochemical model developed with the Ecological Modular Mechanistic Modelling (Eco3M) numerical tool. The three-dimensional coupled model was applied to the NW Mediterranean Sea to study the dynamics of the key biogeochemical processes in the area in relation with hydrodynamic constraints. In particular, we focused on the temporal and spatial variability of intracellular contents of living and non-living compartments. The conceptual scheme of the biogeochemical model accounts for the complex food web of the NW Mediterranean Sea (34 state variables), using flexible plankton stoichiometry. We used mechanistic formulations to describe most of the biogeochemical processes involved in the dynamics of marine pelagic ecosystems. Simulations covered the period from September 1, 2009 to January 31, 2011 (17 months), which enabled comparison of model outputs with situ measurements made during two oceanographic cruises in the region (Costeau-4: April 27–May 2, 2010 and Costeau-6: January 23–January 27, 2011).

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Acknowledgments

The present research is part of the project COSTAS (“Trophic contaminants in the system: phytoplankton, zooplankton, anchovy, sardine”), funded by the French ANR/CES and IFREMER. One of the objectives of COSTAS was related to the model’s ability to replicate space-temporal dynamics of different functional groups of plankton and associated trophic fluxes. Part of this research is also a contribution to the Labex OT-Med (no. ANR-11-LABX-0061) funded by the French Government “Investissements d’Avenir” program of the French National Research Agency (ANR) through the A*MIDEX project (no ANR-11-IDEX-0001-02). We thank our anonymous reviewers for their helpful comments that allowed us to improve the manuscript.

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Correspondence to Elena Alekseenko.

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This article is part of the Topical Collection on the 16th biennial workshop of the Joint Numerical Sea Modelling Group (JONSMOD) in Brest, France 21-23 May 2012

Responsible Editor: Martin Verlaan

Appendices

Appendix A. Equations of the biogeochemical model

Mesozooplankton (adults).

$$ \frac{ dZ}{ dt}=\underset{\mathrm{growth}}{\underbrace{f_z^{\mu}\cdot Z}}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_z^m\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\;\\ {}\mathrm{higher}\kern0.36em \mathrm{trophic}\;\mathrm{levels}\end{array}}{\underbrace{f_z^{mq}\cdot {Z}^2}} $$
(1)
$$ \begin{array}{l}\frac{d{Z}_C}{ dt}=\underset{\mathrm{grazing}\;\mathrm{on}\kern0.48em \mathrm{CIL},\mathrm{HNF}\mathrm{and}\;\mathrm{PHYL}}{\underbrace{\left(1-{\in}_{sf}\right)\left({f}_z^{g_{\mathrm{CIL}}}\cdot {Q}_C^{\mathrm{CIL}}+{f}_z^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}+{f}_z^{g_{\mathrm{PHYL}}}\cdot {Q}_C^{\mathrm{PHYL}}\right)\cdot {h}_Z^{Q_C}\cdot Z}}-\underset{\mathrm{resp}\mathrm{iration}}{\underbrace{f_z^{\mathrm{resp}}\cdot {Z}_C}}\hfill \\ {}-\underset{\mathrm{natural}\ \mathrm{mortality}}{\underbrace{f_z^m\cdot {Z}_C}}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\;\\ {}\mathrm{high}.\mathrm{troph}.\kern0.36em \mathrm{lev}.\end{array}}{\underbrace{f_z^{mq}\cdot {Z}^2\cdot {Q}_C^Z}}+\underset{\begin{array}{l}\mathrm{grazing}\kern0.48em \mathrm{during}\kern0.36em \mathrm{juvenile}\\ {}\mathrm{stages}\kern0.36em \mathrm{on}\kern0.36em \mathrm{CIL},\mathrm{HNF}\kern0.36em \mathrm{and}\kern0.36em \mathrm{POM}\end{array}}{\underbrace{f_Z^{gjuv}\cdot Z}}\hfill \end{array} $$
(2)

Microzooplankton.

$$ \frac{ dCIL}{ dt}=\underset{\mathrm{growth}}{\underbrace{f_{\mathrm{CIL}}^{\mu}\cdot \mathrm{CIL}}}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{CIL}}^m\cdot CIL}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.36em \mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{CIL}}}\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_1\cdot {f}_Z^{\mathrm{gjuv}}\cdot \frac{Z}{Q_C^{\mathrm{CIL}}}}} $$
(3)
$$ \begin{array}{l}\frac{d{\mathrm{CIL}}_C}{ dt}=\underset{\mathrm{grazing}\;\mathrm{on}\;\mathrm{HNFand}\;\mathrm{PHYS}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}+{f}_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_C^{\mathrm{PHYS}}\right)\cdot {h}_{\mathrm{CIL}}^{Q_C}\cdot CI L}}-\underset{\mathrm{resp}\mathrm{iration}}{\underbrace{f_{\mathrm{CIL}}^{\mathrm{resp}}\cdot CI{L}_C}}\hfill \\ {}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{CIL}}^m\cdot {\mathrm{CIL}}_C}}-\underset{\mathrm{grazing}\;\mathrm{by}\;\mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{CIL}}}\cdot {Q}_C^{\mathrm{CIL}}\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_1\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z}}\hfill \end{array} $$
(4)
$$ \begin{array}{l}\frac{d{\mathrm{CIL}}_N}{ dt}=\underset{\mathrm{grazing}\;\mathrm{on}\;\mathrm{HNFand}\;\mathrm{PHYS}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot {Q}_N^{\mathrm{HNF}}+{f}_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_N^{\mathrm{PHYS}}\right)\cdot {h}_{\mathrm{CIL}}^{Q_N}\cdot \mathrm{CIL}}}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{CIL}}^m\cdot {\mathrm{CIL}}_N}}\hfill \\ {}-\underset{\mathrm{grazing}\;\mathrm{by}\;\mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{CIL}}}\cdot {Q}_N^{\mathrm{CIL}}\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_1\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z\cdot \frac{Q_N^{CIL}}{Q_C^{CIL}}}}\hfill \end{array} $$
(5)
$$ \begin{array}{l}\frac{d{\mathrm{CIL}}_P}{ dt}=\underset{\mathrm{grazing}\;\mathrm{on}\;\mathrm{HNFand}\;\mathrm{PHYS}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot {Q}_P^{\mathrm{HNF}}+{f}_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_P^{\mathrm{PHYS}}\right)\cdot {h}_{\mathrm{CIL}}^{Q_P}\cdot \mathrm{CIL}}}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{CIL}}^m\cdot {\mathrm{CIL}}_P}}\hfill \\ {}-\underset{\mathrm{grazing}\;\mathrm{by}\;\mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{CIL}}}\cdot {Q}_P^{\mathrm{CIL}}\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_1\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z\cdot \frac{Q_P^{\mathrm{CIL}}}{Q_C^{\mathrm{CIL}}}}}\hfill \end{array} $$
(6)

Nanozooplankton.

