Parameterising a microplankton model

Parameterising a microplankton model

Paul Tett

Department of Biological Sciences, Napier University, Edinburgh EH10 5DT

ReportNovember 1998

ISBN 0 902703 60 9

[email protected] corrections, 02/00

Contents

Abstract ............................................................................................................... 2

1. Introduction ..................................................................................................... 32. The microplankton compartment .................................................................... 43. Microplankton equations of state .................................................................... 64. Autotroph (phytoplankton) equations ............................................................. 95. Comparisons with autotroph equations in ERSEM and FDM ........................ 136. Heterotroph equations ..................................................................................... 18

Simplifying the description of heterotrophic processes .......................... 18Growth of the microheterotroph compartment under C or N control ..... 21Respiration .............................................................................................. 23Ingestion and clearance ........................................................................... 24

7. Comparisons with heterotroph equations in ERSEM and FDM..................... 258. Microplankton rate equations.......................................................................... 319. Comparison of MP simulation with microcosm data...................................... 37

Model ...................................................................................................... 38Phosphorus submodel and other non-standard parameter values ........... 40Initialisation and external forcing ........................................................... 40Numerical methods ................................................................................. 41Results ..................................................................................................... 41

10. An unconstrained autotroph-heterotroph model (AH) .................................. 43Nullcline analysis .................................................................................... 45

11. Discussion ..................................................................................................... 4612. References ..................................................................................................... 49

Appendix...............................................................................................................54

Key Tables

Table 4.1. Autotroph equations. ............................................................. 9Table 4.3. Autotroph parameters. ........................................................... 13Table 6.1. Heterotroph equations. ........................................................... 23Table 6.4. Heterotroph parameters. ........................................................ 26Table 8.1. Microplankton equations. ...................................................... 32Table 8.2. Microplankton rate equations and parameters. ..................... 37

MP page 2 Feb 00

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.

grazing pressuremesozooplankton

defecation

DETRITUS

C & N

Sinking

MICROPLANKTON

C & N

Sinking

(a)

Dissolved

NO

3-

& NH

4+

NH

4+

C respiration

implicit microheterotrophs

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'.

MICROPLANKTON

- total biomass

B

mmol C m

-3

growth rate µ, d-1 χ: mg chl (mmol C)-1

heterotroph fraction η = Bh/(Ba+Bh)

respiration

mesozooplankton

grazing

(b)

Phytoplankton

growth rate

µ

a

biomass B a

photosynthesis

α

.

I

mmol C (mg chl)

-1

d

-1

DOM

Microheterotrophs

growth rate

µ

(Protozoa & Bacteria)

biomass

B

h

MP page 3 Feb 00

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 non-

conservative 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 + βY

flux divergence nonconservative processes

where the vertical flux is:

ϕY = <(w + w ' + wY).(Y + Y')>

≈ -Kz.∂Y /∂z + (w + wY).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.

Table 1.1. General forms of symbols common in this document

Symbol General meaning Units (in MP, if variable)

α photosynthetic 'efficiency' mmol C (mg chl)-1 d-1 I-1

B biomass (as carbon) mmol C m-3

b slope factor (e.g. r/ µ) or inefficiency coefficient -

β (total) nonconservative flux for a substance mmol m-3 d-1

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 d-1

ε photosynthetic pigment attenuation cross- section m2 (mg chl)-1

f fraction of (or efficiency with which) input diverted to named output -

Φ photosythetic quantum yield nmol C µE-1

G grazing pressure d-1

η heterotroph fraction (of biomass) -

I PAR µE m-2 s-1

i relative ingestion rate d-1

K (half)-saturation constant, as in KS or KI S or I

k miscellaneous constants various

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 [mmol (mmol C)-1] d-1

S dissolved nutrient concentration mmol m-3

Θ temperature °C

u biomass-related nutrient uptake rate mmol (mmol C)-1 d-1

Table 1.1., continued.

Like most languages, the symbol set and associated conventions used in this report reflectsorigins and history, including failed attempts to impose universal rules.

1. All greek and bold roman letters are symbols for variables or parameters: they may bereplaced by numbers. Subscripted or superscripted light-type letters are identifiers. Subscriptsor superscripts written in bold qualify the preceeding symbol. All other characters written inlight 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 havepreceeding light-text superscripts or following light-text subscripts. As far as possible, ratevariables are given lower case roman letters, whereas symbols for parameters may have lowercase 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 anydissolved nutrient, and hence is, more often than not, accompanied by preceeding superscriptsspecifying the nutrient.

3. The conventions of Tett & Droop (1988) concerning sub- and superscripts have to someextent 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 variablehas been normalised. Default identifiers or normalising variables are ommitted in the interestsof simplicity, or when they may be considered to cancel, or when there is a locally clearcontext. The trailing subscript is used both for qualification (bold) and identification (light).

examples: CrB and NrN written as r; Nr is N excretion relative to C biomass;NqB written q ; PqB written Pq ; XqN is chlorophyll relative to N in biomass;

biomass; Qmax a (parameter) is upper limit to variable Qa, for autotrophs.

4. Identifiers include: a : autotrophic; b: bacterial;h: (micro)heterotrophic; p: protozoan;N: (organic) nitrogen; NH: (diss.) ammonium;NO: (diss.) nitrate; ON: (diss.) organic nitrogen;P: (organic) phosphorus; PO: (diss.) phosphate;Si: silica (any form).

5. Qualifiers include: min(imum) value; max (imum) value;some state variables, as in KNHS ;

0: basal of threshold value ( ≈ minimum).

MP page 4 Feb 00

(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.

(a) ML

NZ

rp

ZP excretion

Phyto CPhyto N

Nitrate

Gpea

ubNea

photosynthesis

GZ2bGp

Ammonium

Proto CProto N

NbGp

rb

grazing–by–zooplankton

NHr

DOC

Bact C

DON

BactN

uptake

NGp

NH–uptake

eZ

NG

Nub

NHub

respiration

MICROPLANKTON

(b) MP

microplankton C

microplankton N

photosynthesis

respiration

nitrate ammonium

uptake NO3 uptake NH4

ammonium oxidation

grazing on N

mineralisation

grazing on C

mesozooplankton & detritus N

MICROPLANKTON

Figure 2-1. Model comparisons. Relevant compartments and flows in (a) ML ofTett & Wilson (2000), and (b) MP, shown according to the conventions of the modellingsoftware STELLA. The large rectangle shows the limits of the MP microplanktoncompartment or its analogue.

MP page 5 Feb 00

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. 2-

1†). 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.

(a) ERSEM

FLAGELLATE C

DIATOM CBACTERIA C

NANNOFLAGELLATE C

MICROZOPLANKTON C

photosynthesis

p f

DISSOLVED NUTRIENTS

excretiondiss ex

Gnf

mzpGnf

rD

respiration

DETRITUS C

particulate excretion

ZOOPLANKTON C

mzpGf

ZGf

ZGd

Zgmzp

r b

r n f

rmzp

e&mmzp

mortal i ty

MICROPLANKTON

(b) FDM

DON

PHYTOPLANKTON N

BACTERIA N

excretion & uptake

uptake DON

leakage

NITRATE AMMONIUM

uptake NO3 uptake NH4

messy feeding

grazing on N

excretionDETRITUS N

defecation & feeding

MESOZOOPLANKTON N

mortal i ty

decay

grazing on bacteria

MICROPLANKTON

Figure 2-2. Model comparisons, continued. Relevant parts of (a) ERSEM and (b)FDM.

MP page 6 Feb 00

(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

and organic nitrogen N :

(3.2) ∂N /∂t = -∂ϕN/∂z + βN mmol N m-3 d-1

Two other variables are linked to the above; the nitrogen quota:

(3.3) Q = N /B mmol N (mmol C)-1

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 page 7 Feb 00

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 = Na + Nh = Qa.Ba + qh.Bh = ( Qa.(1-η) + qh.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) βB = βBa + βBh = (µa - ch.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 page 8 Feb 00

(3.9) βB = (µ - G).B mmol C m-3 d-1

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 = βNa + βNh mmol N m-3 d-1

= (ua - (ch.Bh + G).Qa).Ba + (ch.Qa.Ba - Nrh - G .qh). Bh

where:

ua 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) βN = u .B - G.N = (u - G.Q).B mmol N m-3 d-1

where:

(3.11.a) Q = Qa.(1-η) + qh.η mmol N (mmol C)-1

(3.11.b) u = ua.(1-η) - Nrh.η mmol N (mmol C)-1 d-1

Sections 4 and 6 expand the autotroph and heterotroph rate terms in these equations.

