Parameterising a microplankton model
Paul Tett
Department of Biological Sciences, Napier University, Edinburgh EH10 5DT
Report
November 1998
ISBN 0 902703 60 9
[email protected]
with corrections, 02/00
Contents
Abstract ............................................................................................................... 2
1. Introduction ..................................................................................................... 3
2. The microplankton compartment .................................................................... 4
3. Microplankton equations of state .................................................................... 6
4. Autotroph (phytoplankton) equations ............................................................. 9
5. Comparisons with autotroph equations in ERSEM and FDM ........................ 13
6. Heterotroph equations ..................................................................................... 18
Simplifying the description of heterotrophic processes .......................... 18
Growth of the microheterotroph compartment under C or N control ..... 21
Respiration .............................................................................................. 23
Ingestion and clearance ........................................................................... 24
7. Comparisons with heterotroph equations in ERSEM and FDM ..................... 25
8. Microplankton rate equations.......................................................................... 31
9. Comparison of MP simulation with microcosm data...................................... 37
Model ...................................................................................................... 38
Phosphorus submodel and other non-standard parameter values ........... 40
Initialisation and external forcing ........................................................... 40
Numerical methods ................................................................................. 41
Results ..................................................................................................... 41
10. An unconstrained autotroph-heterotroph model (AH) .................................. 43
Nullcline analysis .................................................................................... 45
11. Discussion ..................................................................................................... 46
12. References ..................................................................................................... 49
Appendix...............................................................................................................54
Key Tables
Table 4.1. Autotroph equations. ............................................................. 9
Table 4.3. Autotroph parameters. ........................................................... 13
Table 6.1. Heterotroph equations. ........................................................... 23
Table 6.4. Heterotroph parameters. ........................................................ 26
Table 8.1. Microplankton equations. ...................................................... 32
Table 8.2. Microplankton rate equations and parameters. ..................... 37
Abstract
This report describes and assesses the parameterisation of MP, the microplankton
compartment of the carbon-nitrogen microplankton-detritus model of Tett (1990b). The
compartment is 'the microbial loop in a box' and includes pelagic bacteria and protozoa as
well as phytoplankton. The report presents equations and parameter values for the autotroph
and microheterotroph components of the microplankton. Equations and parameter values for
the microplankton as a whole are derived on the assumption of a constant 'heterotroph
fraction' η.
The autotroph equations of MP allow variation in the ratios of nutrient elements to
carbon, and are largely those of the 'cell-quota, threshold-limitation' algal growth model of
Droop (1968; 1983), which can deal with potential control of growth by several nutrients and
light. The heterotroph equations, in contrast, assume a constant elemental composition.
These autotroph and heterotroph equations are compared with those used for more detailed
parameterisation of the microbial loop in the European Regional Seas Ecosystem model
(Baretta, et al. 1995) and the model of Fasham et al. (1990).
Nitrogen is used as the limiting nutrient in most of the model description, and is
special in that MP links chlorophyll concentration to the autotroph nitrogen quota. However,
phosphorus dynamics were added to a version of MP used to simulate the results of a
microcosm experiment by Jones et al. (1978a). Using standard parameter values with a single
value of η taken from the observed range, the best simulation successfully captured the main
features of the time-courses of chlorophyll and particulate organic carbon, nitrogen and
phosphorus, with RMS error equivalent to 29% of particulate concentration. The standard
version of MP assumes complete internal cycling of nutrient elements; adding a term for
ammonium and phosphate excretion by microheterotrophs did not significantly improve
predictions.
Relaxing the requirement for constant η resulted in an autotroph-heterotroph model
AH, with dynamics resembling those of a Lotka-Volterra predator-prey system. AH fitted the
microcosm data worse than did MP, justifying the suppression of Lotka-Volterra dynamics in
MP. The report concludes with a discussion of possible reasons for the success of the simple
bulk dynamics of MP in simulating microplankton behaviour.
MP
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(a)
mesozooplankton
grazing pressure
MICROPLANKTON
C&N
defecation
DETRITUS
C&N
Dissolved
NH 4 +
implicit microheterotrophs
NO3 - & NH4 +
C respiration
Sinking
Sinking
photosynthesis
α . I mmol C (mg chl)-1 d-1
(b)
MICROPLANKTON - total biomass B mmol C m-3
χ: mg chl (mmol C)-1
growth rate µ, d-1
heterotroph fraction η = B h/(B a+B h)
mesozooplankton
biomass Ba
Phytoplankton
growth rate µa
grazing
biomass
Bh
Microheterotrophs
(Protozoa & Bacteria)
growth rate µ
DOM
respiration
Figure 1-1. The microplankton-detritus model.
(a) The complete microbiological model of Tett (1990b), with slowly-mineralising detritus compartment.
(b) The microplankton compartment (MP), showing the implicit 'microbial loop in a box'.
1. Introduction
Descriptions of biological-physical interactions in the sea must take account of physical
transports (which conserve the total quantity of transported variables) and of nonconservative biological or chemical processes (which convert one variable into another). An
equation which summarises both causes of variation, and which forms the basis of (Eulerian)
vertical-process models, is:
(1.1)
∂Y /∂t
=
-∂ϕY/∂z
flux divergence
where the vertical flux is:
ϕY
=
≈
+
βY
nonconservative processes
<(w + w ' + wY).(Y + Y')>
-Kz.∂ Y /∂z
+
(w + w Y).Y
eddy mixing water advection & particle sinking
The equation describes the rate of change at a given point in the sea of a generalised
biological variable Y which is a function of time t and height z above the sea bed. The
equation must be repeated, and numerically solved, for every state variable in a model. It is
desirable to minimise the list of such variables, not only in order to minimise computer
storage & calculation (especially in 2D or 3D models, or during the repeated simulations
necessary for parameter optimisation), but also because of Occam's Razor ('do not
unnecessarily multiply explanations') and on account of practical and theoretical difficulties
in estimating parameters (which, in general, increase with variable number).
Taking into account this need for parsimony in state variables, Tett (1990b) proposed a
microbiological model (Figure 1-1(a) † ) with three pelagic compartments and 6 independent
state variables:
• microplankton organic carbon and nitrogen (and non-independent chlorophyll);
• detrital organic carbon and nitrogen;
• dissolved ammonium and nitrate (and non-independent oxygen);
The microbiological (or microplankton-detritus) model was embedded in a 3-layer physical
framework, and the combination named L3VMP. The microplankton compartment was
defined as including planktonic microheterotrophs (protozoa and bacteria) as well as
photoautotrophic phytoplankton. Mesozooplankton were represented as a grazing pressure
rather than a dynamic compartment. The microplankton-detritus model has been used in a
number of studies (Huthnance, et al. 1993; Hydes, et al. 1997; Smith & Tett, in press; Tett &
Grenz 1994; Tett, et al. 1993; Tett & Smith 1997; Tett & Walne 1995; Wilson & Tett 1997)
†
Figure 1-1. The microplankton-detritus model.
MP
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Table 1.1. General forms of symbols common in this document
General meaning
Units (in MP, if variable)
α
photosynthetic 'efficiency'
mmol C (mg chl)-1 d-1 I -1
B
biomass (as carbon)
b
slope factor (e.g. r/µ) or inefficiency coefficient
β
(total) nonconservative flux for a substance
c
clearance rate or transfer coefficient
m3 (mmol C)-1 d-1
χ
chlorophyll:carbon ratio
mg chl (mmol C)-1
e
relative (organic) excretion rate
ε
photosynthetic pigment attenuation cross- section
f
fraction of (or efficiency with which) input diverted to named output
Φ
photosythetic quantum yield
G
grazing pressure
η
heterotroph fraction (of biomass)
I
PAR
i
relative ingestion rate
K
(half)-saturation constant, as in K S or KI
k
miscellaneous constants
m
relative mortality rate
d-1
µ
relative growth rate
d-1
N
nitrogen associated with biomass
mmol N m-3
P
phosphorus associated with biomass
mmol P m-3
p
preference (e.g. for one diet compared with other)
Q
nutrient quota or content in biomass
mmol nutrient (mmol C)-1
q
fixed quota
mmol nutrient (mmol C)-1
r
biomass-related mineralisation/respiration rate
S
dissolved nutrient concentration
Θ
temperature
u
biomass-related nutrient uptake rate
Symbol
mmol C m-3
mmol m-3 d-1
d-1
m2 (mg chl)-1
-
nmol C µE -1
d-1
µE m -2 s-1
d-1
S or I
various
-
[mmol (mmol C)-1] d-1
mmol m-3
°C
mmol (mmol C)-1 d-1
Table 1.1., continued.
Like most languages, the symbol set and associated conventions used in this report reflects
origins and history, including failed attempts to impose universal rules.
1. All greek and bold roman letters are symbols for variables or parameters: they may be
replaced by numbers. Subscripted or superscripted light-type letters are identifiers. Subscripts
or superscripts written in bold qualify the preceeding symbol. All other characters written in
light type and at the standard size are operators.
operators include:
exp(..) for exponentiate, and ln(..) for natural logarithm;
min{..,..} and max{..,..} for the smallest or largest value of a set;
ƒ(..) means 'function of', to be expanded.
2. Model state variable are identified by upper case roman letters, which may have
preceeding light-text superscripts or following light-text subscripts. As far as possible, rate
variables are given lower case roman letters, whereas symbols for parameters may have lower
case greek letters.
exceptions:
Φ for quantum yield (to allow ϕ to be used for transport fluxes);
G for grazing pressure (to avoid confusion with gravity);
K in e.g. KS for saturation constant (to allow k for minor constants)
µ for growth rate (following microbiological convention)
Θ for temperature (to avoid confusion with time);
Following microbiological convention, S is used to refer to the concentration of any
dissolved nutrient, and hence is, more often than not, accompanied by preceeding superscripts
specifying the nutrient.
3. The conventions of Tett & Droop (1988) concerning sub- and superscripts have to some
extent been adopted. The preceeding (light-text) superscript is always used for an identifier,
and the following (bold) superscript always used for the variable to which the main variable
has been normalised. Default identifiers or normalising variables are ommitted in the interests
of simplicity, or when they may be considered to cancel, or when there is a locally clear
context. The trailing subscript is used both for qualification (bold) and identification (light).
examples:
biomass;
C rB and NrN written as r; Nr is N excretion relative to C biomass;
Nq B written q ; Pq B written P q ; Xq N is chlorophyll relative to N in
Qmaxa (parameter) is upper limit to variable Qa, for autotrophs.
4. Identifiers include:
a : autotrophic;
h: (micro)heterotrophic;
N: (organic) nitrogen;
NO: (diss.) nitrate;
P: (organic) phosphorus;
Si: silica (any form).
5. Qualifiers include:
min(imum)
b: bacterial;
p: protozoan;
NH: (diss.) ammonium;
ON: (diss.) organic nitrogen;
PO: (diss.) phosphate;
value;
max(imum) value;
some state variables, as in KNHS ;
0 : basal of threshold value ( ≈ minimum).
biomass;
(also Hydes, et al. 1996; Soetaert, et al., in press) with some variations in the process
equations and parameter values.
This paper concerns the microplankton compartment (Figure 1-1(b)), which was proposed by
Tett (1987) as a relatively simple way of parameterising the most important (for water quality
models) of the processes in the 'microbial loop' (Azam, et al. 1983; Williams 1981). The first
aim of the paper is to describe this parameterisation in more detail than given by Tett (1990b)
or than was possible in subsequent papers, and to discuss more fully the basis and
consequences of the assumptions employed to obtain a simple description. Earlier versions of
the microplankton equations were given as part of the model L3VMP (Tett 1990b; Tett &
Grenz 1994; Tett & Walne 1995), and by Smith & Tett (in press) as part of the model
SEDBIOL. 'MP' will denote the equation set here proposed. It will be compared with the
relevant parts of two well-documented and widely-used models: the model (hereafter called
FDM) of (Fasham, et al. 1990) and the first version of ERSEM (the European Regional Seas
Ecosystem Model) (Baretta, et al. 1995). The objective of these comparisons is to
demonstrate how process descriptions differ and to bring out similarities in underlying
biological parameters. Table 1.1* lists common symbols used throughout this report for all
the models.
Earlier accounts of MP (Tett, et al. 1993; Tett & Walne 1995) tested the microplankton
equations as part of the physical-microbiological model L3VMP, and hence the second aim
of the present work was to test MP on its own, using data from a microcosm experiment
(Jones, et al. 1978a; Jones, et al. 1978b). MP assumes a constant relation between autotroph
and heterotroph processes in the microplankton compartment, and so the tests also involved
simulations with a model AH in which this assumption was relaxed, allowing
microautotrophs and microheterotrophs to vary independently.
2. The microplankton compartment
The microplankton compartment of MP is deemed to contain all pelagic micro-organisms less
than 200 µm, including heterotrophic bacteria and protozoa (zooflagellates, ciliates,
heterotrophic dinoflagellates, etc) as well as photo-autotrophic cyanobacteria and micro-algae
(diatoms, dinoflagellates, flagellates, etc). This definition (Tett 1987) employs an earlier use
(Dussart 1965) of the term 'microplankton' than that suggested by Sieburth (1979), who
contrasted microplankton to picoplankton and nanoplankton. Flows of energy and materials
through and between the organisms of the microplankton link them in the 'microbial loop'
(Azam, et al. 1983; Williams 1981). Microplankters reproduce mainly by binary division, in
contrast to mesozooplankton, such as copepods, which typically have relatively long periods
* Table 1.1. General forms of the symbols used in this document.
MP
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(a) ML
MICROPLANKTON
photosynthesis
ea
rp grazing–by–zooplankton
respiration
Phyto C
Phyto N
Gp
NGp
bGp
Bact C
uptake
Proto C
Proto N
GZ2
NG
DOC
BactN
ub
Nea
rb
DON
NbGp
Nub
NHub
NH–uptake
Nitrate
ZP excretion
Ammonium
eZ
NZ
NHr
(b) MP
MICROPLANKTON
respiration
grazing on C
microplankton C
mesozooplankton & detritus N
photosynthesis
microplankton N
uptake NO3
grazing on N
uptake NH4
nitrate
ammonium
ammonium oxidation
mineralisation
Figure 2-1. Model comparisons. Relevant compartments and flows in (a) ML of
Tett & Wilson (2000), and (b) MP, shown according to the conventions of the modelling
software STELLA. The large rectangle shows the limits of the MP microplankton
compartment or its analogue.
of individual growth after the laying of many eggs by those few females that survive to
maturity. Many of the microbial populations in the loop have turnover rates of order 10-1 d-1
during the productive season, and can be parameterised as a unit without unduly distorting
the response of the model on time-scales of a few days or longer.
Models of the microbial loop can be recognised by their explicit inclusion of compartments
for heterotrophic bacteria and their consumers. The loop as thus defined has been modelled
by several authors, including Baretta-Bekker et al. (1995) as part of ERSEM, Fasham et al.
(1990) in FDM, and Taylor and co-workers (Taylor, et al. 1993; Taylor & Joint 1990).
Wilson & Tett (ms) compare Microbial Loop (ML) and Microplankton (MP) models (Fig. 21†). Their version of MP has 2 independent state variables and 12 parameters in the
description of the microplankton compartment itself; the analogous compartments of ML
(phytoplankton, bacteria, protozoa and dissolved organic matter) have 6 independent state
variables and 27 parameters. These numbers demonstrate the relative simplicity of MP, which
is 'the microbial loop in a box'.
The microplankton compartment of MP contains several trophic levels, and thus would seem
to confuse the distinction between autotrophs and heterotrophs. However, the distinction is
not clear-cut in microplanktonic organisms. Not only do phytoplankton respire, but some
protozoans retain ingested chloroplasts and some micro-algae can grow heterotrophically or
mixotrophically. The microplankton may thus be seen from a functional viewpoint as a
suspension of chloroplasts (and cyanobacteria) and mitochondria (and heterotrophic bacteria)
with associated organic carbon and nitrogen. Tett (1987) proposed a model in which bulk
"photoautotrophic processes are made simple functions of chlorophyll concentration,
representing algal biomass ...; heterotrophic processes are made simple functions of ATP
concentration, representing total microplankton biomass ... and including the algal
component." The model was used to estimate carbon fluxes in an enclosed coastal
microplankton (Tett, et al. 1988), and the concept of the microplankton in MP developed
from that work. The microplankton model distinguishes autotrophic from heterotrophic
processes rather than autotrophic from heterotrophic organisms.
Originally (Tett 1990b), the single microplankton compartment of MP was seen as
predominantly algal in character, with a heterotroph 'contamination' merely exaggerating the
effect of the heterotrophic processes of the algae themselves - for example, implicitly
increasing the microplankton's respiration rate. In the treatment that follows, however, the
effects of an autotroph-heterotroph mixture are taken explicitly into account. This involves
the parameter η, the ratio of microheterotroph to microplankton biomass:
† Figure 2-1. Model comparisons.
MP
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(a) ERSEM
DISSOLVED NUTRIENTS
MICROPLANKTON
DIATOM C
BACTERIA C
diss ex
rD
ZGd
photosynthesis
excretion
rb
Gnf
ZGf
FLAGELLATE C
ZOOPLANKTON C
rnf
pf
respiration
NANNOFLAGELLATE C
mzpGf
e&mmzp
Zgmzp
MICROZOPLANKTON C
mzpGnf
rmzp
DETRITUS C
mortality
particulate excretion
(b) FDM
MICROPLANKTON
DON
uptake DON
BACTERIA N
messy feeding
grazing on bacteria
excretion & uptake
PHYTOPLANKTON N
leakage
grazing on N
MESOZOOPLANKTON N
uptake NO3
uptake NH4
mortality
NITRATE
AMMONIUM
DETRITUS N
decay
excretion
defecation & feeding
Figure 2-2. Model comparisons, continued. Relevant parts of (a) ERSEM and (b)
FDM.
(2.1)
η
=
Bh/(Ba + Bh)
where subscripts a and h indicate autotrophs (phytoplankton) and (pelagic micro-)
heterotrophs (protozoa and heterotrophic bacteria). In the equations hereunder, terms without
subscripts refer to the microplankton as a whole. It is a crucial assumption of this version of
MP that the value of the heterotroph fraction does not change during a simulation. Variation
in η has been addressed by models (Tett & Smith, 1997) using two MP compartments that
vary in their relative contribution to total biomass and which differ in η. Tett & Wilson
(2000) and Wilson & Tett (1997) examine the effect of treating η as a forcing variable. The
alternative model AH, considered in Section 10, allows η to vary dynamically.
It is further assumed that organic matter excreted by phytoplankton or leaked during 'messy
feeding' by protozoans is re-assimilated by microplankton bacteria so rapidly that the
turnover times of pools of labile dissolved organic matter are less than a day. Thus the
existence of such pools can be ignored. It is, similarly, assumed that inorganic nutrients
(ammonium, phosphate) excreted by microheterotrophs are rapidly re-assimilated by the
autotrophs and thus retained within the microplankton. This assumption is further examined
in Sections 8 and 9.
The treatment of these matters by ERSEM and FDM can be seen in Figure 2-2† and is
discussed in Sections 5 and 7.
3. Microplankton equations of state
The microplankton compartment has two independent state variables, organic carbon B:
(3.1)
∂B/∂t
=
-∂ϕB /∂z
+
βB
mmol C m-3 d-1
=
-∂ϕN/∂z
+
βN
mmol N m-3 d-1
and organic nitrogen N:
(3.2)
∂N /∂t
Two other variables are linked to the above; the nitrogen quota:
(3.3)
Q
=
mmol N (mmol C)-1
N /B
and the chlorophyll concentration:
(3.4)
X
=
χ.B
mg chl m-3
where χ is the variable ratio of chlorophyll to microplankton carbon, with a value that
depends on Q .
† Figure 2-2. Model comparisons, continued.
MP
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Microplankton carbon is the sum of autotroph and heterotroph contributions:
(3.5)
B
=
Ba + Bh
=
B.(1-η) + B.η
mmol C m-3
In the case of microplankton nitrogen,
(3.6)
N = N a + Nh = Qa.Ba + qh.Bh = ( Qa.(1-η) + q h.η).B
mmol N m -3
where Q a is the variable autotroph nutrient quota and qh is a constant heterotroph nutrient
quota (mmol N (mmol C)-1).
The flux divergences in Equations (3.1) and (3.2) are not of concern here. The
nonconservative term for microplankton carbon rate of change in Eqn. (3.1) can be
expanded:
(3.7)
β B = βBa + βBh
=
(µa - c h.Bh - G).Ba + (ch.Ba - rh- G).Bh mmol C m-3 d-1
where:
µa is the relative growth rate (d -1) of the autotrophs;
G is mesozooplankton grazing pressure (the instantaneous probability per unit time
that a microplankter will be consumed by a copepod or similar animal); it is assumed that this
pressure applies equally to heterotrophs and autotrophs;
ch. Bh is the 'transfer pressure' (d -1) on autotrophs due to microplankton heterotrophs;
it is analogous to the mesozooplankton grazing pressure;
rh is the relative respiration rate (d-1) of microplankton heterotrophs.
If the heterotrophic component were solely herbivorous protozoa, then ch might be seen as a
clearance rate (volume of water made free of autotrophs by the grazers, per unit grazer
biomass and time). However, there are other ways in which organic matter is transferred from
producers to consumers, including photosynthetic and grazing-induced leakage of Dissolved
Organic Matter (DOM) which is assimilated by bacteria, in turn grazed by protozoa (Fig.
6-1). ch.Bh.Ba (mmol C m-3 d-1) is thus best understood as the total organic carbon flux from
autotrophs to microplankton heterotrophs. Details of the routes are of no importance so long
as the transfer term appears identically (except for opposite sign) in βBa and βB h, and so
cancels. It is thus assumed that all DOM produced by leakage is labile and rapidly
assimilated by other microplankters. It is also assumed by MP that protozoans do not
defecate, and hence the only intrinsic loss of carbon experienced by the microheterotrophs is
through their respiration:
(3.8)
µh
=
ch.Ba - rh
d-1
It may be noted that microheterotroph growth rate µh must be the same as microplankton
growth rate µ if the heterotrophs are to remain a constant fraction of the microplankton.