$$ \begin{array}{l}\frac{ dHNF}{ dt}=\underset{\mathrm{growth}}{\underbrace{f_{\mathrm{HNF}}^{\mu}\cdot \mathrm{HNF}}}-\underset{\mathrm{natural}\kern0.36em \mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot \mathrm{HNF}}}-\underset{\mathrm{grazing}\;\mathrm{by}\;\mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{HNF}}}\cdot Z}}\hfill \\ {}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.36em \mathrm{CIL}}{\underbrace{f_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot \mathrm{CIL}}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_2\cdot {f}_Z^{\mathrm{gjuv}}\cdot \frac{Z}{Q_C^{\mathrm{HNF}}}}}\hfill \end{array} $$
(7)
$$ \begin{array}{l}\frac{d{\mathrm{HNF}}_C}{ dt}=\underset{\mathrm{grazing}\;\mathrm{on}\kern0.48em \mathrm{PHYS}\;\mathrm{and}\kern0.36em \mathrm{BAC}}{\underbrace{\left({f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_C^{\mathrm{PHYS}}+{f}_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot {Q}_C^{\mathrm{BAC}}\right)\cdot {h}_{\mathrm{HNF}}^{Q_C}\cdot \mathrm{HNF}}}-\underset{\mathrm{resp}\mathrm{iration}}{\underbrace{f_{\mathrm{HNF}}^{\mathrm{resp}}\cdot {\mathrm{HNF}}_C}}-\underset{\mathrm{natural}\kern0.36em \mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot {\mathrm{HNF}}_C}}\hfill \\ {}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.36em \mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}\cdot Z}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.36em \mathrm{CIL}}{\underbrace{f_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}\cdot \mathrm{CIL}}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.36em \mathrm{stages}\end{array}}{\underbrace{\alpha_2\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z}}\hfill \end{array} $$
(8)
$$ \begin{array}{l}\frac{d{\mathrm{HNF}}_N}{ dt}=\underset{\mathrm{grazing}\kern0.36em \mathrm{on}\kern0.36em \mathrm{PHYS}\;\mathrm{and}\kern0.36em \mathrm{BAC}}{\underbrace{\left({f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_N^{\mathrm{PHYS}}+{f}_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot {Q}_N^{\mathrm{BAC}}\right)\cdot {h}_{\mathrm{HNF}}^{Q_N}\cdot \mathrm{HNF}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot {\mathrm{HNF}}_N}}\hfill \\ {}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.36em \mathrm{meso}-\mathrm{Z}\kern0.36em \mathrm{and}\kern0.36em \mathrm{CIL}}{\underbrace{\left({f}_z^{g_{\mathrm{HNF}}}\cdot Z+{f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot \mathrm{CIL}\right)\cdot {Q}_N^{\mathrm{HNF}}}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_2\cdot {f}_Z^{\mathrm{gjuv}}\cdot \frac{Q_N^{\mathrm{HNF}}}{Q_C^{\mathrm{HNF}}}}}\hfill \end{array} $$
(9)
$$ \begin{array}{l}\frac{d{\mathrm{HNF}}_P}{ dt}=\underset{\mathrm{grazing}\;\mathrm{on}\;\mathrm{PHYS}\;\mathrm{and}\;\mathrm{BAC}}{\underbrace{\left({f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_P^{\mathrm{PHYS}}+{f}_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot {Q}_P^{\mathrm{BAC}}\right)\cdot {h}_{\mathrm{HNF}}^{Q_P}\cdot \mathrm{HNF}}}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot {\mathrm{HNF}}_P}}\hfill \\ {}-\underset{\mathrm{grazing}\;\mathrm{by}\;\mathrm{meso}-\mathrm{Z}\;\mathrm{and}\kern0.48em \mathrm{CIL}}{\underbrace{\left({f}_z^{g_{\mathrm{HNF}}}\cdot Z+{f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot \mathrm{CIL}\right)\cdot {Q}_P^{\mathrm{HNF}}}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.24em \mathrm{by}\kern0.24em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.36em \mathrm{juvenile}\kern0.24em \mathrm{stages}\end{array}}{\underbrace{\alpha_2\cdot {f}_Z^{\mathrm{gjuv}}\cdot \frac{Q_P^{\mathrm{HNF}}}{Q_C^{\mathrm{HNF}}}}}\hfill \end{array} $$
(10)

Phytoplankton.