Table 4.1: Autotroph rate equations used in MP and AH

growth µa = minµa(I ), µa(Qa)

where:µa(Qa) = µmax a(1 - (Qmina /Qa)): Qa≥Qmina

µa(I ) = α.χa.I - ra whereα = k.ε.Φ

χa = Xq Na.Qa

d-1

mmol C (mg chl)-1 d-1

(µE m-2 s-1)-1

mg chl (mmol C)-1

respir-ation

ra = r0 a + ba.µa : µa > 0

r0 a : µa ≤ 0

d-1

nutrientuptake

ua = umax a.ƒ(S ).ƒin(Qa)[.ƒin(NHS)]where: ƒ(S ) = (S/(kS + S)) : S ≥ 0ƒin(NHS) = (1/(1 + (NHS/k in))) : NHS ≥ 0ƒin(Qa) = (1 - (Qa/Qmax a)) : Qa ≤ Qmax a

mmol N (mmol C)-1 d-1

[inhibits NO3- uptake]

temp-erature

µmax a = µmax a[20°C].ƒ(θ)

umax a = umax a[20°C].ƒ(θ)

where:ƒ(θ) = exp(kθ.(θ- 20°C))

θ = temperature

d-1

d-1

°C

MP page 9 Feb 00

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 = ua - 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 Cell-

Quota function for growth is

(4.2) µa = µmax a(1 - (Qmina /Qa)) : Qa ≥ Qmina

where Qmina 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 = µmax a((Qa - Qmi na)/(kQa + Qa - Qmi na)) : Q a ≥ Qmina

but, as argued by Droop (1983), this is the same as Eqn. (4.2) when kQa = 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 Qmin a 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.

Table 4.2. Estimates of respiration parameters

species ba r0 a, d-1 source

Dunaliella tertiolecta 0.74 0.028 (Laws and Wong 1978)

Dunaliella tertiolecta 0.24 (Richardson, Beardall &Raven 1983)

Gonyaulax polyedra 0.41 (Richardson, Beardall &Raven 1983)

Leptocylindrus danicus 0.03 (Richardson, Beardall &Raven 1983)

Pavlova lutheri 0.18 0.082 (Laws and Caperon 1976)

Pavlova lutheri 0.15 (Droop et al. 1982)

Pavlova lutheri 0.48 0.028 (Laws and Wong 1978)

Skeletonema costatum 0.04 (Richardson, Beardall &Raven 1983)

Thalassiosira alleni 0.20 0.037 (Laws and Wong 1978)

MP page 10 Feb 00

(4.3) µa = minµa(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 ) = p (I) - r a d-1

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 = k .ε.Φ.χa.I - ra d-1

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 (m2 (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 light-

limited 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 + ba.µa : µa > 0

r0 a : µa ≤ 0 d-1

Values of the parameters r0 a and ba 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 11 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 = umax a.(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 KS gives the nutrient concentration at which uptake is

half the maximum rate. Some authors (e.g. Droop 1974; Paasche 1973) added a parameter

S0, the threshold for uptake:

ua = umax a.((S - S0)/(K S + (S - S0))) : 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:

una = umax a.(S /(KS + S)).ρ

where the 'coefficient of luxury': ρ = (Rm/(Λ .(Rm - 1) + 1))

and, for light in control of growth: Λ = (S /KS).(KI /I) : (I /KI) ≤ (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 Rm 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, Qmax a:

(4.7) ua = umax a.ƒ(S ).ƒ(Qa) mmol nutrient (mmol C)-1 d-1

where:

ƒ(S ) = (S/(KS + S)) : S ≥ 0ƒ(Qa) = (1 - (Qa/Q max a)) : Qa ≤ Qmax a

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. Qmax a 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

0-1 1-2 2-3 3-4 4-5 5-6 6-7 - 7-170

10

20

30

40

mg chl/mmol N

frequency of reported values

Figure 4-1. Chorophyll:nitrogen yield. Histogram of the frequency of values ofthe ratio of chlorophyll (g) to nitrogen (moles) in cultured marine algae growing atirradiances less than 300 µE m-2 s-1. Literature data for Chaetoceros gracilis,Dunaliella tertiolecta, Gymnodinium sanguineum, Pavlova lutheri, Skeletonemacostatum, and Thalassiosira pseudonana, mostly found by Vivien Edwards.

MP page 12 Feb 00

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 .

Table 4.3: Autotroph parameters used in MP and AH with nitrogen as

potentially limiting nutrient

ref value range units

umax a maximum relative rate of:nitrate uptakeammonium uptake

1 at 20°C0.51.5

0.2-0.8?

mmol N(mmol C)-1 d-1

KS half-saturation concentration foruptake of: nitrate

ammonium

10.320.24

0.2-50.1-0.5

mmol N m-3

Kin ammonium concentration givinghalf-inhibition of nitrate uptake

2 0.5 0.5-7 mmol N m-3

Qmax a maximum cell nitrogen content 1, 3 0.20 0.15-0.25-

mmol N(mmol C)-1

Qmina minimum cell nitrogen content 1 0.05 0.02-0.07 mmol N(mmol C)-1

µmax amaximum (nutrient-controlled)growth rate

1 2.0 at20°C

1.2-2.9 d-1

ε PAR adsorption cross-section 4 0.02 0.01-0.04 m2 (mg chl)-1

Φ photosynthetic quantum yield 5 40 40-60 nmol C µE-1

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

r0 a basal respiration rate 6 0.05 0.03-0.41 d-1

ba rate of increase of respirationwith growth rate (∆ra/∆µa)

6 0.5 0.2- 0.7 -

Xq Na ratio of chlorophyll to nitrogen 7 2.2 0.5 - 7 mg chl(mmol N)-1

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 bySetiapermana (1990) .(4) Standard value for moderately clear coastal water; range related to water type (lowest inturbid 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.

MP page 13 Feb 00

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 µmax a and umax a 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:

(4.10) ƒ(Θ) = exp(kΘ.(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(kA.(Θref -1 - Θ-1)) ≈ exp(kA.(Θ - Θref )/Θref 2)

where the temperature is given in degrees Kelvin and Θref is the reference temperature. kΘin Eqn (4.10) is equivalent to kA/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 Na is (in my symbols):

(5.1.) βNa = (µa - ma - 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 - fea).µmax a.ƒ(I ).ƒ(NHS, NOS) d-1

* Table 4.3. Autotroph parameter values.

Table 5.1. Standard values for phytoplankton parameter in FDM

local symbol Fasham et al.symbol

description std value units

fea g1 phytoplankton DON excretionfraction (of production)

0.05 -

KI = µmax a/αa 'saturation irradiance' 116 W m-2

Kin ψ ammonium inhibition (of nitrateuptake) parameter

1.5 mmolNH4+-N m-3

KNHS K1 half-saturation concentration forammonium uptake

0.5 mmol N m-3

KNOS K1 half-saturation concentration fornitrate uptake

0.5 mmol N m-3

ma µ1 specific mortality rate 0.05 d-1

qa phytoplankton N:C, inverseRedfield ratio

1/6.63 mmol N(mmol C)-1

Xq N = χa/qa phytoplankton chlorophyllyield from N

1.59 mg chl(mmol N)-1

αa α P-I curve initial slope, maximum'photosynthetic efficiency'

0.025 d-1

(W m-2)-1

χachlorophyll:carbon ratio 12/50 mg chl

(mmol C)-1

ε

kC (coefficient of) PAR attenuation byphytoplankton

==> chl-related version (kC/Xq N)

0.03

0.02

m2

(mmol N)-1

m2 (mgchl)-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), photosyntheticquantum yield

18 nmol CµE-1

µmax a Vp maximum growth rate 2.9* d-1

* for Bermuda station S; but in general based on Eppley (1972) according to Fasham et al.

