Harvard Reports in Physical/Interdisciplinary Ocean Science
#67
A GENERALIZED PROGNOSTIC MODEL
OF MARINE BIOGEOCHEMICAL-ECOSYSTEM DYNAMICS:
STRUCTURE, PARAMETERIZATION AND ADAPTIVE MODELING
May 2004
Rucheng C. Tian, Pierre F.J. Lermusiaux,
James J. McCarthy and Allan R. Robinson
Harvard University
Division of Engineering and Applied Sciences
Department of Earth and Planetary Sciences
29 Oxford Street, Cambridge MA 02138
Contents
Abstract
1. Introduction
2. Review of biogeochemical-ecosystem modeling up to date
3. Structure
4. Parameterization
4.1. Phytoplankton
4.1.1. Light forcing on phytoplankton growth rate
4.1.2. Temperature forcing on phytoplankton growth rate
4.1.3. Nutrient limitation on phytoplankton growth rate
4.1.4. DOM exudation and mortality of phytoplankton
4.2. Zooplankton
4.2.1. Grazing functions
4.2.2. Temperature forcing on zooplankton growth
4.2.3. Zooplankton mortality, excretion and respiration
4.2.4. Zooplankton diel vertical migration
4.3. Bacteria
4.4. Dissolved organic matter (DOM)
4.5. Biogenic detritus
4.6. Nutrients
4.7. Auxiliary state variables
3.7.1. Chlorophyll a
3.7.2. Bioluminescence
5. Adaptive modeling and real-time ecosystem forecasting
6. Conclusions
References
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ABSRACT
In this report, we present a generalized and flexible prognostic model for marine biogeochemicalecosystem processes. This generalized model currently focuses on pelagic dynamics and consists of 7
functional groups of state variables: nutrients, phytoplankton, zooplankton, detritus, dissolved organic
matter, bacteria and auxiliary state variables. The number of components in each functional group is
flexible varying from 1 to n and their corresponding biological, chemical or detrital features are defined by
users. Primarily, categories of dissolved organic matter are classified by their bioavailability, bacteria by
their ability to respond to environmental variations, nutrients by their chemical composition, and
phytoplankton, zooplankton and detritus by their size. A different classification can be used, however, such
as species or stages of zooplankton. Auxiliary state variables represent oceanic properties that can depend
directly on other biological features and are thus not always independent prognostic components in food
web structure and trophic dynamics. Such auxiliary variables can include for example chlorophyll,
bioluminescence, dissolved oxygen, CO2 and phycotoxins. The model was constructed so as to allow easy
identification and adaptation, i.e., the state variables, model structures and parameter values can be changed
in response to field measurements, ecosystem functions and different scientific objectives. The model is
currently coupled with the Harvard Ocean Prediction System (HOPS) for real-time ocean forecasting.
1. INTRODUCTION
Marine ecosystems are usually represented by compartment models, each
compartment representing a trophic level or taxonomic group such as phytoplankton,
zooplankton or nutrient. In the real ocean, however, there are organisms of unnumbered
species, at various stages and of different weights at each trophic level. The structural
design of an ecological model is thus critical in determining its adequacy and capability
to approach real ecosystems.
Since the first marine ecological model of phytoplankton (Riley, 1946), the variety of
ecological dynamics being modeled has widen and there has been a trend towards
increasing complexity in model structure. For example, Andersen et al. (1987) found it
necessary to partition phytoplankton into diatoms and flagellates, and zooplankton into
copepods and appendicularia. Moloney and Field (1991) partitioned the autotrophic level
into pico-, nano- and net-phytoplankton, and the heterotrophic trophic level into bacteria,
heterotrophic flagellates, microzooplankton and mesozooplankton. Moisan and Hofmann
(1996) simulated gelatinous zooplankton, copepods and euphausiids whereas Pace et al.
(1984) considered bacteria, protozoa, grazing zooplankton, carnivorous zooplankton,
mucous net feeders and pelagic fishes at the heterotrophic level. Tian et al. (2000)
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coupled the microbial food web with the traditional mesoplankton food chain.
Wroblewski (1982) distinguished five life stages of copepods: eggs, nauplii, early
copepodites (CI-III), copepodites (CIV-CV) and adult. Armstrong (1994) proposed a
model with n parallel food chains, each consisting of a phytoplankton species Pi and its
dedicated zooplankton predator Zi classified by size. Nihoul and Djenidi (1998) presented
a conceptual food web model in which phytoplankton are divided into cynobacteria,
ultraphytoflagellates, phytoflagellates, diatoms and autotrophic dinoflagellates, and
zooplankton are divided into heterotrophic flagellates, ciliates, heterotrophic
dinoflagellates, arthropods and gelatinous zooplankton. Although many of the biological
models are nitrogen driven, Lancelot et al. (2000) considered five types of nutrients (Si,
PO4-, NO3-, NH4+ and Fe). Cousins (1980) suggested a conceptual trophic continuum
model in which an ecosystem is divided into three basic components (autotrophs,
heterotrophs and detritus), with each component representing a size continuum from
small to large organisms or particles. Armstrong (1999) argued that biological models of
pelagic ecosystems should reflect both the taxonomic and size structure of the plankton
community. Considering all of these advances as a whole, the main modeling challenge
relates to the complexity of the physiological, trophic and ecological dynamics. This
complexity prohibits the development of a single model with fixed structure that can be
applied to various oceanic ecosystems and be used for various scientific objectives.
Ideally, generalized and flexible models can be more efficient and useful. Developing
such a model is the main objective of the present research.
Another challenge relates to the lack of a common set of equations that represent the
source and sink terms of the biological state variables (e.g., Robinson and Lermusiaux,
2002). There are currently no standard parameterizations in marine ecological modeling.
A wide variety of mathematical formulations have been developed to describe
fundamental biological processes and forcing functions, such as those of light, nutrient
and temperature on phytoplankton growth rate and on zooplankton grazing and predation.
These formulations are usually based on empirical relationships, for example expressing
a correlation with measurable variables. The real ecological or physiological processes
underlying the observed correlation are not often explicit in these relationships. Sound
statistical or physiological bases to reject one or another parameterization are rare
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(Sakshaug et al., 1997), but the choice among them can be critical with respect to the
model functionality (Franks et al., 1986; Smith et al., 1989; Steele and Hendersen, 1992;
Gao et al., 2000; Gentleman et al., 2003). Although we can infinitely divide an ecosystem
model, our knowledge of biological dynamics and rates is limited, but evolving.
Ecological modeling largely relies on field data to validate simulations. An ideal and
intelligent modeling system should thus learn from data and model misfits, and
quantitatively identify the most adequate formulations. The present model development is
simply a prerequisite to such automated system identification (e.g. Duda et al, 2000).
Importantly, oceanic states evolve and go through transitions. In the coastal
environment, properties can be highly variable and intermittent on multiple scales. The
biogeochemical-ecosystem variables, processes and interactions that matter vary with
time and space. For efficient forecasting, prognostic models should thus have the same
behavior and adapt to the ever-changing dynamics (Lermusiaux et al., 2004). This is
especially important for marine ecosystems. With adaptive modeling, model switches and
improvements are dynamic and aim to continuously select the best model structures and
parameters among different physical or biogeochemical parameterizations. It is a system
identification that is dynamic and sustained in time and space.
In summary, the properties of the modeling system that cope with various ecosystems,
multiple scientific objectives, unknown equations and ever-changing dynamics are
generality, flexibility, identification and adaptation, respectively. This report is mostly on
generality and flexibility of the biogeochemical-ecosystem model. System identification
and adaptive modeling, and the corresponding automations, are only touched upon. Of
course, even though the model formulation and its computational structure aim to be
general, to initiate the functional implementations and survey the relevant literature, some
focus is necessary. First, the oceanic region for which the model is being developed is the
pelagic (littoral, neritic and open-oceanic) mid-latitude ocean, including coastal-deep sea
interactions. Benthic processes and some specifics of low and high latitudes are for
example not yet explicitly implemented (even though the new model structure could
accommodate the corresponding variables and functionalities). Second, the scales
considered are from hundreds of meters to thousands of kilometers, and hours to decades.
The Eulerian formulation is thus preferred to the Lagrangian formulation and to the
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Individual Based Models (IBMs, e.g., DeAngelis and Gross, 1992) and ordinary
population dynamics formalisms. All prognostic equations are of the partial differential,
advective-diffusive-reactive type.
Our generalized model has been scientifically constructed based on a study of all
possible functional groups and parameterizations relevant for mid-latitude coastal
ecosystems such as Massachusetts Bay and Monterey Bay. A general set of state
variables and of mathematical structures representing the interactions of these variables
was selected, based on importance, completeness, efficiency and accuracy. This led to a
generalized model with the following functional groups of state variables: nutrients,
phytoplankton, zooplankton, detritus, dissolved organic matter, bacteria and auxiliary
variables. Firstly, the generalized model is flexible on three levels: (i) the functional
groups that are relevant can be determined and selected; (ii) within each functional group,
the number of state variables is not fixed but is defined by the user; and (iii) the
interactions among state variables within and across functional groups are also variable.
Secondly, the generalized model is modular: one can select the model structures and
parameterizations that are most adequate for the application of interest. In some cases,
changes at each trophic level can also result in automatic changes in specific
parameterizations. With this flexibility, the generalized biogeochemical model will be
able to adapt to different ecosystems, scientific objectives and availability of field
measurements.
Such a generalized model has several advantages. It can simulate various ecosystems
by using a subset of its generalized state variables. It can adapt to the changing ecosystem
dynamics by changing its state variables, structures and parameters. Importantly, it
provides an efficient numerical tool for real-time ecosystem forecasting. This is because
one can downsize the model configuration to the simplest one that is sufficiently accurate
to achieve the specific forecasting goals. In general, we expect the utilization of the
generalized model for a given application to proceed as follows. First, an a priori model,
i.e. the state variables, their interactions, the parameterizations and the parameter values,
is chosen. As new data are collected, as events occur, as the dynamics changes or as the
scientific objectives are modified, the configuration and specificities of the a priori
model are updated, tuned and evolved. The result of this identification and adaptation is a
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fitted a posteriori model. This process can be manual or automated. The latter case is an
advanced application of data assimilation (Robinson and Lermusiaux, 2002; Gregoire et
al, 2003). By data assimilation, (non)-observed state variables and parameters are
classically constrained or estimated from the observed data, respectively by field
estimation and parameter estimation. Similarly, to obtain the a posteriori model, specific
parameterizations and model structures are selected among a set of possible choices,
based on quantitative criteria that are a function of model-data misfits. Importantly, just
