Journal of Marine Systems 49 (2004) 105 – 122
www.elsevier.com/locate/jmarsys
Incorporating turbulence into a plankton foodweb model
Aisling M. Metcalfe a,*, T.J. Pedley a, T.F. Thingstad b
a
Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge, CB3 0WA, UK
b
University of Bergen, Department of Microbiology, Jahnebakken 5, N-5020 Bergen, Norway
Received 15 October 2002; accepted 9 July 2003
Available online 19 March 2004
Abstract
Small-scale fluid motions in the ocean affect the rate of nutrient uptake by bacteria and phytoplankton and the predation
rates of zooplankton. The magnitude of the effect depends on the size and swimming speed of the organisms. Theory predicts
that nutrient uptake will be increased by turbulence and that zooplankton – phytoplankton encounter and capture rates will be
increased at low turbulent intensity but the capture rate will be decreased at high turbulent intensity.
We present a mathematical model of an enclosure experiment carried out in Norway in July 2001. In the experiment the
enclosed plankton communities were subjected to various levels of turbulence, generated using oscillating grids, and to different
initial nutrient conditions. We predict that, for the experimental conditions, the rate of nutrient uptake by diatoms and grazing by
copepods will increase with turbulence but other parameters will be unaffected. We present results of simulations in which these
parameters are increased separately and together. Comparison of computational and experimental results suggests that the
dominant effect of turbulence is the increase in nutrient uptake by diatoms.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Nutrient uptake; Turbulence; Zooplankton – phytoplankton encounter
1. Introduction
The oceans are continually disturbed by wind and
waves, generating small-scale fluid motion or turbulence, which has been shown both experimentally and
theoretically to affect nutrient uptake by phytoplankton and predation by zooplankton (see for example the
review by Peters and Marrasé, 2000). The large eddies
generated by wind and wave motion in the ocean are
* Corresponding author. Tel.: +44-1223-765-000; fax: +441223-765-900.
E-mail address: [email protected]
(A.M. Metcalfe).
0924-7963/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmarsys.2003.07.003
unstable and break down to form smaller eddies. On
long length scales the effect of viscosity is unimportant and no kinetic energy is lost between the larger
and smaller length scales but at sufficiently small
length scales, given by the Kolmogorov microscale g
¼ ðm3 =eÞ1=4 , viscosity becomes important and the
kinetic energy in the eddies is dissipated to heat
(Batchelor, 1953). Here e is the turbulent kinetic
energy dissipation rate and m = 10 6 m2 s 1 is the
kinematic viscosity of water (Batchelor, 1967). At
length scales smaller than g rapid fluctuations in fluid
velocity are dissipated by viscosity and the fluid flow
can be regarded as laminar, though still random. For
many realistic values of e the Kolmogorov microscale
lies in the middle of the plankton size spectrum (see
106
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
Table 1
Parameter definitions and values
Parameter
Unit
Definition
Value
e
m 1
3 4
g ¼ me
YH
m2 s 3
m2 s 1
m
dimensionless
0 – 10 4
10 6
1.78 10 3 – 3.16 10 4
0.3
YC
YZ
aBN
aAN
aDN
aHB
aCH
aCA
aZC
aZD
kB
kA
kD
kH
kC
kZ
lB ¼ kB =aBN
lA ¼ kA =aAN
lD ¼ kD =aDN
lH ¼ kH =2aHB
lC ¼ kC =2aC
lZ ¼ kZ =2aZ
dZ
Pe
U
R
D0
Ucrit
Peturb
Sh
qH
Rc
cP
vH, vP
wT
tR 1
sK = me 2
dimensionless
dimensionless
l nmol P 1 h 1
l nmol P 1 h 1
l nmol P 1 h 1
l nmol P 1 h 1
l nmol P 1 h 1
l nmol P 1 h 1
l nmol P 1 h 1
l nmol P 1 h 1
h 1
h 1
h 1
h 1
h 1
h 1
nmol P l 1
nmol P l 1
nmol P l 1
nmol P l 1
nmol P l 1
nmol P l 1
h 1
dimensionless
m s 1
m
cm2 s 1
m s 1
dimensionless
dimensionless
cells m 3
Am
nmol P cell 1
Am s 1
Am s 1
s
s
dissipation rate of turbulent kinetic energy
kinematic viscosity of water
Kolmogorov lengthscale
fraction of nutrient in prey incorporated into
heterotrophic flagellates
fraction of nutrient in prey incorporated into ciliates
fraction of nutrient in prey incorporated into copepods
affinity of bacteria for nutrient
affinity of autotrophic flagellates for nutrient
affinity of diatoms for nutrient
clearance rate of heterotrophic flagellates on bacteria
clearance rate of ciliates on heterotrophic flagellates
clearance rate of ciliates on autotrophic flagellates
clearance rate of copepods on ciliates
clearance rate of copepods on diatoms
maximum nutrient uptake rate for bacteria
maximum nutrient uptake rate for autotrophic flagellates
maximum nutrient uptake rate for diatoms
maximum ingestion rate for heterotrophic flagellates
maximum ingestion rate for ciliates
maximum ingestion rate for copepods
half-saturation coefficient for bacteria
half-saturation coefficient for autotrophic flagellates
half-saturation coefficient for diatoms
half-saturation coefficient for heterotrophic flagellates
half-saturation coefficient for ciliates
half-saturation coefficient for copepods
intrinsic death rate of copepods
Péclet number
characteristic fluid velocity
characteristic lengthscale
molecular diffusivity of phosphate
critical swimming velocity
turbulent Péclet number
Sherwood number
prey density
contact radius
phosphorus content of predator P
swimming speed of prey H, predator P
turbulent velocity scale
handling time
Kolmogorov timescale
0.3
0.2
0.20
0.0048
7.2 10 4 – 9.8 10 4
0.026
0.0039
0.0039
4.5 10 4 – 9.1 10 4
4.5 10 4 – 9.1 10 4
0.40
0.091
0.043
1.6
0.39
0.063
2.0
20
60 – 44
31
50
71 – 35
0.002
6 10 6
The allometric (still water) values of lD and lZ are the largest values of the range shown, i.e., lD = 60, lZ = 71 nmol P l 1. The allometric (still
water) values of aDN, aZD and aZC are the smallest of the range shown, i.e., aDN = 7.2 10 4, aZD = aZC = 4.5 10 4 l nmol P 1 h 1. Note that
ciliates and diatoms have different carbon contents so the half-saturation value for copepods preying on ciliates will be slightly larger than that
for copepods preying on diatoms. The value given here is the average of the two.