(1993), written here as: µmax a[Θ°C] = µmax a[0°C].exp((0.063°C-1).Θ °C), equivalent to

Q10=1.89; µmax a[0°C] = 0.6 d-1.

MP page 14 Feb 00

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/Qa , 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) fea 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 = µmax a/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 = XqN.Na where: Xq N = χa/qa

and qa-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 nutrient-

limitation is expressed through a low chl:C ratio: so, for Qa = Qmin a, 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.

Table 5.2. Standard values for phytoplankton growth parameter in

ERSEM

localsymbol

Varela et al.name; x isreplaced by1:diatoms2:flagellates

description valuefordiatoms

forflag-ellates

units

µmax a

[<Θ>]

sumPxc$ maximum growth rate (at meantemperature)

2.5 2.0 d-1

kΘ =ln(Q10)/10°C 0.347 0.347 °C-1

Q10 q10Px$ increase in rate for 10°C increasein temperature (Q10)

4.0 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.forphosphate control of growth

0.10 0.05 mmol P m-3

KSiS chPxs$ half-saturation conc. for silicatecontrol of growth

0.30 - mmol Si m-3

KI min clPIi$ minimum optimal PAR,equivalent to 'saturationirradiance'

40 40 W m-2

α a P-I curve initial slope, maximum'photosynthetic efficiency'=µmax a[<Θ>]/KI min

0.063 0.050 d-1 (W m-2)-1

α =µmax a[<Θ>]/(χa.KImin) 0.26 0.104 mmol C (mgchl)-1 d-1

(W m-2)-1

(converted at 4.15 µE J-1) 0.062 0.025 mmol C (mgchl)-1 d-1 (µEm-2 s-1)-1

Xq Na - = χa/qa phytoplankton (min.)chlorophyll yield from N

1.4 2.6 mg chl(mmol N)-1

χauhPxc$ chlorophyll:carbon ratio 12/50 12/25 mg chl

(mmol C)-1

ε (coefficient of) PAR attenuationby phytoplankton*

? ? m2 (mg chl)-1

Φ = αa/(k.ε.χa), photosyntheticquantum yield

? ? nmol C µE-1

* not given; there is a reference to Baretta et al. (1988).

MP page 15 Feb 00

phosphorus and silicon. In my notation and units, the main equation for nonconservative

changes in carbon in a given autotroph compartment, is:

(5.3) βB a = (µa - G).Ba mmol C m-3 d-1

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:

(5.4) µa = µmax a.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:

µmax a = µmax a[<Θ>].ƒ(Θ) d-1

where

ƒ(Θ) = Q10((Θ - <Θ>)/10°C)

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 KI min of 40

W (PAR) m-2. The saturation parameter can be related to maximum photosynthetic rate and

photosynthetic activity (Lederman & Tett 1981):

KI = pB max a/αa W m-2

Equating the maximum photosynthetic rate pB max a with ERSEM's maximum growth rate,

allows an estimate of maximum photosynthetic 'efficiency':

αa = µmax a[<Θ>]/(χa.KImin) 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.

Table 5.3. Standard values for phytoplankton loss parameter in ERSEM

localsymbol

Varela et al.name

description valuefor x=1diatoms

for x=2flag-ellates

units

fD pe_R1Cxc$ fraction of phytoplankton lysiswhich becomes detritus (restbecomes labile DOM) [unclear]

0.20 0.50 -

bea pu_eaPx$ coefficient of growth-dependentexcretion

0.05 0.05 -

bla pum_eoPx$ coefficient of nutrient-stress-dependent lysis

0.30 0.30 -

Kl b chB1eP2c$ bacterial concentration for half-maximum effect of bacterialproteases on diatoms (only)

50 - mmol C m-3

r0 a srsPx$ rate of rest respiration (indarkness)

0.25 0.15 d-1

ba pu_raPx$ rate at which (activity)respiration increases with growth

0.10 0.25

b Sa pum_roPx$ rate at which respirationincreases with nutrient stress

0.05 0.05

Table 5.4. Standard values for phytoplankton nutrient parameter in

ERSEM

localsymbol

Varela et al.name

description valuefor x=1diatoms

for x=2flag-ellates

units

NHpa xpref-N4n$ relative preference forammonium (over nitrate) uptake

3 3

KNOS chPxn$ half-saturation concentration fornitrate uptake

0.10 0.05 mmol N m-3

KNHS chPxn$ half-saturation concentration forammonium uptake

0.10 0.05 mmol N m-3

qa qnPxc$ phytoplankton 'maximum'(nominally constant) nitrogencontent

0.172 0.188 mmol N(mmol C)-1

Pqa qpPxc$ phytoplankton 'maximum'phosphorus content

0.0132 0.0073 mmol P(mmol C)-1

Siqa qsPxc$ phytoplankton maximum siliconcontent

0.21 - mmol Si(mmol C)-1

MP page 16 Feb 00

ƒ(NHS,NOS,POS ,SiS) = √(S /(KS+S)).(POS /(KPOS+POS )) [flagellates]

where dissolved inorganic nitrogen concentration

S = NHS + NOS 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.(µmax a.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 bSa.µmax a.ƒ(I )+(ba-bSa).µa will

normally be less than ba.µa in MP, because of the higher value of ba 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 (fD) 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 = bea .µa + bla .(µmax a.ƒ(I ) - µa)[.(1 + ƒ(Bb))] d-1

where bea is the coefficient of activity excretion and bla 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) βPa = (POua - ea.PQa).Ba mmol P m-3 d-1

where biomass-related uptake rate

POua = Pqa.(µa - ra) - m'.max0, ( PQa - Pqa) 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 17 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 (PQa = 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 Pqa. 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 NHpa and the

following equation:

(5.8) NOf = ƒ(NOS)/((NHpa.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.

b3

Protozoa

r p2

mesozooplankton grazing

Phytoplankton

b

1

r

a

Protozoa

r

p1

Bacteriovorous

p

1

Herbivorous

p

2

b

2

r b2

pico

(autotrophs)

b

1

, b

2

and b

3

are bacteria

r is respiration

'messy feeding'

extracellularproduction

Figure 6-1. Conceptual model of interactions between bacteria and other microplankton.

MP page 18 Feb 00

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 carbon-

nitrogen 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 othermicroplankton.

MP page 19 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 ϕBb2/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) cp2.Bp2.Ba2 + cp1.Bp1.Ba1 ≈ ch.Bh.Ba mmol C m-3 d-1

MP page 20 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 m3 (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 ch must lie between cp2.((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 b2 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) rh = r0 h + bh.µ d-1

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 = bb2.(Bb2/Bh).( µb2/µ) + bp1.(Bp1/Bh).( µp1/µ) + bp2.(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 bh

in (6.4) contains the multiplying terms (µb2/µ) and (µp1/µ) which imply that it is necessary

MP page 21 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 bh can be treated as parameters,

with the value of r0 h within the range defined by r0 b2, r0 p1 and r0p2 unless Bb3 is large.