$$ \frac{ dPHYL}{ dt}=\underset{\mathrm{growth}}{\underbrace{f_{\mathrm{PHYL}}^{\mu}\cdot \mathrm{PHYL}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYL}}^m\cdot \mathrm{PHYL}}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{PHYL}}}\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{higher}\\ {}\mathrm{troph}.\kern0.48em \mathrm{levels}\end{array}}{\underbrace{f_{\mathrm{PHYL}}^{mq}\cdot {\mathrm{PHYL}}^2}} $$
(11)
$$ \begin{array}{l}\frac{d{\mathrm{PHYL}}_C}{ dt}=\underset{ primary\; production}{\underbrace{f_{nr}^{\mathrm{PP}}\cdot {h}_{\mathrm{PHYL}}^{Q_C}\cdot \mathrm{PHYL}}}-\underset{\mathrm{resp}\mathrm{iration}}{\underbrace{f_{\mathrm{PHYL}}^{\mathrm{resp}}\cdot {\mathrm{PHYL}}_C}}-\underset{\mathrm{natural}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{PHYL}}^m\cdot {\mathrm{PHYL}}_C}}\hfill \\ {}-\underset{\mathrm{grazing}\;\mathrm{by}\;\mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{PHYL}}}\cdot {Q}_C^{\mathrm{PHYL}}\cdot Z}}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{higher}\\ {}\mathrm{troph}.\kern0.48em \mathrm{levels}\end{array}}{\underbrace{f_{\mathrm{PHYL}}^{mq}\cdot {\mathrm{PHYL}}^2\cdot {Q}_C^{\mathrm{PHYL}}}}\hfill \end{array} $$
(12)
$$ \begin{array}{l}\frac{d{\mathrm{PHYL}}_N}{ dt}=\underset{\mathrm{primary}\;\mathrm{production}}{\underbrace{\left({f}_{\mathrm{PHYL}}^{{\mathrm{upt}}_N}+{f}_{\mathrm{PHYL}}^{{\mathrm{upt}}_{\mathrm{DON}}}\right)\cdot {h}_{\mathrm{PHYL}}^{Q_N}\cdot \mathrm{PHYL}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYL}}^m\cdot {\mathrm{PHYL}}_N}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{PHYL}}}\cdot {Q}_N^{\mathrm{PHYL}}\cdot Z}}\hfill \\ {}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{higher}\\ {}\mathrm{troph}.\kern0.48em \mathrm{levels}\end{array}}{\underbrace{f_{\mathrm{PHYL}}^{mq}\cdot {\mathrm{PHYL}}^2\cdot {Q}_N^{\mathrm{PHYL}}}}\hfill \end{array} $$
(13)
$$ \begin{array}{l}\frac{d{\mathrm{P}\mathrm{HYL}}_P}{ dt}=\underset{\mathrm{primary}\;\mathrm{production}}{\underbrace{\left({f}_{\mathrm{P}\mathrm{HYL}}^{{\mathrm{upt}}_{\mathrm{P}}}+{f}_{\mathrm{P}\mathrm{HYL}}^{{\mathrm{upt}}_{\mathrm{DOP}}}\right)\cdot {h}_{\mathrm{P}\mathrm{HYL}}^{Q_P}\cdot \mathrm{PHYL}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{P}\mathrm{HYL}}^m\cdot {\mathrm{P}\mathrm{HYL}}_P}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{P}\mathrm{HYL}}}\cdot {Q}_P^{\mathrm{P}\mathrm{HYL}}\cdot Z}}\hfill \\ {}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{higher}\\ {}\mathrm{troph}.\kern0.48em \mathrm{levels}\end{array}}{\underbrace{f_{\mathrm{P}\mathrm{HYL}}^{mq}\cdot {\mathrm{P}\mathrm{HYL}}^2\cdot {Q}_P^{\mathrm{P}\mathrm{HYL}}}}\hfill \end{array} $$
(14)
$$ \begin{array}{l}\frac{d{\mathrm{PHYL}}_{\mathrm{chl}}}{ dt}=\underset{\mathrm{chl}\mathrm{orophyll}\kern0.48em \mathrm{synthesis}}{\underbrace{f^{PChl}\cdot {\mathrm{PHYL}}_N}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYL}}^m\cdot {\mathrm{PHYL}}_{chl}}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{meso}-\mathrm{Z}}{\underbrace{f_z^{g_{\mathrm{PHYL}}}\cdot {Q}_{chl}^{\mathrm{PHYL}}\cdot Z}}\hfill \\ {}-\underset{\begin{array}{l}\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{higher}\\ {}\mathrm{troph}.\kern0.48em \mathrm{levels}\end{array}}{\underbrace{f_{\mathrm{PHYL}}^{mq}\cdot \mathrm{PHYL}\cdot {\mathrm{PHYL}}_{\mathrm{chl}}}}\hfill \end{array} $$
(15)
$$ \frac{ dPHYS}{ dt}=\underset{\mathrm{growth}}{\underbrace{f_{\mathrm{PHYS}}^{\mu}\cdot \mathrm{PHYS}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{motrality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot \mathrm{PHYS}}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.48em \mathrm{CIL}}{\underbrace{f_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot \mathrm{CIL}}}-\underset{\mathrm{grazing}\;\mathrm{by}\kern0.36em \mathrm{HNF}}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot \mathrm{HNF}}} $$
(16)
$$ \begin{array}{l}\frac{d{\mathrm{PHYS}}_C}{ dt}=\underset{\mathrm{primary}\;\mathrm{production}}{\underbrace{f_{nr}^{\mathrm{PP}}\cdot {h}_{\mathrm{PHYS}}^{Q_C}\cdot \mathrm{PHYS}}}-\underset{\mathrm{resp}\mathrm{iration}}{\underbrace{f_{\mathrm{PHYS}}^{\mathrm{resp}}\cdot {\mathrm{PHYS}}_C}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot {\mathrm{PHYS}}_C}}\hfill \\ {}-\underset{\mathrm{grazing}\kern0.48em \mathrm{by}\kern0.48em \mathrm{CIL}\kern0.48em \mathrm{and}\kern0.48em \mathrm{NHF}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot \mathrm{CIL}+{f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot \mathrm{HNF}\right)\cdot {Q}_C^{\mathrm{PHYS}}}}\hfill \end{array} $$
(17)
$$ \begin{array}{l}\frac{d{\mathrm{PHYS}}_N}{ dt}=\underset{\mathrm{net}\kern0.36em \mathrm{uptake}\kern0.36em \mathrm{of}\kern0.36em {\mathrm{N}\mathrm{O}}_3\mathrm{and}\ {\mathrm{N}\mathrm{H}}_4\kern0.36em \mathrm{and}\kern0.36em \mathrm{DON}}{\underbrace{\left({f}_{\mathrm{PHYS}}^{{\mathrm{upt}}_{\mathrm{N}}}+{f}_{\mathrm{PHYS}}^{{\mathrm{upt}}_{\mathrm{DON}}}\right)\cdot {h}_{\mathrm{PHYS}}^{Q_N}\cdot \mathrm{PHYS}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot {\mathrm{PHYS}}_N}}\hfill \\ {}-\underset{\mathrm{grazing}\kern0.48em \mathrm{by}\kern0.48em \mathrm{CIL}}{\underbrace{f_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_N^{\mathrm{PHYS}}\cdot \mathrm{CIL}}}-\underset{\mathrm{grazing}\kern0.48em \mathrm{by}\kern0.48em \mathrm{H}\mathrm{NF}\;}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_N^{\mathrm{PHYS}}\cdot \mathrm{HNF}}}\hfill \end{array} $$
(18)
$$ \begin{array}{l}\frac{d{\mathrm{P}\mathrm{HYS}}_P}{ dt}=\underset{\mathrm{net}\kern0.36em \mathrm{uptake}\kern0.36em \mathrm{of}\kern0.36em {\mathrm{P}\mathrm{O}}_4\kern0.36em \mathrm{and}\kern0.36em \mathrm{DOP}}{\underbrace{\left({f}_{\mathrm{P}\mathrm{HYS}}^{{\mathrm{upt}}_{\mathrm{P}}}+{f}_{\mathrm{P}\mathrm{HYS}}^{{\mathrm{upt}}_{\mathrm{DOP}}}\right)\cdot {h}_{\mathrm{P}\mathrm{HYS}}^{Q_P}\cdot \mathrm{PHYS}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{P}\mathrm{HYS}}^m\cdot {\mathrm{P}\mathrm{HYS}}_P}}\hfill \\ {}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{CIL}}{\underbrace{f_{\mathrm{CIL}}^{g_{\mathrm{P}\mathrm{HYS}}}\cdot {Q}_P^{\mathrm{P}\mathrm{HYS}}\cdot \mathrm{CIL}}}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{HNF}\;}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{P}\mathrm{HYS}}}\cdot {Q}_P^{\mathrm{P}\mathrm{HYS}}\cdot \mathrm{HNF}}}\hfill \end{array} $$
(19)
$$ \begin{array}{l}\frac{d{\mathrm{PHYS}}_{\mathrm{chl}}}{ dt}=\underset{\mathrm{chl}\mathrm{orophyll}\kern0.48em \mathrm{synthesis}}{\underbrace{f^{PChl}\cdot {\mathrm{PHYS}}_N}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot {\mathrm{PHYS}}_{\mathrm{chl}}}}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{CIL}}{\underbrace{f_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_{\mathrm{chl}}^{\mathrm{PHYS}}\cdot CIL}}\hfill \\ {}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{HNF}}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_{chl}^{\mathrm{PHYS}}\cdot \mathrm{HNF}}}\hfill \end{array} $$
(20)

Bacteria.

$$ \frac{ dBAC}{ dt}=\underset{\mathrm{growth}}{\underbrace{f_{\mathrm{BAC}}^{\mu}\cdot \mathrm{BAC}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot \mathrm{BAC}}}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{HNF}}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot HNF}} $$
(21)
$$ \begin{array}{l}\frac{d{\mathrm{BAC}}_C}{ dt}=\underset{\mathrm{net}\kern0.36em \mathrm{uptake}\kern0.48em \mathrm{of}\kern0.36em \mathrm{LDOC}\kern0.36em \mathrm{and}\kern0.48em \mathrm{SLDOC}}{\underbrace{\left({f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{LDOC}}}+{f}_{\mathrm{BAC}}^{up{t}_{\mathrm{SLDOC}}}\right)\cdot {h}_{\mathrm{BAC}}^{Q_C}\cdot \mathrm{BAC}}}-\underset{\mathrm{resp}\mathrm{iration}}{\underbrace{f_{\mathrm{BAC}}^{\mathrm{resp}}\cdot {\mathrm{BAC}}_C}}-\underset{\mathrm{natural}\kern0.6em \mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot {\mathrm{BAC}}_C}}\hfill \\ {}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{HNF}}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot \mathrm{HNF}\cdot {Q}_C^{\mathrm{BAC}}}}\hfill \end{array} $$
(22)
$$ \frac{d{\mathrm{BAC}}_N}{ dt}=\underset{\mathrm{net}\kern0.36em \mathrm{uptake}\kern0.18em \mathrm{of}\kern0.18em \mathrm{N}}{\underbrace{\left({f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{N}}}+{f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{DON}}}\right)\cdot {h}_{\mathrm{BAC}}^{Q_N}\cdot \mathrm{BAC}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot {\mathrm{BAC}}_N}}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.48em \mathrm{N}\mathrm{HF}}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot \mathrm{HNF}\cdot {Q}_N^{\mathrm{BAC}}}} $$
(23)
$$ \frac{d{\mathrm{BAC}}_P}{ dt}=\underset{\mathrm{net}\kern0.28em \mathrm{uptake}\kern0.18em \mathrm{of}\kern0.18em \mathrm{P}}{\underbrace{\left({f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{P}}}+{f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{DOP}}}\right)\cdot {h}_{\mathrm{BAC}}^{Q_P}\cdot \mathrm{BAC}}}-\underset{\mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot {\mathrm{BAC}}_P}}-\underset{\mathrm{grazing}\kern0.36em \mathrm{by}\kern0.18em \mathrm{NHF}}{\underbrace{f_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot \mathrm{HNF}\cdot {Q}_P^{\mathrm{BAC}}}} $$
(24)