However, µh need not appear explicitly in the microplankton equation of state, because the
transfer term ch.Bh.Ba cancels in (3.7), which can then be simplified:
MP
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Feb 00
(3.9)
βB
=
mmol C m-3 d-1
(µ - G).B
where microplankton growth rate is:
(3.9.a)
µ
=
µa.(1-η)
rh.η
d-1
As with carbon, so with microplankton nitrogen. The non-conservative term for rate of
change in Eqn. (3.2) can be expanded:
(3.10)
βN
=
β Na + βNh
=
(u a - (ch.Bh + G).Qa).Ba + (ch.Qa.Ba - Nrh - G .qh). Bh
mmol N m-3 d-1
where:
u a is autotroph biomass-related nutrient uptake rate, in the case of nitrogen the sum of
ammonium and nitrate uptakes (mmol N (mmol C)-1 d-1);
Nrh is the heterotroph ammonium excretion rate (mmol N (mmol C) -1 d-1);
As in the case of carbon, the transfer of nitrogen from autotrophs to heterotrophs is not a loss
when included within the microplankton compartment, and (3.10) simplifies to:
=
(3.11) β N
where:
(3.11.a)
(3.11.b)
u .B - G.N
=
Q
u
Qa.(1-η) + q h.η
u a.(1-η) - Nrh.η
=
=
(u - G.Q). B
mmol N m-3 d-1
mmol N (mmol C)-1
mmol N (mmol C)-1 d-1
Sections 4 and 6 expand the autotroph and heterotroph rate terms in these equations.
MP
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Feb 00
Table 4.1: Autotroph rate equations used in MP and AH
growth
d-1
µa = min{µa(I ), µa(Qa)}
where:
µa(Qa) = µmax a(1 - (Qmina /Q a)): Q a≥Qmina
µa(I ) = α.χa.I - ra
where
α = k.ε.Φ
χ a = Xq Na.Qa
mmol C (mg chl)-1 d-1
(µE m-2 s-1)-1
mg chl (mmol C)-1
respiration
ra =
nutrient
uptake
u a = umaxa.ƒ(S ).ƒin(Qa)[.ƒin(NHS)]
where:
ƒ(S ) = (S/(k S + S))
:S ≥0
NH
NH
ƒin( S) = (1/(1 + ( S/k in))) : NHS ≥ 0
: Q a ≤ Qmaxa
ƒin(Qa) = (1 - (Qa/Qmaxa))
mmol N (mmol C)-1 d-1
µmax a = µmaxa[20°C].ƒ(θ)
d-1
temperature
r0 a + ba.µa
r0 a
:
:
µa > 0
µa ≤ 0
u max a = umaxa[20°C].ƒ(θ)
where:
ƒ(θ) =
θ
=
exp(k θ.(θ- 20°C))
temperature
d-1
[inhibits NO3- uptake]
d-1
°C
4. Autotroph (phytoplankton) equations
The autotroph equations used by MP are summarised in Table 4.1*. They are largely derived
from the 'Cell-Quota, Threshold-Limitation' (CQTL) model of Droop (Droop 1968), and
others (Caperon 1968; Fuhs 1969; Paasche 1973; Rhee 1973), reviewed by Droop (1983) and
Rhee (1980). Tett & Droop (1988) review estimates of CQTL parameter values. At the heart
of the model is the concept of a variable cell nutrient quota (Eppley & Strickland 1968),
changing as a result of nutrient uptake and of the dilution of nutrient by biomass during
growth:
(4.1)
dQa/dt
=
u a - µa.Qa
mmol nutrient (mmol C)-1 d-1
Such variability allows 'luxury uptake' of nutrient in excess of immediate need for growth
(Kuenzler & Ketchum 1962; Mackereth 1953), the stored nutrient being available for later
use. Despite the name 'cell quota', the variable is normally measured as the ratio of population
nutrient to biomass (Droop 1979), although the biomass has sometimes been expressed in
terms of numbers of cells. Droop (1968) showed that vitamin B12-limited growth of the
prymnesiophyte Pavlova (=Monochrysis) lutheri was a function of the cell quota of the
vitamin, and Caperon (1968) showed a similar dependency of N-limited growth on the N:C
ratio of the prymnesiophyte Isochrysis galbana. The simplest form (Droop 1968) of the CellQuota function for growth is
(4.2)
µa = µmaxa(1 - (Qmina /Q a))
: Q a ≥ Qmina
where Q mina is the minimum cell quota that supports life (but not growth), and µmax a gives
the notional growth rate at infinite (and therefore, unrealisable) cell quota. Caperon's version
has an extra parameter:
µa = µmaxa((Qa - Qmina )/(k Q a + Qa - Qmina )) : Q a ≥ Qmina
but, as argued by Droop (1983), this is the same as Eqn. (4.2) when k Q a = Qmina , and so
there is little case for introducing the third parameter when observations cannot well
distinguish the value of the internal half-saturation constant kQ a from that of the threshold
quota Qmina . For consistency in notation within MP, I use Q mina for Droop's symbol kQ.
For simplicity, Droop's symbol µ' m for maximum growth rate (at infinite Q) is written as
µmax , but this should not be confused with the use of µmax in the Monod (or similar) growth
equation, where the maximum rate is that at infinite external nutrient concentration.
Droop (1974; 1975) and Rhee (1974) showed that there was a threshold, rather than
multiplicative, relationship between two potentially limiting nutrients, and Droop et al.
(1982) extended threshold-limitation theory to include irradiance I :
* Table 4.1. Autotroph equations for MP.
MP
page 9
Feb 00
Table 4.2. Estimates of respiration parameters
species
Dunaliella tertiolecta
Dunaliella tertiolecta
ba
r0 a, d-1
0.74
0.028
0.24
Gonyaulax polyedra
0.41
Leptocylindrus danicus
0.03
Pavlova lutheri
Pavlova lutheri
Pavlova lutheri
Skeletonema costatum
0.18
Thalassiosira alleni
0.20
0.48
0.082
0.15
0.028
0.04
0.037
source
(Laws and Wong 1978)
(Richardson, Beardall &
Raven 1983)
(Richardson, Beardall &
Raven 1983)
(Richardson, Beardall &
Raven 1983)
(Laws and Caperon 1976)
(Droop et al. 1982)
(Laws and Wong 1978)
(Richardson, Beardall &
Raven 1983)
(Laws and Wong 1978)
(4.3)
µa
=
min{µa(I ), µa(1Qa), µa(2Qa), ....}
d-1
This states that, at a given instant, phytoplankton growth rate depends solely on the factor
that predicts the least growth rate. Standard MP uses only the first two terms: µa(I ) and
µa(Qa) for nitrogen.
The irradiance function includes photosynthesis and respiration:
µa(I ) =
d-1
p (I) - r a
The function p(I ) could be any of the photosynthesis-irradiance equations reviewed by
Jassby & Platt (1976) and Lederman & Tett (1981), but the use of a linear equation simplifies
integration over an optically thick layer (Tett 1990a) and is a good approximation under most
light-limiting conditions (Droop, et al. 1982). Thus MP's light-controlled growth equation is:
(4.4)
µa(I ) =
α.χa.I - ra
=
d-1
k .ε.Φ.χa.I - ra
where photosynthetic 'efficiency' α (a parameter which does not need subscripting when
defined in relation to chlorophyll) is made up from a phytoplankton attenuation cross-section
ε (m 2 (mg chl)-1) and a photosynthetic quantum yield Φ (nmol C µE-1); the constant k serves
to convert units. Droop et al. (1982) showed that photosynthetic efficiency was high in lightlimited Pavlova, and lower when the algae were nutrient-controlled. MP treats the efficiency
parameters ε and Φ as constants (in a given simulation), on the grounds that they are not used
to calculate nutrient-controlled growth rate. However, the algal-biomass-related
photosynthetic efficiency αa (d-1 I -1) decreases with increasing nutrient limitation because of
the relationship, discussed below, between chlorophyll content and the cell quota.
As discussed by Tett & Droop (1988) and Tett (1990a) there is evidence that micro-algal
respiration depends more on growth rate than on temperature (some temperature effect
being expected because µ(Q ) depends in part on µmax.ƒ(Θ) ). In MP, respiration is made up
of a basal and a growth-rate-related component:
(4.5)
ra
=
r0 a
r0 a
+
b a.µa
:
:
µa > 0
µa ≤ 0
d-1
Values of the parameters r0 a and b a were based on measurements made using algal cultures.
However, in the case of r0 a in particular, the literature gives a wide range of estimates (Table
4.2*). The standard value of 0.05 d-1 for basal respiration r0 a was chosen because higher
values would make it difficult for simulated algae to survive under winter conditions. A
respiration slope ba of 0.5 is supposed to take account of the effects of epiphytic bacteria, as
discussed in section 6.
* Table 4.2. Micro-algal respiration.
MP
page 10
Feb 00
MP defines nutrient uptake in relation to carbon biomass. Most studies concerning
nutrient-limited growth have found that the uptake of a limiting nutrient at extracellular
concentration S (mmol m-3) can be described by the Michaelis-Menten equation:
(4.6)
ua
=
u maxa.(S /(KS + S))
mmol nutrient (mmol C)-1 d-1
The parameter umax a specifies the uptake rate at infinite concentration of the external
nutrient; the half-saturation parameter K S gives the nutrient concentration at which uptake is
half the maximum rate. Some authors (e.g. Droop 1974; Paasche 1973) added a parameter
S 0, the threshold for uptake:
ua
=
u maxa.((S - S0)/(K S + (S - S 0)))
: S ≥ S0
Droop (1974; 1975) distinguished between the uptake of the nutrient currently controlling
growth and the uptake of another nutrient, and Droop et al. (1982) proposed an equation for
the uptake of a nutrient when another nutrient, or light, was controlling growth:
=
u maxa.(S /(KS + S)).ρ
u na
where the 'coefficient of luxury':
ρ = (R m/(Λ.(Rm - 1) + 1))
and, for light in control of growth: Λ = (S /KS).(KI /I) : (I/K I ) ≤ (S/KS)
whereas, for another nutrient (2) in control of growth (Droop 1974):
Λ
=
(1S /1Qmin).( 2Qmin/2S 2)
: (2S/2Qmin) ≤ (1S /1Qmin)
KI is a saturation constant for (absorbed) irradiance and R m is the maximum value of ρ.
The autotroph nutrient uptake equation used in MP is simpler than this. It does not distinguish
controlling from non-controlling nutrient. Instead, it regulates uptake so as to prevent the cell
quota from exceeding a realistic upper limit, Qmaxa:
(4.7)
ua
=
u maxa.ƒ(S ).ƒ(Qa)
mmol nutrient (mmol C)-1 d-1
where:
ƒ(S ) = (S/(K S + S))
ƒ(Qa) = (1 - (Qa/Q maxa))
: S ≥0
: Q a ≤ Qmaxa
This equation preserves the essential features (luxury uptake, partial suppression of the
uptake of a currently non-controlling nutrient) of CQTL theory without using the maximum
luxury coefficient Rm , which is difficult to measure. Q maxa is the greatest amount of
nutrient that a phytoplankton cell can store, and may be higher than the value (as given in
Tett & Droop 1988) for a limiting nutrient required by the theory of Droop et al. (1982) or the
maximum observed in a chemostat. The standard value used in MP is a typical maximum
N:C ratio observed in axenic algal batch cultures.
CQTL theory has been shown to apply to many types of micro-algae, and cyanobacteria,
limited by the nutrient elements nitrogen, phosphorus, silicon and iron and the vitamin, B12
(Droop 1983). It is thus general purpose. Nevertheless, each nutrient has special features that
MP
page 11
Feb 00
frequency of reported values
40
30
20
10
0
0-1 1-2 2-3 3-4 4-5 5-6 6-7 - 7-17
mg chl/mmol N
Figure 4-1. Chorophyll:nitrogen yield. Histogram of the frequency of values of
the ratio of chlorophyll (g) to nitrogen (moles) in cultured marine algae growing at
irradiances less than 300 µE m-2 s-1. Literature data for Chaetoceros gracilis,
Dunaliella tertiolecta, Gymnodinium sanguineum, Pavlova lutheri, Skeletonema
costatum, and Thalassiosira pseudonana, mostly found by Vivien Edwards.
may need to be taken into account. In the case of nitrogen, considered here as the most likely
limiting nutrient element in the sea, the special feature is the distinction between oxidised and
reduced forms. This distinction between nitrate (plus nitrite) and ammonium is important in
relation to water quality and the estimation of new, as opposed to recycled, production.
Uptake of nitrate must be followed by its reduction, and so uses more energy and reducing
power than does assimilation of ammonium. MP uses a relatively simple inhibition term
(Harrison, et al. 1987) to describe suppression of nitrate uptake by ammonium:
(4.8)
ƒin(NHS)
=
(1/(1 + (NHS/Kin )))
: NHS ≥ 0
This term is applied to Eqn. (4.6), and may be contrasted with the treatment of the process in
more detailed models (e.g. Flynn, et al. 1997; Flynn & Fasham, 1997).
The next part of the parameterisation involves the ratio of chlorophyll to autotroph organic
carbon, χa. The ratio is known to vary (from 0.04 to 1 mg chl (mmol C)-1) as a function of
temperature, irradiance and nutrient status (Baumert 1996; Cloern, et al. 1995; Geider, et al.
1997; Laws & Bannister 1980; Sakshaug, et al. 1989). However, such variability was not
dealt with explicitly by Droop's CQTL theory, perhaps because the chlorophyll content of
cells of Pavlova lutheri appears to be rather invariant. Droop et al. (1982) showed that
photosynthetic energy yield decreased as P. lutheri became increasingly nutrient-limited.
Although Baumert (1996) suggests that microalgae have several strategies for adapting
photosynthesis to changes in relative supplies of photons and nutrients, this version of MP
deals implicitly with the light-nutrient interaction by means of a simple equation for the
phytoplankton ratio of chlorophyll to carbon:
(4.9)
χa
=
Xq Na.Qa
mg chl (mmol C)-1
In earlier versions of MP (Smith & Tett, in press; Tett & Walne 1995) we allowed the
microplankton chlorophyll:nitrogen ratio Xq N, and by implication Xq Na , to vary between 2
and 1 mg chl (mmol N)-1 as the microplankton nutrient quota varied from maximum to
minimum. A similar approach was taken by Doney et al., (1996), who had 1 mg chl (mmol
N)-1 at saturating irradiance increasing to 2.5 mg chl (mmol N)-1 at zero light. In this version
of MP, however, I treat Xq Na as a constant. Its value was estimated from data for algal
cultures (Figure 4-1† ). The pigment data (Caperon & Meyer 1972; Levasseur, et al. 1993;
Sakshaug, et al. 1989; Sosik & Mitchell 1991; Sosik & Mitchell 1994; Tett, et al. 1985; Zehr,
et al. 1988) were obtained by 'standard' spectrophotometric or fluorometric methods, and
thus overestimate chlorophyll a determined by precise chromatographic methods (Gowen, et
al. 1983; Mantoura, et al. 1997). Nevertheless, they are appropriate for a model intended for
comparison with observations made by the same 'standard' field methods. The culture data
show a wide range of values of the ratio of chlorophyll to nitrogen without any clear overall
† Figure 4-1. Chorophyll:nitrogen yield.
MP
page 12
Feb 00
Table 4.3: Autotroph parameters used in MP and AH with nitrogen as
potentially limiting nutrient
ref
value
range
units
maximum relative rate of:
nitrate uptake
ammonium uptake
1
at 20°C
0.5
1.5
0.2-0.8
?
mmol N
(mmol C)-1 d-1
KS
half-saturation concentration for
uptake of:
nitrate
ammonium
1
0.32
0.24
0.2-5
0.1-0.5
mmol N m-3
Kin
ammonium concentration giving
half-inhibition of nitrate uptake
2
0.5
0.5-7
mmol N m-3
Qmaxa
maximum cell nitrogen content
1, 3
0.20
0.150.25-
mmol N
(mmol C)-1
Qmina
minimum cell nitrogen content
1
0.05
0.02-0.07 mmol N
(mmol C)-1
µmaxa
maximum (nutrient-controlled)
growth rate
1
2.0 at
20°C
1.2-2.9
ε
PAR adsorption cross-section
4
0.02
0.01-0.04 m2 (mg chl)-1
Φ
photosynthetic quantum yield
5
40
40-60
k
converts ε.Φ to typical units of α
0.0864
s d-1
nmol mmol-1
α
photosynthetic efficiency,
(derived from k.ε.Φ)
0.069
mmol C
(mg chl)-1 d-1
(µE m-2 s-1)-1
u maxa
d-1
nmol C µE-1
r0 a
basal respiration rate
6
0.05
0.03-0.41 d-1
ba
rate of increase of respiration
with growth rate (∆ra/∆µa)
6
0.5
0.2- 0.7
-
ratio of chlorophyll to nitrogen
7
2.2
0.5 - 7
mg chl
(mmol N)-1
Xq Na
kθ
temperature coefficient
0.069
°C-1
Sources:
(1) Tett & Droop (1988) with standard values largely after Caperon and Meyer (Caperon &
Meyer 1972a,b) for diatoms and prymnesiophytes.
(2) Standard value from Harrison et al., (1987); higher values from Maestrini et al, (1986)
(3) Maximum value is that observed in a batch cultures of Nannochloropsis atomus by
Setiapermana (1990) .
(4) Standard value for moderately clear coastal water; range related to water type (lowest in
turbid coastal) (Tett 1990).
(5) Tett (1990) and Tett et al. (1993).
(6) See text and Table 4.1.
(7) See text and Fig. 4.1.
pattern in relation to cell size, growth rate or irradiance (below 300 µE m-2 s-1). Ignoring a
few values of more than 7 mg chl (mmol N) -1, the median was 2.2 mg (mmol N)-1, and this
was taken as an initial value for Xq Na. It may be compared with maximum values of 3.6 to
4.8 in a model which allowed the ratio of chlorophyll to nitrogen to vary dynamically
(Geider, et al. 1998).
CQTL theory does not include the effects of temperature (see Tett & Droop 1988), and it is
therefore assumed (as is almost universal, but not self-evident) that the temperature effect is
multiplicative. Thus the maximum rate parameters µmaxa and u maxa were given a Q10 of
2, a little higher than the value of 1.88 given by Eppley (1972). The temperature function in
MP is:
ƒ(Θ)
(4.10)
=
exp(k Θ.(Θ - 20°C))
which can also be written as Q10((Θ - 20°C)/10°C), so that kΘ is (ln(Q10))/10°C. Eqn. (4.10)
is thus essentially the same as Eppley's function. When the difference between the actual and
reference temperature is a small fraction of the Kelvin temperature, then (4.10) is also a good
approximation to the equation of Arrhenius:
ƒΑ(Θ) = exp(k A.(Θref -1 - Θ-1)) ≈ exp(k A.(Θ - Θref )/Θref 2)
where the temperature is given in degrees Kelvin and Θref is the reference temperature. kΘ
in Eqn (4.10) is equivalent to k A/Θref 2 in the Arrhenius equation (and Θref 2 ≈ Θ .Θref ).
Table 4.3* lists autotroph parameter values used in MP and AH.
5. Comparisons with autotroph equations in ERSEM and FDM
FDM (Fasham, et al. 1990) uses a nitrogen currency. There is a single phytoplankton
compartment, which is parameterised for a surface mixed layer. The nonconservative part of
the equation for phytoplankton nitrogen N a is (in my symbols):
(5.1.)
β Na
=
(µa - m a - G).N a
mmol N m-3 d-1
where the parentheticised right-hand terms are phytoplankton relative growth rate, relative
mortality rate, and the grazing pressure to due zooplankton (including microzooplankton).
Mortality results in a direct conversion of phytoplankton to detritus.
Growth rate is
(5.2)
µa
=
(1 - f ea ).µmax a.ƒ(I ).ƒ(NHS, NOS)
d-1
* Table 4.3. Autotroph parameter values.
MP
page 13
Feb 00
Table 5.1. Standard values for phytoplankton parameter in FDM
local symbol Fasham et al. description
symbol
f ea
g1
KI
std value units
phytoplankton DON excretion
fraction (of production)
0.05
-
= µmaxa/αa 'saturation irradiance'
116
W m-2
Kin
ψ
ammonium inhibition (of nitrate
uptake) parameter
1.5
mmol
NH4+-N m-3
KNHS
K1
half-saturation concentration for
ammonium uptake
0.5
mmol N m-3
KNOS
K1
half-saturation concentration for
nitrate uptake
0.5
mmol N m-3
ma
µ1
specific mortality rate
0.05
d-1
phytoplankton N:C, inverse
Redfield ratio
1/6.63
mmol N
(mmol C)-1
qa
= χa/qa phytoplankton chlorophyll 1.59
yield from N
mg chl
(mmol N)-1
P-I curve initial slope, maximum
'photosynthetic efficiency'
0.025
d-1
(W m-2)-1
chlorophyll:carbon ratio
12/50
mg chl
(mmol C)-1
(coefficient of) PAR attenuation by
phytoplankton
0.03
ε
==> chl-related version (kC/Xq N)
0.02
m2
(mmol N)-1
m2 (mg
chl)-1
k
coverts between units of α and Φ
4.15 ×
0.0864
µE J-1 s d-1
nmol mmol-1
Φ
= αa/(k.ε.χa), photosynthetic
quantum yield
18
nmol C
µE-1
maximum growth rate
2.9*
d-1
Xq N
αa
α
χa
kC
µmaxa
Vp
* for Bermuda station S; but in general based on Eppley (1972) according to Fasham et al.