as some variables or parameters cannot always be estimated unambiguously by data
assimilation (e.g. Matear, 1995; Vallino, 2000), some parameterizations and model
structures may appear equivalent to the available data (referred to as equifinality, Duda et
al, 2000). The complete automation of the adaptation requires substantial research.
In what follows, biogeochemical-ecosystem modeling efforts up to date are first
briefly reviewed (Sect. 2). The dynamical structures, equations and parameterizations of
our generalized prognostic biogeochemical-ecosystem model are then defined and
described (Sects. 3 and 4). The presentation aims to be comprehensive conceptually and
mathematically, but the technical details of the computations and software are not
discussed. Finally, the utilization of the generalized and flexible model in an adaptive
data-driven modeling mode is briefly discussed (Sect. 5). Our applications of the
generalized model and the research towards the automation of model identification and
adaptation are not reported.
2. REVIEW OF BIOGEOCHEMICAL-ECOSYSTEM MODELS UP TO DATE
Hofmann and Lascara (1998) and Hofmann and Friedrichs (2002) recently conducted
overall reviews of interdisciplinary modeling for marine ecosystems. They provide a
detailed description of the progress in marine ecological modeling. Riley (1946, 1947a,b)
first used simple mathematical equations to simulate observed seasonal variations in
phytoplankton and zooplankton abundance. Parameterizations of biological dynamics
were not developed at that time and mathematical formulations were based on linear
assumptions that describe a single state variable. Following his earlier efforts, Riley
(1949) developed a trophic dynamics model that included nutrient, phytoplankton and
zooplankton, i.e., the so-called NPZ model. This model structure has been widely used
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since then, as NPZ only (Steele, 1974; Evans and Parslow, 1985; Franks et al., 1986) or
as NPZ and a detritus pool (e.g., Ishizaka 1990; Nihoul et al, 1993; McGillicuddy et al.,
1995a, b). Riley’s earlier efforts have triggered advances in the parameterization of
biological dynamics such as the relationships between photosynthesis and light (e.g.,
Steele, 1962; Vollenweider, 1965; Webb, 1974; Parker, 1974; Bannister, 1979) and
nutrient uptake kinetics (Caperon, 1967; Dugdale 1967, Droop, 1968; 1983; Platt et al.,
1977; Goldman and McCarthy, 1978; McCarthy, 1981).
About a quarter century later, Walsh (1975) developed an ecological model applied to
the Peru upwelling ecosystem. His model included nitrate, phosphate, silicate,
phytoplankton, zooplankton and planktonic fish (anchoveta) and detritus. Light forcing,
ammonium inhibition on nitrate uptake, grazing and current advection (upwelling) were
explicitly implemented. This represented a major advance in marine ecological modeling
although some of the parameterizations were based on idealized assumptions, such as
formulations of phytoplankton growth rate and zooplankton grazing as a sine or cosine
function of time.
Fasham et al (1990) further considered bacteria and dissolved organic nitrogen in their
biogeochemical model, using more advanced parameterizations such as switching grazing
and predation on multiple types of prey. The inclusion of the bacterial pool allowed the
explicit simulation of new production versus regenerated production, which is one of the
bases of carbon biogeochemical cycles and sequestration. This model of biological
dynamics in the surface mixed-layer has been utilized substantially in the past decade and
its application range has been extended, for example to study biogeochemical cycles over
basin scales by coupling with 3D general circulation models (e.g. Sarmiento et al., 1995).
Tian et al. (2000) recently coupled the microbial food web (small phytoplankton or
picophytoplankton, microzooplankton, bacteria, small suspended detritus, DOM and
ammonium) with the traditional mesoplankton food chain (large phytoplankton or
diatoms, mesozooplankton, large sinking detritus and nitrate). The model can estimate
several secondary quantities, including new production versus regenerated production,
DOC versus POC export and active export through zooplankton vertical migration (Tian
et al., 2001, 2004). The explicit coupling of the microbial food web with the
mesozooplankton food wed allows the simulation of seasonal food web succession and
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variability that occurs due to climate changes (Tian et al., 2003a, b).
The European Regional Sea Ecosystem Model (ERSEM) is likely the most
comprehensive ecosystem simulation model up to date (Vichi et al., 2003). ERSEM
consists of 3 sub-models (circulation, pelagic and benthic sub-models). Each sub-model
contains several modules (Baretta-Bekker et al., 1995). The pelagic sub-model is
composed of phytoplankton, zooplankton, microzooplankton, nutrient, bacteria and fish
modules (Baretta et al., 1995). Phytoplankton includes diatoms, flagellates,
picophytoplankton and large phytoplankton. Zooplankton includes carnivorous and
omnivorous mesozooplankton. Microzooplankton includes heterotrophic flagellates and
other microzooplankton. In addition, 5??? nutrients (nitrate, ammonium, phosphate, ???
and silicate), dissolved oxygen, CO2, DOM and POM are all included in the pelagic submodel (Vichi et al., 2003; Baretta et al., 1995).
Several research efforts have focused on different life stages of zooplankton, such as
eggs, nauplii, copepodites and adults (Wroblewski, 1982; Hofmann and Ambler, 1988).
Individual-based models (IBMs) have also been developed to track the life-stage cohorts
of organisms. For example, for the life history of copepods, state variables of IBMs can
include eggs, 6 stages of nauplii, 5 stages of copepodites and adults (Carlotti, and
Sviandra 1989; Carlotti and Nival, 1992). Importantly, IBMs can be stage-, age-, size-, or
weight-structured (Bryant et al., 1997; Heath et al., 1998). For ecological and
biogeochemical studies, IBMs are often coupled with population dynamic models in
order to quantified energy flows between different trophic levels (Carlotti et al., 1996).
3. STRUCTURE
The structure of the generalized biological model is illustrated in Fig. 1. There are 7
functional groups of state variables: Ni (nutrients), Pi (phytoplankton), Zi (zooplankton),
Di (detritus), DOMi (dissolved organic matter, e.g., DOC or DON), Bi (bacterioplankton)
and Ai (auxiliary state variables). The total number of components is flexible and varies
for each functional group, i.e., nn components for nutrients, np for phytoplankton, nz for
zooplankton, nd for biogenic detritus, ns for dissolved organic matter, nb for bacteria and
na for auxiliary state variables (Fig. 1). The number of components (state variables) in
each functional group can be freely assigned from 0 to an any large number. In practice
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however, we expect that these numbers will range from 0 to an order of 10.
The biological and biogeochemical pools corresponding to the symbolic state variables
are also allowed to be freely designed. From the practical point of view, we expect that
the biological and biogeochemical pools which can be quantified by measurements will
be often more useful because the model can then be effectively compared to and
constrained by field data. Nutrient (1) and nutrient (2) designate nitrate and ammonium,
respectively. Additional nutrients and micronutrients can be assigned to number 3 or
higher, e.g., silicate, phosphorus and iron. All other functional groups are designed to be
classified by users for a specific application. For example, phytoplankton (Pi) can range
from picophytoplankton through nanophytoplankton and microphytoplankton to
macrophytes. Pi can also represent autotrophic cyanobacteria, dinoflagellates and
diatoms, or small and large phytoplankton. Zooplankton can range from protozoa through
ciliates and copepods to net mucous feeders, or represent different stages and different
weights of zooplankton. Detritus is usually classified by size, but it can also be grouped
otherwise, e.g., according to chemical composition. DOM is usually divided by chemical
composition (DOC, DON, …). For example, in modeling studies, Vallino (2000) and
Spitz et al. (2001) have used DOC and DON. Alternatively, DOM classification can be
done at the molecule level (e.g., high and low molecular weight DOM; Amon et al.,
1994; Gardner et al., 1996). For biogeochemical cycle studies, the bioavailability of
DOM is important. Theoretically, DOM can then be considered as a continuum from
labile to refractory dissolved organic matter and the number of DOM categories (ns) can
be large. Recently, DOM has been separated into two or three categories, labile, semilabile and refractory DOM, both in data analyses (Carlson and Ducklow, 1995) and
1D/mesoscom modeling studies ( Anderson and Williams, 1999; Vallino, 2000; Tian et
al., 2004). However, these categories reflect bioavailability and as such have limited use
in a generalized model since they cannot be either routinely quantified with existing
analytical methods (data collection) or approximated with confidence. Bacterial
compartments are designed to reflect the response to environmental factors such as
temperature tolerance (Delile and Perret, 1989). Bacteria are assumed to have the same
role in trophic dynamics, i.e., acquiring energy from DOM and being consumed by small
or mucous-net-feeding zooplankton. Auxiliary state variables represent oceanic
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properties that are not independent components in food web structure and trophic
dynamics, and their field depends on other biological pools (e.g. chlorophyll,
bioluminescence, optics, acoustic properties, dissolved oxygen, CO2, etc).
In order that the model be flexible, energy flows between food web components are
not a priori determined. All zooplankton can feed on all phytoplankton and all
zooplankton. The specific energy flow between two biological pools will be determined
by users by assigning a specific value to the corresponding preference coefficient (Fig.
2). For example, omnivorous and carnivorous copepods may have raptorial feeding with
a narrow-size window of prey whereas mucous-net feeders, such as appendicularia, salps,
and doliolids, can collect particles ranging from bacteria (0.2-5 m in diameter) to large
diatoms (8x100 m; Madin and Deibel, 1997). Energy flows are quite different between
these two types of zooplankton although they can be represented by the same symbol in
different applications.
The basic unit (or currency) of the model varies with regard to specific applications.
The commonly used unit in ecological and biogeochemical models is carbon either
nitrogen. Conversion factors from one unit (e.g., carbon) to another (e.g., nitrogen) are
provided in this generalized model by a specific elemental ratio for each trophic
component.
4. PARAMETERIZATION
The generic equation for state variables in coupled physical-biological models can be
written as:
d i
 u   i  ( K i )  f i (1 ,  2 ,...,  n )
dt
(1)
where i represents a biological state variable, u is the current field, K is the turbulence
diffusivity and fi represents biological sources and sinks function for the state variable i
(Anderson et al., 2000; Robinson and Lermusiaux, 2002; Besiktepe et al., 2003). In the
following text, only the biological sources and sinks (Bi) are presented on the right-hand
side of equations.
4.1. Phytoplankton
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The partial differential equation for phytoplankton Pi is written as:
Pi
P nz
m
 (1  rPSi ) i Pi   PDi Pi pi  rPSi Pi  s Pi i   GPji
t
z j 1
(2)
where ri, rPi, PDi, mPi and sPi are, respectively, growth-dependent active and biomassdependent passive exudation of DOM, the mortality coefficient and power and sinking
velocity. GPji is the grazing amount of zooplankton Zj on phytoplankton Pi, and i is the
growth rate of PI which will be discussed in the following sections
All phytoplankton pools are described by the same equation. In fact, the fundamental
biological dynamics governing the phytoplankton temporal variations are the same. For
example, net growth of all phytoplankton is controlled by light forcing, nutrient
availability, temperature influence, mortality, excretion (or respiration) and grazing loss.
What differ are their biological rates, i.e., the parameter values in the equations.
However, the changes in the number of phytoplankton components will modify the
equation of nutrients, zooplankton, DOM, detritus and certain auxiliary state variables