The value of bh, however, will be larger than the component values bb2, b p1 and bp2,

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 = minµC, µN d-1

Under carbon limitation ,

(6.6) µC = ih - rh d-1

where ih ( = 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+bh) d-1

Under nitrogen limitation,

(6.8) µN = (ih.Qa - Nrh)/q h d-1

where Q a is food N:C ratio, qh is heterotroph N:C ratio and Nrh is heterotroph biomass-

related 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 = qh.rh = qh.(r0 h + bh.µN) mmol N (mmol C)-1 d-1

Substituting this in Eqn. (6.8) gives:

(6.10) µN = (ih.(Qa/qh) - r0 h)/(1+bh) 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:

Table 6.1: Heterotroph rate equations used in MP and AH

growth µh = (ih.q * - r0h)/(1+b h)

where q* = 1 : Qa ≥ qh

Qa/qh : Qa < 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.qh = ih.Qa - Nrhmmol N(mmol C)-1

d-1

NH4+ excretion Nrh = ih.Qa - µh.qh

= µh.((Qa-qh)+bh.Qa) + r0h.Qa [C-limited]

µh.bh.qa + r0h.q h [N-limited]

mmol N(mmol C)-1

d-1

respiration rh = ih - µh

= µh.bh + r0h [C-limited]

= µh.(((1+bh).q*)-1) + r0 h/q* [N-limited]

d-1

temperature ƒ(θ) = exp(kθ.(θ- 20°C))

MP page 22 Feb 00

(6.11) µh = (ih.q * - r0 h)/(1+bh) d-1

whereq* = 1 : Qa ≥ qh [C-limiting]

Qa/qh : Qa < qh [N-limiting]

Carbon respiration and ammonium excretion must be set by the difference between food

intake and the growth need:

(6.12) rh = ih - µh d-1

(6.13) Nrh = ih.Qa - µh.qh 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*) d-1

= µh.(((1+bh).qh/Qa)-1) + (r0 h.qh/Qa) [N-limiting]

µh.bh + r0 h [C-limiting]

(6.15) Nrh = µh.(((1+bh).Qa/q*)-qh) + (r0 h.Qa/q*)mmol N (mmol C)-1 d-1

= µh.((1+bh).Qa - qh) + (r0 h.Qa) [C-limiting]

(µh.bh + r0 h).qh [N-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.qh = ih.Qa - Nrh mmol N (mmol C)-1 d-1

The heterotroph N:C ratio qh 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 qh 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/ih = q*/(1+bh), must be less under nutrient-controlled conditions (when

q* = Qa/qh < 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, bh and ch).

* Table 6.1. Heterotroph rate equations used in MP and AH.

Table 6.2: Dependence of growth rate on ingestion rate

(Fuller, 1990)

Grazer/food GGE (s.e.) regression, µ = a + bi .i bp

a , d-1 (s.e.) bi (s.e.)

Euplotes sp. & Dunaliellaprimolecta 0.02 (0.01) *

Euplotes & Oxyrrhismarina 0.04 (0.01) 0.01 0.01 0.023 0.004 *

Euplotes & Oxyrrhis &Dunaliella 0.29 (0.03) 2.5

Oxyrrhis & Dunaliellatertiolecta 0.55 (0.06) 0.05 (0.13) 0.49 (0.10) 1.0

Oxyrrhis & Brachiomonassubmarina 0.47 (0.04) 0.21 0.07 0.25 0.05 1.1, 3.0

Oxyrrhis &Chlamydamonas spreta 0.08 (0.01) 0.05 0.07 0.07 0.03 *

Oxyrrhis &Nannochloropsis oculata 0.42 (0.05) 0.06 0.18 0.33 0.11 1.4, 2.0

Pleurotricha sp. &Brachiomonas 0.07 (0.01) *

Strombidium sp. & Pavlovalutheri 0.24 (0.05) 3.2

Uronychia sp. & Dunaliella 0.02 (0.01) *

Uronychia & Oxyrrhis 0.10 (0.03) *

Uronychia & Oxyrrhis &Dunaliella 0.32 (0.03) 2.1

GGE is 'Gross Growth Efficiency'; the slope coefficient bp = GGE-1-1. Additionally, whereregression coefficients a and bi were given by Fuller, a second value of the slope wasestimated from bp = bi-1 - 1. Low values of bp were assumed to represent N-limited

conditions, in which bi ≈ (Qa/qp)/(1+bp); hence these bp values are not given as Fuller didnot report food nitrogen.

MP page 23 Feb 00

Respiration

Equation (6.7) embodies the assumption of linearly growth-dependent respiration for carbon-

limited microheterotrophs. Here is the growth equation with protozoan or bacterial subscripts:

(6.17a) µCp = (ip - r0 p)/(1+bp) d-1

(6.17b) µCb = (ib - r0 b)/(1+bb) d-1

where ip 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 + bi .i. These may be compared with eqn. (6.17a) for the

carbon-limiting case, when bi = 1/(1+bp). When nitrogen limits growth, bi should be

smaller. Thus, it is Fuller's higher values of bi that point to the most appropriate values of bp

for MP: between 1 and 3. The data cannot be used to estimate r0p, 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 bp = 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 ( ≈ bi when r0 p small) in the range

* Table 6.2. Fuller's data on growth efficiency.

Table 6.3: Ingestion and related parameters

(Fuller, 1990)

Grazer/food Imax p,d-1

KB f,mmolm-3

B0 f,mmolm-3

KB f+B0 f,mmol m-3

cp, dm3

mmol-1 d-1

Euplotes sp. & Dunaliellaprimolecta 3.32 9.5 6.4 16 348

Euplotes & Oxyrrhismarina 4.81 172 0 172 26

Euplotes & Oxyrrhis &Dunaliella 2.24 3.2 33 36 694

Oxyrrhis & Dunaliellatertiolecta 2.02 9.7 1.6 11 208

Oxyrrhis & Brachiomonassubmarina 2.10 9.9 3.7 14 211

Oxyrrhis & Chlamydamonasspreta 4.80 22 5.1 27 215

Oxyrrhis & Nannochloropsisoculata 2.26 8.5 0.8 9 268

Pleurotricha sp. &Brachiomonas 8.32 2.5 65 68 ?

Strombidium sp. & Pavlovalutheri 2.98 - - - -

Uronychia sp. & Dunaliella 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

MP page 24 Feb 00

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 bp, bacterial bb 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 bb 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, bb = 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 bp and bb are likely to be the lowest observed values, of about 1.0;

however, bh 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 ch 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 ip should be a saturation function of food concentration Bf:

(6.18) ip = cp.Bf.(1/(1+(Bf/K B 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 < KB f) 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 >> KB f:

cp = imax p/KB 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.

Table 6.4: Heterotroph parameters used in MP and AH

std value range units

ch (maximum) transfer rate, expressed asclearance, relative to heterotroph biomass[AH only]

[temperature: ch = ch[20°C].ƒ(θ)]

0.2 at20°C

0.01 -4.5

m3 (mmolC)-1 d-1

KB a concentration of food that half-saturatesingestion [AH only]

50 10170

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 µh1.5 1 - 3

kθ temperature coefficient 0.069 °C-1

MP page 25 Feb 00

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 half-

saturation 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 m3 (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 m3 (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 microplankton-

detritus 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 .

(1-f

a

).f

e

.u

excretion, or'messy feeding'

u

, uptake

activityrespiration

(1-f

a

).(1-f

e

).u

DIM pools

assimilation

respiration

r

0

, rest

f

a

.u

the 'STANDARD ORGANISM'

G

,grazing

m

, mortality

DOM

Detritus

Figure 7-1. The ERSEM 'standard organism' in the case of microheterotrophs.

MP page 26 Feb 00

"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):

(7.1) βB i = (µi - m i - Gi).Bi mmol C m-3 d-1

where mi 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 = Cu i - ei- ri d-1

Ingestion is:

(7.3) ii = u i - ei = k1 .Cu i d-1

where :

k1 = (1-fei.(1-fa i))

Here, fe is the fraction of unassimilated food that is excreted (lost into the detrital and

implicit DOM pools), and fa 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-fa i).Cu i = ei + (ri - r0 i) d-1

assimilation efficiency, fa 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-fei).(1-fa i) d-1

and so,

(7.5) µi = fa i.Cu i - r0 i d-1

Combining Eqns. (7.3) and (7.5) gives:

µi = (fa i/k1 i).ii - r0 i d-1

† Figure 7-1. The ERSEM 'standard organism' in the case ofmicroheterotrophs.

Table 7.1 : ERSEM (and derived MP or AH) parameters for heterotrophs

(at 10°C)

ERSEMname*

B1 Z6 Z5

Localsymbol

Description bacteria nano-flagellates

microzoo-plankton

units(here)

fa puST$ assimilation efficiency 0.3 0.2 0.5 -

fe pu_eaST$ fraction excreted ofunassimilated food; 1-feis 'activity' respired

(0) 0.5 0.5 -

fD pe_R1ST$ particulate fraction ofexcretion

(0) 0.5 0.5 -

k1 = 1-fe.(1-fa) 0.7 0.6 0.75

r0 srsST$ basal respiration rate 0.01 0.05 0.02 d-1

Cumax sumST$ maximum food uptakerate

8.38 10.0 1.2 d-1

KB f chuSTc$ food conc. half-saturating uptake

- 29 6.7 mmol Cm-3

Q10 q10ST$ temperature coeff. 2.95 2.0 2.0 (10°C)-1

c = k1 .Cumax /KB f,clearance rate

- 0.21 0.13 m3 (mmolC)-1 d-1

b = (k1 .fa)-1, slope ofrespiration on growth

1.3 2.0 0.5 -

p suP1_ST$ availability (diatoms)for ....