(1993), written here as: µmaxa[Θ°C] = µmaxa[0°C].exp((0.063°C-1).Θ °C), equivalent to
Q10=1.89; µmaxa[0°C] = 0.6 d-1.
In contrast to the CQTL growth parameterisation in MP, that of FDM (in 5.2) uses
multiplicative growth kinetics with Monod-type dependency on external nutrient
concentration. Eqn. (4.1) shows that Cell Quota and Monod parameterisations can be
equated when dQ /dt = 0, i.e. assuming steady state conditions (see Droop 1983). Then,
µa = ua/Q a , with the Monod parameterisation assuming that 'yield' Q a-1 is constant. When
biomass is measured in units of the limiting nutrient, and there are no growth-related losses of
nutrient after uptake (that is, growth efficiency is 100%, which is plausible for nitrogen), then
the quota is 1, and growth and uptake may be equated, as in e.g. Dugdale (1967).
In Eqn. (5.2) f ea is the fraction of phytoplankton production that is excreted as dissolved
organic matter. The irradiance function is that of Smith (1936) and Talling (1957):
ƒ(I )
=
I /√(KI 2 + I2)
in which the 'saturation irradiance' is in FDM written as a function of maximum growth rate
(d-1) and biomass-related photosynthetic efficiency αa (d-1 I -1):
KI
= µmaxa/αa
The nutrient function includes inhibition of nitrate uptake by ammonium:
ƒ(NHS, NOS) = (NHS/(KNHS + NHS)) + (NOS.e-Kin.NHS/(KNOS + NOS))
Nitrogen is linked to carbon and chlorophyll by fixed ratios, so that
X = Xq N.N a
where:
Xq N
= χa/qa
and q a-1 is the Redfield N:C ratio.
Standard parameter values are summarised in Table 5.1 * . The most significant difference
from MP is in the low value of αa of 0.025 d-1 (W m-2)-1 used as standard in FDM and
implying a photosynthetic quantum yield of 18 nmol C per µEinstein absorbed. The FDM
value was based on measurements in the Sargasso Sea and may indicate nutrient-limited
conditions here, although Fasham et al. (1993) concluded that the value "could be considered
reasonably representative of a large part of the North Atlantic Ocean". In MP, such nutrientlimitation is expressed through a low chl:C ratio: so, for Qa = Qmina, χa = 0.1 mg chl
(mmol C)-1, giving α.χa (equivalent to FDM's αa) of 0.029 d-1 (W m-2)-1. Finally, the
absence of a respiration term in Eqn. (5.2) is what would be expected of a model using a
nitrogen currency, as micro-algae do not normally mineralise organic nitrogen.
ERSEM (Varela, et al. 1995) contains two autotroph compartments, representing the
functional groups, diatoms (> 20 µm cell size) and autotrophic flagellates (< 20 µm cell size).
Biomass is quantified as carbon, although the model also deals with the cycling of nitrogen,
* Table 5.1. FDM phytoplankton parameters.
MP
page 14
Feb 00
Table 5.2. Standard values for phytoplankton growth parameter in
ERSEM
local
symbol
µmax a
[<Θ>]
Varela et al.
name; x is
replaced by
1:diatoms
2:flagellates
description
value
for
for
flagdiatoms ellates
units
sumPxc$
maximum growth rate (at mean
temperature)
2.5
2.0
d-1
=ln(Q10)/10°C
0.347
0.347
°C-1
kΘ
Q10
q10Px$
increase in rate for 10°C increase 4.0
in temperature (Q10)
4.0
-
KS
chPxn$
half-saturation conc. for diss.
nitrogen control of growth
0.10
0.05
mmol N m-3
KPOS
chPxp$
half-saturation conc.for
phosphate control of growth
0.10
0.05
mmol P m-3
KSiS
chPxs$
half-saturation conc. for silicate
control of growth
0.30
-
mmol Si m-3
KI min
clPIi$
minimum optimal PAR,
equivalent to 'saturation
irradiance'
40
40
W m-2
0.050
d-1 (W m-2)-1
mmol C (mg
chl)-1 d-1
(W m-2)-1
mmol C (mg
chl)-1 d-1 (µE
m-2 s-1)-1
αa
P-I curve initial slope, maximum 0.063
'photosynthetic efficiency'
=µ max a[<Θ>]/KI min
α
=µ max a[<Θ>]/(χa.KI min)
0.26
0.104
(converted at 4.15 µE J -1)
0.062
0.025
= χa/qa phytoplankton (min.)
chlorophyll yield from N
1.4
2.6
mg chl
(mmol N)-1
chlorophyll:carbon ratio
12/50
12/25
ε
(coefficient of) PAR attenuation
by phytoplankton*
?
?
mg chl
(mmol C)-1
m2 (mg chl)-1
Φ
= αa/(k.ε.χa), photosynthetic
quantum yield
?
?
nmol C µE-1
Xq Na
χa
uhPxc$
* not given; there is a reference to Baretta et al. (1988).
phosphorus and silicon. In my notation and units, the main equation for nonconservative
changes in carbon in a given autotroph compartment, is:
βB a
(5.3)
=
mmol C m-3 d-1
(µa - G).Ba
Grazing pressure G (d -1) is that due to mesozooplankton in the case of diatoms, to
mesozooplankton and microzooplankton (see section 7) in the case of flagellates.
Growth rate is:
µa
(5.4)
=
µmaxa.ƒ(I ).ƒ(NHS,NOS,POS ,SiS) - ra - ea
d-1
Maximum growth rate varies with the difference between the actual temperature Θ and <Θ>,
the annual mean temperature for a given region:
µmaxa
=
µmaxa[<Θ>].ƒ(Θ)
=
Q10((Θ - <Θ>)/10°C)
d -1
where
ƒ(Θ)
Q10 is 4.0, much higher than the value given by Eppley (1972), or used in MP or FDM. The
irradiance function is that of Steele (1962):
ƒ(I )
=
(I /KI ).exp(-(1/e).I /KI )
Varela et al. give this in a thick-layer version which also allows for the lack of photosynthesis
at night. They make the saturation parameter KI , which they call the 'optimum irradiance', a
function of mixed-layer PAR, thus allowing for some measure of adaptation to changing
light. However, the variation in KI is relatively small, with a minimum value KImin of 40
W (PAR) m-2. The saturation parameter can be related to maximum photosynthetic rate and
photosynthetic activity (Lederman & Tett 1981):
KI
=
p B maxa/αa
W m-2
Equating the maximum photosynthetic rate p B maxa with ERSEM's maximum growth rate,
allows an estimate of maximum photosynthetic 'efficiency':
αa
= µmaxa[<Θ>]/(χa.KI min)
mmol C (mg chl)-1 d -1 (W m-2)-1
equivalent to 0.062 mmol C (mg chl)-1 d-1 (µE m-2 s-1)-1 in the case of diatoms. This value is
close to that of MP autotrophs. These and other growth parameters are listed in Table 5.2 *.
Autotroph nutrient-limitation in this version of ERSEM I is both multiplicative and external:
ƒ(NHS,NOS,POS ,SiS) = 3√(S/(KS+ S)).(POS /(KPOS+ POS )).(SiS/(KSiS +SiS)) [diatoms]
* Table 5.2. Phytoplankton growth parameters in ERSEM.
MP
page 15
Feb 00
Table 5.3. Standard values for phytoplankton loss parameter in ERSEM
local
symbol
fD
Varela et al.
name
pe_R1Cxc$
description
value
for x=2 units
for x=1 flagdiatoms ellates
fraction of phytoplankton lysis
which becomes detritus (rest
becomes labile DOM) [unclear]
0.20
0.50
-
b ea
pu_eaPx$
coefficient of growth-dependent
excretion
0.05
0.05
-
b la
pum_eoPx$
coefficient of nutrient-stressdependent lysis
0.30
0.30
-
Kl b
chB1eP2c$
bacterial concentration for halfmaximum effect of bacterial
proteases on diatoms (only)
50
-
mmol C m-3
r0 a
srsPx$
rate of rest respiration (in
darkness)
0.25
0.15
d-1
ba
pu_raPx$
rate at which (activity)
0.10
respiration increases with growth
0.25
b Sa
pum_roPx$
rate at which respiration
increases with nutrient stress
0.05
0.05
Table 5.4. Standard values for phytoplankton nutrient parameter in
ERSEM
local
symbol
NHp a
Varela et al.
name
description
value
for x=2 units
for x=1 flagdiatoms ellates
xpref-N4n$
relative preference for
ammonium (over nitrate) uptake
3
3
KNOS
chPxn$
half-saturation concentration for
nitrate uptake
0.10
0.05
mmol N m-3
KNHS
chPxn$
half-saturation concentration for
ammonium uptake
0.10
0.05
mmol N m-3
qa
qnPxc$
phytoplankton 'maximum'
(nominally constant) nitrogen
content
0.172
0.188
mmol N
(mmol C)-1
Pq a
qpPxc$
phytoplankton 'maximum'
phosphorus content
0.0132
0.0073
mmol P
(mmol C)-1
Siq a
qsPxc$
phytoplankton maximum silicon 0.21
content
-
mmol Si
(mmol C)-1
ƒ(NHS,NOS,POS ,SiS) = √(S /(KS+S)).(POS /(KPOS+ POS ))
where dissolved inorganic nitrogen concentration
NHS + NOS
S
=
[flagellates]
mmol m-3
The value of the nitrogen half-saturation concentration is low compared with that of MP and
FDM. It is, however, a value for growth, whereas that in MP is for uptake, and it can be
shown (Tett & Droop 1988) that half-saturation constants for growth are less than those for
nutrient uptake.
Respiration includes rest, growth (called 'activity') and nutrient-stress components:
(5.5)
ra
=
[r0 a] + ba.µa + bS a.(µmaxa.ƒ(I ) - µa)
d -1
The terms are subject to temperature effects, either by way of maximum growth rate or
directly. The rest term is zero during periods of illumination, having the values of Table 5.3*
only during darkness. The diatom rest rate, 0.13 d-1 at mean temperature and for 12 hours of
daylight in each 24 hours, implies a 24-hr mean rate of 0.065 d-1, only a little more than the
MP autotroph value. The effect of the 'slope' terms b S a.µmaxa.ƒ(I )+(b a-b S a).µa will
normally be less than b a.µa in MP, because of the higher value of b a in MP. Nutrient stress
is quantified in ERSEM as the difference between actual growth rate and potential rate
µmax a.ƒ(I ) under light-limitation alone
Excretion includes 'lysis', a part of which (f D) goes to detritus, the remainder to DOC, and
growth-related 'activity excretion', all of which goes to DOC. The total rate is:
(5.6)
ea
=
b ea .µa + bla .(µmaxa.ƒ(I ) - µa)[.(1 + ƒ(Bb))]
d-1
where b ea is the coefficient of activity excretion and b la is the coefficient of lysis. Lysis is
held to be proportional to the extent of nutrient stress. In the case of diatoms only, the activity
of bacterial protease is supposed to increase the lysis rate, according to a saturation function
of bacterial biomass Bb:
ƒ(Bb) =
Bb/(K lb + Bb)
Changes in phytoplankton nutrient in ERSEM are the result of uptake less excretion.
Parameters for nitrogen, phosphorus and silicon kinetics are given in Table 5.4*. I will start
this account with phosphorus, which has no special features. For a given autotroph type,
(5.7)
(POu a - ea.P Qa).Ba
mmol P m-3 d-1
β Pa =
where biomass-related uptake rate
POu a = Pq a.(µa - ra) - m'.max{0, ( P Qa - P q a)} mmol P (mmol C)-1 d-1
* Table 5.3. Phytoplankton loss parameters in ERSEM.
* Table 5.4. Phytoplankton nutrient parameters in ERSEM.
MP
page 16
Feb 00
The equation implies that uptake is driven mainly by the demand resulting from the carbon
assimilation of net production. There may, however, also be excretion, which occurs when
the actual phytoplankton phosphorus quota (P Qa = Pa/Ba) exceeds the constant P:C ratio
assigned to each type of phytoplankton. In writing Eqn. (5.7), I have slightly simplified the
ERSEM equations, and have supplied the rate constant m' (d-1) in the correction term, in
order to emend an apparent dimensional error in equation (22) of Varela et al. The value of
m' would need to be of the same order as µa. It is, however, not clear why the correction
term is needed (except perhaps immediately after initialisation), since the actual quota should
remain always the same as P q a. Finally, phosphorus lost in excretion is distributed between
detritus and DOM in a way that assumes different P:C ratios in particulate and soluble cell
fractions.
The silicon equation is similar to that for phosphorus, but applies only to diatoms. That for
nitrogen is also similar, but the total uptake of dissolved inorganic nitrogen is partitioned
between nitrate and ammonium according to an ammonium preference factor NHp a and the
following equation:
(5.8)
NOf
=
ƒ(NOS)/((NHp a.ƒ(NHS)) + ƒ(NOS))
where the saturation function is, for either nutrient
ƒ(S ) =
S /(KS + S)
The fraction NOf of the required total nitrogen uptake flux is taken from nitrate, and the
remainder from ammonium. The half-saturation constant has the same value for ammonium
as for nitrate, and the same value as that used in the nutrient-limited growth equation.
Varela et al. remark that this ERSEM module was developed at NIOZ (Baretta, et al. 1988)
before they joined the project. They suggest that an internal-nutrient model would describe
phytoplankton growth better than does the external-limitation of Eqn. (5.3). This
improvement is implemented in ERSEM II (Baretta-Bekker, et al. 1997), although the later
version continues to assume that light and nutrients effects multiply. Apart from these
differences in the growth function, the ERSEM autotroph model differs from the autotroph
component of MP by including excretory and lytic losses. Such losses are assumed to take
place largely independently of grazing or other interactions with heterotrophs. The losses of
soluble organic material provide food for bacteria and hence support the microbial loop (see
Section 7). MP implicitly includes these losses, and the corresponding heterotroph gains,
completely within the microplankton compartment. The transfer of lysed autotroph
particulate material to detritus in ERSEM, which corresponds to autotroph mortality in FDM,
has no analogue in MP.
MP
page 17
Feb 00
mesozooplankton grazing
Phytoplankton
(autotrophs)
Herbivorous
'messy
feeding'
pico
b1
extracellular
production
ra
Protozoa
p2
rp2
b2
b3
rb2
Bacteriovorous
Protozoa
p1
rp1
b1 , b2 and b3 are bacteria
r is respiration
Figure 6-1. Conceptual model of interactions between bacteria and other microplankton.
6. Heterotroph equations
The microheterotrophs in MP include several functional types of marine pelagic bacteria and
protozoa, and the most general difficulty concerns how to make a simple parameterisation of
the key processes in which these organisms are involved. A second difficulty is that their
growth physiologies are less well known than those of algae. Although there have been many
studies of feeding and growth efficiencies, there have been few that have examined carbonnitrogen dynamics under well-controlled conditions, because of the difficulty of maintaining
well-defined populations of delicate protozoa, or bacteria with largely unknown nutritional
needs, under such conditions in the laboratory. The solution to these problems adopted by MP
is to treat the heterotrophic component as (a) a single compartment, with properties (b)
constrained (mathematically) by the need to conserve totals of elements and (physiologically)
by the assumed tendency of the microheterotrophs to maintain an optimal elemental
composition (Caron, et al. 1990; Goldman, et al. 1987).
Simplifying the description of heterotrophic processes
In the conceptual model for the implicit contents of MP (Figure 6-1†), bacteria are allocated
to one of three categories:
• b1, bacteria assimilating DOM leaked by phytoplankton during photosynthesis or
cell lysis, and returning this DOM, less a respiratory tax, to the microplankton pool;
• b2, bacteria assimilating DOM leaked from microplankters during messy feeding by
protozoa, and returning this DOM, less a respiratory tax, to the microplankton pool;
• b3, dormant bacteria, making no contribution to any microplankton process, but
adding carbon and nitrogen to the microplankton pool.
The argument that a given size of predator can ingest only a restricted range of prey sizes
suggests considering (at least) two categories of protozoa:
• p1, bacterivorous protozoa (zooflagellates and smaller ciliates) feeding on
picoplanktonic autotrophs and heterotrophs;
• p2, herbivorous and carnivorous protozoa (dinoflagellates and larger ciliates)
feeding on autotrophs larger than picoplanktonic size and on the bacterivorous protozoa.
Microbial loop models represent several of these components by explicit compartments. The
aim here is, however, to consider how this diversity of organisms might be approximated by a
†
Figure 6-1. Conceptual model of interactions between bacteria and other
microplankton.
MP
page 18
Feb 00
single microheterotroph compartment which can itself be subsumed into the microplankton
compartment of MP.
The b1 bacteria are, mainly, those attached to cell walls or found in the viscous, perhaps
glycosaminoglycan (= mucopolysaccharide) enriched, layer surrounding each algal cell. In
either case they will be consumed by grazers at the same time as their hosts. 'Extracellular
production', or DOM excreted by micro-algae, characteristically at high irradiances, is
thought to be largely glycollate (Fogg 1991) and so is a source of carbon with little nitrogen.
If bacterial metabolism tends to preserve a balanced N:C ratio of 0.22 (Goldman & Dennett
1991), then bacterial respiration is likely to take a high proportion of this DOM, unless the
bacteria have access to another source of organic matter rich in nitrogen, or unless they
compete with algae for ammonium. The excreted DOM is assumed by MP to be immediately
assimilated by b1 bacteria, and is thus ignored because these bacteria are treated as part of
the autotroph component, giving it a somewhat higher relative respiration rate than is
measured in axenic cultures of micro-algae.
Another source of DOM is cell lysis after viral attack or during 'messy feeding' by
protozoans. This DOM, with the C:N ratio of its source, is assumed in MP to pass efficiently
to the microheterotroph component, although at the cost of a respiratory tax. The route, by
way of b2 bacteria and p1 protozoans, re-assimilates labile DOM before any can diffuse away
from the region of production. The assumption of efficient transfer requires that bacterial
uptake rates are relatively high and do not saturate at the concentrations of labile DOM that
might be generated by realistic rates of protozoan feeding. It is also assumed that the relative
physical fluxes, given by terms such as ϕB b2/Bb2, are the same for each microheterotroph
component. This might be the case if all the components of Fig. 6.1 are conceived of as
occurring together within packets of water defined by the Kolmogorov scale (Lazier & Mann
1989) and dominated by viscous rather than inertial (including eddy-diffusive) forces.
The significance of the transfers shown in Fig. 6-1 can be examined by listing the processes
which result in net gain or loss to the microheterotrophs as a whole. These processes are 'net
ingestion', 'net mesozooplankton grazing loss', and 'total respiration'. Conversions which take
place (or are assumed to take place) wholly within the heterotroph compartment can be
ignored in constructing a unitary parameterisation of this compartment. Such conversions
include: interchanges between b2 and b3, as some active bacteria become dormant, and vice
versa; and the leakage of DOM during messy feeding and its subsequent rapid re-assimilation
by b2 bacteria.
'Net ingestion' is obtained by summing the protozoan consumption of autotrophs:
(6.1)
MP
cp2.Bp2.Ba2 + cp1.Bp1.Ba1
≈
page 19
ch.Bh.Ba
mmol C m-3 d-1
Feb 00
where the suffixes a2 and a1 stand, respectively, for large (i.e. > 2 µm) autotrophs (including
associated heterotrophic b1 bacteria) and small (picoplanktonic) autotrophs. The coefficients
cp refer to protozoan volume clearance rates, in m 3 (mmol protozoan C)-1 d-1. The
simplification in (6.1) results from the definitions that Ba = Ba2 + Ba1, and
Bh = Bp1 + Bp2 + Bb2 + Bb3, and so requires that
(6.2)
ch
=
cp1.(Bp1/Bh).( Ba2/Ba) + cp2.(Bp2/Bh).( Ba1/Ba)
The bulk heterotroph transfer coefficient c h can be treated as a parameter if the proportions of
the two kinds of protozoa, and the two kinds of autotroph, remain constant. Because of the
effect of the value of Ba1/Ba on ch, it seems likely that the value of cp2 normally has the
strongest effect on ch, giving way to cp1 only when picophytoplankton are the dominant
autotrophs. Thus, the value of c h must lie between c p2.((Bp1+ Bp2)/Bh) and
cp1.((Bp1+ Bp2)/Bh).
'Net mesozooplankton grazing losses' from the microheterotrophs were assumed in
equations (3.7) and (3.10) to be G.Bh for carbon and G.q h.Bh for nitrogen, where G is the
grazing pressure due to copepods and other mesozooplankton. This formulation assumes
grazing without selection, a contestable assumption. It was discussed by Huntley (1981), who
reported studies on Calanus spp. He concluded that for "phytoplankton > 20 µm in diameter
there appeared to be no selective ingestion according to the size, shape or species." It is,
however, likely that bacteria and small zooflagellates will be grazed less effectively than
larger microplankton by mesozooplankton, and thus that the grazed flux from the
heterotrophs will be less than G.Bh in the ratio Bp' /Bh, where Bp' includes Bp2 and some or
none of Bp1. This 'undergrazing' of microheterotrophs may need further consideration.