because the number of phytoplankton items in these equations will differ (see the
following sections)..
4.1.1. Light forcing on phytoplankton growth rate
Based on early experiments, Blackman (1905) described the relationship between
phytoplankton growth and light (-E relationship) as a rectilinear function: the
phytoplankton growth rate increases proportionally with light intensity up to a certain
level beyond which the growth rate ceases to increase (Table 1 Eq. 1). He interpreted the
saturation light level as a result of other limiting factors that overwhelmed the effect of
light. Field and laboratory observations later showed that the -E relationship is a
hyperbolic curve and can be expressed by the Michaelis-Menten function (Table 1 Eq 2;
Baly, 1935; Tamiya et al., 1953). The Michaelis-Menten function was developed to
describe enzymatic activities (Michaelis and Menten, 1913). Its application to light
limitation was chosen to fit experimental results and is without fundamental
physiological underpinnings. Smith (1936) modified the Michaelis-Menten function
while trying to improve the fitting of experimental data. Under high light intensity,
however, photosynthesis is photoinhibited, most likely through photo-oxidation reactions,
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i.e., over excited antenna chl a can be combined with oxygen to become chemically
altered. (Rabinowitch, 1945; Steele, 1962; Prezelin, 1981). Neither the Michaelis-Menten
nor the Smith function parameterizes photoinhibition. Vollenweider (1965) and Peeters
and Eilers (1978) further modified the Michaelis-Menten function to take into account
photoinhibition (Table 1 Eqs. 4 and 5). Webb (1974) used an exponential function to
reproduce the observed -E hyperbolic relationship (Table 1 Eq. 6) and Platt et al. (1980)
added a second term to the exponential function to represent photoinhibition observed in
field studies. Steele (1962) combined the linear and exponential functions (Table 1 Eq. 8)
and Paker (1974) modified the Steele function by adding a power parameter to increase
flexibility for data fitting (Table 1 Eq. 9). Jassby and Platt (1976) first used a hyperbolic
tangent function (Table 1 Eq. 10) and Bissett et al. (1999) modified this function by
adding an exponential photoinhibition term (Table 1 Eq. 11). Finally, Saksaug and Kiefer
(1989) developed a mechanistic function for the -E relationship (Table 1 Eq. 12) in
which  is the C:chlorophyll ratio, a represents the specific absorption coefficient for
chlorophyll a, max is the maximum quantum yield,  is the mean absorption cross section
of photosystem II and  is the minimal turnover time of the photosystem. The last
exponential term represents the Poisson probability that a photosynthetic unit being hit is
open.
Except for the last mechanistic parameterization, all others are empirical and
constructed from curve fitting to data. All these empirical formulations have been used in
modelling applications and there is no strong evidence to reject any one of them. It
should be pointed out, however, that different formulations may yield different parameter
values when fit to the same data set (Sakshaug et al., 1997). Mechanistic models, on the
other hand, are based on accepted knowledge about the mechanisms of specific
processes. Their parameters are generally interpretable and their application can be
extended further than empirical models. Moreover, the Sakshaug and Kiefer’s
mechanistic parameterization can be transformed into the Webb exponential formulation.
Assuming that the composite term amax represents the initial slope  of the  curve
and the composite term  equals to :Pm (Cullin, 1990), Eq. 9 becomes Eq. 6 (Table 1).
However, neither of these two formulations simulates photoinhibition. The Platt function
(Table 1 Eq. 7) has a specific photoinhibtion term and its parameter values can be derived
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from other measurable parameters such as the specific absorption coefficient, maximum
quantum yield and the mean cross section of photosystem II, which are commonly used
in remote-sensing studies. The theoretical maximum growth rate of phytoplankton in the
Platt function is higher than the maximum quantum yield. All these formulations have
been developed to simulate the same -E relationship without specific environmental
conditions or phytoplankton species.
4.1.2. Nutrient limitation on phytoplankton growth rate
Brandt (1899, 1902) first called attention to the importance of phosphate and nitrate as
limiting factors for phytoplankton growth in the ocean and Ketchum (1939) established
the of hyperbolic nature of the relation between nutrient absorption and concentration.
Monod (1941) introduced the Michaelis-Menten function in marine biology to describe
bacterial growth rate as a function of substrate concentration. Based on a review of
laboratory and field measurements and following Monod (1941), Caperon (1967) and
Dugdale (1967) argued that nutrient uptake can be described by Michaelis-Menten
enzyme kinetics (Table 2, Eq. 1). Thus the Michaelis-Menten function is also an
empirical formulation that can accommodate experimental data in marine biology.
Field observations and laboratory experiments showed that nutrient uptake does not
start at zero concentration (Caperon and Myer, 1972; Paasche, 1973). Droop (1973, 1983)
interpreted this to indicate the presence of an unreactive intercellular nutrient quota below
which phytoplankton growth rate is zero. Consequently, Droop (1973, 1983) suggested
the Droop equation (Table 2 Eq. 3) for nutrient uptake. Some authors use a sigmoid
function to parameterize nutrient thresholds (Table 2 Eq. 2; Fennel, 1995; Newmann,
2002) whereas others suggest a combination of Michaelis-Menten and Droop functions
(Table 2 Eq.4; Caperon and Myer, 1972; Paasche, 1973; Dugdale, 1977; Droop, 1983;
Flynn et al., 1999). The Droop equation is applicable for minor nutrient elements such as
iron, vitamin B12 and phosphorus, but for major nutrient elements such as nitrogen and
silicate, its applicability is limited (Goldman and McCarthy, 1977). It should be pointed
out that the Droop equation models the relationship between phytoplankton growth rate
and the internal cellular nutrient contents whereas the Michaelis-Menten function models
the relationship between phytoplankton growth rate and external nutrient concentrations
14
in sea water. Since the cellular nutrient content and the external nutrient concentration
differ, these two relationships are not equivalent. The interpretation of the direct
combination between the two functions (Table 2 Eq.4) is thus more complex than it has
been believed in certain modeling applications.
There are two main forms of dissolved inorganic nitrogen that can be taken up by
phytoplankton, nitrate (NO3-) and ammonium (NH4+). NH4+ is generally preferred over
NO3+, most likely due to the low cost of NH4+ uptake and because nitrate reductase
activity decreases as NH4+ concentration increases (Dugdale and Goering, 1967; Eppley
et al., 1969). Different functions have been developed to parameterize ammonium
inhibition on nitrate uptake. Walsh (1975) was the first to consider NH4+ inhibition on
NO3- uptake by using the linear function of NH4+ concentration (Table 3, Eq. 1).
Wroblewski (1977) proposed an empirical function with an exponential inhibition term
(Table 3 Eq. 2). However, this equation can generate values higher than 1 and the growth
rate can decrease with increasing NH4+ concentration in combination with NO3(Eigenheer et al., 1996). Hurrt and Armstrong (1996) proposed a formulation based on
the argument that the sum of NH4+ and NO3- uptake should be (NO3-+NH4+ )/(KN+
NH4++NO3-) (Table 3 Eq. 3). This formulation uses the same half-saturation
concentration for both NO3- and NH4+ and thus does not reflect the preference of NH4+
over NO3-. O’Neill et al. (1989) deduced from molecular kinetics a substitute formulation
between two nutrients (Table 3 Eq. 4; Fasham, 1995; Eigenheer et al., 1996). It does not
contain a specific inhibition factor and NO3- can substitute NH4+ uptake in this function.
Also based analysis of substitution between two interacting nutrients, Yajnik and Sharada
(2003) suggested an alternative formulation to simulate NH4+ inhibition on NO3- uptake
(Table 3, Eq. 5). Spitz et al (2001) combined the Wroblewski and O’Neil functions for
nitrogen uptake (Table 3 Eq. 6). Alternatively, while taking into account nitrate reductase
activity, Parker (1993) developed a formulation for nitrogen uptake based on the
Michaelis-Menten function (Table 3 Eq. 7). This function has only the interpretable halfsaturation constants of nitrate and ammonium as controlling parameters.
4.1.3. Temperature forcing on phytoplankton growth rate
Most biological processes are subject to temperature. Various empirical
15
formulations have been used to reproduce observed relationships between growth rates
and temperature. Based on previous studies, Dam and Peterson (1988) used different
functions to fit experimental data between zooplankton ingestion and temperature, and no
significant difference was found among the formulations used (Table 4 Eqs. 1-4). In
many modeling applications, however, temperature forcing on growth rates is described
as an exponential function (including the Arrhenius function; Table 4 Eqs. 5,6), or by the
operational formulation of Q10, i.e., the increase in biological growth rate over 10 °C
(Table 4 Eq. 7). The Arrhenius function is mechanistic (Table 4 Eq. 5), with E presenting
the activation energy of the process of reaction (R is the gas constant (8.3 mol-1 K-1),, and
T is the absolute temperature; Raven and Geider, 1988). The activation energy can be
determined from the corresponding Q10 (E=RTln(Q10)/10; Dixon and Webb, 1979, p.
175).
A major challenge to these monotonic functions is that in many cases biological
growth rates show an optimal temperature above which the rate decreases again
(Pomeroy and Deibel, 1986; Wiebe and Dieck, 1989; Zupan and West, 1990 and Yager
and Deming, 1999). Different formulations have been developed to parameterize the
optimal temperature (Table 4 Eq. 9-11). The formulation proposed by Lancelot et al.
(2002) (Table 4 Eq. 11) is symmetric below and above the optimal temperature, and that
used by Thebault (1985) (Table 4 Eq. 9) is asymmetric. The exponential product
formulation (Table 4 Eq. 10) allows simulation of all these various curves, symmetric or
asymmetric (Kamykowski and McCollum, 1986).
4.1.4 DOC exudation and mortality of phytoplankton
DOC exudation from phytoplankton is generally parameterized as a linear function of
phytoplankton biomass (4th term in Eq. 2). This process may be subject to temperature
influence so that the exudation rate (rpi) can be scaled to temperature. Bannister (1979)
and Laws and Bannister (1980) linked exudation with both phytoplanktom biomass and
growth rate:
DOC
 ( a  b ) P
t
(3)
where  and P are the phytoplankton growth rate and biomass, respectively and a and b
16
are constants.
Both mortality and aggregation of phytoplankton result in biogenic detritus. Mortality
is parameterized as a linear function (mpi term in Eq. 2) whereas aggregation as a
quadratic function of phytoplankton biomass (pdi term in Eq. 2). The grazing term (5th
term in Eq. 2) will be described in the “Zooplankton” section.
4.1.5 Combination of light, temperature and nutrient forcing on phytoplankton growth
When multiple types of nutrient are considered, phytoplankton growth rate is generally
determined by the availability of the nutrient in the shortest supply relative to the
requirement by balanced growth, i.e., the Liebig Law. The Liebig Law of minimum
initially concerned only nutrient availability (Liebig, 1840). However, in addition to the
nutrient supply, Blackman (1905) also considered light and temperature. Following
Blachman’s suggestion, the minimum between light and nutrient limitation is often
chosen as the combined effect on phytoplankton growth in certain modeling applications:
1   (T ) min(  ( E ),  ( N (1,2)),  ( N (3)), , ,  ( N (nn)))
(4)
where i(T), i(E) and i(N(j)) are temperature, light and nutrient-dependent growth rate
of phytoplankton Pi, respectively (Radach and Moll, 1993; Hurrt and Armstrong, 1996;
Denman et al., 1998; Napolitano et al., 2000; Oschlies et al., 2000; Carbonel and
Valentin, 1999; Oschlies and Koeve, 2000). Alternatively, Baule (1918) expressed the
combined effect of limiting factors by a multiplication. Like temperature, light also tends
to influence the maximum growth rate of phytoplankton and, as a result, the products of
light-, temperature- and nutrient-dependent growth rate is an alternative parameterization
for the combination of the three controlling factors:
 2   ( E )  (T ) min(  ( N (1,2)),  ( N (3)), , ,  ( N (nn)))
(5)
(Goldman and Carpenter, 1974; Parsons et al., 1984; Andersen and Nival, 1987;
Hofmann and Ambler, 1988; Moisan and Hofmann, 1996; Doney et al., 1996; Leonard et
al., 1999; Gao et al., 2000; Kawamiya et al., 2000; Tian et al., 2000, 2001; Chifflet et al.,
2001; Fennel et al., 2002). Both formulations are hypothetical. Experimental and field
data often showed that the combined effect lies between the minimum and multiplication
(Rhee and Gotham, 1981a, b; Redalje and Laws, 1983). The minimum and multiplication
formulation can be combined as follows:
17
=1+(1-)2
(6)
where 0<<1. When the Liebig Law is applied to light with nutrient limitation, both
light and nutrients should be parameterized in a similar way, e.g. by the MichaelisMenten function.
4.2 Zooplankton
The general equation for zooplankton Zj is written as:
Z j
t
np
nz
nd
nb
i 1
k 1
l 1
m 1
  eZPji G Pji   (eZZjk G Zjk  G Zkj )   eZDjl G Djl   eZBjm G Bjm
  ZDj Z
mZj
 rZj Z j  w j
Z j
(7)
z
where GPji, GZjk, GDjl and GBjm are the total grazing (or predation) amounts of zooplankton
Zj on phytoplankton Pi, zooplankton Zk, detritus Dl and bacteria Bm, respectively, eZPji,
eZZjk eZDjl and eZBjm are the corresponding gross growth efficiency of Zj, GZkj is the
predation loss of Zj by other zooplankton, mZDj, mZj are the coefficient and power of
zooplankton mortality, rZj is the biomass-based respiration coefficient and wj is the
vertical migration speed of heterotrophs Zj.
As phytoplankton, all zooplankton pools are described by the same equation whereas
parameter values are specific to each zooplankton pools. When the numbers of
components of phytoplankton, zooplankton, detritus and bacteria are changed, the
summation items in the generalized zooplankton equation are modified.
3.2.1. Grazing functions
Zooplankton feeding modes can be broadly divided into suspension filter feeding (e.g.,
herbivorous copepods), raptorial feeding (e.g., carnivorous copepods, euphausiids), and
mucous net feeding (e.g., salps, doliolids, appendicularia). These feeding modes are not
mutually exclusive, and examples of both filter feeding and raptorial feeding can often be
found in one species, especially among planktonic crustaceans. Zooplankton feeding may
be herbivorous, omnivorous, carnivorous, phytophagous, zoophagous, euryphagous or
detritivorous (Parsons et al., 1984). Calanoid copepods are the most abundant and
ecologically significant marine herbivores. They are thought to feed primarily by
filtering, and some species demonstrate selectivity in feeding and predation by switching
18
according to food density (Wilson, 1973). There are two kinds of fundamental predator
response to changes in prey density: “numerical response” (i.e., the number of predators
changes as a function of prey density) and "functional response" (the ingestion rate of the
predator changes as a function of prey density; Solomon, 1949; Murdoch, 1969).
Generally, there are three types of functional response by predators to prey density
(Holling, 1959, 1965, 1969; Real, 1978): The first type of functional response is linear or
rectilinear (e.g., mucous net feeding). In this representation there is assumed to be no
interference between particles in the capture-ingestion mechanisms until the critical
concentration is reached. The rectilinear relationship results from continuous filtration of
water unaffected by the concentration of phytoplankton, so that ingestion increases
linearly with food concentration, up to an initial concentration above which the rate of
passage of food through the gut limits the rate of ingestion. The linear function between
uptake and prey concentration is explained by the unsaturated concentration in the ocean
and food acclimation of predators. The second type of functional response is curvilinear,
which is also called the “invertebrate curve”. Predation is saturated at a certain food
density level after which there is no significant increase in food ingestion. The degree of
interference increases continuously as the concentration of particle increases, so that the
rate of ingestion decelerates. The third type of response is called “switching response” or
“vertebrate curve” and is characterized by a threshold of food density at which predation
becomes negligible (Murdoch, 1969; Gismerik and Andersen, 1997). Holling (1959)
described the phenomenon as a “threshold of security” that stabilizes the prey population,
others referred to a “refuge”, or “learning response”, i.e., zooplankton ignore low-density
food because of high energy cost (Holling, 1965; Mutrdoch and Oaten, 1973; Mullin et
al., 1975). Another hypothesis for the threshold suggests that if the energy cost to the
zooplankton in searching and capturing food is high relative to the nonfeeding metabolic
rate, it might be advantageous to cease feeding when the concentration of food is very
low (Mullin et al., 1975).
When multiple kinds of prey are involved, feeding modes can be divided into three
types: non-selection feeding (i.e., the proportion of different types of prey in the diet is
the same as in the food available), passive selection (i.e., the proportion of different types
of prey in the diet is different from that of the food available, but with constant selectivity
19
or preference), and switching selection (i.e., the preferences or proportion change as a
function of prey density). Predator switching can exert striking effects on the stability of
prey population by preventing extinction of rare species, conserving biodiversity and
favoring coexistence of different predators (Murdoch, 1969; Murdoch, 1973; Murdoch
and Oaten, 1975; Comins and Hassell, 1976; Tansky, 1978; Hutson, 1984). However,
shifting of prey occurs under strong preference conditions. In the weak preference case,
no shift would occur except where there is an opportunity for the predator to become
accustomed to the abundant species.
The Type I functional response is described by a linear or rectilinear function between
prey density and predation (Table 5 Eqs. 1,2). The Type II functional response is usually
parameterized using the Ivlev function (Table 5 Eq. 3), the disc function (Table 5 Eq. 7)
or the Michaelis-Menten function (Table 5 Eq. 8). Ivlev (1955) showed that the quantity
of food eaten increased with the concentration of food available up to some maximum
ration, which could not be further increased by increasing food concentration. Thus the
grazing rate at a given prey concentration P must be proportional to the difference
between the actual and the maximal ration:
g
  ( g max  g )
P
(8)
and the integration of the above equation yields the Ivlev formulation:
g  g max (1  e P )
(9)
where  is a constant and P represents food abundance. The Ivlev functon was based on
fish feeding activities. A number of authors found that this formulation could be applied
to zooplankton feeding (Parsons et al., 1984). Rashevsky (1959) explained the physical
and biological meaning of the initial slope  as the ratio between the rate that the prey
becomes available to the predator and the maximum rate the predator can feed. Thus the
Ivlev function can be modified as:
g  g max (1  e P / g max )
(10)
According to Rashevsky (1959), the Ivlev curve best describes situations in which a
starved animal feeds for a relatively short period, while measurements of well nourished
animals whose ingestion has reached some sort of equilibrium with food supply should
20
generate a similar curve of a somewhat different shape. Wroblewski (1977) modified the
Ivlev function to parameterize threshold response (Table 5 Eq. 5), whereas Mayzaud and
Poulet (1978) combined the linear and Ivlev functions to simulate both Type I and Type
II responses. They attributed the increase in ingestion with increasing prey density to
herbivore acclimation to variation in prey density (Table 5 Eq. 6).
The disc function was based on mechanistic analysis (Table 5 Eq. 7) in which 
represents the rate at which the predator attacks the prey and  represents the handling
time (Holling, 1959). Fenchel (1980) deduced the same equation for suspension feeders
where  represents the feeding current and  represents the handling time or the time that
a particle blocks the mouth of the predator. Assuming 1/=K, the disc function is then
transformed into the Michaelis-Menten function (Table 5 Eq 8). Caperon (1967)
explained the half-saturation constant as the ratio between the rate of freeing the
absorption site (e.g. digestion) and the rate of food uptake. The Michaelis-Menten
function was modified with a specific threshold (Table 5 Eq. 9, 10) or into a sigmoid
function (Table 5 Eq. 11) to mimic the Type III functional response. Real (1977)
suggested a generalized Michaelis-Menten function which can parameterize both Type II
and Type III responses (Table 5 Eq. 13). A variant functional response that was not
included in the Holling’s definition is the toxicity-grazing response, i.e., the grazing rate
decreases with the increased food density (Table 5 Eq. 12).
When multiple food types are involved, formulations of non-switching zooplankton
grazing on multiple types of prey include rectilinear, Ivlev, disc and Michaelis-Menten
functions and their modified forms (Table 6 Eqs. 1-6). There is evidence that some
predators concentrate on the more abundant prey, so that the rarer species is mainly
ignored, i.e., predators switch prey. Parameterization of switching-predation between
different types of prey is essentially based on modified Michaelis-Menten or disc
functions (Table 7 Eq.1-5). Gismervik and Andersen (1997) suggested a generalized
grazing function on multiple types of prey (Table 7 Eq. 5) where the power index m
determines the degree of switching feeding. No switching occurs when m=1 (i.e. Eq. 7 in
Table 7) and the higher m is, the more switching occurs. When m, switching occurs
exclusively. With this flexible parameterization, zooplankton grazing can be
21
parameterized as:
G Pji  g j (T )Z j
p
( Pi  P0i )
mj
Pji
(11)
1 Rj
np
R j   ( p Pji ( Pi  P0i ))
mj
i 1
nb
  ( p Bjm ( Bm  B0 m ))
nz
  ( p Zjk ( Z k  Z 0 k ))
k 1
mj
nd
  ( p Djl ( Dl  D0l ))
mj
l 1
(12)
mj
m 1
where gj(T) is the temperature-forced grazing rate using (see Table 4), pPji, pZjk, and pDjl
and pBjm are the preference coefficients determined as the inverse of the corresponding
half-saturation constants and P0I, Z0k, D0l and B0m are the corresponding thresholds for
each type of prey. The grazing terms GZjk , GDjl and GBjm predation terms in Eq. 11 are
also determined using Eq. 12.
4.2.2. Mortality, excretion and respiration
Zooplankton mortality represents the model closure term and the corresponding
parameterizations are listed in Table 8. Zooplankton mortality consists of natural
mortality, which may be caused by disease and starvation, and mortality due to predation
by higher predators. Both disease and predation are likely density-dependent processes
(i.e., quadratic, Eqs. 2-4, 7 in Table 8). Andersen et al. (1987) and Andersen and Nival
(1988) considered mortality caused by starvation and parameterized as a function of the
food concentration (Table 8 Eqs. 5 and 6). Edwards and Yool (2000) proposed a
generalized mortality formulation, which can simulate linear, quadratic or a higher order
of mortality terms (Table 8 Eq. 8).
Respiration and excretion represent the metabolic losses of carbon and nitrogen
respectively. Metabolic processes consist of different components such as basic
metabolism, locomotion, assimilation, synthesis of somatic and gonad tissue, matter
transformation and etc. (Clarke, 1987; Carlotti et al., 2000). Total metabolism is 2-3
times the basic metabolism at resting (Steele and Mullin, 1977; Parsons et al.,1984). The
simplest formulation of respiration and excretion is parameterized as a linear function of
biomass or food ingestion (Table 9 Eqs. 1,2). It can be divided into basic metabolism
which is linearly linked to zooplankton biomass and active metabolism which is related
to food ingestion (Table 9 Eq. 3; Steele and Mullin, 1977; Carlotti and Sciandra, 1989;
22
Carlotti et al., 2000). In some applications, zooplankton respiration was linked to
temperature and individual dry weight (Table 9 Eqs. 5-8). Wroblewski (1977) suggested
an egestion formulation that was exponentially linked to prey density. In a biomass-based
model without parameterization of population dynamics, both excretion and respiration
can be formulated as a linear function of zooplankton biomass and ingestion (Table 9 Eq.
3).
4.2.4. Zooplankton diel vertical migration
wl and wf are light- and food-induced vertical migration speed of zooplankton. Lightinduced migration speed (wl) is linked to the change in percentage of solar radiation at the
sea surface (I0):
 100 I 0 