- 0 0.5 -

suP2_ST$ availability(phytoflagellates)

- 0.3 0.5 -

suB1_ST$ availability (bacteria) - 1.0 0 -

suZ6_ST$ availability(nanoflagellates)

- 0.2 0.6 -

suZ5_ST$ availability(microzooplankton)

- 0 0.2 -

* The ST component of the ERSEM name is replaced by B1, Z5 or Z6.

MP page 27 Feb 00

which may be compared with the MP equation for C-controlled growth for an example

protozoan:

µC = (ip - r0 p)/(1+bp)

The two versions can be equated if ERSEM fa i/k1 i ≡ 1/(1+bp) in MP, and ERSEM

r0 i ≡ r0 p/(1+bp) in MP. Thus MP bp = (k1 i/fa i) - 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 bh 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 bh 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 = Cumax i.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 >> KB fi gives

ci = imax i/Ki = k1 .Cumax i/KB fi m3 (mmol C)-1 d-1

This allows (maximum) clearance rates to be calculated from the ERSEM parameters for

maximum uptake Cumax and uptake half-saturation KB 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 Cumax ), 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 ERSEM-

analogue 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 qmax h, 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.

Table 7.2. Standard values for zooplankton parameters in FDM

localsymbol

FDMsymbol

description stdvalue

units

na Z,n=a,b,D

β1,β2,β3, assimilation efficiencies for diets ofphytoplankton, bacteria or detritus

0.75 -

imax Z g maximum ingestion rate (relative tozooplankton biomass)

1.0 d-1

cZ = (imax Z/KFZ).qZ, maximumclearance rate

0.15 m3 (mmol C)-1 d-1

KF Z K3 (total) food concentration whichhalf-saturates ingestion

1.0 mmol N m-3

np 0 ,n=a,b,D

p preference for a food type when allfoods equally abundant

??

NrNZ µ2 (N-)relative rate of N excretion 0.1 d-1

NrZ (C-)relative rate of N excretion 0.015 mmol N (mmol C)-1

d-1

ε ammonium fraction of N excretion 0.75

mZ µ5 specific mortality rate 0.05 d-1

Ω fraction of dead animals thatbecome detritus ?? (the remainderminineralising to ammonium ??)*

0.33

qZ − N:C, inverse Redfield, ratio 1/6.63 mmol N (mmol C)-1

Values are for Bermuda station S, but it does not appear from Fasham et al. (1993) that anyare temperature-dependent.*The value and definition of Ω are not quite clear.

MP page 28 Feb 00

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 - mZ).Z mmol N m-3 d-1

where

µZ = ∑(nfZ.niZ) - rZ d-1

The terms nfZ 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/(KFZ + F) d-1

F is the total concentration of food, summed over phytoplanktonic, bacterial and detrital

nitrogen. The maximum relative ingestion rate imax Z (d-1) and the half-saturation food

concentration KFZ (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.

Table 7.3. Standard values for bacterial parameters in FDM

localsymbol

FDMsymbol

description stdvalue

units

uNmax b Vb maximum N-relative ammonium orDON uptake rate

2.0 d-1

umax b = uNmax b.qh, maximum C-relativeuptake rate

0.4 mmol N (mmol C)-1

d-1

KS K4 half-saturation concentration fornutrient uptake

0.5 mmol N m-3

cb = uNmax b/KS , trophic transfercoefficient

0.8 m-3 (mmol C)-1 d-1

NrNb µ3 (N-)relative rate of nitrogenexcretion

0.05 d-1

NHqON η ammonium:DON uptake ratio(assumes DOC:DON = 8)

0.6

qb − nitrogen:carbon ratio 0.20 mmol N (mmol C)-1

MP page 29 Feb 00

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 = ap0 .N a + bp0 .Nb + Dp0 .ND 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:

(7.10) ap = ap0 .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) βNb = µb.Nb - biZ.Z mmol N m-3 d-1

where:

µb = NHuNb + ONuNb - 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 (qb) 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 30 Feb 00

(7.12) NHuNb = uNmax b.S '/(KS + S' + ONS) d -1

ONuNb = uNmax b.ONS/(KS + S' + ONS) d -1

where

S ' = min(NHS, ONqb.ONS) mmol N m-3

ONqb (= qON/qb) 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.

Table 8.1: MP equations derived from equations of state

carbon biomass βB = (µ - G).B mmol C m-3 d-1

where: growth rate µ = µh = µa.(1-η) - r h.η d-1

nitrogen in biomass βN = u .B - G.N = (u - G.Q).B mmol N m-3 d-1

where: nutrient quota Q = Qa.(1-η) + qh.η mmol N (mmol C)-1

uptake rate u = ua.(1-η) - Nrh.η mmol N (mmol C)-1 d-1

MP page 31 Feb 00

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

χa 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 : µa > 0 d-1

r0 a : µa 0

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 0h + bh.µ).η d-1

* Table 8.1. Microplankton equations.

Q

max

a

= 0.20

Q

min

a

= 0.05

q h

= 0.18

µ

a

= f(

I

)

µ

a

= f(

Q

a

)

Q

a

<

q

h

Q

a

≥

q

h

autotrophs N-limited

autotrophs light-limited

axis of

Q

a

, mmol N (mmol C)

-1

, as food for heterotrophs

µ

= 0

µ = - r 0

µ

h

< 0

µ

h

> 0

µ

a

> 0

in this region, autotroph growth in excess of maintenance supplies maintenance needs of heterotrophs

axis of microplankton growth rate

µ

(d

-1

)

heterotrophs C-limited

heterotrophs N-limited

µ

a

< 0

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.

MP page 32 Feb 00

Eqn. (8.1) can be re-arranged to give autotroph growth rate:

µa = (µ + rh.η)/(1-η) d-1

Combining this with (8.4) results in:

µa = (µ.(1+bh.η) + r0 h.η)/(1-η) d-1

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 = χa.(1-η) mg chl (mmol C)-1

= Xq Na.(Qmin - q h.η) : Q ≤ Qmin

Xq Na.(Q - q h.η) : Qmin < Q < Qmax

Xq Na.(Qmax - q h.η) : Q ≥ Qmax

the basal respiration rate, r0 = r0 a.(1-η) + r0 h.η.(1+ba) d-1

and the respiration slope, b = ba.(1 + bh.η) + bh.η : µ > 0

0 : µ ≤ 0

In this equation, the chlorophyll:carbon ratio has been expanded using the two equivalencies:

χa = Xq Na.Qa and Q = Qa.(1-η) + qh.η.

See below for Qmax and Qmin. 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 = µmax a.(1 - (Qmin a/Qa)).(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 - qh.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.

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

0.00

0.50

1.00

Q, mmol N (mmol C)-1

µ, d-1

µ (exact: η=0.3)

µ (approx: η=0.3)

µ (exact: η=0.7)

µ (approx: η=0.7)

Figure 8-2. Growth rate. Comparisons of microplankton nutrient-limited growth

according to the more exact equation (8.8) and the approximation (8.9).