'Total heterotroph respiration' is the sum of contributions from b 2 bacteria and the
protozoans; b3 bacterial respiration is assumed insignificant, and b1 respiration is included in
that of the autotrophs. Thus,
(6.3)
rh.Bh =
rp1.Bp1 + rp2.Bp2 + rb2.Bb2
mmol C m-3 d-1
If it is assumed that the respiration of each component is linearly related to growth rate, then
(6.4)
=
r0 h + bh.µ
d-1
rh
where µ is microheterotroph and microplankton relative growth rate and
r0 h
=
r0 b2.(Bb2/Bh) + r0 p1.(Bp1/Bh)+ r0 p2.(Bp2/Bh)
bh
=
b b2.(Bb2/Bh).( µb2/µ) + b p1.(Bp1/Bh).( µp1/µ) + b p2.(Bp2/Bh)
Some evidence for linearity comes from Fenchel & Findlay (1983), who reviewed published
estimates of protozoan respiratory rates and concluded that during "balanced growth, energy
metabolism is nearly linearly proportional to the growth rate constant". The expansion for b h
in (6.4) contains the multiplying terms (µb2/µ) and (µp1/µ) which imply that it is necessary
MP
page 20
Feb 00
for the intrinsic growth rate of the bacterial population to exceed that of the bacterivore
population, and their rate that of the protistivores, in order that all the microheterotroph
components may in MP change at the same relative rate as the microplankton as a whole.
Given the assumption of fixed ratios of components, r0 h and b h can be treated as parameters,
with the value of r0 h within the range defined by r0 b2, r0 p1 and r 0 p2 unless Bb3 is large.
The value of bh, however, will be larger than the component values bb2, b p1 and b p2,
because of the effects of component growth rates. This additional 'respiratory tax' paid by the
microheterotroph compartment is the main device for parameterising the extra losses
resulting from several trophic transfers.
Growth of the microheterotroph compartment under C or N control
It is assumed that heterotroph (biomass-related) growth is either carbon or nitrogen limited,
(6.5)
µh
=
min{µC, µN}
d-1
ih - rh
d-1
Under carbon limitation,
(6.6)
µC
=
where i h ( = ch.Ba) is food intake rate per unit heterotroph carbon biomass, the result of
ingestion of particles or uptake of dissolved organic matter (DOM). Respiration rate rh has
already been assumed to be related to growth rate (Eqn. 6.4), leading to:
(6.7)
µC
=
(ih - r0 h)/(1+b h)
d-1
(ih.Qa - Nrh)/q h
d-1
Under nitrogen limitation,
(6.8)
µN
=
where Q a is food N:C ratio, q h is heterotroph N:C ratio and Nrh is heterotroph biomassrelated rate of ammonium excretion. It is assumed that , unlike the autotrophs, the
microheterotrophs have a constant N:C ratio, and that, under nitrogen-limited conditions,
ammonium excretion rate is related to growth rate by the same parameters that control
respiration rate under C-limited conditions:
(6.9)
Nrh
=
q h.rh =
q h.(r0 h + bh.µ N)
mmol N (mmol C)-1 d-1
Substituting this in Eqn. (6.8) gives:
(6.10)
µN
=
(ih.(Qa/q h) - r0 h)/(1+b h)
d-1
Which of µN and µC is lowest depends on the N:C ratio of the food in relation to the optimal
N:C ratio of heterotrophs. Carbon limitation must obtain when Q a > qh, and N limitation
when Qa < qh, since in the latter case the food is poorer in N than the microheterotrophs that
ingest it. A form of the growth equation covering both limiting conditions is:
MP
page 21
Feb 00
Table 6.1: Heterotroph rate equations used in MP and AH
growth
µh
=
(ih.q * - r0h)/(1+bh)
where q*
=
1
: Q a ≥ qh
Qa/q h : Q a < qh
d-1
C ingestion
ih
=
=
ch.Ba
[MP - cancels out]
ch.ƒ(θ).(1/(1+Ba/kB a)).Ba [AH]
d-1
net N uptake
uh
=
µh.q h =
ih.Qa - Nrh
mmol N
(mmol C)-1
d-1
NH4+ excretion
Nrh
=
=
ih.Qa
µh.q h
µh.((Qa-q h)+bh.Qa) + r0h.Qa [C-limited]
[N-limited]
µh.b h.q a + r0h.qh
mmol N
(mmol C)-1
d-1
respiration
rh
=
=
=
ih
µh
µh.b h + r0h
[C-limited]
µh.(((1+bh).q *)-1) + r0 h/q* [N-limited]
d-1
temperature
ƒ(θ)
=
exp(k θ.(θ- 20°C))
(6.11)
µh
=
(ih.q * - r0 h)/(1+b h)
q*
=
1
Qa/q h
d-1
where
:
:
Qa ≥ qh
Qa < qh
[C-limiting]
[N-limiting]
Carbon respiration and ammonium excretion must be set by the difference between food
intake and the growth need:
(6.12)
rh
=
ih - µ h
(6.13)
Nrh
=
ih.Qa - µh.q h
d-1
mmol N (mmol C)-1 d-1
Substituting ih by (µh.(1+bh) + r0 h)/q* from eqn. (6.11), gives:
(6.14)
rh
=
=
µh.((1+bh)/q *)-1) + (r0 h/q*)
µh.(((1+bh).q h/Q a)-1) + (r0 h.q h/Q a)
µh.b h + r0 h
(6.15)
Nrh
=
=
µh.(((1+bh).Qa/q *)-q h) + (r0 h.Qa/q *)mmol N (mmol C)-1 d-1
µh.((1+bh).Qa - q h) + (r0 h.Qa)
[C-limiting]
[N-limiting]
(µh.b h + r0 h).q h
d-1
[N-limiting]
[C-limiting]
Eqn. (6.14) shows that more carbon is respired under N-limiting conditions, and (6.15) shows
that more nitrogen is mineralised under C-limiting conditions. In effect, the N:C ratios of
food and heterotroph are harmonised by 'burning off' the unwanted portion of the excess
element. Finally, the equation for nitrogen assimilation, or net uptake, is:
(6.16)
uh
= µh.q h
=
ih.Qa - Nrh
mmol N (mmol C)-1 d-1
The heterotroph N:C ratio q h is an important parameter. Protozoan optima of 0.15 - 0.16
(Caron, et al. 1990; Davidson, et al. 1995) and bacterial optima of 0.22 mol N (mol C)-1
(Goldman & Dennett 1991), suggest a standard value for q h of 0.18 mol N (mol C)-1.
Table 6.1* summarises the microheterotroph equations. Unlike the autotroph equations, these
have not been empirically validated, but follow from assuming that (i) the protozoan-bacterial
mixture has a constant composition, and (ii) exhibits threshold-limited growth with
mineralisation as the only loss of C and N. They predict that growth efficiency, defined (for
ih >> r0 h) as µh/i h = q*/(1+ bh), must be less under nutrient-controlled conditions (when
q * = Q a/q h < 1) than under C-controlled conditions. Evidence cited below suggests that this
is the case for some single-species populations and for mixtures of micro-organisms.
The rest of this section considers the remaining heterotroph parameters (r0 h, b h and c h).
* Table 6.1. Heterotroph rate equations used in MP and AH.
MP
page 22
Feb 00
Table 6.2: Dependence of growth rate on ingestion rate
(Fuller, 1990)
Grazer/food
GGE
(s.e.)
0.02
(0.01)
0.04
(0.01)
0.29
(0.03)
0.55
(0.06)
0.05
(0.13)
0.49
(0.10)
0.47
(0.04)
0.21
0.07
0.25
0.05
1.1, 3.0
0.08
(0.01)
0.05
0.07
0.07
0.03
*
0.42
(0.05)
0.06
0.18
0.33
0.11
1.4, 2.0
0.07
(0.01)
*
0.24
(0.05)
3.2
0.02
(0.01)
*
Uronychia & Oxyrrhis
0.10
(0.03)
*
Uronychia & Oxyrrhis &
Dunaliella
0.32
(0.03)
2.1
Euplotes sp. & Dunaliella
primolecta
Euplotes & Oxyrrhis
marina
Euplotes & Oxyrrhis &
Dunaliella
Oxyrrhis & Dunaliella
tertiolecta
Oxyrrhis & Brachiomonas
submarina
Oxyrrhis &
Chlamydamonas spreta
Oxyrrhis &
Nannochloropsis oculata
Pleurotricha sp. &
Brachiomonas
Strombidium sp. & Pavlova
lutheri
Uronychia sp. & Dunaliella
bp
regression, µ = a + bi .i
bi
(s.e.)
a, d-1 (s.e.)
*
0.01
0.01
0.023
*
0.004
2.5
1.0
GGE is 'Gross Growth Efficiency'; the slope coefficient b p = GGE-1-1. Additionally, where
regression coefficients a and bi were given by Fuller, a second value of the slope was
estimated from b p = bi-1 - 1. Low values of bp were assumed to represent N-limited
conditions, in which bi ≈ (Qa/q p)/(1+b p); hence these b p values are not given as Fuller did
not report food nitrogen.
Respiration
Equation (6.7) embodies the assumption of linearly growth-dependent respiration for carbonlimited microheterotrophs. Here is the growth equation with protozoan or bacterial subscripts:
(6.17a)
µCp
=
(ip - r0 p)/(1+b p)
d-1
(6.17b)
µCb
=
(ib - r0 b)/(1+b b)
d-1
where i p is the relative ingestion rate (d-1) at which protozoa ingest biomass. The analogous
ib is the relative rate at which bacteria assimilate DOM.
Fuller (1990) studied ingestion and growth in 12 combinations of marine protozoans and
algae in batch cultures (Table 6.2*). Gross Growth Efficiency was estimated by dividing
predator specific growth rate by specific food ingestion rate, using maximum rates when
accurately estimated. In 6 cases he obtained significant regressions of growth rate on
ingestion, of general form µ = a + b i .i. These may be compared with eqn. (6.17a) for the
carbon-limiting case, when b i = 1/(1+b p). When nitrogen limits growth, b i should be
smaller. Thus, it is Fuller's higher values of bi that point to the most appropriate values of b p
for MP: between 1 and 3. The data cannot be used to estimate r 0 p, because the regression
intercepts are positive (they should be negative), although in most cases not significantly so.
Hansen (1992) estimated a growth/ingestion yield of 0.36 for the heterotrophic dinoflagellate
Gyrodinium spirale feeding on the autotrophic dinoflagellate Heterocapsa triquetra. This
value corresponds to b p = 1.8, within the range of 1 to 3 taken from Fuller's results. However,
the corrsponding basal rates were large: even Gyrodinium populations previously kept on a
maintenance ration, and then starved, decayed at 0.19 d-1.
Fenchel & Findlay (1983) reviewed published estimates of protozoan respiratory rates. They
concluded that "the data show a surprisingly large variance when similarly sized cells or
individual species are compared. This is attributed to the range of physiological states in the
cells concerned. The concept of basal metabolism has little meaning in protozoa. During
balanced growth, energy metabolism is nearly linearly proportional to the growth rate
constant; at the initiation of starvation, metabolic rate rapidly declines. " Later in the paper
they state that in "small [starved] protozoa the respiratory rate per cell may eventually
decrease to 2-4% of that in growing cells." This contrasts with Hansen's decay rate for
starved Gyrodinium. MP follows Fenchel & Findlay in assuming a low basal respiration rate.
As Fuller's pelagic protozoa (Strombidium, Pleurotricha and Oxyrrhis) had maximum growth
rates of 0.4 to 1.3 d-1, the value of the basal respiratory rate r0 p was taken as 0.02 d-1.
Fenchel & Findlay estimated food conversion efficiency ( ≈ b i when r0 p small) in the range
* Table 6.2. Fuller's data on growth efficiency.
MP
page 23
Feb 00
Table 6.3: Ingestion and related parameters
(Fuller, 1990)
Grazer/food
I maxp,
d-1
KBf,
mmol
m-3
B0f,
mmol
m-3
KBf+B0f,
mmol m-3
cp, dm3
mmol-1 d-1
Euplotes sp. & Dunaliella
primolecta
Euplotes & Oxyrrhis
marina
Euplotes & Oxyrrhis &
Dunaliella
Oxyrrhis & Dunaliella
tertiolecta
Oxyrrhis & Brachiomonas
submarina
Oxyrrhis & Chlamydamonas
spreta
Oxyrrhis & Nannochloropsis
oculata
Pleurotricha sp. &
Brachiomonas
Strombidium sp. & Pavlova
lutheri
Uronychia sp. & Dunaliella
3.32
9.5
6.4
16
348
4.81
172
0
172
26
2.24
3.2
33
36
694
2.02
9.7
1.6
11
208
2.10
9.9
3.7
14
211
4.80
22
5.1
27
215
2.26
8.5
0.8
9
268
8.32
2.5
65
68
?
2.98
-
-
-
-
4.74
66
-1.6
64
72
Uronychia & Oxyrrhis
3.56
58
6.8
65
61
Uronychia & Oxyrrhis &
Dunaliella
2.42
16
14
30
149
Median
3.15
10
5
30
210
0.4 to 0.6 for protozoa growing under good conditions; these values support the use of a value
of about 1.0 for bp.
Caron et al. (1990) tabulate ranges of GGE from published studies of 25 single species and
mixed assemblages. The minima (for nutrient-limited conditions?) range from 1 to 64% with
a median of 12%, the maxima ( for C-limited conditions?) from 11 to 82% with a median of
49%. The latter value corresponds to bp of 1.04.
As in the case of protozoan b p, bacterial b b can be estimated from growth efficiencies if
basal respiration is assumed low. Goldman & Dennett (1991) obtained Gross Growth
Efficiencies of 47% to 60% for natural populations of marine bacteria grown with added
sources of nitrogen and carbon. Calculating b b from (GGE-1 - 1) gives values between 0.7
and 1.2. Danieri et al. (1994) obtained a wide range of efficiencies for bacterial growth in
mesocosms and the Bay of Aarhus, but most of their values were between 15% and 45%.
Growth with added glycine had mean GGE of 32%, whereas growth on added inorganic
nutrients had mean GGE of 16%. Taking 32-45% for C-limited bacteria, b b = 1.2 to 2.1.
The assumption that starved bacteria become dormant is equivalent to supposing that r 0b is
zero. Hence the standard value of r0 h will be taken as the protozoan value of 0.02 d -1. The
'best' estimates of b p and b b are likely to be the lowest observed values, of about 1.0;
however, b h is required (p.21) to be larger than this, and so the standard value is set at 1.5.
Ingestion and clearance
Although terms in c h cancel in the equations of MP, values will be useful in examining
heterotroph-autotroph interactions in the model AH of section 10. On theoretical grounds,
protozoan specific ingestion rate i p should be a saturation function of food concentration Bf:
(6.18)
ip
=
cp.Bf.(1/(1+(Bf/KB f)))
d-1
The explanation is that clearance slows as animals spend more time digesting food. Thus cp
is a maximum clearance rate. Writing ip = cp.Bf is a linearisation which is appropriate for the
low food concentrations (Bf < KBf) that are typical of most natural waters. When cp is not
given explicitly, it can be calculated from the solution of Eqn. (6.18) for Bf >> KBf:
cp
=
imax p/K B f
m3 (mmol protozoan C)-1 d-1
When examined over a wide range of cell sizes, biomass-related rates of clearance and
ingestion decrease with increasing cell size (Fuller 1990). However, there is considerable
variability amongst organisms of similar size. I have relied on the data (Table 6.3*) of Fuller
(1990), who measured ingestion rates of several laboratory-grown marine ciliates, and a
* Table 6.3. Fuller's data for ingestion and clearance.
MP
page 24
Feb 00
Table 6.4: Heterotroph parameters used in MP and AH
std value range
units
0.2 at
20°C
0.01 4.5
m3 (mmol
C)-1 d-1
concentration of food that half-saturates
ingestion [AH only]
50
10
170
mmol
(auto)C m-3
qh
nitrogen: carbon ratio
0.18
0.15 0.22
mmol N
(mmol C)-1
r0 h
basal (biomass-related) respiration rate
0.02
0.00 0.05
d-1
bh
slope of graph of rh on µh
1.5
1-3
kθ
temperature coefficient
0.069
ch
(maximum) transfer rate, expressed as
clearance, relative to heterotroph biomass
[AH only]
[temperature: ch = ch[20°C].ƒ(θ)]
KBa
°C-1
dinoflagellate, feeding on various algae. In some cases Fuller found that food concentration
had to exceed a threshold before grazing took place. Adding this threshold B0 f to the halfsaturation concentration KB f gives estimates of the food concentration above which ingestion
begins to saturate; the median value is 30 mmol C m-3. According to Table 6.3. the median
protistivorous protozoan ingests a maximum of 3.15 times its own biomass daily, and has a
maximum clearance rate of 0.21 m3 (mmol protozoan C)-1 d-1. However, Fuller's values
ranged from 0.026 to 0.69 m3 (mmol C)-1 d-1, which may be understood as demonstrating the
difficulty of making the measurements as well as showing the effects of dependence of
clearance or ingestion rates on predator prey size, prey size, and prey nutritional status
For comparison, the results of a study (Davidson, et al. 1995) of the dinoflagellate Oxyrrhis
marina feeding on the prymnesiophyte alga Isochrysis galbana, gives a (maximum)
clearance rate of 0.009 m 3 (mmol C)-1 d-1, when the data are appropriately converted.
Grazing data (Hansen 1992) for the dinoflagellate Gyrodinium spirale feeding on the
(autotrophic) dinoflagellate Heterocapsa triquetra gives 0.038 m3 (mmol C)-1 d-1 .
Observations of tintinnid ciliates feeding on small autotrophic flagellates give clearance rates
just above 1.0 m3 (mmol C)-1 d-1 (Heinbokel 1978). Caron et al. (1985) report a study of the
zooflagellate Paraphysomonas imperforata. After conversion to my units, clearance rates on
a diet of bacteria ranged from 0.021 to 0.040 m3 (mmol C)-1 d-1, those on a diet of the
anomalous diatom Phaeodactylum tricornutum ranged from 0.024 to 0.061 m3 (mmol C)-1
d-1. Literature reviewed by Caron et al. gave a very wide range of bacterivore clearance rates,
from 0.05 to 4.5 m3 (mmol C)-1 d-1.
Given these ranges, and the uncertainties involved in deriving ch from c p, the standard value
of ch has been taken as 0.2 m 3 (mmol C)-1 d-1, close to the median of Fuller's value.
Table 6.4* lists the full set of microheterotroph parameter values.
7. Comparisons with heterotroph equations in ERSEM and FDM
The ERSEM Microbial Food Web sub-model includes three explicit compartments for
microheterotrophs (Baretta-Bekker, et al. 1995): bacteria, nanoflagellates, and
microzooplankton. Dissolved organic matter is not explicitly represented, as it is assumed
that such material, excreted by phytoplankton and protozoans, is immediately taken up by
bacteria. The bacterial compartment also assimilates carbon from explicitly modelled
particulate detritus, and so includes some of the bacterial activity that in the microplanktondetritus model is ascribed to the detrital compartment. In ERSEM, the bacteria provide the
main food source for heterotrophic nanoflagellates (2-20 µm). Microzooplankton are
* Table 6.4 . Microheterotroph parameters used in MP and AH.
MP
page 25
Feb 00
DIM pools
u, uptake
(1-f a).(1-f e ).u
activity
respiration
r0 , rest
respiration
G,grazing
f a.u
assimilation
(1-f a).f e .u
excretion, or
'messy feeding' the 'STANDARD ORGANISM'
m, mortality
DOM
Detritus
Figure 7-1. The ERSEM 'standard organism' in the case of microheterotrophs.
"heterotrophic planktonic organisms from 20 to 200 µm SED, excluding .... naupliar/larval
stages ... [and comprising] ciliates and other heterotrophic protists ... feeding on
phytoplankton and heterotrophic nanoflagellates ... [and] itself ... [and] grazed by omnivorous
zooplankton."
ERSEM's generalised non-conservative equation for a 'standard organism'† of type i is (in my
symbols):
βB i
(7.1)
=
mmol C m-3 d-1
(µi - m i - Gi ).Bi
where m i is (constant) mortality rate and grazing pressure G i ("specific grazing rate") is the
total for all predators. ERSEM uses neither of the terms 'specific growth rate' or 'ingestion'
rate' employed in MP ("net growth" in ERSEM refers to βBi ). According to my definitions,
however, (specific) growth rate would be the result in ERSEM of food uptake Cu i (mmol C
(mmol C)-1 d-1) less (organic) excretion ei and (mineralising) respiration ri .
(7.2)
µi
=
ii - ri
ii
=
u i - ei =
k1
=
(1-f ei .(1-f ai))
Cu i
=
d-1
- ei - ri
Ingestion is:
(7.3)
k 1 .Cu i
d-1
where :
Here, f e is the fraction of unassimilated food that is excreted (lost into the detrital and
implicit DOM pools), and f a is assimilation efficiency. ERSEM (Fig.7-1) defines
assimilation as taking place after losses of captured carbon by activity respiration as well as
by the 'messy feeding' implied by uptake-related excretion:
assimilation =
(1-f ai).Cu i
assimilation efficiency,
f ai
=
=
ei + (ri - r0 i)
1 - ((ei + (ri - r0 i)/Cu i)
Respiration is a basal rate plus 'activity' respiration, the fraction of food uptake which is
neither excreted or converted to biomass:
(7.4)
ri
=
r0 i + Cu i.(1-f ei ).(1-f ai)
d-1
µi
=
f ai.Cu i - r0 i
d-1
and so,
(7.5)
Combining Eqns. (7.3) and (7.5) gives:
µi
†
=
(f ai/k 1i).ii - r0 i
d-1
Figure 7-1. The ERSEM 'standard organism' in the case of
microheterotrophs.
MP
page 26
Feb 00
d-1
Table 7.1 : ERSEM (and derived MP or AH) parameters for heterotrophs
(at 10°C)
ERSEM
name*
Local
symbol
B1
Description
bacteria
Z6
Z5
nanomicrozoo- units
flagellates plankton (here)
fa
puST$
assimilation efficiency
0.3
0.2
0.5
-
fe
pu_eaST$
fraction excreted of
unassimilated food; 1-f e
is 'activity' respired
(0)
0.5
0.5
-
fD
pe_R1ST$ particulate fraction of
excretion
(0)
0.5
0.5
-
= 1-f e.(1-f a )
0.7
0.6
0.75
srsST$
basal respiration rate
0.01
0.05
0.02
d-1
C u max
sumST$
maximum food uptake
rate
8.38
10.0
1.2
d-1
KB f
chuSTc$
food conc. halfsaturating uptake
-
29
6.7
mmol C
m-3
Q10
q10ST$
temperature coeff.