Table 1) and we expect that turbulence will have a
different effect on species of different sizes.
In this paper we incorporate previous work on the
effect of turbulence on plankton into a model of a
plankton foodweb, which we compare with the results
of a mesocosm experiment. We begin by describing
the experiment and the equations used to model it,
presenting several different choices for the functional
forms in the equations. We then summarise the
theoretical predictions of the effect of turbulence on
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
the different plankton groups and the effect of turbulence on our model parameters. Finally we present the
results of the model and compare them to the experimental results.
2. Materials and methods
The mesocosm experiment aimed to explore the
effect of turbulent fluid motions on plankton and
nutrient dynamics. It was carried out from July 11–
25, 2001 at the Marine Biological Station in Espegrend, Norway. Twelve 2400-l mesocosms were filled
with seawater, collected offshore, which had been
filtered through a 250-Am mesh to remove larger
organisms. At the start of the experiment six of the
mesocosms were perturbed by the addition of
nutrients (1 AM P, 16 AM N, 32 AM Si); the remaining
six had no nutrient addition. Out of each series of six
mesocosms two contained still water and the remaining four were subjected to four different levels of gridgenerated turbulence. The still mesocosms are referred
to as T0, the turbulent mesocosms as T1 – T4 and the
nutrient-enriched mesocosms are NT0 – NT4. Turbulence level T1 has a turbulent kinetic energy dissipation rate of e = 10 7 m2 s 3, T2 has e = 5 10 6 m2
s 3 , T3 has e = 5 10 5 m 2 s 3 , and T4 has
e = 10 4 m2 s 3.
Turbulence was generated using a pair of vertically
oscillating grids 89 cm apart. The level of turbulence
in the tanks was measured with an acoustic doppler
current meter (an NDP from Nortek) and the stroke
length and frequency required for the different levels
of turbulence were calculated according to the method
detailed in Stiansen and Sundby (2001).
An integrated sample was taken every day using
three plastic cylinders 5 cm in diameter. Sample
volume varied according to the day (some parameters
were only measured every 2 days) and the total
volume taken from each mesocosm over the course
of the experiment was less than 170 l.
We present here the results for diatom and copepod
concentration. Enumeration and identification of diatoms were carried out on samples (100 ml) preserved
by formaldehyde (0.4% final concentration) using the
method of Utermöhl (1931). A minimum of 400 cells
were counted using a Zeiss axiovert 100 inverted
microscope. Estimates of diatom carbon biomass were
107
made by converting microscopic size measurements
to cell volumes, which were converted to carbon
biomass (Menden-Deuer and Lessard, 2000). A molar
C/P ratio of 106:1 was used to convert from the
carbon units.
The copepod volume was calculated as detailed in
Alcaraz et al. (in press) and converted to phosphate
using an average C/P conversion factor (HärdstedtRoméo, 1982; Le Borgne, 1975).
3. Model equations
The major groups of organisms present in the
mesocosms were bacteria, autotrophic flagellates, diatoms, heterotrophic flagellates, ciliates and meso-zooplankton (copepods). The growth of bacteria,
autotrophic flagellates and diatoms is limited by the
availability of inorganic nutrient (phosphate), as shown
in previous experiments in the same region (Thingstad
et al., 1999a) and the trophic interactions are shown in
Fig. 1. The properties of the various species will be
important in determining the parameters and for convenience we summarise them in Table 2.
The development of the various populations is
represented by differential equations in the concentrations of bacteria (B), autotrophic flagellates (A),
diatoms (D), heterotrophic flagellates (H), ciliates
(C), copepods (Z) and free nutrient (N), measured
in nmol P l 1. We assume that no nutrient is lost
from the system and that all nutrient released by a
Fig. 1. The model foodweb.
108
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
Table 2
Properties of the different plankton groupings
Length
(Am)
Bacterium
Autotrophic
Flagellate
Diatom
Heterotrophic
Flagellate
Ciliate
Copepod
Swimming
speed (Am s 1)
Carbon content
(pg C cell 1)
Phosphorus content
(nmol P cell 1)
30
200
0.02
7.4
3.3 10 8
5.8 10 6
–
–
40
4
10
200
150
7.4
1.2 10 4
5.8 10 6
–
50
1000
1000
4000
2000
3 106
1.6 10 3
2.4
50
1000
0.5
4
Contact radius
(Am)
Handling
time (s)
–
–
–
15
5
10
0.1
The lengths and swimming speeds of bacteria, ciliates and copepods and the swimming speed of flagellates are taken from Peters and Marrasé
(2000). The size and carbon content of a flagellate were provided by F. Peters (personal communication) and the length and carbon content of a
typical diatom were taken from data provided by A. Jacobsen. The swimming speed given for diatoms is actually a sinking speed, taken from
the experiments of Bienfang (1980), and agrees with the theoretically calculated sinking speed (Karp-Boss et al., 1996). The phosphorus content
of bacteria and ciliates is taken from Thingstad et al. (1999b) and that of copepods from Kiørboe et al. (1985). The carbon and phosphorus
contents are related using a molar C/P ratio of 50 for bacteria and 106 for protists (Thingstad et al., 1999b). The contact radius of a predator is
the distance over which it can perceive its prey and the handling time is the time required for it to locate and capture prey. The contact radius and
handling time for copepods are taken from Jonsson and Tiselius (1990) and Kiørboe and Saiz (1995). The contact radii of ciliates and
heterotrophic flagellates are estimates based on their sizes. The handling time of heterotrophic flagellates is taken from Boenigk and Arndt
(2000), who found times of 2 – 14 s from first contact of a bacterium with the flagellate to the bacterium being completely enclosed in a vacuole,
so the value in Table 2 is an overestimate. The handling time for ciliates is an estimate, which we expect to be an overestimate, but in Section 5
we will see that accurate handling time and contact radius are only necessary for copepods.