MP page 33 Feb 00

µ = µmax .ƒ1(Q) - r h.η d-1

whereƒ1(Q) = (Q - Q min)/(Q - qh.η) : Q ≥ Q min

µmax = µ max a.(1-η) d-1

Qmin = Qmina .(1-η) + qh.η 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 = µh.((1+bh)/q*)-1) + (r0 h/q*) d-1

where q* = Qa/qh

Attempting a treatment in terms of microplankton variables:

(8.8) µ = µmax .ƒ1(Q)/(1 + ((1+bh).ƒ2(Q)) - η) - r0 h.ƒ2(Q) d -1

whereƒ1(Q) = (Q - Q min)/(Q - qh.η) : Q ≥ Qmin

0 : Q < Qmin

ƒ2(Q) = qh.η.(1-η)/(Q - qh.η) : Q ≥ Qmin

qh.η.(1-η)/(Qmin - qh.η) : Q < Qmin

Equation (8.8) is complicated, and can be approximated (Figure 8-2†) as:

(8.9) µ = µmax .ƒ3(Q)where

µmax = µmax a.(1-η)/(1.2 - η) d -1

ƒ3(Q) = (Q - Q min)/Q = 1 - (Qmin/Q ) : Q ≥ Qmin

0 : Q < Qmin

Qmin = Qmin a.(1-η) + qh.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 Qmin 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 34 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 = ua.(1-η) - Nrh.η = (NO ua + NH ua).(1-η) - Nrh.ηmmol 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:

(8.11) NOu = NOua.(1-η) = NOumax .ƒ(NOS).ƒin(NHS).ƒin1(Q)

whereNOumax = NOumax a.(1-η) mmol N (mmol C)-1 d-1

ƒ(NOS) = NOS/(KNOS + NOS)

ƒin(NHS) = 1/(1 + (NHS/Kin))

ƒin1(Q) = (1-((Q - q h.η )/(Qmax - q h.η))) : Q ≤ Q max

0 : Q > Qmax

Qmax = Qmax a.(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 ≥ qh.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 :

(8.12) NHu = NHumax .ƒ(NHS).ƒin(Q) - Nrh.ηwhere

NHumax = NHumax a.(1-η) mmol N (mmol C)-1 d-1

ƒ(NHS) = NHS/(KNHS + NHS) : Q ≤ Qmax

1 : Q > Qmax

ƒin2(Q) = 1-((Q - qh.η)/(Qmax - q h.η)) : Q ≥ Q min

Qmax = Qmax a.(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:

(8.13) Nrh = µ.((1+bh).q† - qh) + r0 h.q†) mmol N (mmol C)-1 d-1

whereq† = (Q - qh.η)/(1-η) : Q ≥ qh

qh : Q < qh

0.0 2.0 4.0 6.0 8.0 10.00.0

2.0

4.0

6.0

8.0

10.0

mg chl m

-3

η

= 0.3

η

= 0.7

MP

MP+

N

r

h

0.0 2.0 4.0 6.0 8.0 10.00.00

0.20

0.40

0.60

0.80

1.00

mmol NH

4+

m

-3

η

= 0.7

η

= 0.3

MP

MP+

N

r

h

0.0 2.0 4.0 6.0 8.0 10.00.00

0.20

0.40

0.60

0.80

1.00

mmol NH

4+

m

-3

d

-1

η

= 0.7

η

= 0.3

MP

MP+

N

r

h

day

(a) microplankton chlorophyll

(b) dissolved ammonium

(c) ammonium uptake flux

Figure 8-3. Test case for ammonium uptake.

Equations (8.12) to (8.14)

examined in a system with state variables

NH

S

(initial concentration 1 mmol m

-3

),

NO

S

(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

µ

E m

-2

s

-1

and all parameters given standard values. There was

a constant mesozooplankton grazing pressure of 0.05 d

-1

, with 50% ammoniumrecycling. Microplankton parameters calculated from standard autotroph and

heterotroph parameters with

η

= 0.3 and 0.7. The equation at issue is:

NH

u

=

NH

u

max

.ƒ(

NH

S

).ƒ

in2

(

Q

) [ -

N

r

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

N

rh, was better for lower values of η.

MP page 35 Feb 00

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 = NHumax .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 forNHS≥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.

Table 8.2 MP rate equations

growth rate

µ = minµI,µQ d-1

µI = (α.I.Xq Na.(Q - q h.η) - r 0)/(1+b ) d-1

µQ = µmax .(1 - (Qmin/Q)) d-1

uptake rateu = (NO u + NHu ) mmol N (mmol

C)-1 d-1

(NO3-) NOu = NOumax .ƒ(NOS).ƒin(NHS).ƒin1(Q)

(NH4+) NHu = NHumax .ƒ(NHS).ƒin2(Q) [- Nrh]where

ƒ(NOS) = NOS/(kNOS + NOS)

ƒ(NHS) = NHS/(kNHS + NHS) : Q ≤ Qmax

1 : Q > Qmax

ƒin(NHS) = 1/(1 + (NHS/k in))

ƒin1(Q) = (1-((Q - q h.η)/(Qmax - q h.η))) : Q ≤ Qmax

0 : Q > Qmax

ƒin2(Q) = 1-((Q - q h.η)/(Qmax - q h.η))

[ optional ][ Nrh = µ.((1+bh).q † - qh) + r0h.q mmol N (mmol C)-1d-1 ]

d-1] [ where ][ q† = (Q - qh.η)/(1-η) : Q ≥ qh ]

[ qh : Q < qh ]

variables

I : PAR experienced by microplankton, µE m-2 s-1

Q : microplankton nitrogen:carbon ratio, mmol N (mmol C)-1

NHS : sea-water ammonium concentration, mmol m-

NOS : sea-water nitrate concentration, mmol m-3

Θ : temperature experienced by microplankton, °C

.... continued

(Table 8.2, continued)

parameters

heterotroph fraction, η = Bh/(Ba + Bh)

photosynthetic efficiency, α = k .ε.Φ mmol C (mg chl)-1 d-1( µE m-2 s-1)-1

yield of chlorophyll from N, Xq N = Xq Na.(1-η) mg chl (mmol N)-1

basal respiration rate, r0 = r0 a.(1-η) + r0 h.η.(1+ba) d-1

respiration slope, b = ba.(1 + bh.η) + bh.η : µ > 0

0 : µ ≤ 0

max. N-limited growth rate, µmax = µmax a.(1-η)/(1.2 - η) d -1

minimum N quota, Qmin = Qmina .(1-η) + q h.η mmol N (mmol C)-1

maximum N quota, Qmax = Qmax a.(1-η) + q h.η mmol N (mmol C)-1

max. NO3 uptake, NOumax = NOumax a.(1-η) mmol N (mmol C)-1 d-1

max. NH4 uptake, NHumax = NHumaxa.(1-η) mmol N (mmol C)-1 d-1

half-sat. conc., NH4 uptake, kNHS = autotroph value mmol N m-3

half-sat. conc., NO3 uptake, kNOS = autotroph value mmol N m-3

half-sat. conc., NH4 inhibition, kin = autotroph value mmol N m-3

temperature effects

max. growth rate, µmax a = µmax a[20°C].ƒ(Θ)d-1

max. uptake rate, umax a = umax a[20°C].ƒ(Θ)d-1

temperature effect: ƒ(Θ) = exp(kΘ.(Θ - 20°C)

MP page 36 Feb 00

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.

0 10 20 30 40

10 1

10 2

days

µ

M organic carbon

POC

MPC

ppC

(a) particulate carbon components

0 10 20 30 400.0

1.0

2.0

3.0

days

µ

g chl :

µ

M MPN

µ

g chl :

µ

M ppC

(b) pigment ratios

pheo : total

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).

MP page 37 Feb 00

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 µm-

screened 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.

0.0000

0.0050

0.0100

0.0150

0.0200

0 10 20 30 400.00

0.20

0.40

0.60

days

0 10 20 30 400.000

0.050

0.100

0.150

0.200

molar ratios to carbon

days

PON/POC

POP/POC

(b) nutrient quotas

(a) heterotroph fraction

η

N:C P:C

Figure 9-2. The microcosm experiment, continued.

(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.

(a) The heterotroph fraction

η

estimated from hetC/MPC for days 0 - 8 (where hetC is microscopically heterotroph carbon), then from (MPC-ppC)/MPC.

MP page 38 Feb 00

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 (MPC-

ppC)/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.PQ).B mmol N m-3 d-1

(9.1.d) nitrate: dNOS/dt = - NOu.B - D.(NOS - NOSr) mmol N m-3 d-1

† Figure 9-2. The microcosm experiment, continued.