2.95
2.0
2.0
(10°C)-1
c
= k1 .Cu max /K B f,
clearance rate
-
0.21
0.13
m3 (mmol
C)-1 d-1
b
= (k1 .f a )-1, slope of
respiration on growth
1.3
2.0
0.5
-
availability (diatoms)
for ....
availability
(phytoflagellates)
availability (bacteria)
availability
(nanoflagellates)
availability
(microzooplankton)
-
0
0.5
-
-
0.3
0.5
-
-
1.0
0.2
0
0.6
-
-
0
0.2
-
k1
r0
p
suP1_ST$
suP2_ST$
suB1_ST$
suZ6_ST$
suZ5_ST$
* The ST component of the ERSEM name is replaced by B1, Z5 or Z6.
which may be compared with the MP equation for C-controlled growth for an example
protozoan:
µC
=
(ip - r0 p)/(1+b p)
The two versions can be equated if ERSEM f ai/k 1i ≡ 1/(1+bp) in MP, and ERSEM
r0 i ≡ r0 p/(1+b p) in MP. Thus MP b p = (k1 i/f ai) - 1 in ERSEM.
Table 7.1* gives values of the relevant ERSEM parameters for each microheterotroph
compartment. The pattern of carbon flow in ERSEM (Fig. 2-1(a)) tends to make bacteria the
most abundant microheterotroph, followed by nanoflagellates. Therefore, the respiration
slope b h of MP heterotrophs constructed with ERSEM organisms is likely to be between 1.3
and 2.0, with r0 h less than 0.03 d-1. However, the nanoflagellates and microzooplankton of
ERSEM are partly cannibalistic, imposing, in effect, an extra respiratory tax. Thus the
ERSEM-analagous b h is likely often to exceed the MP value of 1.5
ERSEM describes food uptake by saturation (Michaelis-Menten or Langmuir isotherm)
kinetics:
(7.6)
Cu i
=
C u maxi.Bf/(K B fi
+ Bf)
[mmol C (mmol C)-1] d-1
where food concentration Bf is the result of totalling over all possible foods the product of
concentration and 'availability' or preference pi,j. A solution of (7.6) for Bf >> KBfi gives
ci
=
imaxi/K i
=
k 1 .Cu maxi/K B fi
m3 (mmol C)-1 d-1
This allows (maximum) clearance rates to be calculated from the ERSEM parameters for
maximum uptake Cu max and uptake half-saturation K B fi (Table 7.1). These rates are for
10°C. Combining them with some preference for the nanoflagellate clearance rate, and
correcting to 20°C (because of the temperature-dependence of ERSEM Cu max), suggests a
value for the ERSEM analogue of ch of about 0.4 at 20°C. However, the 'availability' of
foods is always less than one in ERSEM, and there is some cannibalism. Thus the ERSEManalogue may be closer to the MP standard value of 0.2 m3 (mmol C)-1 d-1.
ERSEM uses carbon as its main currency, but allows for variation in the nutrient:carbon ratio,
or fraction, of the phytoplankton or detrital food of the microheterotrophs. In the model,
bacteria, nanoflagellates or microzooplankton assimilate food without changing its nutrient
ratio. If this ratio exceeds q maxh, the maximum nutrient:carbon ratio for the compartment,
the excess nutrient is immediately returned to the inorganic nutrient pool. However, when
"the difference between the actual and the maximum nutrient fraction becomes negative,
nutrients are retained, until the maximum value is re-attained." Such a nutrient deficit does
* Table 7.1. ERSEM parameters for microheterotrophs.
MP
page 27
Feb 00
Table 7.2. Standard values for zooplankton parameters in FDM
local
symbol
na Z,
n =a,b,D
imax Z
FDM
symbol
std
value
units
0.75
-
maximum ingestion rate (relative to
zooplankton biomass)
1.0
d-1
= (imaxZ/K FZ).q Z, maximum
clearance rate
0.15
m3 (mmol C)-1 d-1
(total) food concentration which
half-saturates ingestion
1.0
mmol N m-3
preference for a food type when all
foods equally abundant
??
(N-)relative rate of N excretion
0.1
d-1
(C-)relative rate of N excretion
0.015
mmol N (mmol C)-1
d-1
ammonium fraction of N excretion
0.75
µ5
specific mortality rate
0.05
Ω
fraction of dead animals that
become detritus ?? (the remainder
minineralising to ammonium ??)*
0.33
−
N:C, inverse Redfield, ratio
1/6.63 mmol N (mmol C)-1
β 1,β 2,β 3, assimilation efficiencies for diets of
phytoplankton, bacteria or detritus
g
cZ
KF Z
K3
np 0 ,
p
NrNZ
µ2
n =a,b,D
NrZ
ε
mZ
qZ
description
d-1
Values are for Bermuda station S, but it does not appear from Fasham et al. (1993) that any
are temperature-dependent.
*The value and definition of Ω are not quite clear.
not otherwise affect microheterotroph rates. The maximum nitrogen fraction is 0.20 mol N
(mol C)-1 for protozoans and 0.25 mol N (mol C)-1 for bacteria.
Finally, an important difference from MP is that in ERSEM the products of excretion and
mortality "are partitioned over dissolved and particulate organic matter". Thus, ERSEM
microheterotrophs contribute directly to the Detritus compartment, whereas MP
microplankton biomass passes to Detritus only by way of mesozooplankton grazing.
FDM (Fasham, et al. 1990) includes compartments for DON (made by phytoplankton
excretion), bacterial nitrogen, and zooplankton nitrogen, and most rates are relative to
nitrogen. The "zooplankton compartment describes an animal which is a combined herbivore,
bacterivore and detritivore … [with] parameters that are more typical of the herbivorous
copepod part of this 'portmanteau' animal than the bacterivorous flagellate part."
Nevertheless, the describing equation gives the bulk dynamics of a homogenous population
and does not allow for delays between generations of animals. In my terms, it is:
(7.7)
βZ
=
(µZ - m Z).Z
µZ
=
∑(nf Z.niZ) - rZ
mmol N m-3 d-1
where
d-1
The terms nf Z and niZ are, respectively, assimilation efficiency and relative ingestion rate
(d-1) for food type n. Assimilation efficiency, relative excretion rate rZ
(mmol N (mmol N)-1 d-1) and relative mortality rate mZ (d-1) are assumed constant. The
latter "parameterises both natural and predator mortality." The variable term is that for
ingestion rate, exemplified for phytoplankton (nitrogen concentration N a) as food:
(7.8)
ai
=
imax Z.ap.Na/(K FZ + F)
d-1
F is the total concentration of food, summed over phytoplanktonic, bacterial and detrital
nitrogen. The maximum relative ingestion rate i maxZ (d-1) and the half-saturation food
concentration K FZ (mmol N m -3) can be used to compute a maximum clearance rate
equivalent to 0.15 m3 (mmol zooplankton C)-1 d-1, given qZ of 0.16 mmol N (mmol C -1), the
Redfield ratio (see Table 7.2* ). As well as being close to the MP value for ch, this is similar
to copepod clearance rates obtained (per animal) by Paffenhöfer (1971) and Paffenhöfer &
Harris (1976), and thus a little controversial. Paffenhöfer's rates, obtained from animals
cultivated in the laboratory, are an order of magnitude greater than measured by workers
using 'wild' animals, and when related to biomass are of the same order as those for
protozoans. Nevertheless, the Paffenhöfer rates seem correct, in that they, unlike the lower
rates, will allow copepods to feed themselves at concentrations of phytoplankton encountered
under typical conditions in the sea. So far as FDM is concerned, the similarity of biomass-
* Table 7.2. Zooplankton parameters in FDM.
MP
page 28
Feb 00
Table 7.3. Standard values for bacterial parameters in FDM
local
symbol
u Nmaxb
FDM
symbol
Vb
u maxb
KS
K4
cb
description
std
value
units
maximum N-relative ammonium or
DON uptake rate
2.0
d-1
= uNmaxb.qh, maximum C-relative
uptake rate
0.4
mmol N (mmol C)-1
d-1
half-saturation concentration for
nutrient uptake
0.5
mmol N m-3
= uNmaxb/K S , trophic transfer
coefficient
0.8
m-3 (mmol C)-1 d-1
d-1
NrNb
µ3
(N-)relative rate of nitrogen
excretion
0.05
NHq ON
η
ammonium:DON uptake ratio
(assumes DOC:DON = 8)
0.6
qb
−
nitrogen:carbon ratio
0.20
mmol N (mmol C)-1
related clearance rates for copepods and protozoans would seem to help justify their inclusion
in the same model compartment.
Total food concentration is:
(7.9)
F
=
ap 0 .N a
+ bp 0 .N b + Dp 0 .N D
mmol N m-3
The terms in p0 are the standardised preferences of the zooplankton for phytoplankton,
bacteria and detritus, when these potential foods are presented in equal amounts. The actual
preference for phytoplankton, for example, given a set of food abundances, is:
ap
(7.10)
=
ap 0 .N a/F
and this has the consequence that the simulated zooplankton exert the greatest grazing
pressure (i.Z) on the most abundant type of food, "equivalent to assuming that the
zooplankton actively select the most abundant food organisms, or, as we are dealing with an
aggregated entity, that particular sub-groups of zooplankton will develop to crop the most
abundant organisms." Conversely, this description of grazing relieves grazed populations of
grazing pressure when their abundance is low, and hence avoids driving phytoplankton or
bacteria to extinction during simulations. Following Fasham et al., I will call this a 'switching
model' (although the change-over from one diet to another is continuous rather than abrupt),
and it may have the advantage of stabilising the trophic network of FDM. Eqn. (7.8) seems
equivalent to a Holling (1959) type III grazing function for any given prey type, a point
discussed in section 11. Fasham et al. report that the use of (7.10), ascribed to Hutson (1984),
"increased the likelihood of a zooplankton population surviving the winter, thereby producing
a more robust model."
The bacterial equation in FDM is
(7.11)
β Nb
=
µb.N b - biZ.Z
µb
=
NHu Nb
mmol N m-3 d-1
where:
+ ONu Nb - NrNb
d-1
(The suffix N is added to the symbol u to emphasise that these uptake rates are relative to
nitrogen biomass, in contrast to the carbon-related rates of MP and ERSEM. The FDM rates
thus have dimensions of time-1.) Parameter values are listed in Table 7.3*. The bacteria are
supposed to have a composition (q b) of 5 mole carbon per mole of nitrogen. The uptake
equations postulate a requirement for ammonium to supply the relative nitrogen deficiency of
dissolved organic matter, assumed to have a C:N ratio (qON) of 8. Thus,
* Table 7.3. Bacterial parameters in FDM.
MP
page 29
Feb 00
(7.12)
NHu Nb
ONu Nb
=
=
u Nmaxb.S '/(KS + S' + ONS)
u Nmaxb.ONS/(KS + S' + ONS)
=
min(NHS, ONq b.ONS)
d -1
d -1
where
S'
mmol N m-3
ONq b
(= qON/q b) is 0.6, the ratio of contents of nitrogen (relative to carbon) in DOM and
bacteria. In the absence of ammonium, uptake (and hence growth) depends on DON
concentration only; when ammonium is present the additional uptake of nitrogen allows
faster growth. In the absence of DON, uptake is zero and hence growth is negative.
Finally, it may be noted in Table 7.3. that the transfer coefficient for bacteria has the deduced
value of 0.8 m3 (mmol C)-1 d-1. There are no equivalent values in MP or ERSEM for
comparison but this high rate does not seem unreasonable given the large surface:volume
ratio and potentially fast metabolic rate of bacteria.
MP
page 30
Feb 00
Table 8.1: MP equations derived from equations of state
carbon biomass
where:
growth rate
βB
nitrogen in biomass β N
where:
nutrient quota
uptake rate
mmol C m-3 d-1
=
(µ - G).B
µ
=
µh
=
µa.(1-η) - r h.η
=
u .B - G.N
=
(u - G.Q). B mmol N m-3 d-1
Q
u
=
=
Qa.(1-η) + q h.η
u a.(1-η) - Nrh.η
d-1
mmol N (mmol C)-1
mmol N (mmol C)-1 d-1
8. Microplankton rate equations
The microplankton equations derived in Section 3 are summarised in Table 8.1*. My purpose
now is to derive versions of the equations for growth and nutrient uptake which do not
include explicit autotroph or heterotroph variables such as µa or Nrh, but only microplankton
variables and parameters (which can be recognised by the absence of a or h subscripts). In this
version of MP, with explicit η, most microplankton parameters will be derived from
autotroph and heterotroph parameters, using the assumption of constant η.
The starting point is the equation for the intrinsic growth rate of the microplankton. This rate
must be the same as that of the microheterotrophs if autotroph and heterotroph biomasses are
to remain in constant proportion:
(8.1)
µ
=
µh
µa.(1-η) - r h.η
=
d-1
Under light-limiting conditions, µa is given by (4.4) and so (8.1) is
(8.2)
µ
=
(α.χa.I - ra).(1-η) - r h.η
d-1
where α is photosynthetic efficiency (mmol C fixed (mg chl)-1 d-1 (unit of irradiance)-1) and
χ a is the ratio of chlorophyll to autotroph carbon. (Photosynthetic efficiency is not
subscripted when related to chlorophyll: the same value applies to both autotrophs and the
microplankton because heterotrophs do not add chlorophyll to the microplankton total). For
light-limited autotrophs, respiration rate is a function of growth rate (Eqn. 4.5):
(8.3)
ra
=
r0 a + ba.µa
r0 a
: µa > 0
: µa 0
d-1
It is convenient to assume that autotrophs and heterotrophs are always in analogous
physiological states - i.e. both are either simultaneously nitrogen-limited or simultaneously
carbon-limited at a given time. This excludes some possible combinations of limitation states
(Fig. 8-1). It is not a fundamental requirement, but serves to simplify MP equations, avoid
discontinuities, and reduce the number of logical tests to be made during numerical
simulations. Thus, assuming that heterotroph carbon-limitation always corresponds to
autotroph light-limitation, the simplest form
(8.4)
rh
=
r0 h + bh.µ
d-1
of Eqn. (6.14) can be used for respiration in Eqn. (8.2):
(8.5)
µ
=
(α.χa.I - (r0 a + ba.µa)).(1-η) - (r 0 h + bh.µ).η
d-1
* Table 8.1. Microplankton equations.
MP
page 31
Feb 00
Qmax a = 0.20
heterotrophs C-limited
Qa ≥ qh
autotrophs light-limited
µa = f(I)
qh = 0.18
autotrophs N-limited
µa = f(Qa)
heterotrophs N-limited
Qa < qh
Qmin a = 0.05
axis of Qa, mmol N (mmol C)-1, as food for heterotrophs
µh > 0
µa > 0
µ=0
µh < 0
µa < 0
µ = -r0
axis of microplankton growth rate µ (d-1)
in this region,
autotroph growth in excess of maintenance
supplies maintenance needs of heterotrophs
Figure 8-1. Limitation states of MP . The diagram concerns the assumption that heterotrophs
and autotrophs are always in the same state of limitation. The shaded regions show combinations
of limitation states that are excluded in the interests of simplicity.
Eqn. (8.1) can be re-arranged to give autotroph growth rate:
µa
=
d-1
(µ + rh.η)/(1-η)
Combining this with (8.4) results in:
µa
=
d-1
(µ.(1+bh.η) + r0 h.η)/(1-η)
which can be substituted in (8.5) to give:
µ = (α.χa.I - (r0 a + ba.((µ.(1+bh.η) + r0 h.η)/(1-η)))).(1-η) - (r0 h + bh.µ).η
This is not as complicated as it seems. The autotroph and heterotroph terms can be re-written
as microplankton parameters, leading to:
(8.6)
µ
=
(α.I.χ - r0)/(1+b )
d-1
where the terms are:
the ratio of chlorophyll to microplankton carbon,
χ
=
χ a.(1-η)
mg chl (mmol C)-1
=
 Xq Na.(Qmin - q h.η)
: Q ≤ Q min
 Xq Na.(Q - q h.η) : Q min < Q < Qmax
 Xq Na.(Qmax - q h.η)
: Q ≥ Q max
the basal respiration rate,
r0 = r0 a.(1-η) + r0 h.η.(1+b a)
d-1
and the respiration slope,
b = ba.(1 + bh.η) + b h.η
: µ>0
0
: µ≤0
In this equation, the chlorophyll:carbon ratio has been expanded using the two equivalencies:
χ a = Xq Na.Qa
and
Q = Qa.(1-η) + q h.η.
See below for Qmax and Q min. Finally, the condition when µ ≤ 0 is a simplification of two
conditions (µ ≤ 0 and µa ≤ 0) (see Fig. 8-1†). As a result of the simplifications, Eqn. (8.6) has
essentially the same form as the autotroph equation (4.4), but its parameters take account of
the heterotroph contributions to respiration and biomass.
Under nitrogen-limiting conditions, Eqn. (8.1) becomes:
(8.7)
µ
=
µmaxa.(1 - (Qmin a/Q a)).(1-η) - r h.η
d-1
by inclusion of Eqn. (4.2). Q a is the (variable) autotroph cell quota. Rearrangement of the
Eqn. (3.11.a) for Q gives:
Qa
=
(Q - q h.η)/(1-η)
mmol N (mmol C)-1
Substituting this into (8.7), and writing the result in terms of microplankton parameters,
results in:
†
Figure 8-1. Limitation states of MP.
MP
page 32
Feb 00
µ, d-1
µ (exact: η=0.3)
1.00
µ (approx: η=0.3)
µ (exact: η=0.7)
µ (approx: η=0.7)
0.50
0.00
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
Q, mmol N (mmol C)-1
Figure 8-2. Growth rate. Comparisons of microplankton nutrient-limited growth
according to the more exact equation (8.8) and the approximation (8.9).
µ
=
µmax.ƒ1(Q) - r h.η
d-1
where
ƒ1(Q) =
µmax =
Qmin =
(Q - Q min)/(Q - qh.η)
µ maxa.(1-η)
Qmina.(1-η) + q h.η
: Q ≥ Q min
d -1
mmol N (mmol C)-1
Qmin is the minimum microplankton nutrient quota. µ max is the limit for microplankton
relative growth rate as Q becomes infinite. The final steps are the expansion of the limits and
rh. The simplifying assumption that heterotrophs as well as autotrophs are nitrogen-limited,
allows use of a cut-down version of Eqn. (6.14):
rh
where
=
µh.((1+bh)/q *)-1) + (r0 h/q*)
q*
=
d-1
Qa/q h
Attempting a treatment in terms of microplankton variables:
µ = µmax.ƒ1(Q)/(1 + ((1+b h).ƒ2(Q)) - η) - r0 h.ƒ2(Q)
(8.8)
d -1
where
ƒ1(Q) =
ƒ2(Q) =
(Q - Q min)/(Q - qh.η)
0
q h.η.(1-η)/(Q - q h.η)
q h.η.(1-η)/(Qmin - q h.η)
:Q
:Q
:Q
:Q
≥ Q min
< Q min
≥ Q min
< Q min
Equation (8.8) is complicated, and can be approximated (Figure 8-2 †) as:
µ
(8.9)
=
µmax.ƒ3(Q)
where
µmax =
ƒ3(Q) =
=
Qmin =
µmaxa.(1-η)/(1.2 - η)
d -1
(Q - Q min)/Q
: Q ≥ Q min
1 - (Qmin/Q )
0
: Q < Q min
Qmina.(1-η) + q h.η
mmol N (mmol C)-1
This is the form used in earlier versions of MP (Smith & Tett , in press; Tett 1990b; Tett &
Grenz 1994; Tett & Walne 1995), although the definitions of µmax and Q min were, in most
cases, implicit (i.e. parameter values for algae were adjusted for the presence of
microheterotrophs without taking explicit account of η). ƒ3(Q) is the cell-quota function of
Droop (1968). The term (1.2 - η) in the definition of µmax was obtained empirically, and the
term r0 h.ƒ2(Q) ) for the effect of microheterotroph basal respiration on growth was omitted
as insignificant. No upper limit is applied in ƒ3(Q) because the function's value
asymptotically approaches 1.
†
Figure 8-2. Growth rate.
MP
page 33
Feb 00
The final step is to deal with uptake of nutrients. In MP, autotrophs can take up both nitrate
and ammonium:
(8.10)
u
=
u a.(1-η) - Nrh.η
(NO u a + NH u a).(1-η) - Nrh.η
=
mmol N (mmol C)-1 d-1
It is assumed that nitrate uptake can be inhibited by the presence of ammonium as well as by
a high cell quota, but never involves excretion. Substituting microplankton for autotroph
parameters in Eqn. (4.7) results in:
NOu
(8.11)
=
NOu a.(1-η)
=
NOu max .ƒ(NOS).ƒin(NHS).ƒin1(Q)
where
NOu max
ƒ(NOS)
ƒin(NHS)
ƒin1(Q)
=
=
=
=
Qmax
=
NOu
mmol N (mmol C)-1 d-1
maxa.(1-η)
NOS/(KNOS + NOS)
1/(1 + (NHS/Kin ))
(1-((Q - q h.η )/(Qmax - q h.η))) : Q ≤ Q max
0
: Q > Q max
Qmaxa.(1-η) - q h.η mmol N (mmol C)-1
Qmax is the maximum nitrogen:carbon ratio which should occur in the microplankton. The
nitrate and ammonium functions are meaningful only for S ≥ 0, and ƒin1 (Q) only for
Q ≥ q h.η.
Ammonium uptake is not inhibited by nitrate, but should include the heterotroph excretion
term. It could also be negative if the microplankton nitrogen quota exceeds a maximum.