process (e.g. the death of copepods) is immediately
remineralised to inorganic forms. The equations
are:
dB
¼ uBN B gHB H
dt
ð3:1aÞ
dA
¼ uAN A gCA C
dt
ð3:1bÞ
dD
¼ uDN D gZD Z
dt
ð3:1cÞ
dH
¼ YH gHB H gCH C
dt
ð3:1dÞ
dC
¼ YC ðgCA þ gCH ÞC gZC Z
dt
ð3:1eÞ
dZ
¼ YZ ðgZD þ gZC ÞZ dZ Z
dt
ð3:1f Þ
3.1. Linear model
The simplest model is to assume that the nutrient
uptake and grazing rates are linear, i.e.
uBN ¼ aBN N ;
dN
¼ dZ Z þ ð1 YH ÞgHB H þ ð1 YC ÞðgCA þ gCH ÞC
dt
þ ð1 YZ ÞðgZD þ gZC ÞZ uBN B uAN A
uDN D:
Here uBN, uAN and uDN are nutrient uptake rates,
measured in units of h 1, for bacteria, autotrophic
flagellates and diatoms and gIJ is the grazing rate of
species I feeding on species J. dZ is the copepod death
rate. The uptake and grazing rates and copepod death
rate are measured in h. YH, YC and YZ are dimensionless
yields which represent the inefficiency of predator
feeding. Copepod mortality is sometimes represented
by a quadratic closure term dZZ2 (Edwards and Brindley, 1999).This is intended to model predation by
higher predators, whose population is assumed to be
proportional to Z, so for the mesocosm experiment
(where there is no higher predator) we choose a linear
death function.
ð3:1gÞ
uAN ¼ aAN N ;
uDN ¼ aDN N ;
ð3:2aÞ
gHB ¼ aHB B;
gCA ¼ aCA A;
gZC ¼ aZC C;
gZD ¼ aZD D;
gCH ¼ aCH H;
ð3:2bÞ
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
where the parameters a are affinity constants, with
units of 1 nmol P 1 h 1 (see Table 1). Models of this
type have been used in modelling previous mesocosm
experiments (Thingstad et al., 1999a,b) and it is easy
to incorporate a description of the effect of turbulence
on nutrient uptake and predation into such models.
However, we find that there is only a narrow range of
parameters in which the three phytoplankton species
can coexist. A slight variation in the parameters, for
example to model the effect of turbulence, may cause
some or more of the species to become extinct, an
effect which is not seen in the experiments. This
instability makes the model a very poor tool for
prediction.
3.2. Linear saturation model
The simplest improvement to the linear model of
Section 3.1 is to include the maximum rate at which
phytoplankton can take up nutrient and the maximum rate at which predators can ingest prey. We
replace the uptake and grazing rates given in Eq.
(3.2a,b) by
uBN ¼ minðaBN N ; kB Þ;
uDN
uAN ¼ minðaAN N ; kA Þ;
ð3:3aÞ
¼ minðaDN N ; kD Þ;
gHB ¼ minðaHB B; kH Þ;
ð3:3bÞ
aCA AkC
gCA ¼ min aCA A;
;
aCA A þ aCH H
aCH HkC
;
gCH ¼ min aCH H;
aCA A þ aCH H
ð3:3cÞ
gZC
gZD
aZC CkZ
¼ min aZC C;
;
aZC C þ aZD D
aZD DkZ
¼ min aZD D;
;
aZC C þ aZD D
ð3:3dÞ
where kB, kA and kD are the maximum nutrient
uptake rates for bacteria, autotrophic flagellates and
diatoms and kH , kC and kZ are the maximum
ingestion rates for heterotrophic flagellates, ciliates
and copepods. The affinity constants, a, are the same
as those used in the linear model. In this formulation
the ciliates and copepods do not favour one type of
prey over the other and consume prey in proportion
109
to the relative prey concentration when there is an
excess of prey.
3.3. Holling type III grazing functions
An alternative to the linear model of Section 3.2 is
to use nonlinear uptake and grazing functions, an
approach which has been widely used. Nutrient uptake is represented by Michaelis– Menten functions
uBN ¼
kB N
;
lB þ N
uAN ¼
kA N
;
lA þ N
uDN ¼
kD N
;
lD þ N
ð3:4Þ
where kB, kA and kD are the maximum nutrient uptake
rates as in Section 3.2 and lB, lA and lD are the halfsaturation coefficients for nutrient uptake by bacteria,
autotrophic flagellates and diatoms.
The grazing terms are modeled using a sigmoidal
function, often referred to as a Holling type III
function (Steele and Henderson, 1981; Edwards and
Brindley, 1999; Palmer and Totterdell, 2001). The
important difference between this model and that of
Section 3.2 is that the grazing rate function has zero
gradient when the prey concentration is zero. There is
experimental evidence to suggest that copepods cease
feeding at low food concentrations (Frost, 1975;
Mullin et al., 1975; Price and Pafenhofer, 1986),
which is theoretically appealing as we would expect
predation to cease if prey are so sparse that the
energetic cost of finding and capturing prey outweighs
the benefit, but there is little evidence that protozoa
exhibit such feeding thresholds. The main benefit of
using a sigmoidal function is that the zero gradient at
zero prey concentration stabilizes the model by providing a refuge for rare phytoplankton and allowing
coexistence of phytoplankton species (see Strom et
al., 2000, for a fuller discussion).
For the grazing of heterotrophic flagellates on
bacteria, we use a standard Holling type III function
gHB ¼
kH B2
;
l2H þ B2
ð3:5aÞ
where kH is the maximum ingestion rate as in Section
3.2 and lH is the half-saturation coefficient for heterotrophic flagellates.
110
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
In the model of Fig. 1 ciliates and copepods each
prey on two different species and this requires a
modification to the function given in Eq. (3.5a), which
we model in the same way as previous authors
(Palmer and Totterdell, 2001; Pitchford and Brindley,
1999; Fasham et al., 1990), assuming that a predator
with a choice of prey will preferentially feed on the
more plentiful prey species. We use
gCA ¼
C on A and H as an example. If we assume that
aCA = aCH = aC the total grazing by C has half its
maximum value, k2C , when the total food available,
kC
A + H, is 2a
. Using the Holling type III functions of
C
Eqs. (3.5b,c), total grazing by C is
2
A þ H2
kC
AþH
2
;
A þ H2
lC þ
AþH
gCH
which has half its maximum value,
2
þH 2
food available, AAþH
, is lC. Hence
kC A2
;
lC ðA þ HÞ þ A2 þ H 2
kC H 2
¼
;
lC ðA þ HÞ þ A2 þ H 2
ð3:5bÞ
kC
¼ lC ;
2aC
2
kZ C
;
lZ ðD þ CÞ þ D2 þ C 2
kZ D2
¼
;
lZ ðD þ CÞ þ D2 þ C 2
gZC ¼
gZD
ð3:5cÞ
3.4. Relating the parameter values
In order to compare the results of the different
models we need a consistent way of relating the
different parameters. If we assume that the initial
slope of the Michaelis – Menten uptake rates Eq.