Dissolved organic carbon material.

$$ \begin{array}{l}\frac{ dLDOC}{ dt}=-\underset{\mathrm{gross}\kern0.26em \mathrm{uptake}\kern0.18em \mathrm{by}\kern0.18em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{LDOC}}}\cdot \mathrm{BAC}}}+\underset{\mathrm{BAC}\kern0.26em \mathrm{natural}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot {\mathrm{BAC}}_C}}+\underset{\mathrm{PHYS}\kern0.36em \mathrm{natural}\kern0.48em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot {\mathrm{PHYS}}_C}}\hfill \\ {}+\underset{\mathrm{HNF}\kern0.28em \mathrm{natural}\kern0.26em \mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot {\mathrm{HNF}}_C}}+\underset{\mathrm{POC}\;\mathrm{mineralization}}{\underbrace{f_{\mathrm{BAC}}^{\mathrm{rem}}\cdot {\mathrm{BAC}}_C}}\hfill \end{array} $$
(25)
$$ \begin{array}{l}\frac{ dSLDOC}{ dt}=-\underset{\mathrm{net}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{BAC}^{{\mathrm{upt}}_{\mathrm{SLDOC}}}\cdot {h}_{\mathrm{BAC}}^{Q_C}\cdot \mathrm{BAC}}}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{SLDOC}}}\cdot \left(1-{h}_{\mathrm{BAC}}^{Q_C}\right)\cdot \mathrm{BAC}}}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{f_{nr}^{\mathrm{PP}}\cdot \left(1-{h}_{\mathrm{PHYL}}^{Q_C}\right)\cdot \mathrm{PHYL}}}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.4em \mathrm{PHYS}}{\underbrace{f_{nr}^{\mathrm{PP}}\cdot \left(1-{h}_{\mathrm{PHYS}}^{Q_C}\right)\cdot \mathrm{PHYS}}}+\\ {}+\underset{\mathrm{DOC}\kern0.26em \mathrm{release}\kern0.3em \mathrm{by}\kern0.28em \mathrm{meso}-\mathrm{Z}}{\underbrace{\varepsilon \left(1-{\in}_{sf}\right)\left({f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_C^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}+{f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_C^{\mathrm{PHYL}}\right)\cdot \left(1-{h}_Z^{Q_C}\right)\cdot Z}}+\underset{\mathrm{DOC}\kern0.28em \mathrm{release}\kern0.28em \mathrm{by}\kern0.28em \mathrm{CIL}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}+{f}_Z^{g_{\mathrm{PHYS}}}\cdot {Q}_C^{\mathrm{PHYS}}\right)\cdot \left(1-{h}_{\mathrm{CIL}}^{Q_C}\right)\cdot \mathrm{CIL}}}+\hfill \\ {}+\underset{\mathrm{DOC}\kern0.28em \mathrm{release}\kern0.28em \mathrm{by}\kern0.28em \mathrm{HNF}}{\underbrace{\left({f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_C^{\mathrm{PHYS}}+{f}_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot {Q}_C^{\mathrm{BAC}}\right)\cdot \left(1-{h}_{\mathrm{HNF}}^{Q_C}\right)\cdot \mathrm{HNF}}}\end{array} $$
(26)
$$ \begin{array}{l}\frac{ dDON}{ dt}=-\underset{\mathrm{gross}\kern0.26em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.248em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{DON}}}\cdot \mathrm{BAC}}}+\underset{\mathrm{mineralization}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{\mathrm{rem}}\cdot BA{C}_N}}-\underset{\mathrm{net}\kern0.26em \mathrm{uptake}\kern0.26em \mathrm{of}\kern0.236em \mathrm{DON}\kern0.26em \mathrm{by}\kern0.26em \mathrm{PHYS}}{\underbrace{f_{\mathrm{PHYS}}^{{\mathrm{upt}}_{\mathrm{DON}}}\cdot {h}_{Q_N}^{\mathrm{PHYS}}\cdot \mathrm{PHYS}}}+\underset{\mathrm{BAC}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot {\mathrm{BAC}}_N}}+\underset{\mathrm{PHYS}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot {\mathrm{PHYS}}_N}}+\underset{\mathrm{HNF}\;\mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot {\mathrm{HNF}}_N}}+\\ {}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}\kern0.28em \mathrm{and}\kern0.28em \mathrm{PHYS}}{\underbrace{\gamma \cdot \left({f}_{\mathrm{PHYL}}^{{\mathrm{upt}}_N}\cdot \left(1-{h}_{\mathrm{PHYL}}^{Q_N}\right)\cdot \mathrm{PHYL}+{f}_{\mathrm{PHYS}}^{{\mathrm{upt}}_N}\cdot \left(1-{h}_{\mathrm{PHYS}}^{Q_N}\right)\cdot \mathrm{PHYS}\right)}}\end{array} $$
(27)
$$ \begin{array}{l}\frac{ dDOP}{ dt}=-\underset{\mathrm{upt}\mathrm{ake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{DOP}}}\cdot \mathrm{BAC}}}+\underset{\mathrm{mineralization}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{\mathrm{rem}}\cdot {\mathrm{BAC}}_P}}-\underset{\mathrm{net}\kern0.36em \mathrm{uptake}\kern0.26em \mathrm{of}\kern0.26em \mathrm{DOP}\kern0.26em \mathrm{by}\kern0.26em \mathrm{PHYS}}{\underbrace{f_{\mathrm{PHYS}}^{{\mathrm{upt}}_{\mathrm{DOP}}}\cdot {h}_{Q_P}^{\mathrm{PHYS}}\cdot \mathrm{PHYS}}}+\underset{\mathrm{BAC}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{BAC}}^m\cdot {\mathrm{BAC}}_P}}+\underset{\mathrm{PHYS}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYS}}^m\cdot {\mathrm{PHYS}}_P}}+\underset{\mathrm{HNF}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{HNF}}^m\cdot {\mathrm{HNF}}_P}}+\hfill \\ {}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}\kern0.28em \mathrm{and}\kern0.28em \mathrm{PHYS}}{\underbrace{\gamma \cdot \left({f}_{\mathrm{PHYL}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \left(1-{h}_{\mathrm{PHYL}}^{Q_P}\right)\cdot \mathrm{PHYL}+{f}_{\mathrm{PHYS}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \left(1-{h}_{\mathrm{PHYS}}^{Q_P}\right)\cdot \mathrm{PHYS}\right)}}\end{array} $$
(28)