Rewriting Eqn. (4.7) for ammonium, and with microplankton parameters :
NHu
(8.12)
=
NHu max .ƒ(NHS).ƒin(Q)
- Nrh.η
where
NHu max
ƒ(NHS)
=
=
ƒin2(Q)
Qmax
=
=
NHu maxa.(1-η)
mmol N (mmol C)-1 d-1
 NHS/(KNHS + NHS) : Q ≤ Qmax
1
: Q > Q max
1-((Q - q h.η)/(Qmax - q h.η)) : Q ≥ Q min
Qmaxa.(1-η) + q h.η mmol N (mmol C)-1
The term for ammonium excretion by microheterotrophs can be taken from Eqn. (6.15) by
replacing q* defined with autotroph parameters by q † defined with microplankton
parameters:
Nrh
(8.13)
=
µ.((1+bh).q † - qh) + r0 h.q †)
q†
=
mmol N (mmol C)-1 d-1
where
MP
(Q - q h.η)/(1-η)
qh
page 34
: Q ≥ qh
: Q < qh
Feb 00
10.0
(a) microplankton chlorophyll
η = 0.3
8.0
mg chl m-3
6.0
MP
MP+Nrh
4.0
2.0
η = 0.7
0.0
0.0
1.00
2.0
4.0
6.0
8.0
10.0
MP
0.80
MP+Nrh
(b) dissolved ammonium
0.60
mmol NH4+ m-3
0.40
η = 0.7
η = 0.3
0.20
0.00
0.0
1.00
2.0
4.0
6.0
8.0
10.0
mmol NH4+ m-3 d-1
MP
0.80
MP+Nrh
η = 0.3
0.60
(c) ammonium uptake flux
0.40
0.20
η = 0.7
0.00
0.0
2.0
4.0
6.0
8.0
day
10.0
Figure 8-3. Test case for ammonium uptake. Equations (8.12) to (8.14)
examined in a system with state variables NHS (initial concentration 1 mmol m-3),
NOS (6 mmol m -3), B (1 mmol m-3) and N (0.15 mmol m-3). Growth according to
Table 8.2 with I = 50 µE m-2 s-1 and all parameters given standard values. There was
a constant mesozooplankton grazing pressure of 0.05 d-1, with 50% ammonium
recycling. Microplankton parameters calculated from standard autotroph and
heterotroph parameters with η = 0.3 and 0.7. The equation at issue is:
NHu
=
NHu
NH
max .ƒ( S ).ƒ in2(Q)
[-
Nr ]
h
the ammonium respiration term being included in MP+Nrh and
excluded from MP. Ammonium uptake flux is shown in the third panel, and was
always positive. The standard simplification, omitting Nrh, was better for lower values of η.
This is complicated because Nrh is a function of two variables, Q and µ(I ,Q). Test cases
(Figure 8-3†) show that uptake predicted by Eqn. (8.12), with Nrh defined by (8.13), does not
becomes negative. This is because excreted ammonium leads to an increase in the ambient
concentration and so gives rise to additional uptake. Thus, in accord with the view that
excreted material (but not respired carbon) is recycled within the microplankton, the default
option in MP assumes that there is no ammonium excretion by the microplankton (except in
special circumstances, see below), and so the definitive equation for ammonium uptake is:
(8.14)
NHu
=
NHu max .ƒ(NHS).ƒin2(Q)
where the parameters and included functions are as in (8.12). This equation does allow for
ammonium to be excreted, but only when Q > Qmax (when ƒin2(Q)< 0). Such a condition is
possible, for example when irradiance is very low and as a result light-controlled growth is
negative because of the effects of respiration. Without allowing (in the model) for nitrogen
excretion, the autotroph (and hence microplankton) cell quota would, in these circumstances,
continue to increase far above the maximum quota. Finally, ƒ(NHS) is meaningful only for
NHS≥0, and ƒ in2(Q) only for Q≥ qh.η. These points, however, are matters to be considered
during numerical simulation rather than as part of the model.
†
Figure 8-3. Test case for ammonium uptake.
MP
page 35
Feb 00
Table 8.2 MP rate equations
growth rate
µ
=
min{µI,µQ}
d-1
µI
=
d-1
µQ
=
(α.I.Xq Na.(Q - q h.η) - r 0)/(1+b )
µmax .(1 - (Qmin/Q))
=
(NO u
NOu
=
=
d-1
uptake rate
u
+ NH u )
mmol N (mmol
C)-1 d-1
(NO3-)
(NH4+)
NHu
NOu max .ƒ(NOS).ƒin(NHS).ƒin1(Q)
NHu max .ƒ(NHS).ƒin2(Q)
[- Nrh]
where
ƒ(NOS)
=
NOS/(k NOS
ƒ(NHS)
=
 NHS/(k NHS + NHS)
1
ƒin(NHS)
=
1/(1 + (NHS/k in))
ƒin1(Q)
=
(1-((Q - q h.η)/(Qmax - q h.η)))
0
ƒin2(Q)
=
1-((Q - q h.η)/(Qmax - q h.η))
[
[ Nrh =
-1
d ] [ where
[
q†
[
+ NOS)
optional
µ.((1+bh).q † - qh) + r0h.q
=
(Q - q h.η)/(1-η)
qh
: Q ≤ Q max
: Q > Q max
: Q ≤ Q max
: Q > Q max
]
mmol N (mmol C)-1d-1 ]
]
: Q ≥ qh
]
: Q < qh
]
variables
I
:
PAR experienced by microplankton,
µE m-2 s-1
Q
NOS
:
:
:
microplankton nitrogen:carbon ratio,
sea-water ammonium concentration,
sea-water nitrate concentration,
mmol N (mmol C)-1
mmol mmmol m-3
Θ
:
temperature experienced by microplankton, °C
NHS
.... continued
(Table 8.2, continued)
parameters
heterotroph fraction,
η
=
Bh/(Ba + Bh)
photosynthetic efficiency,
α
=
k .ε.Φ
yield of chlorophyll from N, Xq N
=
Xq Na.(1-η)
basal respiration rate,
r0
=
r0 a.(1-η) + r0 h.η.(1+b a)
respiration slope,
b
=
b a.(1 + bh.η) + b h.η
0
mmol C (mg chl)-1 d-1( µE m-2 s-1)-1
mg chl (mmol N)-1
d-1
: µ>0
: µ≤0
max. N-limited growth rate, µmax =
minimum N quota,
Qmin =
µmaxa.(1-η)/(1.2 - η)
d -1
Qmina .(1-η) + q h.η
mmol N (mmol C)-1
maximum N quota,
Qmaxa.(1-η) + q h.η
mmol N (mmol C)-1
Qmax =
NOu max
=
NOu maxa.(1-η)
mmol N (mmol C)-1 d-1
NHu max
max. NH4 uptake,
half-sat. conc., NH4 uptake, k NHS
half-sat. conc., NO3 uptake, k NOS
half-sat. conc., NH4 inhibition, kin
=
=
=
=
NHu maxa.(1- η)
mmol N (mmol C)-1 d-1
mmol N m-3
mmol N m-3
mmol N m-3
max. NO3 uptake,
autotroph value
autotroph value
autotroph value
temperature effects
max. growth rate,
d-1
µmaxa
=
µmaxa[20°C].ƒ(Θ)
max. uptake rate,
d-1
u maxa
=
u maxa[20°C].ƒ(Θ)
temperature effect:
ƒ(Θ)
=
exp(k Θ.(Θ - 20°C)
This concludes the derivation of the microplankton equations that commenced in Section 3.
These growth and uptake equations are summarised in Table 8.2*, together with the
definitions of microplankton parameters in terms of autotroph and heterotroph parameters.
These definitions allow parameter values to be calculated from Tables 4.3 and 6.4, given a
value of η. Under the assumption of constant η, a microplankton parameter has a constant
value so long as there is constancy in the algal and heterotroph values from which it is
derived. Numerical simulations of micorplankton rgowth thus require micorplankton
parameter values to be calculated once only, before the start of numerical integration. This is
exemplified in the Appendix.
* Table 8.2. MP rate equations and parameter definitions.
MP
page 36
Feb 00
µM organic carbon
(a) particulate carbon components
102
POC
MPC
101
ppC
0
10
20
30
days
40
3.0
(b) pigment ratios
2.0
µg chl : µM MPN
1.0
µg chl : µM ppC
pheo : total
0.0
0
10
20
30
40
days
Figure 9-1. Results from a microcosm experiment.
Total particulate organic carbon and nitrogen (POC and PON) were determined after combustion.
Photosynthetic pigments (pheo = 'pheopigment' and chl = 'chlorophyll') measured by fluorescence.
Phytoplankton organic carbon (ppC) was estimated by microscopy.
Microplankton organic matter (MPC and MPN) estimated from corresponding POM × chl/(chl + pheo).
9. Comparison of MP simulation with microcosm data
Jones et al., (1978a; 1978b) and Jones (1979) report results from a microcosm experiment
('H' of a series) which allows a test of MP. The microcosm was filled with 19 dm3 of 200 µmscreened water taken from the Scottish sea-loch Creran in July 1975, enriched with nitrate,
silicate, vitamins and trace minerals, and incubated in a water bath at 10°C, corresponding to
sea water temperature at the time. It was exposed, through a green filter, to light from a
north-facing window, approximating in amount and colour balance the irradiance a depth of
4 m in the loch. The contents of the microcosm were gently stirred, and continuously diluted
at 0.21 d-1 with 0.45 µm-filtered, nutrient-enriched, water taken from the loch at intervals
during the experiment. The aim of the dilution was to reduce wall effects and remove detritus
and populations of non-dividing cells.
Samples were taken regularly for measurement of pigments and organic carbon, nitrogen and
phosphorus retained on glass fibre filters. 'Chlorophyll' and chlorophyll-equivalent
pheopigments were extracted into 90% acetone and measured by fluorescence before and
after acidification (Holm-Hansen, et al. 1965). What was called 'chlorophyll' includes
chlorophyll a and chlorophyll-a equivalent amounts of some other pigments, especially
chlorophyllide a (Gowen, et al. 1983). C and N were measured after combustion in an
elemental analyser, and P by wet oxidation to phosphate (Tett, et al. 1985). Concentrations of
dissolved nutrients were measured at the start of the experiment and in the vessel supplying
the diluent. Water samples were preserved with Lugol's iodine and analysed with an inverted
microscope for abundance and mean size of phytoplankton and protozoa (Tett 1973).
Phytoplankton abundances given in Jones (1979) have been converted to carbon
concentrations using mean cell volumes observed during the experiment, and carbon contents
of 0.12 pg C µm-3 for diatoms and 0.18 pg C µm -3 for other microplankters .
The experiment had three phases during 6 weeks (Figure 9-1 †). During the first phase (up to
day 8), initial, relatively high, concentrations of detritus were diluted, most microplankters
decreased in abundance and some species disappeared. During phase 2, at least 5 species of
diatoms increased in parallel as the growth-limiting element switched from nitrogen to
phosphorus. At the end of this phase there remained 10 taxa from the 15 recorded at the start
of the experiment, the most important loss being ciliates, replaced by zooflagellates. At the
start of phase 3, on day 30, the microcosm reactor and reservoir were enriched with
phosphate. The resulting increase in biomass confirmed control by phosphorus.
†
Figure 9-1. Results from a microcosm experiment.
MP
page 37
Feb 00
0.60
0.40
0.20
(a) heterotroph fraction η
0.00
0
10
20
30
40
days
0.200
0.0200
P:C
N:C
PON/POC
0.150
0.0150
0.100
0.0100
POP/POC
0.050
0.0050
(b) nutrient quotas
molar ratios to carbon
0.000
0
10
20
0.0000
40
30
days
Figure 9-2. The microcosm experiment, continued.
(a) The heterotroph fraction η estimated from hetC/MPC for days 0 - 8 (where hetC is microscopically
heterotroph carbon), then from (MPC-ppC)/MPC.
(b) Microplankton nutrient quotas, estimated as Q = PON/POC and P Q = POP/POC.
Total particulate organic carbon and nitrogen (POC and PON) were determined after combustion.
Total particulate organic phosphorus (POP) was determined by wet oxidation.
On the basis of other work in Creran, detritus probably made up at least half the initial
particulate organic carbon (POC). Although about 80% of this detritus should have been
removed by dilution during phase 1, the continued presence of pheopigments, and their
increase during phase 2, suggests that new detritus was formed, either as a result of protozoan
grazing or of cell death. Nevertheless, a continuing decrease in the ratio of pheopigment to
total pigment (Fig. 9-2† ) indicated a diminishing ratio of detritus to microplankton. Estimates
of microplankton particulates( carbon, MPC; nitrogen, MPN; and phosphorus, MPP) were
made by multiplying POC, PON or POP by the ratio of 'chlorophyll' to total pigments
('chlorophyll' plus pheopigments) on the assumption that fresh detritus had the same
C:N:P:pigment ratio as microplankton and contained pheopigment instead of 'chlorophyll'.
The methods used for microscopy revealed only some photosynthetic and heterotrophic
picoplankton, and might have underestimated protozoans because of failure to preserve the
most fragile naked dinoflagellates and ciliates. Other studies (Tett, et al. 1988) have,
however, shown that picophytoplankton make only a small contribution to phytoplankton
biomass in these waters, which are, generally, dominated by diatoms. Microscopic estimates
of phytoplankton carbon (ppC) may thus be considered reliable. Microscopic estimates of
microheterotroph carbon (mhC) were, however, mostly lower than estimates from the
difference between MPC and ppC, and the heterotroph fraction η was estimated as (MPCppC)/MPC from day 8 onwards. Values of η decreased from about 0.4 to a minimum of 0.2,
rising again towards 0.6 by day 35.
The ratio of 'chlorophyll' to microscopically estimated phytoplankton carbon remained
constant (apart from measurement error) at about 0.5 mg chl (mmol C)-1 during phases 1 and
2, increasing to more than 1 mg mmol-1 after day 35. It is possible that the biomass
contributions of the dominant diatoms (Cerataulina pelagica and Leptocylindrus danicus)
were underestimated at this time. If so, the values of η after day 35 would be unreliable. Data
used for comparison with simulation are thus taken from days 8 to 35 (inclusive) only.
Model
The model needed to take account of P limitation. The full set of equations for state variables
in version MP+P was:
(9.1.a) carbon:
dB/dt =
(µ - D ).B
mmol C m-3 d-1
(9.1.b) nitrogen:
dN /dt =
(NOu - D.NQ).B
mmol N m-3 d-1
(9.1.c) phosphorus:
dP/d t =
(POu - D.P Q).B
mmol N m-3 d-1
(9.1.d) nitrate:
dNOS/dt =
- NOu.B - D.(NOS - NOS r) mmol N m-3 d-1
†
Figure 9-2. The microcosm experiment, continued.
MP
page 38
Feb 00
(9.1.e) ammonium:
dNHS/dt =
- NHu.B - D.(NHS - NHS r) mmol N m-3 d-1
(9.1.f) phosphate:
dPOS /dt =
- POu .B - D.(POS - POS r)
(9.1.g) quota:
NQ
=
N /B
(9.1.h) chlorophyll:
X
=
Xq N.N
PQ
and
= P/ B
mmol P m-3 d-1
mmol (mmol C)-1
mg chl m-3
where D is dilution rate and Sr is the concentration in the reservoir from which diluent is
drawn. Because silicate was added in excess, its dynamics were ignored. There was no
grazing term because mesozooplankton had been removed by the 200 µm screen. Because of
the absence of simulated grazing, there was no formation of detritus during simulations. The
mineralisation of the observed detritus was assumed to be slow, and hence was ignored.
Extra, or modified, rate equations were
µ
(9.2)
=
min{µ(I ), µ(NQ), µ(P Q)}
d-1
where
POu
(9.3)
µ(P Q) =
µmax .(1 - (P Qmin/P Q ))
PQ
=
P/ B
=
POu max .ƒ(POS ).ƒin1(P Q)
d-1
mmol P (mmol C)-1
[- Prh]
mmol P (mmol C)-1 d-1
where
ƒ(POS ) =
ƒin1(P Q) =
 POS /(KPOS + POS )
: PQ ≤ PQmax
1
: PQ > P Qmax
1-((P Q - P q h.η)/( P Qmax - P q h.η))
The phosphate uptake equation allows excretion when P Q > P Qmax. In addition, an optional
'phosphorus respiration' term in (9.3) was expanded in the same way as (8.13) for Nrh:
P rh
(9.4)
=
µ.((1+bh).q † - qh) + r0 h.q †)
q†
=
mmol P (mmol C)-1 d-1
where
(P Q - P q h.η)/(1-η)
Pq h
: PQ ≥ P q h
: PQ < P q h
Finally, it was thought that the strong phosphorus-limitation designed into the experiment
might influence nitrate uptake, and hence an optional modification was added to Eqn. (8.11):
(9.5)
NOu
where
ƒin3(P Q)
MP
=
NOu max .ƒ(NOS).ƒin(NHS).ƒin1(Q)
=
(P Q - P Qmin)/(P Qmax - P Qmin)
1
page 39
[.ƒin3(P Q)]
: µ(P Q)<µ(NQ)<µ(I )
: µ(P Q)>µ(NQ)
Feb 00
Table 9.1. Phosphorus submodel, and other non-standard, parameters in
MP+P
Parameter
Jones et al.
(1978b) value,
from total
particulates
P Qmax
P Qmin
1.02
Pq h
Recalculated
from MP
particulates
After fitting
units
10.5
15
mmol P (mol C)-1
1.3
1.5
mmol P (mol C)-1
mmol P (mol C)-1
P Qmin/η
P u max
2.4
@ 10°C 5
@ 20°C 6
@ 20°C mmol P (mol C)-1 d-1
10(with excr.)
11(with excr.)
KPO
0.21
-
Xq N
2.16 - 3.04
(Gowen et al.,
1992)
2.8
Xq Na
0.21
mmol P m-3
mg chl (mmol N)-1
3.0
3.7 (η = 0.25)
mg chl (mmol N)-1
Table 9.2. Forcing for the microcosm simulations
symbol for
quantity
description of quantity
value(s) or equation
units
NOS r
reservoir nitrate
100
µM
POS r
reservoir phosphate
0.28 (day 0-19), 0.20 (day 2030.4), 0.70 (day 30.5-40)
µM
I
reactor PAR
I 0 .(1 - exp(-ζ))/ζ
µE m-2 s-1
I0
external PAR
50
µE m-2 s-1
ζ
reactor mean optical
pathlength
z.(λ w + ε.X)
z
reactor mean photon
pathlength
0.1
m
λw
seawater attenuation
coefficient
0.2
m-1
D
reactor dilution rate
0.21
d-1
Θ
temperature in reactor
10
°C
Phosphorus submodel and other non-standard parameter values
Initial values of PQmin and PO u max were estimated from microplankton particulates,
following the procedures used by Jones et al. (1978a) (for total particulates). PQmax was
taken as the largest observed ratio of POP to POC. KPOS had the value given by Jones et al..
The only parameter that caused difficulty was P q h, the microheterotroph ratio of P to C. The
Redfield ratio is 9.4 mmol P (mol C)-1, and the results of Caron et al. (1990) suggest
optimum values of 13.3 mmol P (mol C)-1 for protists. Substituting such values into
PQ
mina = ( P Qmin - P q h.η)/(1-η)
together with the initial value of 1.3 used for PQmin, gave P Qmina < 0 for most values of η.
As the measurements of POP and POC used to estimate the phosphorus subsistence quota
seem reliable, the implication is that some microheterotrophs must have had phosphorus
content much below the Redfield value. There is evidence that bacteria can grow with PQb
less than 1 mmol P (mol C)-1 (Currie & Kalff 1984). Whatever the explanation, the difficulty
for the simulations was that (P Q - P q h.η) in equations such as (9.3) could not be allowed to
go negative. The solution was to set Pq h.η equal to P Qmin, implying qh of 4.1 mmol P (mol
C)-1 for η of 0.25.
Some phosphorus parameters were adjusted to minimise the MP+P prediction error with η of
0.25. A plot of 'chlorophyll' against MPN for days 8-35 had a slope of 2.8, which provided
the initial estimate of XqNa ( approximated by XqN/(1-η)) of 3.7 mg chl (mmol N)-1,
adjusted to 3.0 by fitting. The adjusted values (Table 9.1*) were used in all other simulations.
Initialisation and external forcing
Simulations were initialised on day 8, at the beginning of phase 2, with values for the
particulate state variables B, N and P set to MPC, MPN and MPP estimated from observed
particulates. Values of external forcing variables are given in Table 9.2*. The simulations
used a retrospectively estimated approximate 24-hr mean value of 50 µE m-2 s-1 for I ; no
account was taken of natural variation in light, although self-shading, which brought about a
maximum reduction in mean irradiance of 10%, was simulated. However, PAR was not
limiting in any simulation reported here.
* Table 9.1. Phosphorus and other non-standard parameters in MP+P.
* Table 9.2. External forcing.
MP
page 40
Feb 00
µM MPC
250
(a) microplankton carbon
η = 0.20
observations
200
η = 0.25
with rh
150
η = 0.30
100
50
0
10
20
30
µM MPN
days
40
25.0
(b) microplankton nitrogen
20.0
η = 0.20
observations
η = 0.25
η = 0.30
with rh
15.0
10.0
5.0
0.0
10
20
30
days
40
Figure 9-3. MP+P simulations of particulates in the microcosm.
The lines show concentrations of microplankton components predicted during days 8-35
by MP+P simulations with η = 0.20, 0.25 and 0.30, and also by MP+P+rh with η = 0.25.
The points show concentrations deduced from observations.
a) Simulated B and observed (MPC) microplankton organic carbon.
(b) Simulated N and observed (MPN) microplankton organic nitrogen.