(3.4) is the same as the linear uptake rates Eq.
(3.2a) we have
kA
¼ aAN
lA
kD
¼ aDN :
lD
ð3:6Þ
We will also assume that the half-saturation
values of the Holling type III functions Eqs.
(3.5a) – (3.5c) are the same as the half-saturation
values of the corresponding linear saturation functions Eqs. (3.3b) – (3.3d). For the predation of H on
B this gives
kH
¼ aHB :
2lH
when the total
ð3:8Þ
The forms of the functions can be seen in Fig. 2.
where kC and kZ are the maximum ingestion rates for
ciliates and copepods as in Section 3.2 and lC and lZ
are the half-saturation coefficients for ciliates and
copepods.
kB
¼ aBN
lB
kZ
¼ lZ :
2aZ
kC
2,
ð3:7Þ
The cases of C and Z, which prey on two species,
require more thought and we consider the predation of
3.5. Parameter values
We calculate the parameter values using the
empirical, allometric formulae of Moloney and
Field (1989, 1991), using a molar C/N/P ratio of
106:16:1 to convert to our units where necessary.
The maximal nutrient uptake rates for phytoplankton and bacteria are given by k = 0.15 M 1/4 and
the maximal ingestion rates for particle feeding
heterotrophs are given by k = 2.625 M 1/4 where
M is the mass of one organism in pgC and k is in
h 1. Half-saturation values for nutrient uptake are
given in nmolP l 1 by l = 8.93 M 0.38. For ingestion the half-saturation value depends on prey size:
l = 42.45 Mp0.08, where Mp is the mass of a prey
organism in pgC. Table 1 gives the values of k, l
and a (calculated from Eqs. (3.6), (3.7), and (3.8))
for the species in our model, using the masses
given in Table 2. Similar values for maximum
uptake and ingestion rates and half-saturation values
have been used by previous authors (Palmer and
Totterdell, 2001) or found experimentally (Saiz and
Kiørboe, 1995).
The yields are dimensionless ratios, which should
be positive and less than 1. The values we use are
given in Table 1 and agree with experimental and
theoretical estimates (Thingstad et al., 1999a,b; Jackson, 1980; Nagata, 1988). We take a copepod death
rate, dZ, of 5% per day, or 0.002 h 1 (Palmer and
Totterdell, 2001; Baretta-Bekker et al., 1998).
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
4. The effect of turbulence on nutrient uptake
A microorganism absorbs nutrient from the surrounding fluid and transports it across its cell membrane. A nutrient-depleted boundary layer therefore
develops around the microorganism and in the absence of fluid motion nutrient in this depleted layer
is replenished only by diffusion. However, in most
cases the fluid around a microorganism is moving,
either because of turbulence, or because the microorganism is sinking or actively swimming, or a
combination of these. This fluid motion may increase the flow of fresh, nutrient-rich fluid towards
the cell, thus reducing the thickness of the nutrientdepleted layer around the cell and increasing the
nutrient uptake rate (although if there is rotation
there may be regions of recirculating fluid close to
the cell so that the fluid in the boundary layer is not
replenished as frequently and the uptake rate is
reduced). Most of the nutrient transfer to the cell
will take the path of least resistance so a process
which causes thinning of some part of the boundary
layer will generally increase the nutrient uptake rate
even if it increases the boundary layer thickness
elsewhere. A review of the different effects of fluid
motion can be found in Karp and Boss et al. (1996)
and we summarize the relevant results here and
apply them to our model phytoplankton.
The relative importance of advection and diffusion is given by the Péclet number Pe = UR/D0,
where U is the characteristic fluid velocity, R is a
characteristic length scale (here taken to be the cell
radius) and D0 is the molecular diffusivity. Diffusion
dominates if Pe < 1 and advective effects dominate if
Pe>1. Using a molecular diffusivity D0 = 6 10 6
cm2 s 1 for phosphate in water and the lengths and
swimming speeds from Table 2, we find that the
Péclet number for bacteria PeB = 0.0125. This indicates that bacteria are so small that the nutrient
transport around them is dominated by diffusion,
advection is unimportant and their nutrient uptake
rate will not be affected by increased turbulence.
Various experiments have confirmed that the nutrient
uptake rate of isolated bacteria is unaffected by fluid
motion, although bacteria which aggregate may have
an effective size large enough that they can benefit
from fluid motion (Logan and Dettmer, 1990; Peters
et al., 1998). Autotrophic flagellates have PeA = 0.67
111
and diatoms have PeD = 0.33 so advective effects
will be moderately important for these species and
we need to consider them further.
The flow round a microorganism swimming in
turbulent fluid is the superposition of its translational
and stirring motion and the varying shear due to
turbulence. Whether the effect of swimming or of
turbulence dominates depends on the size and swimming speed of the microorganism and on the turbulent
intensity. For a fast swimmer the flow due to swimming will dominate. Batchelor (1980) defined a critical velocity
Ucrit ¼
R2 34
pffiffiffiffi
:
m
D0
If the swimming or sinking velocity of a microorganism is less than Ucrit the effect of turbulence
dominates whereas if the velocity is greater than
Ucrit the effect of swimming or sinking dominates.
For an autotrophic flagellate in the highest experimental turbulence level T4, Ucrit = 5.2 Am s 1,
which is much less than the average swimming
speed of a flagellate (200 Am s 1) indicating that
the effect of turbulence is negligible for them. For a
diatom in T1, Ucrit = 2.9 Am s 1, less than the
average diatom sinking speed. However, for T2,
T3 andT4 Ucrit = 55, 310 and 520 Am s 1, much
greater than the average diatom sinking speed.
Therefore for T1 the effect of sinking dominates
the effect of turbulence, but for T2 – T4 the effect
of turbulence dominates. Experiments have confirmed that the nutrient uptake rate of a diatom is
increased by fluid motion (Pasciak and Gavis,
1975).