Detrial material.

$$ \begin{array}{l}\frac{ dPOC}{ dt}=\underset{\mathrm{meso}-\mathrm{Z}\;\mathrm{mortality}}{\underbrace{f_Z^m\cdot {Z}_C}}-\underset{\begin{array}{l}\mathrm{meso}-\mathrm{Z}\;\mathrm{grazing}\;\mathrm{by}\;\\ {}\mathrm{higher}\kern0.48em \mathrm{trophic}\kern0.28em \mathrm{levels}\end{array}}{\underbrace{f_Z^{mq}\cdot {Q}_C^Z\cdot {Z}^2}}+\underset{\mathrm{CIL}\kern0.28em \mathrm{mortality}}{\underbrace{f_{CIL}^m\cdot CI{L}_C}}+\underset{\mathrm{PHYL}\kern0.28em \mathrm{mortality}}{\underbrace{f_{PHYL}^m\cdot PHY{L}_C}}-\underset{\mathrm{remineralization}}{\underbrace{f^{rem}\cdot BA{C}_C}}-\underset{\begin{array}{l}\mathrm{grazing}\kern0.26em \mathrm{during}\\ {}\mathrm{juvenile}\kern0.26em \mathrm{stages}\end{array}}{\underbrace{\alpha_3\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z}}\hfill \\ {}+\underset{\mathrm{sloppy}\kern0.26em \mathrm{feeding}}{\underbrace{\in_{sf}\cdot \left({f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_C^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}+{f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_C^{\mathrm{PHYL}}\right)\cdot Z}}+\underset{\mathrm{fecal}\kern0.36em \mathrm{pellets}}{\underbrace{\left(1-\varepsilon \right)\left(1-{\in}_{sf}\right)\left({f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_C^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_C^{\mathrm{HNF}}+{f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_C^{\mathrm{PHYL}}\right)\left(1-{h}_Z^{Q_C}\right)\cdot Z}}\end{array} $$
(29)
$$ \begin{array}{l}\frac{ dPON}{ dt}=\underset{\mathrm{meso}-\mathrm{Z}\;\mathrm{mortality}}{\underbrace{f_Z^m\cdot {Z}_N}}+\underset{\begin{array}{l}\mathrm{meso}-\mathrm{Z}\kern0.26em \mathrm{grazing}\kern0.26em \mathrm{by}\;\\ {}\mathrm{higher}\kern0.28em \mathrm{trophic}\kern0.28em \mathrm{levels}\end{array}}{\underbrace{f_Z^{\mathrm{mq}}\cdot {Q}_N^Z\cdot {Z}^2}}+\underset{\mathrm{CIL}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{CIL}}^m\cdot {\mathrm{CIL}}_N}}+\underset{\mathrm{PHYL}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYL}}^m\cdot {\mathrm{PHYL}}_N}}-\underset{\mathrm{rem}\mathrm{ineralization}}{\underbrace{f^{\mathrm{rem}}\cdot {\mathrm{BAC}}_N}}+\underset{\begin{array}{l}\mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}\hbox{--} \mathrm{Z}\\ {}\mathrm{during}\kern0.26em \mathrm{juvenile}\kern0.26em \mathrm{stages}\end{array}}{\underbrace{\alpha_1\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z\cdot \frac{Q_N^{\mathrm{CIL}}}{Q_C^{\mathrm{CIL}}}}}\hfill \\ {}+\underset{\begin{array}{l}\mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.26em \mathrm{juvenile}\kern0.26em \mathrm{stages}\end{array}}{\underbrace{\alpha_2\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z\cdot \frac{Q_N^{\mathrm{HNF}}}{Q_C^{\mathrm{HNF}}}}}+\underset{\mathrm{sloppy}\kern0.36em \mathrm{feeding}}{\underbrace{\in_{sf}\cdot \left({f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_N^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_N^{\mathrm{HNF}}+{f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_N^{\mathrm{PHYL}}\right)\cdot Z}}+\underset{\mathrm{implicit}\kern0.26em \mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}}{\underbrace{\beta \cdot \left({f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_N^{\mathrm{PHYL}}+{f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_N^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_N^{\mathrm{HNF}}\right)\cdot Z}}\end{array} $$
(30)
$$ \begin{array}{l}\frac{ dPOP}{ dt}=\underset{\mathrm{meso}-\mathrm{Z}\;\mathrm{mortality}}{\underbrace{f_Z^m\cdot {Z}_P}}-\underset{\begin{array}{l}\mathrm{meso}-\mathrm{Z}\kern0.24em \mathrm{grazing}\kern0.26em \mathrm{by}\\ {}\;\mathrm{higher}\kern0.28em \mathrm{trophic}\kern0.28em \mathrm{levels}\end{array}}{\underbrace{f_Z^{\mathrm{mq}}\cdot {Q}_P^Z\cdot {Z}^2}}+\underset{\mathrm{CIL}\kern0.28em \mathrm{mortality}}{\underbrace{f_{\mathrm{CIL}}^m\cdot {\mathrm{CIL}}_P}}+\underset{\mathrm{PHYL}\kern0.26em \mathrm{mortality}}{\underbrace{f_{\mathrm{PHYL}}^m\cdot {\mathrm{PHYL}}_P}}-\underset{\mathrm{rem}\mathrm{ineralization}}{\underbrace{f^{\mathrm{rem}}\cdot {\mathrm{BAC}}_P}}+\underset{\begin{array}{l}\mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.26em \mathrm{juvenile}\kern0.26em \mathrm{stages}\end{array}}{\underbrace{\alpha_1\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z\cdot \frac{Q_P^{\mathrm{CIL}}}{Q_C^{\mathrm{CIL}}}}}+\underset{\begin{array}{l}\mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}\\ {}\mathrm{during}\kern0.26em \mathrm{juvenile}\kern0.26em \mathrm{stages}\end{array}}{\underbrace{\alpha_2\cdot {f}_Z^{\mathrm{gjuv}}\cdot Z\cdot \frac{Q_P^{\mathrm{HNF}}}{Q_C^{\mathrm{HNF}}}}}\hfill \\ {}+\underset{\mathrm{sloppy}\kern0.26em \mathrm{feeding}}{\underbrace{\in_{\mathrm{sf}}\cdot \left({f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_P^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_P^{\mathrm{HNF}}+{f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_P^{\mathrm{PHYL}}\right)\cdot Z}}+\underset{\mathrm{implicit}\kern0.26em \mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}}{\underbrace{\beta \cdot \left({f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_P^{\mathrm{PHYL}}+{f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_P^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_P^{\mathrm{HNF}}\right)\cdot Z}}\end{array} $$
(31)

Nutrients.