Numerical methods
The equations of MP+P were numerically integrated using repeated Euler forward difference
solutions, in the form of a Pascal program given (in part) in the Appendix. Standard timestep was 0.01 day; results were not meaningfully different with ∆t of 0.005 and 0.02 day.
Making allowance for dilution, the simulations conserved total nitrogen and phosphorus
exactly.
Agreement between values of a simulated Y' and observed Y variable was assessed by
calculating a sum of squares of the differences between corresponding logarithms:
SOSD(Y) = ΣnY( log10(Y' [t j]) - log10(Y [t j]) ) 2
where there were j = 1 to nY observations in the time-series Y (t). The logarithmic
transformation was used to normalise and homogenise errors, and to allow the assessment of
an overall goodness of fit to concentrations varying with time by an order of magnitude and
between variables by several orders of magnitude. A root-mean-square prediction error was
calculated for each variable from
s (Y) = √(SOSD(Y)/n Y)
The observations were those of concentrations of MPC, MPN, MPP and 'chlorophyll',
corresponding to model state variables B, N, P, X. An overall error was calculated from
s = √((SOSD(B)+SOSD(N )+SOSD(P )+SOSD(X))/(n B +nN+ nP +nX))
Results
Figures 9-3† and 9-4† shows the results of running simulations for standard MP+P with η of
0.20, 0.25 and 0.30.
†
Figure 9-3. MP+P simulations of particulates in the microcosm.
†
Figure 9-4. Simulations of particulates, continued.
MP
page 41
Feb 00
µM MPP
0.50
(c) microplankton phosphorus
observations
0.40
η = 0.25
with rh
0.30
η = 0.20
0.20
η = 0.30
0.10
0.00
10
20
30
days
40
µg chl l-1
(d) chlorophyll
60
η = 0.20
observations
η = 0.25
with rh
η = 0.30
40
20
0
10
20
30
days
Figure 9-4. Simulations of particulates, continued.
(a) Simulated P and observed (MPP) microplankton organic phosphorus.
(b) Simulated X and observed (chl) microplankton chlorophyll.
40
d-1
0.60
from observed ∆ln(MPC)/∆t
with rh
η = 0.25
η = 0.20
0.40
η = 0.30
0.20
(a) microplankton growth rate, µ
0.00
10
20
30
days
40
standardised cell quota
6.0
P Q/P Q
min
(b) standardised cell quotas at η = 0.25
4.0
2.0
NQ/NQ
min
0.0
10
20
30
days
40
Figure 9-5. MP+P simulations of µ and Q in the microcosm .
MP+P and MP+P+rh simulations compared with values deduced from microcosm observations:
(a) microplankton growth rate µ compared with values estimated from ∆ (ln(MPC))/∆ t + D;
(b) standardised cell quotas (Q/Qmin ) compared with estimates from ratios of particulate elements:
Qmin = 0.0825 mmol N (mmol C)-1 and P Qmin = 1.5 mmol P (mol C)-1.
value of statistic
0.40
B and [MPC]
all
N and [MPN]
0.30
X and [chl]
P and [MPP]
B and [MPC]
X and [chl]
0.20
all
N and [MPN]
P and [MPP]
0.10
all (+rh)
0.00
0.00
0.10
0.20
0.30
0.40
η
0.50
Figure 9-6. Fit of MP+P with changes in η.
Effect of the assumed value of the heterotroph fraction η on the fit of MP+P simulations
to observations. Additional points show best fit for option MP+P+rH with heterotroph excretion.
Figure 9-5† shows extent of agreement between observed and simulated microplankton
growth rate.
Figure 9-6† gives the fitting errors as a function of η, showing best overall fit, with prediction
error of 0.108, at η of 0.27. Results for B and N were especially satisfactory (with s(B)
falling to 0.079 at η = 0.27 and s(N ) to 0.068 at η = 0.29). Despite adjustment to Xq Na,
chlorophyll prediction were less good, with s(X ) showing a minimum of 0.105 (η = 0.25).
This may reflect greater day-to-day variability in pigments.
Whereas the fits for carbon, nitrogen and chlorophyll show clear minima, that for phosphorus
did not. This was a result of using η-independent parameters in the phosphorus sub-model. It
is clear that this sub-model is less satisfactory than that for carbon and nitrogen, since s (P)
was 0.148 at η = 0.25, despite preliminary adjustment of most P-model parameters.
Figures 9-3 through 9-6 also show the effect of adding terms (equations 8.14 and 9.4) for
heterotroph mineralisation of nitrogen and phosphorus. Simulation with MP+P+rh, including
excretion, required recalculation (Table 9.1) of the value for maximum uptake rate of
phosphate, since the previous calculation did not take account of remineralisation losses. Best
fit was now obtained with η of 0.25, giving a slightly improved overall error of 0.107. s(N )
reached a minimum of 0.066 at η = 0.27. Simulated free ammonium reached a maximum of
0.04 µM.
The standard runs of MP+P included suppression of nitrate uptake by limiting phosphorus
(Eqn. 9.5). Removing this coupling gave a best fit at η of 0.27 and increased the overall error
to 0.114.
†
Figure 9-5. MP+P simulations of µ and Q in the microcosm.
†
Figure 9-6. Fit of MP+P with changes in η.
MP
page 42
Feb 00
Table 10.1. Phosphorus submodel parameters for autotrophs and
heterotrophs in AH
Parameter
value
P Qmaxa
15
mmol P (mol C)-1
same as P Qmax
P Qmina
1.5
mmol P (mol C)-1
same as P Qmin
Pq h
6
mmol P (mol C)-1
P Qmin/η
P u maxa
KPO
14.7 @ 20°C
0.21
units
mmol P (mol C)-1
d-1
mmol P m-3
how calculated (with η = 0.25)
P u max /(1-η)
including excretion
10. An unconstrained autotroph-heterotroph model (AH)
The behaviour of MP is constrained by the constant ratio of autotrophs to heterotrophs. This
constraint was relaxed in the model AH, which used the autotroph rate equations of Table 4.1
and the heterotroph rate equations of Table 6.1. The differential equations for AH were
(10.1.1.a) autotroph carbon:
dBa/dt =
(µa - ch'.Bh - D ).Ba
(10.1.1.h) heterotroph carbon:
dBh/dt =
(ch'.Ba - rh - D ).Bh
(10.1.2.a) autotroph nitrogen:
dN a/dt =
(u a/Q a - ch'.Bh - D ).Na
(10.1.2.h) heterotroph nitrogen:
dN h/dt =
(ch'.N a - Nrh).Bh - D .Nh
(10.1.3.a) autotroph phosphorus:
dPa/dt =
(P u a/PQa - ch'.Bh - D ).Pa
(10.1.3.h) heterotroph phosphorus:
dPh/dt =
(ch'.Pa - P rh).Bh - D .Ph
(10.1.4) dissolved ammonium:
dNHS/dt =
- NHu a.Ba + Nrh.Bh + D.(NHS r - NHS)
(10.1.5) dissolved nitrate:
dNOS/dt =
- NOu a.Ba + D.(NOS r - NOS)
(10.1.5) dissolved phosphate:
dPOS /dt =
- POu a.Ba + P rh.Bh + D.(POS r - POS )
(10.1.6) chlorophyll:
X
Xq Na.N a
=
mg m-3
All the rates of change have units of mmol m-3 d-1. These equations, and the autotroph and
heterotroph rate equations, were included as alternatives in the Pascal numerical simulation
program used for MP+P, and drew on the same autotroph and heterotroph parameter values
(Tables 4.3 and 6.4). AH simulations were initialised as for MP+P, assuming an initial
heterotroph fraction of 0.25. This fraction was also used to calculate values of autotroph and
heterotroph parameters for phosphorus dynamics (Table 10.1 *) in cases where microplankton
parameter values had been obtained or fitted directly, as described in section 9.
* Table 10.1. Phosphorus parameters in AH.
MP
page 43
Feb 00
µM MPC
250
c h = 0.05 m3 (mmol C)-1 d-1
(a) microplankton carbon
200
observations
150
c h = 0.10
100
50
c h = 0.20 m3 (mmol C)-1 d-1
0
10
20
30
days
40
µg chl l-1
(b) chlorophyll
60
observations
c h = 0.05 m3 (mmol C)-1 d-1
40
20
c h = 0.10
c h = 0.20 m3 (mmol C)-1 d-1
0
10
20
30
days
40
Figure 10-1. AH simulations of particulates in the microcosm.
Simulations with the unconstrained autotroph-heterotroph model AH, for ch = 0.05, 0.10 and 0.20
m3 (mmol C)-1 d-1 (at 20°C), compared with values deduced from observations. Time-series of:
(a) simulated microplankton biomass (B = B a + B h) compared with MPC;
(b) simulated chlorophyll (X = X a) compared with [chl].
The only parameter value not already used by MP+P was the heterotroph transfer coefficient,
or volume clearance rate, c h'. The equation pair 10.1.1 resembles a set of Lotka-Volterra
predator-prey equations. In order to increase the possibility of a stable equilibrium in such a
system (Hastings 1996), the transfer coefficient was given a Holling (Holling 1959) type II
functional response:
(10.2)
ch'
=
ch/(1 + (Ba/K Ba))
where c h is the maximum clearance rate at the given temperature (Section 6).
The food concentration KBa for half saturation of ingestion was taken as 50 mmol autotroph
C m-3. Figure 10-1† compares the results of simulations using ch of 0.20, 0.10 and 0.05
m3 (mmol C)-1 d-1 (at 20°C) with values deduced from the microcosm observations described
in Section 9. The 'standard' value of 0.2 for maximum clearance rate (actually close to 0.1 m3
(mmol C)-1 d-1 at the temperature of the microcosm) resulted in a very poor fit.
†
Figure 10-1. AH simulations of particulates in the microcosm.
MP
page 44
Feb 00
µM heterotroph C (model B h or hetC)
observations
(a) Plots of B h against B a simulated for days 8-35
with the unconstrained autotroph-heterotroph model AH,
for c h = 0.05, 0.10, 0.20 m3 (mmol C)-1 d-1 (at 20°C).
Continuous line shows results from best fit MP+P
with η ( = B h/(B a+B h)) = 0.25.
102
MP+P: η = 0.25
101
c h = 0.10
c h = 0.20
100
c h = 0.05
100
101
µM autotroph C (model B a or observed ppC)
102
heterotroph biomass B h in µM C
60
b) Nullcline analysis of AH .
nullclines for c h=0.10 (0.20 at 20°C)
50
nullclines for c h=0.025 (0.05 at 20°C)
40
30
20
10
0
0
40
80
120
160
autotroph biomass B a in µM C
Figure 10-2. Phase plots of AH and MP simulations and microcosm results.
Observations shown by small squares in (a): autotroph biomass from microscopic observations (ppC);
heterotroph biomass from MPC-ppC. In (b), B maxa was taken as 186 mmol C m-3,
from POS r/P Qmin a, and µmax a was 1.0 d-1 at 10°C
Figure 10-2(a)† shows plots of heterotroph biomass against autotroph biomass for simulations
with the same values of c h. The diagonal line is the corresponding plot from MP+P with η =
0.25, with overall fitting error (for MPC, MPN, MPP and chlorophyll) of 0.108. Even the best
fit of AH, with c h of 0.09 m3 (mmol C)-1 d-1 (at 20°C), gave an overall error of 0.241.
Nullcline analysis
Equations (10.1.1) can be made the subject of nullcline analysis (Hastings 1996), given the
replacement of µa(I ,Q) by a simplified density dependence:
(10.3.a)
dBa/dt
= 0 =
(µmaxa.(1 - (Ba/Bmax a)) - ch'.Bh - D ).Ba
(10.3.h)
dBh/dt
= 0 =
(a.ch'.Ba - D ).Bh
where a = 1/(1+b h), which neglects r0 h. Taking into account the saturation function of Eqn.
(10.2) for ch', these equations lead to the nullclines:
Ba = 1/((a.c h/D) - (1/K Ba))
Bh =
: dBh/dt =0
(µmax a.(1 - (Ba/Bmax a)) - D ).(1 + (Ba/K Ba))/c h : dBa/dt =0
which are plotted in Figure 10-2(b) for several values of ch. The nullclines intersect on the
rising part of that for Bh, which suggests (Hastings 1996) that any equilibrium should be
unstable. Although this does not take account of the other processes described by equations
10.1.2 to 10.1.5, it does help to explain the oscillatory behaviour shown in Figure 10-2,
especially that displayed by the simulation with ch of 0.20 m3 (mmol C)-1 d-1 (at 20°C).
†
Figure 10-2. Phase plots of AH and MP simulations and microcosm
results.
MP
page 45
Feb 00
11. Discussion
The most crucial assumption of MP is that of a constant ratio of heterotrophs to
autotrophs. A comparison of the results of the simulations in sections 9 and 10 would
seem to justify this assumption. The unconstrained autotroph-heterotroph model AH failed to
simulate, adequately, the time-courses of state variables during the microcosm experiment.
In contrast, the microplankton model MP was able to describe the observations quite well,
despite - or, perhaps, because of - assuming a fixed ratio of autotrophs to heterotrophs.
AH may be too simple to represent, properly, the heterogeneous nature of the experimental
microplankton. A system comprising several species at each trophic level may form an more
stable trophic web than implied by the quasi-Lotka-Volterra dynamics of AH, so long as the
protozoan consumers are catholic in their diet and able to switch between favoured foods.
Fasham et al. (1990) were able to increase the robustness of FDM by assuming that the
zooplankton compartment grazed more, at a given time, on the more abundant of
phytoplankton, bacteria or detritus. "This assumption leads to a positive switching… which
has a stablising effect on the predator-prey interaction …" (Fasham, et al. 1993). (Taylor &
Joint 1990) fitted a steady state microbial loop model to data from the Celtic Sea in summer,
finding change in steady-state parameters during the course of the summer but support for the
use of the steady state for any given time.
AH and MP are fairly compared because both models drew on the same set of parameter
values for autotrophs and heterotrophs. They combined them, however, in different ways. In
the case of MP, the ratio of autotroph to heterotroph biomass was fixed during any one
simulation, resulting in particular values of the microplankton parameters that were derived
from the autotroph and heterotroph parameters. The value of η was varied between model
runs to improve the fit of simulations to observations. In the case of AH, the relative
biomasses of autotrophs and heterotrophs were free to change. Thus, autotroph and
heterotroph parameter values were, in effect, combined dynamically, as the implicit value of
η was changed by the model during a simulation. In contrast with MP, which had to be
supplied with a value of η but made no use of the trophic transfer coefficient c h, AH
simulations used η only to calculate initial values of autotroph and heterotroph biomasses but
required a value for c h. The clearance coefficient's value was varied in order to improve the
fit of AH simulations to observations, but gave no fit as good as that obtained with MP.
MP
page 46
Feb 00
Table 11.1. Estimates of the heterotroph fraction
Site
Season
autotroph
mmol C
m-3
heterotroph
mmol C
m-3
microplankton
mmol C
m-3
Scottish coastal:
Easdale Quarry
Scottish coastal:
Loch Creran
English
Channel: mixed
English
Channel:
stratified
Canadian
coastal:
CEPEX
May August
whole
year
July
8.1
3.6
11.7
0.31
6.6
3.8
10.4
0.36
6.8
1.6
8.2
0.19
July
1.4
2.0
3.4
0.58
July
-August
10.4
1.3
11.7
0.11
(Williams 1982)
mmol C
m-2
mmol C
m-2
mmol C
m-2
71
262
0.27
Gasol et al.
(1997)
135
299
0.45
Gasol et al.
(1997)
Area-integrated
gobal means
'Coastal'
euphotic zone or
similar
'Open Ocean'
euphotic zone or
similar
mean of 191
all data (n
82)
mean of 164
all data (n
119)
η
Reference
Tett et al.
(1988)
Tett et al.
(1988)
Holligan et al.
(1984)
Holligan et al.
(1984)
The value of η = 0.25 giving the best fit of MP+P was within the range (0.15 - 0.52)
calculated from the microcosm observations between days 8 and 35. It may also be
compared with estimates derived from the literature (Table 11.1*).
There is, clearly, no universal value of η. Furthermore, the data in Table 11.1, and the
existence of several sets of trophic pathways amongst plankton (Legendre & Rassoulzadegan
1995), suggest that the value of η should change seasonally in temperate waters, with low
values during the early stages of diatom-dominated Spring bloom and higher values when a
recycling, Microbial-Loop, community is established in Summer. There is analagous
variability in space (Holligan, et al. 1984; Richardson, et al. 1998). MP as it presently stands
cannot deal with such seasonal or spatial changes, but Tett & Smith (1997) have described a
'two-microplankton' model, which allows the value of η to change dynamically over a range
set by the value in each of the two microplanktons.
The second most important assumption of MP is that of complete internal recycling.
Under this assumption it is implicit that DOM is excreted by phytoplankton, or leaked by
protozoa, and that protozoa excrete ammonium. However, these processes are not described
explicitly because the excreted material is supposed to be rapidly and completely reassimilated by microplankton components. This assumption was explored for the microcosm
case by including the '+rh' option (for ammonium, and phosphate, excretion and re-uptake) in
some simulations. In these cases there was insufficient improvement in the fit of the
simulations to the observations to justify the inclusion of the excretion option. The more
general results in Fig. 8-3 suggest, however, that the option might be worth including in
simulations with a higher value of η than used in the microcosm case.
The comparisons with FDM and ERSEM (Sections 5 and 7) show that, although these
models differ substantially on structure from each other and from MP, some of the more
important microplankton parameters have broadly similar values in all the models, when
expressed in comparable terms. The models' differences in the control of autotroph growth
are obvious. The other striking difference proves to be the direct production of detritus by
microplankton in FDM and ERSEM. The observations in the microcosm by Jones et al.
suggest that this process should be included in the Microplankton-Detritus model. Although
its absence did not prevent MP from well fitting observed microplankton carbon and nitrogen
during the microcosm experiment, it may be important for detrital production in the sea.
Did the microcosm data provide a good test of MP? Jones et al. (1978b) argued that the
behaviour of the experimental microplankton was natural in respect of maintaining "the
occurrence of typical sea-loch species, their growth in parallel with the same species in the
* Table 11.1. The heterotroph fraction estimated from published data.
MP
page 47
Feb 00
loch [from which they were innoculated] …, the coherence of growth curves … and …
moderate diversity". The model was successful in simulating a biomass increase of an order
of magnitude during an experiment in which growth rates varied as a result of changes in
available nutrient. Although the phosphorus submodel added to MP had some deficiencies,
MP predictions for carbon and nitrogen remained accurate during the switch from N to P
limitation. In these respects, MP has been severely tested. Against this claim, it might be
argued that the range of biomasses observed in the microcosm lay at the upper end of the
natural range, and that a scenario involving a shift from N to P limitation would be unusual
for natural marine plankton. The most important switch in limitation for microplankton in
temperate waters is from light to nutrient control during the transition from Spring to Summer
conditions. The model L3VMP, which combines MP, detritus and a physical sub-model, has
been successfully used to simulate the Spring Bloom in the central North Sea (Tett & Walne
1995). Even so, it will be desirable to test MP with experimental data which includes a shift
from light to nitrogen control of growth.
Physical models of the sea are based on well-known equations of motion, even in cases when
the products of chaotic fluctuations in velocity and concentration are approximated by
Fickian diffusion. Although the equations describing simple biological systems also show
chaotic tendencies, it is as yet uncertain how much of the variability in natural marine
ecosystems derives from sum of the 'simple chaos' of many subsystems, and how much from
fluctuating physical forcing. As Hastings (1996) remarks, many biological systems have
been observed to be more stable than would be expected if they behaved according to LotkaVolterra dynamics. One explanation for such stability would be a system with many
alternative pathways, for which the Internet would be a more suitable metaphor than the
billiard table of classical physics. If this be the case, then the relatively simple mathematics
which have proven successful in describing physical systems, cannot be used with biological
systems. Nevertheless, there may be a bulk level of analysis at which some important
properties of marine pelagic ecosystems can be captured by small sets of relatively simple
differential equations. The ability of MP to simulate events during the microcosm experiment
of Jones et al. (1978a), or during the seasonal cycle in the central North Sea (Tett, et al. 1993;
Tett & Walne 1995), encourages this view.
Demonstrating such points about marine ecosystems made up of pelagic micro-organisms
was not the original aim of the work reported here, and I do not wish to make strong claims
on the basis of the ability of MP to fit some sets of observations. Nevertheless, it is a
convenience to be able to use a simple model to simulate important bulk properties of these
ecosystems.
MP
page 48
Feb 00
12. References
Azam, F., Fenchel, T., Field, J. G., Gray, J. S., Meyer-Reil, L. A. & Thingstad, F. (1983). The
ecological role of water-column microbes in the sea. Marine Ecology - Progress
Series, 10 , 257-263.
Baretta, J. W., Admiraal, W., Colijn, F., Malschaert, J. F. P. & Ruardij, P. (1988). The
construction of the pelagic submodel. In Tidal flat estuaries. Simulation and analysis
of the Ems estuary (ed. Baretta, J. W. & Ruardij, P.), Ecological Studies, 71 , 77-104.
Springer-Verlag, Heidelberg.
Baretta, J. W., Ebenhöh, W. & Ruardij, P. (1995). The European Regional Seas Ecosystem
Model, a complex marine ecosystem model. Netherlands Journal of Sea Research,
33 , 233-246.
Baretta-Bekker, J. G., Baretta, J. W. & Ebenhöh, W. (1997). Microbial dynamics in the
marine ecosystem model ERSEM II with decoupled carbon assimilaion and nutrient
uptake. Journal of Sea Research, 38 , 195-211.
Baretta-Bekker, J. G., Baretta, J. W. & Rasmussen, E. K. (1995). The microbial food web in
the European Regional Seas Ecosystem model. Netherlands Journal of Sea Research,
33 , 363-379.