The Sherwood number is defined for a particular
organism and fluid regime as
Sh ¼
total flux in presence of fluid motion
:
diffusional flux in absence of fluid motion
Karp-Boss et al. (1996) present a graph (their Fig.
2) of Sherwood number against Péclet number for a
microorganism swimming or sinking in still water.
From this, the Sherwood number for diatoms is
ShD = 1.1.
112
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
Fig. 2. Rate of nutrient uptake rate by bacteria, uBN, and of grazing by heterotrophic flagellates on bacteria, gHB, showing the different functional
forms investigated. For uBN, kB = 0.40, lB = 2.0, aBN = 0.20. For gHB, kH = 1.6, lH = 31, aHB = 0.026. (See text and Table 1 for definitions of these
quantities.). Dotted line, linear model; solid line, saturation model; dashed line, nonlinear model.
For turbulent flow the Péclet number is defined as
Peturb ¼
R2 e 12
;
D0 m
the value of Peturb for diatoms for T2 –T4 is given in
Table 3. Karp-Boss et al. (1996) also give a graph of
Sh against Peturb (their Fig. 6) for non-motile cells in
turbulent flow and Table 3 gives the values of Sh that
correspond to Peturb for T2 –T4. For the range of
Peturb which we are considering, Karp-Boss et al.
(1996) actually give a narrow range of values for
Sh; Table 3 contains the mean of their upper and lower
bounds for Sh. We can then calculate the increase in
aDN due to turbulence as aDNturb ¼ ShShturb
aDN , where
D
ShD = 1.1 is the Sherwood number due only to sinking
by diatoms. The values of aDN and the corresponding
values of lD are given in Table 3; the values for T0
and T1 are the allometric values and are included here
for completeness.
Table 3
Values of turbulent Péclet and Sherwood number for the T2 – T4
turbulence levels and the values of aDN and lD for all turbulence
levels
Peturb
Shturb
aDN ( 10 4)
lD
T0
T1
T2
T3
T4
–
–
7.2
60
–
–
7.2
60
1.5
1.3
8.3
52
4.7
1.4
9.4
46
6.7
1.5
9.8
44
aDN is given in l nmol P 1 h 1 and lD in nmol P l 1.
5. The effect of turbulence on predator/prey
capture rates
The rate at which a predator captures prey depends
on the rate at which it encounters prey and on the
probability of it catching the prey once encountered.
We expect that both the encounter rate and the capture
probability will be affected by fluid motion; the
encounter rate will increase as turbulence increases
because increased fluid motion brings more prey to
the predator, but the capture probability will decrease
with increased turbulence as prey spend less time in
close proximity to the predator.
The theoretical encounter-rate problem for swimming organisms was considered by Lewis and Pedley
(2000) following the work of Rothschild and Osborn
(1988). They calculated the contact rate when the
swimming velocity distributions of a predator P and a
prey H are independent, three-dimensional, isotropic
Gaussian distributions with zero mean and the organisms swim linearly with no changes of direction. They
used a swept volume approach and calculated the
contact rate, CPH, (the number of prey encountered by
a single predator in unit time) in turbulent fluid in
terms of prey density qH (cells per unit volume),contact radius Rc (the distance over which a predator can
perceive its prey), the expected values of the swimming speeds of prey and predator, vH and vP, and the
turbulent velocity scale, wT. The turbulent velocity,
wT, depends on Rc and Lewis and Pedley (2000)
clarified a point about which there had been some
debate, that the appropriate length scale to use in wT is
the contact radius rather than the length scale of the
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
predator or the mean distance between prey as had
been suggested by some previous authors (see Visser
and McKenzie, 1998, for a further discussion). The
affinity constant of predator P for prey H, aPH, is
related to the contact rate CPH by
aPH ¼
12
CPH
pR2 8 2
wT þ v2H þ v2P ;
¼ c
q H cP
cP p
ð5:1Þ
where cP is the phosphorus content of one predator.
Using Eq. (5.1) and the swimming speeds and
contact radii of Table 2 we can calculate the affinity
constants for the predators in our model under still and
turbulent conditions and these are shown as the first
four lines of Table 4. We have shown the results to
three significant figures so that the effect of turbulence
can be seen, although the contact rates cannot actually
be known to this accuracy. The swimming speeds of
autotrophic and heterotrophic flagellates are the same
so aCA = aCH = aC, but aZD and aZC are shown separately because ciliates and diatoms have different
swimming or sinking speeds.
Comparing the calculated as of Table 4 with the
allometric as of Table 1 we see that the calculated
encounter-rates a are similar to or larger than the
allometric a. This is what we would expect since a
predator will not consume all the prey it encounters.
We also see that the only significant effect of turbulence is on the contact rate of copepods with their prey
and then only at the higher turbulence levels. From
Table 4 we can see that the contact rate of copepods
with diatoms is not significantly different from that of
copepods with ciliates and we are justified in using
one parameter, aZ, to describe both.
Table 4
The first four lines give the values of the affinities a (l nmol P 1
h 1) in still and turbulent conditions as calculated from the
encounter rate (Eq. (5.1))
T0
T1
T2
T3
T4
aHB
0.00986 0.00986 0.00987 0.009949 0.0100
aC
0.0180
0.0180
0.0181
0.0187
0.0193
0.0188
0.0189
0.0206
0.0312
0.0390
aZD
aZC
0.0194
0.0195
0.0211
0.0315
0.0393
aZ ( 10 4) 4.5
4.5
4.9
7.3
9.1
lZ
71
71
64
43
35
The last two lines give the values of aZ and lZ to be used in the
simulations.
113
To find the values of aZ to use in modelling the
turbulent mesocosms we use the calculated encounter
rates aZD of Table 4 to calculate the increase in
encounter rate compared to that in the still mesocosm
and then multiply this by the allometric value of aZ as
given in Table 1. The corresponding values of lZ are
calculated using Eq. (3.8) and the values of aZ and lZ
are given as the final two lines of Table 4. The results
using aZC instead of aZD differ only in the third
significant figure of aZ.