MP page 39 Feb 00

(9.1.e) ammonium: dNHS/dt = - NHu.B - D.(NHS - NHSr) mmol N m-3 d-1

(9.1.f) phosphate: dPOS /dt = - POu .B - D.(POS - POS r) mmol P m-3 d-1

(9.1.g) quota: NQ = N /B and PQ = P/B mmol (mmol C)-1

(9.1.h) chlorophyll: X = Xq N.N 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

(9.2) µ = minµ(I ), µ(NQ), µ(PQ) d-1

where

µ(PQ) = µmax .(1 - (PQmin/P Q )) d-1

PQ = P/B mmol P (mmol C)-1

(9.3) POu = POumax .ƒ(POS ).ƒin1(PQ) [- Prh] mmol P (mmol C)-1 d-1

whereƒ(POS ) = POS /(KPOS + POS ) : PQ ≤ PQmax

1 : PQ > PQmax

ƒin1(PQ) = 1-((PQ - Pqh.η)/(PQmax - Pqh.η))

The phosphate uptake equation allows excretion when PQ > PQmax . In addition, an optional

'phosphorus respiration' term in (9.3) was expanded in the same way as (8.13) for Nrh:

(9.4) Prh = µ.((1+bh).q† - qh) + r0 h.q†) mmol P (mmol C)-1 d-1

whereq† = (PQ - Pqh.η)/(1-η) : PQ ≥ Pqh

Pqh : PQ < Pqh

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 = NOumax .ƒ(NOS).ƒin(NHS).ƒin1(Q) [.ƒin3(PQ)]

whereƒin3(PQ) = (PQ - PQmin)/(P Qmax - PQmin) : µ(PQ)<µ(NQ)<µ(I )

1 : µ(PQ)>µ(NQ)

Table 9.1. Phosphorus submodel, and other non-standard, parameters in

MP+P

Parameter Jones et al.(1978b) value,from totalparticulates

Recalculatedfrom MPparticulates

After fitting units

PQmax 10.5 15 mmol P (mol C)-1

PQmin 1.02 1.3 1.5 mmol P (mol C)-1

Pqh PQmin/η mmol P (mol C)-1

Pumax 2.4 @ 10°C 5 @ 20°C10(with excr.)

6 @ 20°C11(with excr.)

mmol P (mol C)-1 d-1

KPO 0.21 - 0.21 mmol P m-3

Xq N 2.16 - 3.04(Gowen et al.,1992)

2.8 mg chl (mmol N)-1

Xq Na 3.7 (η = 0.25) 3.0 mg chl (mmol N)-1

Table 9.2. Forcing for the microcosm simulations

symbol forquantity

description of quantity value(s) or equation units

NOSr reservoir nitrate 100 µM

POSr reservoir phosphate 0.28 (day 0-19), 0.20 (day 20-30.4), 0.70 (day 30.5-40)

µM

I reactor PAR I0 .(1 - exp(-ζ))/ζ µE m-2 s-1

I0 external PAR 50 µE m-2 s-1

ζ reactor mean opticalpathlength

z.(λw + ε.X)

z reactor mean photonpathlength

0.1 m

λ w seawater attenuationcoefficient

0.2 m-1

D reactor dilution rate 0.21 d-1

Θ temperature in reactor 10 °C

MP page 40 Feb 00

Phosphorus submodel and other non-standard parameter values

Initial values of PQmin and PO umax 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 Pqh, 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

PQmin a = ( PQmin - Pqh.η)/(1-η)

together with the initial value of 1.3 used for PQmin, gave PQmin a < 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 (PQ - Pqh.η) in equations such as (9.3) could not be allowed to

go negative. The solution was to set Pqh.η equal to PQmin, implying q h 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 Xq Na ( approximated by Xq N/(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.

10 20 30 400

50

100

150

200

250

days

µ

M MPC

(a) microplankton carbon

η

= 0.20

η

= 0.30

with

r

h

η

= 0.25

observations

10 20 30 400.0

5.0

10.0

15.0

20.0

25.0

days

µ

M MPN

η

= 0.20

η

= 0.30

(b) microplankton nitrogen

with

r

h

η

= 0.25

observations

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.

MP page 41 Feb 00

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 time-

step 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' [tj]) - log10(Y [tj]) )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)/nY)

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))/(nB+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.

10 20 30 400.00

0.10

0.20

0.30

0.40

0.50

days

µ

M MPP

η

= 0.20

(c) microplankton phosphorus

with

r

h

η

= 0.25

observations

η

= 0.30

10 20 30 400

20

40

60

days

µ

g chl l

-1

η

= 0.20

(d) chlorophyll

with

r

h

η

= 0.25

observations

η

= 0.30

Figure 9-4. Simulations of particulates, continued.

(a) Simulated

P

and observed (MPP) microplankton organic phosphorus.

(b) Simulated

X

and observed (chl) microplankton chlorophyll.

10 20 30 400.00

0.20

0.40

0.60

days

d

-1

(a) microplankton growth rate,

µ

from observed

∆

ln(MPC)/

∆

t

η

= 0.20

η

= 0.30

η

= 0.25 with

r

h

10 20 30 400.0

2.0

4.0

6.0

days

standardised cell quota

P

Q

/

P

Q

min

N

Q

/

N

Q

min

(b) standardised cell quotas at

η

= 0.25

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

/

Q

min

) compared with estimates from ratios of particulate elements:

Q

min

= 0.0825 mmol N (mmol C)

-1

and

P

Q

min

= 1.5 mmol P (mol C)

-1

.

0.00 0.10 0.20 0.30 0.40 0.500.00

0.10

0.20

0.30

0.40

η

value of statistic

B

and [MPC]

X

and [chl]

all

N

and [MPN]

P

and [MPP]

all

all (+r

h

)

P

and [MPP]

B

and [MPC]

N

and [MPN]

X

and [chl]

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 simulationsto observations. Additional points show best fit for option MP+P+rH with heterotroph excretion.

MP page 42 Feb 00

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.

Table 10.1. Phosphorus submodel parameters for autotrophs and

heterotrophs in AH

Parameter value units how calculated (with η = 0.25)

PQmax a 15 mmol P (mol C)-1 same as PQmax

PQmina 1.5 mmol P (mol C)-1 same as PQmin

Pqh 6 mmol P (mol C)-1 PQmin/ηPumax a 14.7 @ 20°C mmol P (mol C)-1

d-1Pumax /(1-η) including excretion

KPO 0.21 mmol P m-3

MP page 43 Feb 00

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 = (ua/Qa - ch'.Bh - D ).Na

(10.1.2.h) heterotroph nitrogen: dNh/dt = (ch'.N a - Nrh).Bh - D .Nh

(10.1.3.a) autotroph phosphorus: dPa/dt = (Pua/PQa - ch'.Bh - D ).Pa

(10.1.3.h) heterotroph phosphorus: dPh/dt = (ch'.Pa - Prh).Bh - D .Ph

(10.1.4) dissolved ammonium: dNHS/dt = - NHua.Ba + Nrh.Bh + D.(NHSr - NHS)

(10.1.5) dissolved nitrate: dNOS/dt = - NOua.Ba + D.(NOSr - NOS)

(10.1.5) dissolved phosphate: dPOS /dt = - POua.Ba + Prh.Bh + D.(POS r - POS )

(10.1.6) chlorophyll: X = Xq Na.Na 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.

10 20 30 400

50

100

150

200

250

days

µ

M MPC

(a) microplankton carbon c

h

= 0.05 m

3

(mmol C)

-1

d

-1

observations

c

h

= 0.20 m

3

(mmol C)

-1

d

-1

c

h

= 0.10

10 20 30 400

20

40

60

days

µ

g chl l

-1

(b) chlorophyll

c

h

= 0.05 m

3

(mmol C)

-1

d

-1

c

h

= 0.20 m

3

(mmol C)

-1

d

-1

c

h

= 0.10

observations

Figure 10-1. AH simulations of particulates in the microcosm.

Simulations with the unconstrained autotroph-heterotroph model AH, for c

h

= 0.05, 0.10 and 0.20

m

3

(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].

MP page 44 Feb 00

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 ch 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.

10 0 10 1 10 2

10 0

10 1

10 2

µ

M autotroph C (model

B

a

or observed ppC)

µ

M heterotroph C (model

B

h

or hetC)

observations

MP+P:

η

= 0.25

c

h

= 0.10

c

h

= 0.05

c

h

= 0.20

(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 m

3

(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.