$$ \begin{array}{l}\frac{d{\mathrm{NH}}_4}{ dt}=-\underset{\mathrm{gross}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{f_{\mathrm{PHYL}}^{{\mathrm{upt}}_{{\mathrm{NH}}_4}}\cdot \mathrm{PHYL}}}-\underset{\mathrm{gross}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYS}}{\underbrace{f_{\mathrm{PHYS}}^{{\mathrm{upt}}_{{\mathrm{NH}}_4}}\cdot \mathrm{PHYS}}}-\underset{\mathrm{net}\kern0.26em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{{\mathrm{NH}}_4}}\cdot \mathrm{BAC}\cdot {h}_{\mathrm{BAC}}^{Q_N}}}+\underset{\mathrm{mineralization}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{DON}}}\cdot \left(1-{h}_{\mathrm{BAC}}^{Q_N}\right)\cdot \mathrm{BAC}}}\hfill \\ {}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{\left(1-\gamma \right)\cdot {f}_{\mathrm{PHYL}}^{{\mathrm{upt}}_N}\cdot \left(1-{h}_{\mathrm{PHYL}}^{Q_N}\right)\cdot \mathrm{PHYL}}}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYS}}{\underbrace{\left(1-\gamma \right)\cdot {f}_{\mathrm{PHYS}}^{up{t}_N}\cdot \left(1-{h}_{\mathrm{PHYS}}^{Q_N}\right)\cdot \mathrm{PHYS}}}+\underset{\mathrm{excretion}\kern0.28em \mathrm{by}\kern0.28em \mathrm{CIL}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{HNF}}\cdot {Q}_N^{\mathrm{HNF}}+{f}_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_N^{\mathrm{PHYS}}\right)\cdot \mathrm{CIL}\cdot \left(1-{h}_{\mathrm{CIL}}^{Q_N}\right)}}+\\ {}+\underset{\mathrm{excretion}\kern0.28em \mathrm{by}\kern0.28em \mathrm{HNF}}{\underbrace{\left({f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_N^{\mathrm{PHYS}}+{f}_{\mathrm{HNF}}^{g_{BAC}}\cdot {Q}_N^{\mathrm{BAC}}\right)\cdot \mathrm{HNF}\cdot \left(1-{h}_{\mathrm{HNF}}^{Q_N}\right)}}-\underset{\mathrm{nitrif}\mathrm{ication}}{\underbrace{f_{{\mathrm{NH}}_4}^{\mathrm{nitrif}}\cdot {\mathrm{NH}}_4}}+\underset{\mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}}{\underbrace{\left(1-\beta \right)\cdot \left({f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_N^{\mathrm{PHYL}}+{f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_N^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_N^{\mathrm{HNF}}\right)\cdot Z}}\end{array} $$
(32)
$$ \frac{d{\mathrm{NO}}_3}{ dt}=-\underset{\mathrm{gross}\kern0.28em \mathrm{uptake}\kern0.26em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{f_{\mathrm{PHYL}}^{{\mathrm{upt}}_{{\mathrm{NO}}_3}}\cdot \mathrm{PHYL}}}-\underset{\mathrm{gross}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.26em \mathrm{PHYS}}{\underbrace{f_{\mathrm{PHYS}}^{{\mathrm{upt}}_{{\mathrm{NO}}_3}}\cdot \mathrm{PHYS}}}-\underset{\mathrm{net}\kern0.26em \mathrm{uptake}\kern0.24em \mathrm{by}\kern0.26em \mathrm{PHYL}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{{\mathrm{NO}}_3}}\cdot \mathrm{BAC}\cdot {h}_{\mathrm{BAC}}^{Q_N}}}+\underset{\mathrm{nitrif}\mathrm{ication}}{\underbrace{f_{{\mathrm{NH}}_4}^{\mathrm{nitrif}}\cdot {\mathrm{NH}}_4}} $$
(33)
$$ \begin{array}{l}\frac{d{\mathrm{PO}}_4}{ dt}=-\underset{\mathrm{gross}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{f_{\mathrm{PHYL}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \mathrm{PHYL}}}-\underset{\mathrm{gross}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYS}}{\underbrace{f_{\mathrm{PHYS}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \mathrm{PHYS}}}-\underset{\mathrm{net}\kern0.28em \mathrm{uptake}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \mathrm{BAC}\cdot {h}_{\mathrm{BAC}}^{Q_P}}}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYL}}{\underbrace{\left(1-\gamma \right)\cdot {f}_{\mathrm{PHYL}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \left(1-{h}_{\mathrm{PHYL}}^{Q_P}\right)\cdot \mathrm{PHYL}}}\hfill \\ {}+\underset{\mathrm{exudation}\kern0.28em \mathrm{by}\kern0.28em \mathrm{PHYS}}{\underbrace{\left(1-\gamma \right)\cdot {f}_{\mathrm{PHYS}}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}\cdot \left(1-{h}_{\mathrm{PHYS}}^{Q_P}\right)\cdot \mathrm{PHYS}}}+\underset{\mathrm{mineralization}\kern0.28em \mathrm{by}\kern0.28em \mathrm{BAC}}{\underbrace{f_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{DOP}}}\cdot \left(1-{h}_{\mathrm{BAC}}^{Q_P}\right)\cdot \mathrm{BAC}}}+\underset{\mathrm{excretion}\kern0.28em \mathrm{by}\kern0.28em \mathrm{CIL}}{\underbrace{\left({f}_{\mathrm{CIL}}^{g_{\mathrm{HNF}}}\cdot {Q}_P^{\mathrm{HNF}}+{f}_{\mathrm{CIL}}^{g_{\mathrm{PHYS}}}\cdot {Q}_P^{\mathrm{PHYS}}\right)\cdot \left(1-{h}_{\mathrm{CIL}}^{Q_P}\right)\cdot \mathrm{CIL}}}\\ {}+\underset{\mathrm{excretion}\kern0.28em \mathrm{by}\kern0.26em \mathrm{HNF}}{\underbrace{\left({f}_{\mathrm{HNF}}^{g_{\mathrm{PHYS}}}\cdot {Q}_P^{\mathrm{PHYS}}+{f}_{\mathrm{HNF}}^{g_{\mathrm{BAC}}}\cdot {Q}_P^{\mathrm{BAC}}\right)\cdot \left(1-{h}_{\mathrm{HNF}}^{Q_P}\right)\cdot \mathrm{HNF}}}+\underset{\mathrm{grazing}\kern0.26em \mathrm{by}\kern0.26em \mathrm{meso}-\mathrm{Z}}{\underbrace{\left(1-\beta \right)\cdot \left({f}_Z^{g_{\mathrm{PHYL}}}\cdot {Q}_P^{\mathrm{PHYL}}+{f}_Z^{g_{\mathrm{CIL}}}\cdot {Q}_P^{\mathrm{CIL}}+{f}_Z^{g_{\mathrm{HNF}}}\cdot {Q}_P^{\mathrm{HNF}}\right)}}\end{array} $$
(34)

Mortality

The natural mortality is represented through a first-order kinetic law:

$$ {f}_m={k}_m, $$
(35)

where k m in s − 1 is the specific mortality rate. The quadratic mortality function is used as a closure term for an implicit representation of the grazing of mesozooplankton by higher trophic levels:

$$ {f}_{mq}={k}_{mq}, $$
(36)

where  k mq in ind−1  s − 1 is the specific quadratic mortality rate.

Intracellular quota and growth

As usually done in flexible stoichiometry models, the model is based on the assumption that for each organism, intracellular X content (where X stands for C, N, P) must be comprised within a minimum (Q min X )  and maximum (Q max X )  value. Q min X can be interpreted as the amount of element X used in cellular structure and machinery and everything else can be seen as storage for future growth. Since Droop and his work on vitamin B12 (Droop 1968), this concept has been widely used, especially to simulate change in organism’s stoichiometry (Klausmeier et al. 2008). We therefore used the classical Droop formulation (Eq. 37) combined with the Leibig’s law of the minimum to describe the specific growth rate f μ:

$$ {f}^{\mu }=\overline{\mu}\cdot \underset{X}{ min}\left[1-\frac{Q_X^{min}}{Q_X}\right]. $$
(37)

In this formulation, Q X represents the actual cell (or individual) content for a given element X, and \( \overline{\mu} \)—the maximum theoretical growth rate of the organism.