wl j  wb 
 I 0 t 
3
(13)
where wbj is a constant representing the slope between migration speed and light change
(Anderson and Nival, 1991; Tian et al., 2003). The absolute value of wlj is set to wmaxj
when Eq. 38 gives values >wmaxj and to -wmaxj (i.e. upward) when ∂I0/∂t is undetermined
during the night. To preclude zooplankton ascending to the euphotic zone with solar
radiation decreasing (e.g. in the afternoon), wlj is set to zero when light intensity in the
water column is higher than a critical value (I(z)>0.01I0). Light-induced upward
migration velocity is also set to zero when the food gradient is positive, i.e. food
abundance increases with depth (Ft/z>0). When upward migrating zooplankton reach
the chlorophyll maximum, for example, they are thus assumed to stop migrating further
to surface layers where foods are scarce. In this case, the migration speed is determined
only by food abundance. Food-dependent vertical migration speed (wfj) is calculated
using an exponential function:

wf j  ard wmax j 1  e
 k fj Ft j

(14)
where ard is randomly equal to 1 or –1 with 50% probability each at each migration time
step, and kfj is a constant describing the slope between migration speed and total food
abundance (Ft). Seasonal migration and overwintering was modeled using a critical food
23
abundance (Ftmin) below which light-induced migration speed is set to zero. When food
abundance in winter is lower than Ftmin, mesozooplankton overwinter at depth.
4.3 Bacteria
Bacterial trophic dynamics are parameterized in the same way as those of zooplankton,
i.e., using the Michaelis-Menten function for substrate uptake and a linear function for
respiration rate. Both growth and respiration rates are scaled to temperature. The general
equation for bacterial state variables is:
B j
t
ns
nh
i
k 1
  eBSjiU DOMji  eBNjU NH 4 j   GBkj  rBj B j
(15)
where UDOMji and UNH4j are the uptake amount of DOMi and NH4+ by the bacterial pool
Bj, eBSji and eBNj are the corresponding gross growth efficiency, GBkj is the consumption of
bacteria Bj by heterotrophs Hk and rBj is the resiration rate of bacteria Bj.
When the content of dissolved organic nitrogen (DON) in the DOM pools is low,
bacteria can take up ammonium to obtain sufficient nitrogen to synthesize cell protein.
Thus the amount of NH4+ taken up by bacteria primarily depends on the C:N ratio of the
DOMi pools.
 ji 
eBSji ( N : C ) Bj
eZNj ( N : C ) DOMi
1
(16)
where ηji is the uptake ratio between NH4+ and DONi by bacteria Bj, eBSji and eBNj are the
gross growth efficiencies of DOCi and DONi, respectively, and (N:C)Bj and (N:C)DOMi are
the N:C ratio of bacteria Bj and DOMi pool, respectively (Fasham et al., 1990; Bissett et
al., 1999; Tian et al., 2004). ji is zero if the calculated value is negative. In that case,
NH4+ will be released from DON uptake. Spitz et al. (2001) split DOM into DOC and
DON pools to estimate DON versus NH4+ uptake by bacteria.
4.4 DOM
The general equation for DOM state variables is:
24
np
nd
nb
DOM i
  ( rPSji  rPSji  j ) Pj   DSmi Dm  U DOMki
t
j 1
m 1
k 1
nz
nd
nb
 np




G


G


G


G





ZPSkji
Pkj
ZZSkli
Zkl
ZHDSkmi
Dkm
ZBSkni
Bkn


k 1  j 1
l 1
m 1
n 1

  SSii1 DOM i   SSi1i DOM i 1
nz
(17)
where rPSji and rPSji are the coefficients of biomass-based and growth-based exudation of
DOMi from autotroph Pj. The sum of rPSji and the sum of rPSji (i=1 to ns) equal the terms
rPSj and rPSj in Eq. 1. The second term on the right side of Eq. 17 represents the feeding
losses to DOMi from zooplankton including sloppy feeding, defecation and excretion.
The third and forth terms are detritus dissolution and bacterial uptakes. The fifth and
sixth terms represent transformation between DOM pools, such as aging when DOM is
classified according to their bioavailability (Keil and Kirchman, 1994). All DOM pools
are described by the same equation (Eq. 17) and changes in the number of DO
components can modify the bacterial equation.
The elemental ratio between nutrients and the unit element (e.g. carbon or nitrogen) in
living organisms (i.e. phytoplankton, zooplankton and bacteria) are prescribed with
specific value for each organism pools. The elemental ratio in dissolved and detrital pools
(i.e., DOM and detritus) are computed according the elemental ratio in each of the source
and sink terms. The elemental ratio (aSji) between nutrient Ni and the unit element in
DOMj is determined as:
a Sji
t
np
nd
nb
 (  a Pji ( rPSji  rPSji  j ) Pj   a Dmi DSmi Dm  a Sji  U DOMkj
j 1
m 1
k 1
nb

 (18)


a

G

a

G

a

G


  ZZkji ZPSkji Pkj  ZZkli ZZSkli Zkl  ZDkmi ZDSkmi Dkm  a ZPkni ZBSkni G Bkn 
k 1  j 1
l 1
m 1
n 1

DOM j
 a Sj SSjj1 DOM j  a Sj 1i SSj1 j DOM j 1 ) /( DOM j 
)  a Sji
t
nz
na
nz
nd
where aSji, aPji, aDmi, aZPkji, aZZkli, aZDkmi, aZPkni are the elemental ratio between nutient Ni
and carbon in DOMj, in phytoplankton Pj, in detritus Dm, and in feeding losses to DOMj
from zooplankton Zk feeding on phytoplankton Pj, on zooplankton Zl, on detritus Dm and
on bacteria Bn respetively. The elemental ratios in feeding losses aZPkji from zooplankton
Zk feeding on prey Pj is calculated by:
25
 ZPkji 
 Pji  e ZPkj a Zki
(19)
1  e ZPkj
where aPji and aZki and aZPkji are the elemental ratio between nutrient Ni and the unit
element (e.g. carbon or nitrogen) in prey Pj and zooplankton Zk, and eZPkj is the
corresponding growth efficiency (Landry et al., 1993; Tian et al., 2004).
4.5 Detritus
The general equation for detritus is:
nz
Di np
m
   PDji Pj j    ZDki Z kmk   DDi1 Di21   DDi Di2   DDi 1 Di 1   DD1 Di   DSi Di
t
j 1
k 1
 s Di
nz  np
nz
nd
nb

Di
     ZPDkji G Pkj    ZZDkli G Zkl   Z ZDDkmi G Dkm   Z ZBDkni G Bkn  G Dki 
z
k 1  j 1
l 1
m 1
n 1

(20)
The first two terms on the right-hand side of Eq. 20 are the mortality of phytoplankton
and zooplankton which lead to the formation of biogenic detritus. The third and fourth
terms represent aggregation gain and loss formulated as a quadratic function. The fifth
the sixth terms are the gain and loss due to particle breakage. The seventh and eighth
terms are particle dissolution and sinking and the last term represents zooplankton
feeding losses to detritus including both sloppy feeding and defecation and zooplankton
consumption. All detritus pools share the same equation with different parameter values
such the sinking velocity which is dependent on the size of detritus. The elemental ratio
in detritus pools are computed in the same way as for DOM pools.
4.6 Nutrients and micronutrients
The general equation for nutrients Ni is:
np
nz
nz
nd
nb


N i
   a Zji rZj Z j   rZPjhiG Pjh  rZZjki GZjk   rZDjliG Djl   rZBjmiG Bjm 
t
j 1 
h 1
k 1
l 1
m 1

 ns
 np
    a BSkli (1  e BSkl )U DOMkl  e BNkU NH 4 k  a Bki rBk Bk    a Pmi  m Pm
k 1  l
 m
nb
(21)
where aZji, aBki and aPmi are the ratio between element Ni and the unit element in
zooplankton Zj, bacteria Bk and phytoplankton Pm respectively. rHj and rBk are the
26
biomass-based respiration of zooplankton Zj and bacteria Bk. rZPjhi, rZZjkh,, rZDjli, rZBjmi
are the active respiration coefficient of zooplankton Zj based on grazing amount or
assimilation. The active respiration coefficient of nutrient Ni from zooplankton Zj grazing
on phytoplankton Ph is determined by:
rZPjhi  rZj
 Phi  e ZPjh a Zji
(22)
1  e ZPjh
where rHj is the active respiration rate of zooplankton Zj, aPhi and aZji are the ratio
between element Ni and the unit element in phytoplankton Ph and zooplankton Zj, and
eZPjh is the gross growth efficiency of zooplankton Zj grazing on phytoplankton Ph. The
active respiration coefficient of other terms including the active respiration of bacteria
aBSkli are also determined by Eq. 22 by using the corresponding terms.
When nitrogen is concerned, NH4+ is released from biological metabolism and
respiration. Consequently, Eq. 21 applies to NH4+ and other nutrients except NO3-. NO3is formed through nitrification of NH4+ as source term and consumed by phytoplankton
photosynthesis. The proportion between NH4+ and NO3- uptakes is governed by equations
in Table 3.
Ammonium is nitrified to nitrate through nitrifying bacteria, which is known to be
photoinhibited in surface waters (Olson, 1981). Nitrification rate (QAN) is thus linked to
light intensity in the model:
Q AN
0, E (t , z )  0.1E max

  0.1E  E (t , z )
max
 AN NH 4 ,


0.1E max
(23)
E (t , z )  0.1E max
where AN is the nitrification coefficient and Emax is the maximum solar radiation in
surface waters (i.e. at noon; Tian et al., 2000).
4.7 Auxiliary state variables
Auxiliary state variables represent oceanic features that depend on other biological
pools, e.g., chlorophyll, bioluminescence, dissolved oxygen, CO2, optics, color,
acoustical properties, etc. Only chlorophyll a and bioluminescence are presented below.
27
3.7.1. Chlorophyll a
The chlorophyll a concentration depends on phytoplankton biomass and the
chlorophyll-a:carbon ratio of phytoplankton. Chl:C ratio can be influenced by daylength
(D), irradiance (E), nutrient (N) and temperature (T), i.e., the DENT model. It can also be
influenced by other factors such as species and life history (Cullen, 1993; Claustre et al.,
1994). Based on 219 paired data from various regions, Cloern et al. (1995) obtained the
following empirical relationship between Chl:C ratio and the controlling factors:
Chl : C  0.003  Ae (T  I )
N
N  KN
(24)
where A,  and  and empirical constants and N represent nutrients. Laws and Bannister
(1980) formulated the Chl:C ratio as a linear function of the phytoplankton growth rate
():
Chl : C  a  b
(25)
Based on mechanistic analysis, Geider and MacIntyre (1996) deduced a function of
Chl:C ratio of phytoplankton photosynthesis as:
Chl : C   max i
i
E i
(26)
where i is the growth rate of phytoplankton Pi, which is forced by nutrient, light (PAR)
and temperature (Eq. 2),  is the initial slope between PAR and phytoplakton growth rate
(Eq. 5) and max and  are the maximum and actual Chl:C ratio (Geider and MacIntyre,
1996; Geider et al., 1997; Spitz et al., 2001). With this algorithm as the a priori
parameterization, the general equation for chlorophyll a can be written as:
nz