0 40 80 120 1600

10

20

30

40

50

60

autotroph biomass

B

a

in

µ

M C

heterotroph biomass

B

h

in

µ

M C

nullclines for c h =0.10 (0.20 at 20 ° C)

nullclines for c h =0.025 (0.05 at 20 ° C)

b) Nullcline analysis of AH

.

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

ma

xa

was taken as 186 mmol C m

-3

,

from

PO

S

r

/

P

Q

min

a

, and

µ

max

a

was 1.0 d

-1

at 10

°

C

MP page 45 Feb 00

Figure 10-2(a)† shows plots of heterotroph biomass against autotroph biomass for simulations

with the same values of ch. 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 ch 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 = (µmax a.(1 - (Ba/Bmax a)) - ch'.Bh - D ).Ba

(10.3.h) dBh/dt = 0 = (a .ch'.Ba - D ).Bh

where a = 1/(1+bh), 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/KBa)) : dBh/dt =0

Bh = (µmax a.(1 - (Ba/Bmax a)) - D ).(1 + (Ba/KBa))/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 microcosmresults.

MP page 46 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 ch, 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.

Table 11.1. Estimates of the heterotroph fraction

Site Season autotrophmmol Cm-3

hetero-trophmmol Cm-3

micro-planktonmmol Cm-3

η Reference

Scottish coastal:Easdale Quarry

May -August

8.1 3.6 11.7 0.31 Tett et al.(1988)

Scottish coastal:Loch Creran

wholeyear

6.6 3.8 10.4 0.36 Tett et al.(1988)

EnglishChannel: mixed

July 6.8 1.6 8.2 0.19 Holligan et al.(1984)

EnglishChannel:stratified

July 1.4 2.0 3.4 0.58 Holligan et al.(1984)

Canadiancoastal:CEPEX

July-August

10.4 1.3 11.7 0.11 (Williams 1982)

Area-integratedgobal means

mmol Cm-2

mmol Cm-2

mmol Cm-2

'Coastal'euphotic zone orsimilar

mean ofall data (n 82)

191 71 262 0.27 Gasol et al.(1997)

'Open Ocean'euphotic zone orsimilar

mean ofall data (n 119)

164 135 299 0.45 Gasol et al.(1997)

MP page 47 Feb 00

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 re-

assimilated 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 48 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 Lotka-

Volterra 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 49 Feb 00

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MP Appendix page i Feb 00

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 = recordB, 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 = recordTEMP, 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 = recordZE, 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 page ii Feb 00

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 dobegin

DILUENT.POS := RESERVOIR_POS(DAY);REACTOR.POS := REACTOR.POS + SPIKE_REACTOR(SPIKE, DAY);for T := 1 to TDAY do

beginDAY := DAY + DT;with PHYSICS do

beginTF := TEMPCH(TEMP);if SHADING then

beginOPTHICK := (WATER_ATTENUATION + EPSBIO *

MICRO.X) * THICKNESS;PAR := XPAR * (1 - EXP(-OPTHICK)) / OPTHICK;

endelse

PAR := XPAR;end;

if not LV thenMICROPLANKTON_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 page iii 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 page iv 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);

varUPTAKE_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 dowith REACTOR do

with CONDITIONS dowith GRAZERS do

beginQ := 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 *)

beginUPTAKE_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 v 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 thenQUOTA := QMIN

elsebegin

THIS_Q := N / B;if THIS_Q < QMIN then

QUOTA := QMINelse

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 vi Feb 00

(* regression of microplankton respiration on growth; *)(* XQNA is autotroph chl:N ratio. *)varCHL_TO_C, GL: REAL;begin

CHL_TO_C := XQNA * QMOD;GL := ((ALPHAL * XPAR * CHL_TO_C) - RB0);if GL < 0.0 then

G_LIGHT := GLelse

G_LIGHT := GL / (1 + RMU);end;

varGNUTRIENT, GLIGHT: REAL;

beginGNUTRIENT := GRMAX * TEMP_CORR * (1 - (QMIN / Q));GLIGHT := G_LIGHT(QMOD, XPAR);if GNUTRIENT < GLIGHT then

GROWTH := GNUTRIENTelse

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 *)varUMAX: REAL;

function FNHS (QMOD, NHS: REAL): REAL;(* Michaelis-Menten uptake with half-saturation CUNHS, plus some excretion*)begin

if QMOD > QMAXMOD thenFNHS := 1.0

else if NHS > 0.0 thenFNHS := NHS / (CUNHS + NHS)

elseFNHS := 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 vii Feb 00

function UPTAKENO (TEMP_CORR, QMOD, NOS, NHS: REAL): REAL;(* Ammonium-inhibited nitrate uptake, without excretion. *)varUMAX: REAL;

function FNOS (NOS: REAL): REAL;(* Michaelis-Menten uptake, with half-saturation CUNOS.*)begin

if NOS > 0.0 thenFNOS := NOS / (CUNOS + NOS)

elseFNOS := 0.0;

end;function FINNH (NHS: REAL): REAL;(* Ammonium inhibition, half effective at NHS=HIAMM. *)begin

if NHS > 0.0 thenFINNH := (1.0 / (1.0 + (NHS / HIAMM)))

elseFINNH := 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 thenFINQ := 0.0

elseFINQ := (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. *)varGNUTRIENT, 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. *)varCHL_TO_C, GL: REAL;begin

CHL_TO_C := XQNA * QMOD;

MP Appendix page viii Feb 00

GL := ((ALPHAL * XPAR * CHL_TO_C) - RB0);if GL < 0.0 then

G_LIGHT := GLelse

G_LIGHT := GL / (1 + RMU);end;

beginTHIS_GRMAX := GRMAX * TF;GN := THIS_GRMAX * (1 - (QMIN / Q));GP := THIS_GRMAX * (1 - (QMINP / QP));GLIGHT := G_LIGHT(QMOD, XPAR);if GN < GP then

beginGNUTRIENT := GN;CONTROL := CN;

endelse

beginGNUTRIENT := GP;CONTROL := CP;

end;if GNUTRIENT < GLIGHT then

GROWTHwithP := GNUTRIENTelse

beginGROWTHwithP := 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 *)typeOPTION = (NONE, UPTAKE, QUOTA);varUMAX: REAL;SUPRESSION: REAL;SUPRESSION_BY: OPTION;

function FNOS (NOS: REAL): REAL; (* Michaelis-Menten uptake, with half-saturationCUNOS.*)

beginif 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 thenFINNH := (1.0 / (1.0 + (NHS / HIAMM)))

elseFINNH := 1.0;

end;function FINQ (QMOD: REAL): REAL; (* Cell-quota inhibition, to avoid overfilling

with N; *)

MP Appendix page ix Feb 00

(* QMOD is Q - QH*ETA; QMAXMOD is QMAX - QH*ETA .*)begin

if QMOD > QMAXMOD thenFINQ := 0.0

elseFINQ := (1.0 - (QMOD / QMAXMOD));

end;function FINP (GLF: LF; POS: REAL): REAL; (* uptake supression when P in control *)begin

if GLF = CP thenFINP := POS / (CUPOS + POS)

elseFINP := 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) thenFINQP := (QP - QMINP) / (QMAXP - QMINP)

elseFINQP := 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;

Printing Guide

For the benefit of users who print this document, it may beworth pointing out that the numbered, text, pages are designedto be right-hand pages, with diagrams and tables intended asleft-hand pages.

Readers may thus wish to:

1. print the document on A4 paper;

2. reverse the diagrams and tables and sort them so thatthey lie face-down on top of the text page which has theappropriate footnote;

3. ring-bind the document.

It should then be possible to open on, for example, page 5, andfind 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 moreattention. They are:

• the two-page Tables 1.1 and 8.2, which could each bemade a pair of left- and right- facing pages, Table 1.1being placed between pages 3 and 4 and Table 8.2between pages 35 and 36;

• Figures 9-3 to 9-6: it is suggested that 9-3 be placedopposite page 41, and 9-6 be placed opposite page 42,while 9-4 and 9-5 make a pair of left- and right- facingpages between page 41 and the back of Fig. 9-6.