Lewis and Pedley (2001) extended their earlier
work, using similar ideas to those of McKenzie et
al. (1994), to include the effect of turbulence on
capture probability. They postulated that the capture
probability Pcap depends on the length of time a prey
particle would spend in the capture sphere if the
predator made no attempt at capture and that this time
is affected by turbulence. They calculated the probability that the distance of closest approach of a prey
particle to the predator is r, multiplied this by the
probability of capturing the prey given that its closest
approach is r and integrated over r, to find the number
of prey N likely to be captured in a time T.
The capture probability is assumed to be given by
Pcap ¼
tðrÞ2
2
tðrÞ þ ð stRK Þ2
ð5:2Þ
where tR, the handling time, is the length of time the
predator needs to 1successfully handle and capture the
prey and sK ¼ ðmeÞ2 is the Kolmogorov timescale. The
non-dimensionalised time a prey particle spends in
the contact sphere if the closest distance of approach
is r, t(r), is given in terms of r, Rc, vH, vP and wT (see
(see Lewis and Pedley, 2001, for full details).
The affinity constant aPH is related to N, the
number of prey likely to be captured in time T, by
N
aPH ¼
qH c P T
pffiffiffi
Z Rc
8p g3 g
2 drU
Pcap 2rrU þ r
¼
dr; ð5:3Þ
cP sK 0
dr
where rU(r), the variance of the relative velocity of
predator and prey, depends also on Rc, vH, vP and wT.
The still and turbulent capture rates can be calculated from Eq. (5.3) using the values given in Table 2
and they are given in the first four lines of Table 5. As
in Table 4, aCA = aCH = aC but aZD and aZC are shown
114
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
Table 5
Values of the predator – prey capture rates a (in l nmol P 1 h 1) in
still and turbulent conditions as calculated from Eq. (5.3)
T0
T1
T2
T3
T4
8
aHB ( 10 ) 6.05
6.05
6.05
6.04
6.00
aC ( 10 7)
9.79
9.79
9.77
9.66
9.53
0.01626 0.01628 0.0173 0.0226 0.0256
aZD
aZC
0.0166
0.0167
0.0176 0.0228 0.0257
aZ ( 10 4)
4.5
4.5
4.8
6.3
7.1
lZ
71
71
66
50
44
The last two lines give the values of aZ and lZ to be used in the
simulations.
separately and the results are shown to 3 or 4
significant figures so that the effect of turbulence
can be seen.
For aHB and aC turbulence causes a slight decrease
in the capture rate. We would expect to see a large
decrease in capture rate if the handling time were
large, so the handling times for heterotrophic flagellates and ciliates have been deliberately overestimated, but even so the decrease is very slight so we
can see that turbulence does not affect the capture rate
for heterotrophic flagellates and ciliates.
The difference between aZD and aZC is negligible.
As before, the corresponding values of aZ and lZ for
use in the computations are calculated from the percentage decrease in aZD (last two lines of Table 5). We
see that aZD andaZC increase with turbulence, indicating that the dominant effect is the increase in encounter rate due to turbulence, although this increase is
smaller than that found based on encounter rate alone
(Table 4). If we used a longer copepod handling time
(e.g. 1 s rather than 0.1 s) we would find that
turbulence causes the capture rate to decrease.
From the calculations in this section we see that
turbulence increases the rate of predation by copepods
and does not affect the rate of predation of ciliates or
flagellates. In the computations we use the values of
aZ and lZ given in Table 4.
6. Results
The equations are integrated using a fifth-order
embedded Cash-Karp Runge-Kutta method with
adaptive step sizing (Press et al., 1992). The parameter values for the still mesocosms T0 and NT0 are
given in Table 1. The only parameters affected by
turbulence are aDN, lD, aZ and lZ. The values of aDN
and lD in the turbulent mesocosms T1 – T4 and
NT1 –NT4 are given in Table 3 and the values of
aZ and lZ are given in Table 4. For the computations
we use the parameter values correct to two significant figures, which means that the slightly turbulent
mesocosms T1 and NT1 are identical to the still
mesocosms T0 and NT0.
We assume that, at this time of year the sea, and
hence the initial state of the non-enriched mesocosms,
can be represented by a steady-state solution of the
model equations. For the nutrient-enriched mesocosms we increase the initial value of N.
We present the experimental results in Section 6.1,
in Section 6.2 we discuss the results of the linear
models of Sections 3.1 and 3.2 and in Section 6.3 we
present sample results from the nonlinear model of
Section 3.3.
6.1. Experimental results
When comparing the results of the model with
those of the experiment, it should be remembered that
the model should not be interpreted as a precise
quantitative simulation, but rather as a qualitative
indication of the effects of turbulence.
We consider first the results of the non-enriched
mesocosms. The clearest response to turbulence is
seen in the populations of diatoms and copepods
and we show the experimental concentrations of
diatoms and copepods in Fig. 3a. For both diatoms and copepods the concentrations in the turbulent mesocosms increase more than in the still
mesocosms.
In the nutrient-enriched mesocosms the clearest
response to turbulence is again seen in the populations
of diatoms and copepods and these results are shown
in Fig. 3b. For both diatoms and copepods the peak in
concentration increases under turbulent conditions
and the peak in copepod concentration occurs earlier
under the highest turbulence level (NT4) than under
the others.
6.2. Linear models
Outside a narrow range of parameter values and
initial conditions, the linear models of Sections 3.1
and 3.2 predict that the population of one (or more
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
115
Fig. 3. Experimental results. Concentrations of diatoms (D) and copepods (Z) in the nonenriched mesocosms (a) and in the enriched mesocosms
(b). The different lines represent different experimental conditions: dotted line T0/NT0, short dashed line T1/NT1, long dashed line T2/NT2,
dash-dot line T3/NT3, solid line T4/NT4. Note that for the copepods there are two lines for the still mesocosms, one for each of the replicates
T0A/T0B and NT0A/NT0B. The population concentrations are given in nmol P l 1.
commonly two) of the phytoplankton species will
fall rapidly to zero. This means that the system is no
longer the one we wish to consider and makes the
linear models unsuitable as predictive tools. Previous
authors (Moloney and Field, 1991, for example)
have also found this problem with linear models. It
can be overcome by introducing a threshold value
into the grazing terms, so that grazing ceases once
the prey population falls below a threshold value, or
by introducing nonlinear grazing terms as we have
done.