i
P
Chl np 
m
  (1  rPSi ) max i
Pi   i  PDi Pi i  rPSi Pi  s Pi i   G Pji  


t
 i i E
z
i 1 
j 1


(27)
The four terms on the right-hand side represents changes in chlorophyll concentration due
to phytoplankton growth, aggregation (or mortality), respiration (or exudation), cell
sinking and zooplankton grazing, respectively. All the symbols are defined in Eqs. 2 and
26.
28
4.7.2. Bioluminescence
Bioluminescence is the emission of visible light by living organisms through
chemiluminescent reaction. Bioluminescent organisms range from bacteria through
protozoa, fungi and copepods to fishes. However, the percentage of bioluminescent to
total marine organisms is relatively low (<10% in most cases), and varies depending on
exogenous factors (e.g., light intensity, depth, temperature and dissolved oxygen),
interspecies reaction (e.g., predation, defense) and biological cycles (e.g., foraging,
mating, nursing and schooling; Deheyn et al., 2000). As a primary approximation,
bioluminescence potential (BLi) can be represented as a linear function of bacteria and
zooplankton biomass, using a specific bioluminescence coefficient for each bacterial
(BLj) and zooplankton pool (ZLi):
B j
Z i nb
L nz
   ZLi
   BLj
t i 1
t
t
j 1
(27)
5. ADAPTIVE MODELING AND REAL TIME ECOSYSTEM FORECASTING
Flexible and adaptive modeling implies that models can change in response to
ecosystem functioning, with respect to scientific objectives and according to data
availability from field observations. Presently, a model is considered to be adaptive if its
formulation, classically assumed constant, is made variable as a function of data flows.
Adaptive changes can be made in model structure, equations and/or parameter values. For
example, a marine ecosystem can be nitrogen, silicate, phosphorus or iron limited.
Ecosystem models need to parameterize explicitly the limiting factor in order to simulate
adequately the function of the ecosystem under investigation. Seasonal succession in food
web structure and ecosystem shift due to climate change and anthropogenic stress require
subsequent adaptation from numerical models.
Marine ecosystems consist of a complex network comprising inorganic nutrients,
phytoplankton, bacteria, zooplankton, fish and marine mammals. There is a huge number
of species at each trophic level that interact with each other through various physical,
chemical and biological processes. It is unrealistic to include all chemical elements and
biological populations in a single numerical model. One challenge in ecological modeling
is to integrate state variables and to combine processes. Even within a single ecosystem,
29
integration of state variables and combination of processes can be carried out in different
ways in respect to different scientific objectives. A specific scientific question can also be
addressed at different levels, using minimum, optimal and maximum critical state
variables and processes.
Our generalized model can be adaptive at different levels. For example, when ns
(number of DOM state variables), nb (bacterial state variables), nd (detritus state
variables) and na (auxiliary state variables) are assigned to zero and nn (nutrients), np
(phytoplankton) and nz (zooplankton) are assigned to 1, the generalized biological model
becomes the classical NPZ model. If nn is assigned to 2, the generalized model represents
the McGillicuddy model (McGillicuddy et al., 1995), and if nd is assigned to 1, the
generalized represents the Anderson model (Anderson et al., 2000). The Dussenberry
model (Besiktepeet al., 2003) necessitates the addition of the chlorophyll a auxiliary state
variable and the Fasham model (Fasham et al., 1990) requires the addition of one DOM,
bacteria and detritus state variable, respectively (i.e. ns=nb=nd=1). When np, nz and nd
are assigned to 2 (i.e., two state variables of phytoplankton, zooplankton and detritus,
respectively), the generalized model becomes the Tian model (Tian et al., 2000, 2001).
Four types of functional response to prey density have been observed and
parameterized in previous studies: rectilinear, hyperbolic, sigmoid or threshold and preytoxic responses. Different taxonomic groups have different feeding modes and the
feeding mode of one taxonomic group can change with respect to prey type and density.
Thus, different parameterization of zooplankton feeding can be used when there is a shift
in zooplankton communities and prey availability. Mucous net feeders (e.g., salps,
doliolids and appendicularia) passively collect food available in the feeding current
without any selection so that a rectilinear response can be applied. Herbivorous grazers
(e.g., calanoid copepods) are typically suspension feeders, which can be described by a
hyperbolic function, i.e., the Ivlev or Michaelis-Menten function. Carnivorous predators
(e.g., Euphausiids) can be more adequately described by a sigmoid function (Moisan and
Hofmann, 1996). Field studies suggest that certain copepods are opportunistic feeders,
concentrating their feeding efforts on the most abundant prey (Wilson, 1973; Richman et
al., 1977; Pollet, 1978 and Gismerik and Andersen, 1997), although no-selectivity
grazing of copepods has also been reported (Mullin et al., 1975; Kleppel, 1993). Several
30
copepod species have turned out to be omnivores, switching between a secondary and a
tertiary position in the food web (Stoecker and Capuzzo, 1990). Kiorboe et al. (1996)
observed that the copepod Acartia tonsa has two feeding modes: generating a feeding
current for immobile prey (diatom) and ambushing mobile prey (ciliates). These different
feeding behaviors provide the biological background for adaptive modeling at the
parameterization level.
Real-time forecasting of marine ecosystems is of increasing importance in marine
sciences. Climate changes and anthropogenic stresses are becoming more and more
acute, and these coupled with the potential destabilization resulting from extreme
weather events have the potential to disturb ecosystem functioning and health. Mesoscale
eddies can generate episodic and complex distribution of biological features. Harmful
algal development (redtides), aquaculture farms, oil spills, environmental perturbation,
marine resource management, conservation and exploitation and certain military
applications all necessitate real-time ecosystem forecasting.
A generalized biological model can be specifically conceived and developed for
this purpose. Our version of this generalized model is coupled with the Harvard Ocean
Prediction System (HOPS). HOPS is an integrated system of data assimilation schemes
and a suite of coupled interdisciplinary (physical, optical, biogeochemical –ecosystem)
dynamic models (Robinson et al., 1998; Robinson, 1999). Real-time and at sea
forecasting have been conducted for more than a decade at twenty sites in the Atlantic
and Pacific Oceans and the Mediterranean Sea (Robinson, 1996; Robinson et al., 1996).
The flexibility and adaptability of this generalized model will enable significant progress
in real-time forecasting of marine ecosystems.
6. CONCLUSIONS
Data-driven adaptive modeling and real-time forecast of marine ecosystems is an
increasing challenge in marine sciences. In the context of global climate warming and
increasing anthropogenic stress, marine ecosystems are becoming more and more
vulnerable and uncertain. Eutrophication, harmful algal blooms, red tide, oil spills, toxic
element pollution can all deteriorate the health and functioning of marine ecosystems.
Traditionally marine ecosystems are modeled with simulation models of fixed
31
structure and static data inputs. However, forecasting evolving marine ecosystems, in
space and time, in response to environmental perturbation, necessitate rapid response of
dynamic data-driven adaptive simulation models.
We have developed the first version of a generalized and flexile prognostic model for
marine biogeochemical-ecosystem processes. The model structure was specifically
designed for adaptive modeling and real-time ecosystem forecast, with an initial focus on
pelagic mid-latitude oceans. Marine ecosystems function through a series of highly
integrated interactions between biota and the habitat and dynamic links among food web
components. Based on the trophic and biogeochemical dynamics, the generalized model
is composed of 7 functional groups: nutrients, phytoplankton, zooplankton, detritus,
dissolved organic matter, bacteria and auxiliary state variables. In most biological
models, the number of compartments is fixed, with each compartment representing a
specific biological community or species. In our generalized model however, the number
of components in each functional group is a variable and can be chosen by users for each
application. By using a subset of the state variables, one can simulate various ecosystems.
The potential combinations and actual structures of the generalized model are large.
Importantly, the model can be adaptive, i.e. the state variables, model structures and
parameter values can change in response to field measurements, ecosystem function and
scientific objectives. The computational version of the model has been implemented and
coupled with HOPS. The resulting coupled forecasting system is currently applied to the
Monterey Bay area to study biological response to upwelling events at the ecosystem
level.
Acknowledgments
We are grateful to Patricia Moreno and Dr. Constantinos Evangelinos for multiple
discussions on marine biological dynamics and distributed computing, respectively. We
thank Dr. Pat Haley, W.G. Leslie and Prof N.M. Patrikalakis for their inputs, and the
AOSN-II teams for providing the Monterey Bay data sets and dynamical context. The
present work was supported by NSF/ITR (grant 5710001319) and in part by ONR (grant
N00014-01-1-0771, N00014-02-1-0989 and N00014-97-1-0239).
32
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Table 1. Parameterization of Light forcing on Phytoplankton growth rate ((E):
phytoplankton growth rate; E: PAR; Pm: Maximum growth rate; Eopt: optimal PAR; :
Chlorophyll a: carbon ratio; : initial slope; a, ,  , n and  are constants). (N.B.: Pm/ a
=Eopt so both were used in the literature; Drawback: More parameters, more flexibles,
but more unknowns).
Function
References
(1) Rectilinear function:
E , forE  Pm /  
( E)  

Pm , forE  Pm /  
(2) Michaelis-Menten (MM) function:
E
 ( E )  Pm
Pm  E
(3) Smith function:
E
 ( E )  Pm
2
Pm   2 E 2
(3) Generalized MM function:
E
 ( E )  Pm
n
( Pm  (E ) n )1 / n
(4) Vollenweider function:
E
1
 ( E )  Pm
2
2
Pm   2 E 2 1  E / Pm 
(5) Peeters and Eilers function:
E
2 
 ( E )  Pm
E opt 1  E / E opt  ( E / E opt ) 2
(6) Webb function:

Blackman, 1905; Riley, 1946; Jassby and Platt,
1976; Platt et al., 1977
Baly, 1935; Tamiya et al., 1953; Caperon;
1967; Hofmann and Ambler, 1988; Kiefer and
Mitchell, 1993; Semovski et al., 1996;
Smith, 1936; Laws and Bannister, 1980; Evans
and Parslow, 1985; Fasham et al., 1990; Hurrt
and Armstrong, 1996; Spitz et al., 2001; Fennel
et al., 2002
Bannister, 1979; Laws and Bannister, 1980
(n=1 eqivalent to Baly, n= eqivalent Backman
linear)

n/2
Vollenweider, 1965; Fee, 1969; Wroblewski,
1977; Parsons et al., 1984
Peeters and Eilers, 1978; Andersen and Nival,
1987; Vichi, ERSEM report
 ( E )  Pm (1  e E / P )
Webb et al., 1974; Doney et al., 1996; Denman
et al., 1998; Gao et al., 2000
(7) Platt function:
 ( E )  Pm (1  e E / P )e  E / P
Platt et al., 1980; Moisan and Hofmann, 1996;
Leonard et al., 1999; Lancelot et al., 2000;
Chifflet et al., 2001
m
m
m
(8) Steele function:
E 1 E / Eopt
 ( E )  Pm
e
Eopti
(9) Parker function:
Steele, 1962; Radach and Moll, 1993;
Backhaus et al., 1999; Kawamiya et al., 2000