Baumert, H. (1996). On the theory of photosynthesis and growth in phytoplankton. Part I:
light limitation and constant temperature. Internationale Revue der gesamten
Hydrobiologie, 81 , 109-139.
Caperon, J. (1968). Population growth response of Isochrysis galbana to a variable nitrate
environment. Ecology, 49 , 866-872.
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Appendix
Parts of the Pascal program MPwithP used to make the simulations. The program was
written in 'Think' Pascal v.4 on a Macintosh computer, but everything here is standard Pascal.
Type definitions
LF = (CI, CN, CP); (* to identify limiting factor as light, nitrogen or phosphorus *)
CONCENTRATIONS = record (* of dissolved nutrients *)
NHS, NOS, POS: REAL;
end;
PARTICULATES = record
B, N, P, Q, QP, X: REAL; (* state variables *)
GROWTH_RATE: REAL;
GROWTH_LF: LF;
UPTAKE_RATE_N, UPTAKE_RATE_P: REAL;
ETA: REAL;
end;
ENVIRONMENT = record
TEMP, TF, DIL: REAL;
XPAR, PAR, THICKNESS, OPTHICK, WATER_ATTENUATION: REAL;
(* units of irradiance XPAR and PAR must be compatable with units of ALPHAL *)
end;
ZOOPLANKTON = record
ZE, ZG, ZGN, ZGP: REAL; (* ZGN and ZGP acumulate grazed nutrient *)
end;
Main program
program MPwithP; (* 23/6/98 *)
const
ETA = 0.25; (* heterotroph fraction of total MP biomass *)
RH = FALSE; (* TRUE includes heterotroph respiration terms; if TRUE, set LV false *)
LV = FALSE; (* TRUE unconstrains A and H and so allows ETA to vary *)
SHADING = TRUE; (* true allows self-shading *)
PER_DAY = 10; (* number of output lines per day *)
DT = 0.01; (* day, timestep *)
STARTDAY = 8; (* start day of simulation *)
MAXDAY = 36; (* end day of simulation *)
var
(* variables *)
MICRO, AUTO, HETERO: PARTICULATES;
REACTOR: CONCENTRATIONS;
TOTN, TOTP: REAL;
(* conditions *)
PHYSICS: ENVIRONMENT;
DILUENT: CONCENTRATIONS;
SPIKE: INJECTION;
GRAZING: ZOOPLANKTON;
(* intermediate values *)
QSTAR, QSTARP: REAL;
(* counters *)
D, DURATION, T, TDAY: INTEGER;
DAY: REAL;
MP Appendix
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begin (* edited main program *)
MICRO.ETA := ETA;
DAY:=STARTDAY;
(* attach input and output *)
SET_MP_PARAMETERS(ETA);
SET_P_PARAMETERS(ETA, RH, LV);
INITIALISE(DAY, MICRO, AUTO, HETERO, REACTOR, DILUENT, SPIKE,
GRAZING, LV);
SET_CONDITIONS(PHYSICS, GRAZING);(* once only, since forcing constant *)
(* begin time-loops *)
DURATION := ROUND((MAXDAY - DAY) * PER_DAY);
TDAY := ROUND(1.0 / (DT * PER_DAY));
for D := 1 to DURATION do
begin
DILUENT.POS := RESERVOIR_POS(DAY);
REACTOR.POS := REACTOR.POS + SPIKE_REACTOR(SPIKE, DAY);
for T := 1 to TDAY do
begin
DAY := DAY + DT;
with PHYSICS do
begin
TF := TEMPCH(TEMP);
if SHADING then
begin
OPTHICK := (WATER_ATTENUATION + EPSBIO *
MICRO.X) * THICKNESS;
PAR := XPAR * (1 - EXP(-OPTHICK)) / OPTHICK;
end
else
PAR := XPAR;
end;
if not LV then
MICROPLANKTON_MODEL(MICRO, REACTOR, DILUENT,
PHYSICS, GRAZING, RH, DT)
else
LOTKA_VOLTERRA_MODEL(MICRO, AUTO, HETERO, REACTOR,
DILUENT, PHYSICS, GRAZING, DT);
end;
(* make output *)
end;
end.
Microplankton parameter values for MP
The procedure SET_MICROPLANKTON_PARAMETERS is called once at the
beginning of each simulation. Assignment of parameter values for the phosphorus submodel
is done by a separate procedure, not included here.
const
(* autotrophs *)
EPSBIO = 0.02; (* m2/mg chl, attenuation cross-section *)
PHI = 40; (* nmol C/muE, photosynthetic quantum yield *)
R0A = 0.05; (* d-1, basal respiration *)
BA = 0.5; (* slope of respiration/growth relationship*)
QMINA = 0.05; (* mmol N/mmol C, minimum nutrient quota *)
MP Appendix
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Feb 00
QMAXA = 0.20; (* mmol N/mol C, maximum nutrient quota *)
MUMAXA = 2.0; (* d-1 at 20 °C , maximum growth rate *)
NHUMAXA = 1.5; (* mmol N/mmol C.d at 20°C, max. NH4 uptake *)
NOUMAXA = 0.50; (* mmol N/mmol C.d at 20°C, max. NO3 uptake *)
KNHSA = 0.24; (* mmol/m3, half-saturation conc. for NH4 uptake *)
KNOSA = 0.32; (* mmol/m3, half-saturation conc. for NO3 uptake *)
KINA = 0.5; (* mmol/m3 , NH4 for half inhibition of NO3 uptake *)
XQNA = 3.0; (* mg chl/mmol N, yield of chlorophyll from N *)
(* heterotrophs *)
CH = 0.20; (* max. clearance, m3/mmol hetero C.d *)
R0H = 0.02; (* d-1, basal respiration *)
BH = 1.5; (* slope of respiration/growth relationship*)
QH = 0.18; (* mmol N/mmol C, constant nutrient quota *)
(* general *)
Q10 = 2.0; (* per 10°C, factor for temperature-dependence *)
REFTEMP = 20.0; (* ° C , standard temperature *)
var
(* true (microplankton) parameters *)
ALPHAL, CUNHS, CUNOS, GRMAX, HIAMM, QMIN, QMAX: REAL;
RB0, RMU, UMAXNH, UMAXNO: REAL;
(* computationally efficient parameters *)
QHETA, QMINMOD, QMAXMOD, TQ10LN: REAL;
procedure SET_MP_PARAMETERS (ETA: REAL);
(* ETA is heterotroph fraction of total MP (carbon) biomass *)
const
UCK1 = 0.0864; (* s d-1 mmol nmol-1 *)
EJK2 = 4.15; (* muEinstein per Joule for PAR *)
begin
(* calculated microplankton parameters *)
ALPHAL := UCK1 * PHI * EPSBIO; (* photosyn. eff., mmol C (mg chl)-1 d-1 PAR-1 *)
(* where PAR is in muE m-2 s-1; include EJK2 if PAR in W m-2 *)
CUNHS := KNHSA; (* half-sat for ammonium, mmol m-3 *)
CUNOS := KNOSA; (* half-sat for nitrate, mmol m-3 *)
GRMAX := MUMAXA*(1-ETA)/(1.2-ETA); (* max. growth rate, d-1 *)
HIAMM := KINA; (* const in amm. inhib. of nitrate, mmol m-3 *)
QMIN := QMINA * (1 - ETA) + (QH * ETA); (* minimum mmol N (mmol C)-1 *)
QMAX := QMAXA * (1 - ETA) + (QH * ETA); (* maximum mmol N (mmol C)-1 *)
RB0 := R0A * (1 - ETA) + (R0H * ETA * (1+BA)); (* basal resp., d-1 *)
RMU := BA * (1 + (BH * ETA)) + (BH * ETA); (* respiration slope *)
UMAXNH := NHUMAXA * (1 - ETA); (* max. NH4 uptake, mmol N (mmol C)-1 d-1 *)
UMAXNO := NOUMAXA * (1 - ETA); (* max. NO3 uptake, mmol N (mmol C)-1 d-1 *)
(* computationally efficient parameters *)
QHETA := QH * ETA;
QMINMOD := QMIN - QHETA;
QMAXMOD := QMAX - QHETA;
TQ10LN := LN(Q10) / 10.0;
end;
MP Appendix
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Feb 00
Main procedure for simulation of MP+P
This is called once each time-step.
procedure MICROPLANKTON_MODEL (var MICROPLANKTON:
PARTICULATES; var REACTOR:
CONCENTRATIONS; DILUENT:
CONCENTRATIONS; CONDITIONS:
ENVIRONMENT; GRAZERS: ZOOPLANKTON; RH:
BOOLEAN; DT: REAL);
var
UPTAKE_FLUX_NHS, UPTAKE_FLUX_NOS, UPTAKE_FLUX_POS, ZOOGN, ZOOGP:
REAL; (* fluxes *)
BETA_B, BETA_N, BETA_P, BETA_NHS, BETA_NOS, BETA_POS: REAL; (* change
terms *)
QMOD, QMODP: REAL; (* computational *)
begin
with MICROPLANKTON do
with REACTOR do
with CONDITIONS do
with GRAZERS do
begin
Q := QUOTA (B, N);
QMOD := Q - QHETA;
QP := QUOTAP(B, P);
QMODP := QP - QPHETA;
if QMODP < 0.0 then
QMODP := 0.0;
GROWTH_RATE := GROWTHwithP(TF, Q, QMOD, QP,
QMODP, PAR, GROWTH_LF);
if RH then (* MP+P+rh *)
begin
UPTAKE_RATE_N := UPTAKENH(TF, QMOD, NHS) - ETA
* NRH (Q, GROWTH_RATE, ETA);
UPTAKE_RATE_P := UPTAKEPO(TF, QMODP, POS) - ETA *
PRH(QP, GROWTH_RATE, ETA);
end
else (* standard MP+P *)
begin
UPTAKE_RATE_N := UPTAKENH(TF, QMOD, NHS);
UPTAKE_RATE_P := UPTAKEPO(TF, QMODP, POS);
end;
UPTAKE_FLUX_NHS := UPTAKE_RATE_N * B;
UPTAKE_FLUX_NOS := UPTAKENOP(TF, QMOD, NOS, NHS,
QP, POS, GROWTH_LF) * B;
UPTAKE_RATE_N := UPTAKE_RATE_N + (UPTAKE_FLUX_NOS
/ B);
UPTAKE_FLUX_POS := UPTAKE_RATE_P * B;
ZOOGN := ZG * N;
ZOOGP := ZG * P;
(* calculation and application of of source-sink flux terms *)
BETA_B := B * (GROWTH_RATE - ZG) + DIL * (-B);
BETA_N := UPTAKE_FLUX_NOS + UPTAKE_FLUX_NHS ZOOGN + DIL * (-N);
BETA_P := UPTAKE_FLUX_POS - ZOOGP + DIL * (-P);
BETA_NOS := -UPTAKE_FLUX_NOS + DIL * (DILUENT.NOS NOS);
MP Appendix
page iv
Feb 00
BETA_NHS := -UPTAKE_FLUX_NHS + ZE * ZOOGN + DIL *
(DILUENT.NHS - NHS);
BETA_POS := -UPTAKE_FLUX_POS + ZE * ZOOGP + DIL *
(DILUENT.POS - POS);
B := B + BETA_B * DT;
N := N + BETA_N * DT;
P := P + BETA_P * DT;
X := XQNA * QMOD * B; (* the chl:carbon ratio is XQNA*QMOD *)
NOS := NOS + BETA_NOS * DT;
NHS := NHS + BETA_NHS * DT;
POS := POS + BETA_POS * DT;
ZGN := ZGN + (1 - ZE) * ZOOGN * DT;
ZGP := ZGP + (1 - ZE) * ZOOGP * DT;
end;
end;
Rate functions in MP (nitrogen as sole nutrient)
These are called from the procedure MICROPLANKTON_MODEL; functions for the
phosphorus submodel are not included. The functions GROWTH and UPTAKENO are
not used in MPwithP: they are included in order to provide a complete set of algorithms for
the 'standard' (nitrogen-only) version of MP.
function TEMPCH (TEMP: REAL): REAL;
(* outputs factor to correct for temperature relative to REFTEMP *)
begin
TEMPCH := EXP((TEMP - REFTEMP) * TQ10LN);
end;
function QUOTA (B, N: REAL): REAL;
(* checks for divide by zero and unrealistically low Q *)
const
SMALLEST = 0.0001;
var
THIS_Q: REAL;
begin
if B < SMALLEST then
QUOTA := QMIN
else
begin
THIS_Q := N / B;
if THIS_Q < QMIN then
QUOTA := QMIN
else
QUOTA := THIS_Q;
end;
end;
function GROWTH (TEMP_CORR, Q, QMOD, XPAR: REAL): REAL;
(* Cell Quota Threshold Limitation microplankton growth, *)
(* with calculation of chl:C ratio *)
function G_LIGHT (QMOD, XPAR: REAL): REAL;
(* Growth controlled by photosynthetic light XPAR *)
(* used with efficiency ALPHAL; QMOD is Q-QH*ETA; *)
(*RMU and RBO are slope and intercept of (notional) *)
MP Appendix
page v
Feb 00
(* regression of microplankton respiration on growth; *)
(* XQNA is autotroph chl:N ratio. *)
var
CHL_TO_C, GL: REAL;
begin
CHL_TO_C := XQNA * QMOD;
GL := ((ALPHAL * XPAR * CHL_TO_C) - RB0);
if GL < 0.0 then
G_LIGHT := GL
else
G_LIGHT := GL / (1 + RMU);
end;
var
GNUTRIENT, GLIGHT: REAL;
begin
GNUTRIENT := GRMAX * TEMP_CORR * (1 - (QMIN / Q));
GLIGHT := G_LIGHT(QMOD, XPAR);
if GNUTRIENT < GLIGHT then
GROWTH := GNUTRIENT
else
GROWTH := GLIGHT;
end;
function UPTAKENH (TEMP_CORR, QMOD, NHS: REAL): REAL;
(* ammonium uptake or excretion *)
(* QMOD is Q - QH*ETA, QMAXMOD is QMAX - QH*ETA *)
var
UMAX: REAL;
function FNHS (QMOD, NHS: REAL): REAL;
(* Michaelis-Menten uptake with half-saturation CUNHS, plus some excretion*)
begin
if QMOD > QMAXMOD then
FNHS := 1.0
else if NHS > 0.0 then
FNHS := NHS / (CUNHS + NHS)
else
FNHS := 0.0;
end;
begin
UMAX := UMAXNH * TEMP_CORR;
UPTAKENH := UMAX * FNHS(QMOD, NHS) * (1 - (QMOD / QMAXMOD));
end;
function NRH (Q, GMU, ETA: REAL): REAL;
(* (Option for) ammonium excretion by microheterotrophs, *)
(* in mmol N (mmol C)-1 d-1; GMU is microplankton growth rate *)
(* Q is microplankton quota and QH is heterotroph quota; *)
(* BH and ROH are slope and intercept of heterotroph regression of respiration on GMU. *)
var
QF: REAL;
begin
if Q > QH then (* C-limited *)
QF := (Q - QHETA) / (1 - ETA)
else (* N-limited *)
QF := QH;
NRH := GMU * ((1 + BH) * QF - QH) + R0H * QF;
end;
MP Appendix
page vi
Feb 00
function UPTAKENO (TEMP_CORR, QMOD, NOS, NHS: REAL): REAL;
(* Ammonium-inhibited nitrate uptake, without excretion. *)
var
UMAX: REAL;
function FNOS (NOS: REAL): REAL;
(* Michaelis-Menten uptake, with half-saturation CUNOS.*)
begin
if NOS > 0.0 then
FNOS := NOS / (CUNOS + NOS)
else
FNOS := 0.0;
end;
function FINNH (NHS: REAL): REAL;
(* Ammonium inhibition, half effective at NHS=HIAMM. *)
begin
if NHS > 0.0 then
FINNH := (1.0 / (1.0 + (NHS / HIAMM)))
else
FINNH := 1.0;
end;
function FINQ (QMOD: REAL): REAL;
(* Cell-quota inhibition, to avoid overfilling with N; *)
(* QMOD is Q - QH*ETA; QMAXMOD is QMAX - QH*ETA .*)
begin
if QMOD > QMAXMOD then
FINQ := 0.0
else
FINQ := (1.0 - (QMOD / QMAXMOD));
end;
begin
UMAX := UMAXNO * TEMP_CORR;
UPTAKENO := UMAX * FNOS(NOS) * FINNH(NHS) * FINQ(QMOD);
end;
Rate functions in MP+P
Here are given the two functions for nitrogen which take account of the presence of
phosphorus, which are called from MICROPLANKTON_MODEL in MPwithP, and
which are the alternative versions of the sole-nitrogen functions GROWTH and
UPTAKENO.
function GROWTHwithP (TF, Q, QMOD, QP, QPMOD, XPAR: REAL;
var CONTROL: LF): REAL;
(* Cell Quota Threshold Limitation microplankton growth for light, N and P, *)
(* with calculation of chl:C ratio; CONTROL shows limiting factor. *)
var
GNUTRIENT, GN, GP, GLIGHT, THIS_GRMAX: REAL;
function G_LIGHT (QMOD, XPAR: REAL): REAL;
(* Growth controlled by photosynthetic light XPAR used with efficiency ALPHAL; *)
(* RMU, RBO are slope, intercept of plot of microplankton respiration on growth; *)
(* QMOD is Q-QH*ETA; XQNA is autotroph chl:N ratio. *)
var
CHL_TO_C, GL: REAL;
begin
CHL_TO_C := XQNA * QMOD;
MP Appendix
page vii
Feb 00
GL := ((ALPHAL * XPAR * CHL_TO_C) - RB0);
if GL < 0.0 then
G_LIGHT := GL
else
G_LIGHT := GL / (1 + RMU);
end;
begin
THIS_GRMAX := GRMAX * TF;
GN := THIS_GRMAX * (1 - (QMIN / Q));
GP := THIS_GRMAX * (1 - (QMINP / QP));
GLIGHT := G_LIGHT(QMOD, XPAR);
if GN < GP then
begin
GNUTRIENT := GN;
CONTROL := CN;
end
else
begin
GNUTRIENT := GP;
CONTROL := CP;
end;
if GNUTRIENT < GLIGHT then
GROWTHwithP := GNUTRIENT
else
begin
GROWTHwithP := GLIGHT;
CONTROL := CI;
end;
end;
function UPTAKENOP (TEMP_CORR, QMOD, NOS, NHS, QP, POS: REAL;
GLF: LF): REAL;
(* Ammonium-inhibited nitrate uptake, without excretion. *)
(* with uptake supression when phosphate or cell phosphorus limiting *)
type
OPTION = (NONE, UPTAKE, QUOTA);
var
UMAX: REAL;
SUPRESSION: REAL;
SUPRESSION_BY: OPTION;
function FNOS (NOS: REAL): REAL; (* Michaelis-Menten uptake, with half-saturation
CUNOS.*)
begin
if NOS > 0.0 then
FNOS := NOS / (CUNOS + NOS)
else
FNOS := 0.0;
end;
function FINNH (NHS: REAL): REAL; (* Ammonium inhibition, half effective at
NHS=HIAMM. *)
begin
if NHS > 0.0 then
FINNH := (1.0 / (1.0 + (NHS / HIAMM)))
else
FINNH := 1.0;
end;
function FINQ (QMOD: REAL): REAL; (* Cell-quota inhibition, to avoid overfilling
with N; *)
MP Appendix
page viii
Feb 00
(* QMOD is Q - QH*ETA; QMAXMOD is QMAX - QH*ETA .*)
begin
if QMOD > QMAXMOD then
FINQ := 0.0
else
FINQ := (1.0 - (QMOD / QMAXMOD));
end;
function FINP (GLF: LF; POS: REAL): REAL; (* uptake supression when P in control *)
begin
if GLF = CP then
FINP := POS / (CUPOS + POS)
else
FINP := 1.0;
end;
function FINQP (GLF: LF; QP: REAL): REAL;
(* alternative uptake supression when P is in control *)
(* A disadvantage of this function is that its returned value can alternate,
*)
(* during successive timesteps, between 1, when nitrogen is in control,
*)
(* to << 1, when phosphorus is in control.
*)
begin
if (GLF = CP) then
FINQP := (QP - QMINP) / (QMAXP - QMINP)
else
FINQP := 1.0;
end;
begin
UMAX := UMAXNO * TEMP_CORR;
SUPRESSION_BY := QUOTA; (* QUOTA is standard *)
case SUPRESSION_BY of
NONE:
SUPRESSION := 1.0;
UPTAKE:
SUPRESSION := FINP(GLF, POS);
QUOTA:
SUPRESSION := FINQP(GLF, QP);
end;
UPTAKENOP := UMAX * FNOS(NOS) * FINNH(NHS) * FINQ(QMOD) *
SUPRESSION;
end;
MP Appendix
page ix
Feb 00
Printing Guide
For the benefit of users who print this document, it may be
worth pointing out that the numbered, text, pages are designed
to be right-hand pages, with diagrams and tables intended as
left-hand pages.
Readers may thus wish to:
1. print the document on A4 paper;
2. reverse the diagrams and tables and sort them so that
they lie face-down on top of the text page which has the
appropriate footnote;
3. ring-bind the document.
It should then be possible to open on, for example, page 5, and
find Figure 2-1 on the left hand page facing this text.
The document will print such that little sorting is necessary.
However, a few diagrams and tables require a little more
attention. They are:
•
the two-page Tables 1.1 and 8.2, which could each be
made a pair of left- and right- facing pages, Table 1.1
being placed between pages 3 and 4 and Table 8.2
between pages 35 and 36;
•
Figures 9-3 to 9-6: it is suggested that 9-3 be placed
opposite page 41, and 9-6 be placed opposite page 42,
while 9-4 and 9-5 make a pair of left- and right- facing
pages between page 41 and the back of Fig. 9-6.