If we consider the linear model of Section 3.1
we find that no steady state exists for the parameter
values given in Table 1 in which the concentrations
of all the species are positive. If we use the same
initial state as for the nonlinear model of Section
3.3 (Eq. (6.3)) the populations of A and D fall
rapidly towards zero and the system becomes a
simple chain N-B-H-C-Z with only one species at
each trophic level. This system does behave as one
would expect. Increasing aZ causes D to fall more
rapidly towards zero and the population of C and
then Z to decrease. The population of H increases,
A falls towards zero more slowly and B decreases.
Decreasing aZ has the opposite effect and increasing aD has no effect (because the system contains
no D).Although the system shows a response to
turbulence it is not the system we are trying to
model and the results are not applicable to our
experiment. In reality a system with no phytoplankton would be doomed to extinction since no carbon
is being fixed. Our model considers only inorganic
nutrient and hence allows such a system to persist.
The nutrient-enriched enclosures give similar
results, moving rapidly to an unrealistic state with
A and D zero.
Even if we consider a parameter range for which
the linear model does possess a steady-state solution
116
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
with the populations of all species positive we do not
fare any better. The parameters
aBN ¼ 0:09; aAN ¼ 0:05; aDN ¼ 0:03;
aHB ¼ 0:005; aCA ¼ aCH ¼ 0:001;
aZD ¼ aZC ¼ 0:0002;
ð6:1aÞ
YH ¼ YC ¼ YZ ¼ 0:3;
ð6:1bÞ
dZ ¼ 0:002;
have a steady state
N ¼ 0:410; B ¼ 13:677; A ¼ 33:646; D ¼ 12:818;
H ¼ 7:386; C ¼ 20:516; Z ¼ 61:547:
ð6:2Þ
Using Eq. (6.2) as the initial state and varying aD
and aZ in the same proportions as previously we find
that, for example, for a small increase in aZ (T2,
aZ = 2.2 10 4) the populations of all species remain
nonzero, but for a larger increase in aZ the populations of B and D fall to zero, which means that again
we are not modelling the system we intend to. The
nutrient-enriched mesocosms give similar results,
with one or two of the phytoplankton populations
falling to zero.
The linear saturation model gives similarly unrealistic results with the population of one or two of the
phytoplankton species always falling to zero. For
some values of the parameters the model also predicts
rapid fluctuations in population, which are unlikely to
be found in reality.
6.3. Holling type III model
Taking a total nutrient concentration of 250 nmol P
l 1, the non-enriched steady state (mesocosm T0) is
given by
N ¼ 11:6; B ¼ 5:67; A ¼ 80:3; D ¼ 1:52;
H ¼ 37:4; C ¼ 14:6; Z ¼ 98:9;
ð6:3Þ
where the concentrations are given in nmol P l 1. We
use this as the initial condition for the computations
for the non-enriched mesocosms.
We consider the non-enriched mesocosms first. As
lD decreases the population of D increases, leading to
an increase in Z and also in C, presumably because of
reduced predation on C due to the increased availability of D (Fig. 4). The increase in C leads to a
decrease in H and A and an increase in B; this is
probably due to decreased predation of H on B or to
decreased competition for nutrients (Fig. 4). The
population changes are small, although the percentage
change in D is quite large.
The effect of decreasing lZ is more marked, as
shown in Fig. 5. A small decrease in lZ (T2) leads
initially to a slight increase in the population of Z,
which causes a decrease in the population of D and C.
The decrease in C in turn leads to an increase in A and
H, and hence to a decrease in B, due either to
increased predation of H on B or to increased competition for nutrient from A. After the initial increase,
Z decreases because of the decreased populations of D
and C. The population of C then recovers, as does D,
the populations of A and H start to fall again and the
population of B increases again. For larger decreases
in lZ (T3 and T4) the solution is qualitatively different
and increased predation by Z causes the populations
of D and C to fall almost to zero. The populations of A
and H therefore increase and the population of B
decreases. After an initial slight increase the population of Z also decreases.
When lD and lZ are varied together the dominant effect seems to be the change in lZ. Decreasing lD and lZ, so that T0/T1 is lD = 60, lZ = 71; T2 is
is lD = 52, lZ = 64; T3 is lD = 46, lZ = 43; T4 is
lD = 44, lZ = 35, gives results that are almost indistinguishable from those of Fig. 5.
In the experiment the enriched mesocosms have
nutrient added as 1 Amol P l 1 (with N and Si in an
appropriate ratio). Since an added nutrient level of 1
Amol P l 1 may not correspond to an effective
increase of 1 Amol P l 1 in the ‘free nutrient’ of the
model, we considered the effect of different initial
values of N, ranging from N = 61.6 to N = 1011.6
nmolP l 1 (notionally equivalent to adding 50 –
1000 nmol P l 1).
For a small value of added nutrient (N = 61.6 nmol
P l 1) the populations fluctuate. Increased nutrient
causes an initial increase in the population of A, which
results in an increase in H (due to reduced predation
on H) and a later increase in C. The increase in C then
causes a decrease in A and H and an increase in B. As
the populations of A and H decrease, C starts to
decrease, but the decrease in C allows A and H to
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
117
Fig. 4. Results of computations for the non-enriched mesocosms for various values of lD. Decreasing lD corresponds to increasing the diatom
nutrient uptake rate. Dotted line lD = 60 (steady state, T0/T1), dashed line lD = 52 (T2), dash-dot line lD = 46 (T3), solid line lD = 44 (T4). The
population concentrations are given in nmol P l 1.
recover, leading to the population fluctuations. As in
the non-enriched case, the graphs remain qualitatively
similar as lD is decreased, but the decrease in lD
results in an increase in D, B, C and Z and a decrease
in A and H, with the largest percentage change in D
(Fig. 6a).
118
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
Fig. 5. Results of computations for the non-enriched mesocosms for various values of lZ. Decreasing lZ corresponds to increasing the rate of
grazing by copepods. Dotted line lZ = 71 (steady state, T0/T1), dashed line lZ = 64 (T2), dash-dot line lZ = 43 (T3), solid line lZ = 35 (T4). The
population concentrations are given in nmol P l 1.
As lZ decreases we see two qualitatively different solutions in the same way as in the non-enriched
case. For NT0/NT1 and NT2 the populations fluc-
tuate in a similar way to that discussed above (NT0/
NT1 is of course the same) although for NT2 the
fluctuations are slower and of smaller amplitude.