 E 1 E / Eo p t 

 ( E )  Pm 
e

E
 opti

(10) Hyperbolic tangent function:
 ( E )  Pm tanh( E / Pm )
(11) Modified hyperbolic function:
Parker, 1974
Jassby and Platt, 1976; Keller and Riebesell,
1989; Frost, 1993
Bissett et al., 1999
45
 ( E)  Pm tanh(  ( E  E0 ) / Pm )e
(12) Mechanistic function:
1  e  E
 ( E )  a max E
E
 ( Eopt  E )
Saksaug and Kiefer (1989); Green et al.,
1991Proceedingd of Indian Academy of
Sciences
46
Table 2. Parameterization of nutrient limitation on Phytoplankton growth rate (N: nutrient
concentration in sea water; K: half-saturation constant; Q: internal nutrient content in
cells (or nutrient cell quota); KQ: threshold of internal nutrient content below which
phytoplankton growth rate is zero; N0: threshold of nutrient concentration is sea water
below which phytoplankton growth rate is zero;). The Michaelis-Menten function (Eq. 1)
calculates the phytoplankton growth rate based on nutrient concentration in sea water
whereas the Droop function (Eq. 3) is based on internal nutrient content in cells.
Function
References
(1) Michaelis-Menten (MM) function:
Caperon, 1967; Dugdale (1967); Kiefer and Mitchell,
1993; Semovski et al., 1996; Evans and Garc Marra et
al., 1990; Radach and Moll, 1993; Davidson, 1996;
Flynn, 1998; Backhaus et al., 1999; Napolitano et al.,
2000; Chifflet et al., 2001; Franks an Chen, 2001on,
1997; Gao et al., 2000
Fennel, 1995; Newmann, 2002
N
1 ( N ) 
NK
(2) Modified MM function:
2 (N ) 
N2
N2  K2
(3) Droop function :
 3 (Q )  1 
KQ
Q
Droop, 1973; Marra et al., 1990; Haney and Jackson,
1996; Lange, K. and F.J. Oyarzun 1992; Oyarzun, F.J.
and K. Lange 1994
(4) Combined MM and Droop function: Caperon and Myer, 1972; Paasche; 1973; Dugdale,
4 (N ) 
N j  N0
1977; Lynn et al., 1999
N  K  N0
47
Table 3. Parameterization of ammonium inhibition on nitrate uptake (KNH4+ and KNO3- are
half-saturation constants and NH0 and NO0 are thresholds for NH4+ and NO3- uptake,
respectively).
Function
References
Walsh, 1975
(1) Walsh fucntion:
( N ) 


4
NO3
NH

(1  aNH 4 )


NH 4  K NH 4 NO3  K NO 
3
Wroblewski, 1977; Hofmann and
Ambler, 1988; Fasham et al., 1990;
Bissett et al., 1999; Leonard et al.,
1999; Napolitano et al., 2000;
Chifflet et al., 2001; Skliris et al.,
2001
(2) Wroblewski:

(N ) 
NH 4
NO3 e
NH 4


NH 4  K NH 4 NO3   K NO 
3
(3) Hurrt and Armstrong:
(N ) 
K N NO3
NH 4

NH 4  K N ( NH 4  K N )( NO3  NH 4  K N )
Hurrt and Armstrong, 1996
(4) O’Neil et al.:
(N ) 
NO3 / K NO   NH 4 / K NH 
3
O’Neill et al., 1989; Fasham, 1995;
Eigenheer et al., 1996
4
1  NO3 / K NO   NH 4 / K NH 
3
4
(5) Spitz:
(N ) 
NH 4 / KNH 4  NO3 / K NO  e NH 4
Spitz et al. (2001)
3
1  NO3 / K NO   NH 4 / KNH 4
3
(6) Parker:
( N ) 
NH 4
NH 4  K NH 

4
K NH 
4
NO3

NH 4  K NH  NO3   K NO 3
4
Parker, 1993; Loukos et al., 1997;
Christian et al.,202; Moore et al.,
2002; Tian et al., 2001
3
(7) Yajnik:
( N ) 
NH 4
NH 4  K NH 
4

1  aNH 4
NO3

Yajnik and Sharada, 2003
1  bNH 4 NO3   K NO 3
3
48
Table 4. Parameterization of temperature forcing on biological rate (Topt: optimal
temperature; T0 (T0L, T0H): temperature (low, high) under which the corresponding
biological rate is zero; Q10: the rate at which biological rate increases over 10 degree
increase of temperature; a, b, c R: contants).
Function
References
(1) Linear function:
(T) =a+bT
(2) Log linear function;
(T) =a+blog(T+c)
(3) Power function:
(T) =a(T+c)b
(4) Exponential:
i=a(b)T
(5) Arrhenius function:
 (T )  e
E 1 1 

 
R  Topt T 
(8) Exponential function:
 (T )  e aT
(7) Q10 :
 (T )  Q10
(
Dam and Peterson, 1988
Dam and Peterson, 1988
Eppley, 1972; Dam and Peterson, 1988
Dam and Peterson, 1988; Anderson et al., 2000
Monod, 1941; Packard et al., 1971; Goldman
and Capenter, 1974; Dugdale, 1977; Raven and
Geider 1988
…………………………………………………
……………….
Riley, 1944; Huntley and Lopez, 1992; Radach
and Moll, 1993; Bissett et al., 1999; Leonard et
al., 1999; Kawamiya et al., 2000; Tian et al.,
submitted
Toda et al., 1987; Doney et al., 1996; Gao et
al., 2000
T
)
10
(9) Temperature inhibition:
 i (T )  2(1  a)
T  T0
s
, s
Topti  T0
s 2  2as  1
(6) Beta function:
(T)=(T-T0L)a(T0H-T)b
(10) Exponential product:
 (T )  (1  e  a (T T0 L )(1  e b(T0 H T ) )
Thebault (1985), Andersen and Nival (1988)
and Skliris et al. (2001)
Carlotti et al., 2000
Kamykowski and McCollum, 1986.
(11) Modified exponential:
 (T )  e
 T To p t 



 dT 
2
Lancelot et al. (2002)
49
Table 5. Parameterization of grazing on a single type of prey (gmax: maximum grazing
rate; P0 threshold below which grazing is zero; , : constant).
Function
References
(1) Linear function:
g  P
Riley, 1946; Gamble, 1978; Dagg and Grill 1980; Klein
and Steele, 1985; Gao et al., 2000
(2) Rectilinear function:
P


, forP  K 
 g max
g
K



g
,
forP

K
 max

(3) Ivlev function:
g  g max (1  e
P
)
(4) Modified Ivlev function:
g  g max (1  e P / g max )
(5) Ivlev function with threshold:
g  g max (1  e  ( P  P0 )
Frost, 1972; Armstrong, 1994; Mayzaud et al., 1998
Ivlev, 1955; Andersen and Nival, 1988; McGillicuddy et
al., 1995; Doney et al., 1996; Robinson, 1996;
Besiketepe et al., 2003
Rashevsky, 1959; Kiorboe et al., 1982
Worblewski, 1977; Parsons, 1984;
(6) Combined linear and Ivlev function: Rogers, 1972; Mayzaud and Poulet, 1978; Franks et al.,
1986
g  g max P(1  e  P )
(7) Mechanistic disc function:
N
1  N
g  g max
(8) Michaelis Menten Function:
g  g max
P
PK
Holling, 1959; Chesson, 1983; Verity, 1991; Gentleman
et al., 2003
Monod, 1949; Caperon and Myer, 1872; 1967; Radach
and Moll, 1993; Strom and Loukos, 1998; Gao et al.,
2000; Napolitano et al., 2000
(9) Threshold MM function:
g  g max
P  P0
P  P0  K
Walsh, 1975; Evans, 1988; Frost, 1993
(10) Weight dependent function:
g  g max
P  P0
W
P  P0  K
(11) Modified MM function:
2
P2
or g  gmax 2 P
g  g max 2
2
K P
K  P  P 2
(12) Prey toxicity grazing
P
g  g max
K  P  P 2
(13) Generalized MM function:
Pn
g  g max
K  Pn
Steele and Mullin, 1977
Denman et al., 1998; Oschilies et al., 2000
Van Gemerden, 1974; Gentleman et al., 2003
Real, 1977; Steele and Henderson 1981; Fasham, 1995
50
Table 6. Parameterization of grazing on multiple types of prey with passive selection
(gmax: maximum grazing rate; K: Half-saturation constant (but saturation constant in Eq.
1); P0 threshold below which grazing is zero; pi: preference coefficient; , a, : constant).
Function
References
(1) Rectilinear

 g max

gi  
g
 max
pi Pi

, for R  K 
n
K
, R 
p j Pj


pi Pi
j

1
, for R  K 

R
(2) Combined linear and Ivlev function:
g i  gmax i Pi (1  e i Pi )
(3) Ivlev function with interference between prey types:

g i  g max 1  e R
 pRP , with R   p P
n
i
i
j 1
j
Armstrong, 1994
Leonard et al., 1999
Hofmann and Ambler, 1988
j
(4) Mechanistic disc function:
g i  g max
ai N i
n
1   a j j N j
j 1
Murdoch, 1973; Murdoch and
Oaten, 1975; Holt, 1983; Verity,
1991;
(5) Michaelis Menten Function:
g i  g max
pi Pi
n
K   p j Pj
Moloney and Field, 1991
j 1
(6) Threshold MM function:
n
 R  P0  pi Pi
, with R   p j Pj

g i  g max 
j 1
 K  R  P0  R
Evans, 1988; Lancelot et al., 2000;
Leising et al., 2003
(7) Modified MM function:
g i  g max
pi Pi
n
1   p j Pj
Verity, 1991; Fasham et al. (1999);
Strom and Loukos, 1998; Tian et al.
(2001)
j 1
51
Table 7. Parameterization of grazing on multiple types of prey with active switching
selection (gmax: maximum grazing rate; K: Half-saturation constant; P0 threshold below
which grazing is zero; pi: preference coefficient; , a, : constant).
Function
References
(1) Switching MM predation:
g i  g max
pi Pi
2
n
n
j 1
j 1
K  p j Pj   p j Pj
2
Fasham et al., 1990; Strom and
Loukos, 1998; Pitchford and
Brindley, 1999; Spitz et al., 2001
(2) Mechanistic disc switching predation:
g i  g max
bi N i2
n
b j h j N 2j
j 1
1  c j N 2j
(1  ci N i )(1  
Chesson, 1983
)
(3) Generalized switching function:
g i  g max
 pi Pi m
n
m
  pi Pi 
Tansky, 1978; Matsuda et al., 1986
i 1
(4) Generalized switching function:
g i  g max
 pi Pi m
 n

   pi Pi 
 i 1

Vance, 1978
m
(5) Generalized switching MM function:
g i  g max
 pi Pi m
n
Gismervik and Andersen (1997)
1    pi Pi 
m
i 1
(6) Generalized switching MM function:
g i  g max
 pi ( Pi  P0i ) m
n
1    pi ( Pi  P0i ) 
This work
m
i 1
52
Table 8. Parameterization of zooplankton mortality (m: mortality; m0: minimum
mortality; K: half-saturation constant; P0: threshold below which zooplankton suffer from
starvation).
Function
(1) Linear
Z
 mZ
t
(2) Quadratic
Z
 mZ 2
t
(3) Hyperbolic
Z
Z2
m
t
K Z
(4) Sigmoidal:
Z
Z2
m 2
t
K Z2
References
Evans and Parslow, 1985; Tian et al., 2000
Steele and Henderson, 1981; Denman and Gargett,
1981; Fasham, 1995.
Frost, 1987; Ross et al., 1994; Tian et al., 2003
Malchow, 1994; Edwards and Yool, 2000
(5) Food-dependent rectilinear mortality:


mZ , for P  P0


Z 



t  

  m0  Z , for P  P0 
 P


(6) Food-dependent exponential moartality:
P

Z   Z
  me
 m0  Z
t 

(7) Temperature-dependent:
Z
 m0 eT Z 2
t
(8) Generalized:
Z
 mZ n
t
Andersen and Nival, 1988 (due to starvation)
Andersen et al., 1987
Kawamiya et al., 2000
Edwards and Yool, 2000
53
Table 9. Parameterization of zooplankton excretion/respiration (g(P): ingestion; W: body
dry weight; T: temperature; P0: threshold; Emax and Emin: maximum and minimum of
egestion rate; a, b,and c: constants).
Function
References
(1) Linear function
Z
  aZ
t
(2) Food ingestion dependent:
Z
  ag (P )
t
(3) Linear and ingestion dependent:
Z
  ag ( P )  bZ
t
(4) Food concentration dependent:
Z
 (aP  b) Z
t
(5) Temperature dependent:
Z
  ab T Z
t
(6) Temperature dependent (phyopl.):
Z
 a0 e bT Z
t
(7) Body weight dependent:
Z
 aW b
t
Fasham et al., 1990
(8) Egestion:
Wroblewski, 1977; Ohuchi et al., 1986
Walsh, 1975; Tian et al., 2001
Steele, 1974; Carlotti and Sciandra, 1989
Hofman and Ambler, 1988
Andersen et al., 1987
Kawamiya et al., 2000
Moloney and Field, 1989
Emax Emin e b ( P  P0 )
Z