Grazing

Grazing is represented by a Holling II formulation (Holling 1965) revised by Kooijman (2000)

$$ {f}^{g_{\mathrm{PREY}}}=\frac{I_m\left[\mathrm{PREY}\right]}{\frac{I_m}{F}+\left[\mathrm{PREY}\right]} $$
(38)

and generalized to several (here n PREY ) preys as follows (Gentleman et al. 2003):

$$ {f}^{g_{{\mathrm{PREY}}_i}}=\frac{\phi_i{F}_i\left[{\mathrm{PREY}}_i\right]}{1+{\displaystyle \sum_{k=1}^{n_{\mathrm{PREY}}}\frac{\phi_k{F}_k}{I_m^k}\left[{\mathrm{PREY}}_k\right]}}, $$
(39)

where I m is the maximum ingestion rate of the grazer, F either the clearance (for filter feeding organisms) or attack rate, ϕ i the preference of predator for the i th prey, and [PREY i ] the prey concentration of the ith prey in cell per liter.

Uptake of nutrients

To describe the gross uptake rate of nutrients, the Michaelis–Menten relationship was used (Eq. 42).

$$ {f}^{{\mathrm{upt}}_X}={V}_X^{max}\cdot \frac{\left[X\right]}{\left[X\right]+{K}_X} $$
(40)

[X] is the concentration for element X is the nutrient taken up by osmotrophs. V max X and K X are assumed constant in the model and represent the uptake parameters one would obtain from nutrient starved organisms (e.g., maximum potential uptake rate). Total inorganic N gross uptake rates by phytoplankton and bacteria are also used in the model equations:

$$ {f}_{\mathrm{PHY}}^{{\mathrm{upt}}_N}={f}_{\mathrm{PHY}}^{{\mathrm{upt}}_{{\mathrm{NO}}_3}}+{f}_{\mathrm{PHY}}^{{\mathrm{upt}}_{{\mathrm{NH}}_4}} $$
(41)
$$ {f}_{\mathrm{BAC}}^{{\mathrm{upt}}_N}={f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{{\mathrm{NO}}_3}}+{f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{{\mathrm{NH}}_4}} $$
(42)

Note that \( {f}^{{\mathrm{upt}}_{{\mathrm{NH}}_4}} \) is given by Eq. (45)

In addition, the model considers that ammonium inhibits nitrate uptake by phytoplankton and nitrate and DON uptake by bacteria and that phosphate inhibits DOP uptake by bacteria. This writes:

$$ {f}^{{\mathrm{upt}}_X}={V}_X^{max}\cdot \frac{\left[X\right]}{\left[X\right]+{K}_X}\cdot \frac{1}{1+\frac{\left[\mathrm{INH}\right]}{K_{\mathrm{INH}}}}, $$
(43)

where [INH] is the inhibitor concentration (NH4 or PO4) and K INH the inhibition constant (Frost and Franzen 1992).

Apart from this classical use of inhibition functions, we also added in the model a new function that acts as compensation. Otherwise, the inhibition as it is classically used in models amounts to reduce the total N (or P) uptaken by organisms for a given concentration of nutrients. We therefore deemed it necessary to allow organisms to uptake extra NH4 (PO4) to compensate the amounts of N and P that have not been uptaken as NO3 or DON (DOP) because of inhibition. The corresponding function writes:

$$ {f}^{{\mathrm{cpin}}_X}={V}_{{}_X}^{\max}\cdot \frac{\left[X\right]}{\left[X\right]+{K}_X}\cdot \frac{1}{1+\frac{K_{\mathrm{inh}}^X}{\left[\mathrm{INH}\right]}} $$
(44)

where X = NO3, DON or DOP.

The uptake rates of ammonium and phosphate by Y = PHYL, PHYS or BAC for a given organism among phytoplankton or bacteria are therefore given by:

$$ {f}^{up{t}_{NH4}}={V}_{NH4}^{\max}\cdot \frac{\left[{\mathrm{NH}}_4\right]}{\left[{\mathrm{NH}}_4\right]+{K}_{{\mathrm{NH}}_4}}+\left({V}_{{\mathrm{NO}}_3}^{\max}\cdot \frac{\left[{\mathrm{NO}}_3\right]}{\left[{\mathrm{NO}}_3\right]+{K}_{{\mathrm{NO}}_3}}+{V}_{\mathrm{DON}}^{\max}\cdot \frac{\left[\mathrm{DON}\right]}{\left[\mathrm{DON}\right]+{K}_{{\mathrm{NO}}_3}}\right)\cdot \frac{1}{1+\frac{K_{inh}^{P{O}_4}}{\left[P{O}_4\right]}} $$
(45)
$$ {f}^{{\mathrm{upt}}_{{\mathrm{PO}}_4}}={V}_{{\mathrm{PO}}_4}^{\max}\cdot \frac{\left[{\mathrm{PO}}_4\right]}{\left[{\mathrm{PO}}_4\right]+{K}_{{\mathrm{PO}}_4}}+{V}_{\mathrm{DOP}}^{\max}\cdot \frac{\left[\mathrm{DOP}\right]}{\left[\mathrm{DOP}\right]+{K}_{\mathrm{DOP}}}\cdot \frac{1}{1+\frac{K_{\mathrm{inh}}^{{\mathrm{PO}}_4}}{\left[{\mathrm{PO}}_4\right]}} $$
(46)

Photosynthesis and chlorophyll production

The model uses Han (2002) mechanistic formulation for photosynthesis (see Baklouti et al. 2006a for more details).

$$ {f}_{\mathrm{nr}}^{\mathrm{PP}}=\frac{\phi_{max}^C\cdot {\overline{a}}^{*}\cdot E\cdot \theta }{1+{\sigma}_{\mathrm{PSII}}\cdot E\cdot \tau +\left(\raisebox{1ex}{${k}_d^H$}\!\left/ \!\raisebox{-1ex}{${k}_r$}\right.\right)\cdot {\left({\sigma}_{\mathrm{PSII}}\cdot E\right)}^2\cdot \tau}\cdot {Q}_C^{\mathrm{PHY}} $$
(47)

Chlorophyll production is regulated by the N/Chl ratio of phytoplankton (θ N). The formulation used is extensively described in Baklouti et al. (2006a) and its adaptation to the model including cell densities in Mauriac et al. (2011).

$$ {f}^{\mathrm{PChl}}={\rho}_{\mathrm{chl}}\cdot {f}_{\mathrm{PHY}}^{{\mathrm{upt}}_{\mathrm{N}}}\cdot \frac{1-{\theta}_N/{\theta}_m^N}{\left(1-{\theta}_N/{\theta}_m^N\right)+0.05}, $$
(48)

where \( \theta =\frac{{\mathrm{PHY}}_{\mathrm{Chl}}}{{\mathrm{PHY}}_C} \), \( {\theta}^N=\frac{{\mathrm{PHY}}_{\mathrm{Chl}}}{{\mathrm{PHY}}_{\mathrm{N}}} \), \( {f}^{\mathrm{PP}}={f}_{\mathrm{nr}}^{\mathrm{PP}}\cdot {h}_{\mathrm{PHY}}^{Q_C} \) and \( {\rho}_{\mathrm{chl}}={\theta}_m^N\frac{f^{\mathrm{PP}}}{{\overline{a}}^{*}{\phi}_{max}^C\theta E} \).