Z
t Emax  Emin (e b ( P  P0 )  1)
54
Table A-1. A priori model equations without physical processes.
Phytoplankton
Pi
P nz
m
 (1  rPSi ) i Pi   PDi Pi pi  rPSi Pi  s Pi i   GPji
t
z j 1
where  i   i ( E ) i (T ) min(  i ( N (1,2)),  i ( N (3)), , ,  i ( N (nn)))
i ( E )  Pmi (1  e  E / Pm )e   E / Pm
i
i
i
i
 (T )  (1  e  a (T T )(1  e b(T
0L
 i ( N1,2) 
i ( N j ) 
NH 4
0H



NH 4  K NH 4i
Nj
T )
)
(A1)
K NH 4i
NO3



NH 4  K NH 4i NO3  K NO 3i
N j  K Nij
Zooplankton
Z j
t
np
nz
nd
nb
i 1
k 1
l 1
m 1
  eZPji G Pji   (eZZjk G Zjk  G Zkj )   eZDjl G Djl   eZBjm G Bjm
  ZDj Z
mZj
 rZj Z j  w j
where G Pji  g j (T )Z j
G Zjk  g j (T )Z j
G Djl  g j (T )Z j
Z j
z
 p Pji ( Pi  P0i )m j
1 Rj
p Zjk ( Z k  Z 0 k )
1 Rj
p Djl ( Dl  D0l )
(A2)
1 Rj
p Bjm ( Bm  B0 m )
G Bjm  g j (T )Z j
1 Rj
np
R j   ( p Pji ( Pi  P0i ))
mj
i 1
 100 E0 

wLj  wb 
 E0 t 

wFj  ard wmax j 1  e
nz
  ( p Zjk ( Z k  Z 0 k ))
k 1
mj
nd
  ( p Djl ( Dl  D0l ))
l 1
mj
nb
  ( p Bjm ( Bm  B0 m ))
mj
m 1
3
 kF R j

Bacteria
55
B j
t
ns
nh
i
k 1
  eBSjiU DOMji  eBNjU NH 4 j   GBkj  rBj B j
where U DOMji   Bj (T ) B j
p DOMji ( DOM i  DOM 0i )

S j  min( p NHj ( NH 4  NH 4
1  RBj  S j
; U NH 4 j   Bj (T ) B j
Sj
1  RBj  S j
;
(A3)
ns

0j
), j RDONj ) ; RBj   p DOMji ( DOM i  DOM 0 ji ) ;
i 1
ns
eBSji ( N : C ) Bj
i 1
eBNj ( N : C ) DOMi
RDONj   p DONji ( DON i  DON 0i ) ;  j 
1
DOM
np
nd
nb
DOM i
  ( rPSji  rPSji  j ) Pj   DSmi Dm  U DOMki
t
j 1
m 1
k 1
nz
nd
nb
 np




G


G


G


  ZPSkji Pkj  ZZSkli Zkl  ZHDSkmi Dkm  ZBSkni G Bkn 
k 1  j 1
l 1
m 1
n 1

  SSii1 DOM i   SSi1i DOM i 1
nz
(A4)
Detritus
nz
Di np
m
   PDji Pj j    ZDki Z kmk   DDi1 Di21   DDi Di2   DDi 1 Di 1   DD1 Di   DSi Di
t
j 1
k 1
 s Di
nz  np
nz
nd
nb

Di
     ZPDkji G Pkj    ZZDkli G Zkl   Z ZDDkmi G Dkm   Z ZBDkni G Bkn  G Dki 
z
k 1  j 1
l 1
m 1
n 1

(A5)
Nutrients
np
nz
nz
nd
nb


N i
   a Zji rZj Z j   rZPjhiG Pjh  rZZjki GZjk   rZDjliG Djl   rZBjmiG Bjm 
t
j 1 
h 1
k 1
l 1
m 1

(A6)
nb
 ns
 np
    a BSkli (1  e BSkl )U DOMkl  e BNkU NH 4 k  a Bki rBk Bk    a Pmi  m Pm
k 1  l
 m
Auxiliary state variables
Chlorophyll a
nz


i
P
Chl np 
m
  (1  rPSi ) max i
Pi   i  PDi Pi i  rPSi Pi  s Pi i   G Pji  


t
 i i E
z
i 1 
j 1


(A7)
Bioluminescence
nb
B j
Z
L nz
  ZLi i   BLj
t i 1
t
t
j 1
56
Table A-2. Model variables (listed first) and parameters (units are to be defined for
each application so that they can be different from that listed).
Symbol State variable
Value (unit)
Ai
Bi
Li
Chl
Di
DOMi
Ni
Pi
Zi
To define
mmol C m-3
W
mg Chl m-3
mmol C m-3
mmol C m-3
mmol Ni m-3
mmol C m-3
mmol C m-3
Auxiliary state variables
Bacteria
Bioluminescence
Chlorophyll a
Detritus
Dissolved organic carbon
Nutrients
Phytoplankton
Zooplankton
Symbol Functional variable
Value (unit)
aDi
aSi
Dimensionless
Dimensionless
Dimensionless
Dimensionless

rd
Rj
gj
GBlj
GDOMji
GDji
GPji
GZjj-1
E0
E(z)
Ntf
i
t
T
i
i(0)
i(Nj)
i(E)
i(T)
wF
wL
z
Ratio between nutrient Ni and the unit element in D
Ratio between nutrient Ni and the unit element in DOM
Random number equals to 1 or -1
Total food abundance for mesozooplankton Zj
Grazing rate of zooplankton Zj or Bj
Consumption of Bj by Z1
Consumption of DOMi by Bj
Grazing amount of Zj on Di
Grazing amount of Zj on Pi
Predation amount of Zj on Zj-1
PAR at the sea surface
PAR at depth z
Nitrification
Chl:carbon ratio of phytoplankton Pi
Time
Temperature
Growth rate of phytoplankton Pi
Growth rate of phytoplankton Pi at 0 C
Nutrient Nj limitation of the growth rate of Pi
PAR-dependent growth rate of Pi
Temperature forcing on the growth rate of Pi
Food-induced MeZ vertical migration
Light-induced MeZ vertical migration
Depth
mg C s-1
mg C s-1
mg C s-1
mg C s-1
mg C s-1
W m-2 (watt)
W m-2 (watt)
mmol N m-3 s-1
Dimentionless
s (hour)
°C
mg C mgChl-1 s-1
mg C mgChl-1 s-1
Dimensionless
mg C mgChl-1 s-1
Dimentionless
m s-1
m s-1
m
Symbol Parameter
Value (unit)
aPi
Dimensionless

Ratio between nutrient Ni and the unit element in P
57
aZi
aBi

aPDi
aZDj
DDj
AN
BLj
Ddj
DS11
rPSi1
SSjj+1
ZLj

eBSji
eZDjk
eZPji
eZZjj-1
gmaxj
Kij
nb
nd
nn
np
ns
nz
Pmi
pDjk
pSji
pPji
pZjj-1

maxi
Q10i
QDk
QSi
QPi
QZj-1
rBj
rPi
rZj
j
si
Topti
wbj
wmaxj
Ratio between nutrient Ni and the unit element in Z
Ratio between nutrient Ni and the unit element in B
Initial slope of photosynthesis-radiation function for Pi
Mortality and aggregation of phytoplankton Pi
Mortality of zoooplankton Zj
Aggregation coefficient of Dj
Nitrification rate
Bioluminescence of bacteria Bj
Disaggregation coefficient of Dj
DOM1 formation from dissolution of D1.
DOM1 exudation from phytoplankton Pi.
Aging of DOMj to DOMj+1
Bioluminescence of bacteria Zj
Photoinhibition coefficient of phytoplankton Pi
Gross growth efficiency of Bj on DOMi
Gross growth efficiency of Zj grazing on Dk
Gross growth efficiency of Zj grazing on Pi
Gross growth efficiency of Zj predation on Zj-1
Maximum grazing rate of zooplankton Zj or Bj
Half-saturation constant of Nj for Pi
Total number of bacterial state variables
Total number of detrital state variables
Total number of nutrient state variables
Total number of phytoplankton state variables
Total number of DOM state variables
Total number of zooplankton state variables
Theoretical maximum growth rate of Pi
Preference coefficient of Zj on Dk
Preference coefficient of Bj on DOMi
Preference coefficient of Zj on Pi
Preference coefficient of Zj on Zj-1
Exponential coefficient of NH4+ inhibition on NO3Maximum Chl:carbon ratio of phytoplankton Pi
Q10 of the growth rate of a biological pool
Threshold of Dk for grazing
Threshold of DOMi for bacterial consumption
Threshold of Pi for grazing and mortality
Threshold of Zj-1 for predation and mortality
Metabolism of bacteria Bj.
DOM exudation from phytoplankton Pi
Respiration of zoooplankton Zj
Exponential coefficient of Zj grazing
Sinking velocity of detritus Di
Optimal temperature for Pi (or Zi or Bi)
Slope between light change and Zj migration
Maximum speed Zj vertical migration
Dimensionless
Dimensionless
mgC mgchl-1 s-1 (Wm-2)-1
s-1
s-1
(mg C m-3)-1 s-1
s-1
W (mmol C)-1
s-1
s-1
s-1
s-1
W (mmol C)-1
mgC mgchl-1 s-1 (Wm-2)-1
Dimensionless
Dimensionless
Dimensionless
Dimensionless
s-1
mmol Nj m-3
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
mg C (mg Chl)-1 s-1
(mmol C m-3)-1
(mmol C m-3)-1
(mmol C m-3)-1
(mmol C m-3)-1
(mmol N m-3)-1
dimensionless
dimensionless
mmol C m-3
mmol C m-3
mmol C m-3
mmol C m-3
s-1
s-1
s-1
Dimensionless
m s-1
C
0.04 m h-1
20 m h-1
58
Fig. 1. Generalized biological model. N: Nutrients; P: Phytoplankton; Z: Zooplankton; D:
Biogenic detritus; DOM: Dissolved organic matter; B: Bacteria; A: Auxiliary state
variables; nn, np, nz, nd, ns and na are total numbers of state variables of the
corresponding functional groups.
59
Fig. 2: Example of processes interface: zooplankton grazing on phytoplankton (:
preference coefficient).
60