In Eqs. (47) and (48), ϕ C max represents the maximum quantum yield for carbon fixation, \( {\overline{a}}^{\ast } \), the mean Chl a specific absorption coefficient, E, the irradiance, θ, the chlorophyll to carbon ratio, σ PSII, the PSII cross-section, τ, the electron turnover time, k H d , the PSII damage rate, k r , the PSII repair rate and θ N m the maximum chlorophyll to nitrogen ratio.

Feedback regulation

A feedback regulation from internal cellular status to mediate net primary production, net uptake rate as well as net grazing is in the form of a quota function given by Eq. (49) (Geider et al. 1998) as this is already done in Baklouti et al. (2006a):

$$ {h}^{Q_X}={\left(\frac{Q_X^{max}-{Q}_X}{Q_X^{max}-{Q}_X^{min}}\right)}^{0.06} $$
(49)

In the present version where organisms are represented through biomasses and cellular densities, a dual control is exercised, not only by intracellular quotas but by intracellular ratios. In substance, for the uptake of a given element X, if it happens that the Q X cellular quota is outside the [Q min X ;Q max X ] range, the quota function which is used for the uptake rate regulation is given by Eq. (49).

However, if Q X belongs to the [Q min X ;Q max X ] range, the regulation of the uptake rate is given by the minimum of quota functions similar to that of Eq. (49) in which the intracellular quotas are replaced by intracellular ratios \( {Q}_{X{Y}_i} \) (where Y i represents the biogenic elements apart from X that are handled by the model).

This can be summarized by the following equations:

$$ \mathrm{if}\kern0.5em \left({Q}_X\le {Q}_X^{\min}\right)\kern0.5em \mathrm{or}\kern0.5em \left(Q\ge {Q}_X^{\max}\right)\kern0.5em \mathrm{then} $$

\( {h}^{Q_X} \) is given by Eq. (49)else

$$ {h}_{Q_{X{Y}_i}}=\underset{Y_i}{ \min }{h}_{Q_{X{Y}_i}} $$

where \( {h}_{Q_{{\mathrm{XY}}_i}}=\frac{Q_{{\mathrm{XY}}_i}^{\max }-{Q}_{{\mathrm{XY}}_i}}{Q_{{\mathrm{XY}}_i}^{\max }-{Q}_{{\mathrm{XY}}_i}^{\min }} \)

Finally, the feedback regulation has been described here for the uptake process but the same method is used in the model for the other processes that are regulated through quota functions.

Autotrophic respiration

Several energetic costs associated with phytoplankton activities have been included as in Baklouti et al. (2006a):

$$ {f}_{\mathrm{PHY}}^{\mathrm{resp}}={r}_g{f}^{\mathrm{PP}}+{\displaystyle \sum_{\mathrm{NUT}}{r}_{u_{\mathrm{NUT}}}\cdot {f}^{{\mathrm{upt}}_{\mathrm{NUT}}}},\mathrm{where}\ \mathrm{NUT}\in \left\{{\mathrm{NO}}_3,{\mathrm{NH}}_4,{\mathrm{PO}}_4\right\} $$
(50)

r g and \( {r}_{u_{\mathrm{NUT}}} \) respectively represent the cost associated with growth and net uptake of nutrient NUT (see Table 1 for the units of these parameters).

Table 1 Model’s functions

Bacteria respiration

Heterotrophic bacteria respiration rate is based on cost for DOC acquisition:

$$ {f}_{\mathrm{BAC}}^{\mathrm{resp}}=\left(1-{\omega}_1\right)\cdot {f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{LDOC}}}+\left(1-{\omega}_2\right)\cdot {f}_{\mathrm{BAC}}^{{\mathrm{upt}}_{\mathrm{SLDOC}}} $$
(51)

ω 1 and ω 2 respectively refer to the efficiency of LDOC and SLDOC uptake and Eq. (51) conveys the fact that higher energetic costs are associated with the uptake of semi-labile DOC (SLDOC) than labile one (LDOC). In this way, bacteria are subject to two types of carbon limitation, either a limitation by availability (when DOC resource is scarce) or by lability (when DOC acquisition is costly), both cases resulting in a low bacterial growth rate.

Zooplankton respiration

Zooplankton respiration is proportional to the net growth rate through:

$$ {f}_{\mathrm{zoo}}^{\mathrm{resp}}={r}_g\cdot {Q}_C^{min}\cdot {f}^{\mu } $$
(52)

Carbon acquisition by juvenile stages

Before the adult stage, mesozooplankton (copepods) acquire carbon (and other elements but they are not explicitly represented in our model) through grazing. That means that the carbon which is affected to each new individual in the model does not come from the carbon pool of mesozooplankton but rather from the carbon pool of the different preys grazed by mesozoplankton during juvenile stages. This grazing during juvenile stages is implicitly represented in the model by:

$$ {f}_z^{\mathrm{gjuv}}=0.85\cdot {Q}_C^{\min}\cdot {f}_z^{\mu } $$
(53)

and distributed on three possible food, namely ciliates (arbitrarily 30 % of the food), HNF (50 %) and detrital material (20 %).

Appendix B. Model parameters

Most of the parameter values used in the model come from literature or have been derived from relationships that have been established between parameters. Only very few parameters (typically the mortality constants) have been chosen arbitrarily. The relationship (51) conveys the fact that at very low nutrient concentration, nutrient uptake is limited by molecular diffusion and the diffusion (Kiorboe 2008) and the Michaelis–Menten fluxes can be equalized through:

$$ \frac{V_{\max }}{K_s}=4\pi DR $$
(54)

Where D is the molecular diffusion coefficient and R—the mean equivalent disc radius of cells.

Table 2

Table 2 Model parameters for zooplankton. Parameters values have been taken from Perez et al. (1997); Sherr et al. (1999); Christaki et al. (1999) for ciliates, Christaki et al. (1999, 2009) for HNF and Barquero et al. (1998) for copepods

Table 3

Table 3 Model parameters for phytoplankton. D NO3, \( {D}_{{\mathrm{NH}}_4} \), \( {D}_{{\mathrm{PO}}_4} \), D DON, D DOP are the diffusion coefficient of NO3, NH4, PO4, DON, and DOP, respectively equal to 1.7 · 10−9, 1.9 · 10−9, 7.5 · 10−10, 3 · 10−10, 3 · 10−10 m2s−1. Parameter values have been taken from Thingstad (2005); Baklouti et al. (2006b); Mauriac et al. (2011) and references herein

Table 4

Table 4 Model parameters for heterotrophic bacteria. Parameter values have been taken from Thingstad (2005); Baklouti et al. (2006b); Mauriac et al. (2011)

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Alekseenko, E., Raybaud, V., Espinasse, B. et al. Seasonal dynamics and stoichiometry of the planktonic community in the NW Mediterranean Sea: a 3D modeling approach. Ocean Dynamics 64, 179–207 (2014). https://doi.org/10.1007/s10236-013-0669-2

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Keywords

  • Biogeochemical model
  • Eco3M model
  • MARS3D model
  • Variable stoichiometry
  • Intracellular contents
  • NW Mediterranean Sea
  • Gulf of Lions