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
119
Fig. 6. Results of computations for the nutrient-enriched mesocosms with N = 61.6 nmol P l 1. (a) decreasing lD, (b) decreasing lZ.
Decreasing lD corresponds to increasing the diatom nutrient uptake rate and decreasing lZ corresponds to increasing the rate of grazing by
copepods. Dotted line NT0/NT1, dashed line NT2, dash-dot line NT3, solid line NT4. The population concentrations are given in nmol P l 1.
For NT3 and NT4 the increase in grazing by Z
causes the populations of C and D to fall almost to
zero, resulting in a slow decrease in Z, an increase
in A and H and a decrease in B, with no fluctuations (Fig. 6b).
For larger nutrient increases the population fluctuations are of larger amplitude and occur over a slightly
faster timescale. For high nutrient concentrations there
is little discernible difference in the solutions as lD
decreases, with the largest percentage change still in
D. As lZ decreases we still see two qualitatively
different solutions, with the populations of C and D
falling almost to zero in NT3 and NT4. For an initial
nutrient concentration of N = 1011.6 nmol P l 1 only
the NT4 solution is qualitatively different to the
others.
As for the non-enriched mesocosms, the dominant effect seems to be the change in lZ. Decreasing
lD and lZ simultaneously gives results that are
almost indistinguishable from those found by decreasing lZ alone. For both non-enriched and
enriched mesocosms it is possible to vary lD and
lZ together in such a way that neither effect dominates. However, this requires a smaller change in lZ
and a larger change in lD than that predicted in
Sections 4 and 5.
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A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
7. Discussion
We have described three types of model which
could be used to describe the mesocosm experiment: a
linear model, a linear saturation model and a nonlinear
model. Outside a very narrow range of parameters and
starting conditions the linear and saturation models
give unrealistic results with one or more of the species
concentrations falling to zero. For this reason the
nonlinear model would seem to be the most useful
for modelling the results of the mesocosm experiment.
Our model indicates that turbulence should increase the nutrient uptake rate of diatoms but should
not affect smaller phytoplankton or bacteria. We also
find that the grazing rate of copepods will be affected
by turbulence but that other zooplankton will be
unaffected. The effect of turbulence on the copepod
grazing rate depends on the contact radius and handling time of the copepod. For a short, realistic
handling time turbulence will increase the rate at
which copepods encounter prey without significantly
affecting the capture probability and the copepod
grazing rate will increase. If the handling time is
longer, turbulence will reduce the capture probability
and hence the grazing rate. We can see how important
it is to have a detailed understanding of the dynamics
of copepod hunting.
There is some evidence that smaller plankton can
be affected by turbulence (Peters and Marrasé, 2000;
Dolan et al., 2003). These effects may be due to
indirect effects of turbulence such as physiological
changes to cells, particularly dinoflagellates (Peters
and Marrasé, 2000) or a change in behaviour under
turbulence (as suggested for ciliates by Dolan et al.,
2003).
In the model increased nutrient uptake by diatoms
corresponds to a decrease in lD, which has only a
small effect on the populations, resulting in an increase in the concentrations of diatoms, ciliates,
copepods and bacteria and a decrease in autotrophic
and heterotrophic flagellates. Increased copepod grazing corresponds to a decrease in lZ. This has broadly
the opposite effect to decreasing lD and leads to an
increase in the concentrations of diatoms, ciliates,
copepods and bacteria and an increase in autotrophic
and heterotrophic flagellates. The population changes
for decreasing lZ are more marked than for decreasing
lD and there are two qualitatively different solutions
at low and high turbulence; for a sufficiently large
decrease in lZ (T3 and T4) the populations of diatoms
and ciliates fall almost to zero. When lD and lZ are
varied together the dominant effect seems to be the
change in copepod grazing.
Increasing the initial concentration of free nutrient
generally results in fluctuations in the plankton
populations. As more nutrient is added these fluctuations become slightly more rapid and of larger
amplitude. As in the non-enriched case, there is a
qualitatively different solution when lZ is decreased
in which the populations of diatoms and ciliates fall
almost to zero.
In the experiments turbulence increased the diatom
and copepod concentrations in both non-enriched and
enriched mesocosms. This suggests that the computational results of Figs. 4 and 6a, in which turbulence
causes lD to decrease, reflect the experimental results
more closely than the computational results of Figs. 5
and 6b, in which turbulence causes lZ to decrease.
The dominant effect of turbulence therefore appears to
be an increase in diatom nutrient uptake, which, in
terms of our model parameters, corresponds to a
decrease in lD. In the nutrient-enriched experiments
(Fig. 3b) the peak in copepod concentration occurs
earlier under the highest turbulence level (NT4) than
under the others; this is not explained by the current
model.
Future mesocosm experiments will consider the
effect of subjecting mesocosms to a gradient of added
nutrient and to either still or turbulent conditions.
These should give more insight into the interaction
of nutrients and turbulence and provide further comparison for the model results.
Several microcosm studies on the effect of turbulence on plankton communities have been reported.
The microcosm experiments of Peters et al. (2002)
found an increase in bacterial concentration under
turbulent conditions, which was due to a reduced
grazing pressure on bacteria. This result compares
well with the model results for bacteria shown in
Fig. 4. In similar microcosm experiments Alcaraz et
al. (2002) found an increased phytoplankton bloom in
turbulent conditions; this may correspond to the
increase in diatom concentration predicted in Fig. 4.
It should be remembered that the plankton community
in those microcosm experiments will be different from
that considered here, particularly since they used
A.M. Metcalfe et al. / Journal of Marine Systems 49 (2004) 105–122
seawater from the North West Mediterranean and
filtered though a smaller mesh.
Acknowledgements
We would like to thank everyone who provided
experimental data and biological insight, particularly
M. Alcaraz and M. Parisi for copepod data, A.
Jacobsen for diatom data and C. Marrasé, J. Dolan, J.
Egge, G.-A. Fonnes, O. Guadayol, H. Havskum, A.
Larsen, F. Peters and J. E. Stiansen. We would also
like to thank Dr. J.W. Pitchford for helpful theoretical
discussions.
This research was funded by the shared cost
research project NTAP (Contract No. EVK3-CT2000-00022) of the EU RTD Programme ‘Environment and sustainable development’ and forms part of
the ELOISE projects cluster. It is ELOISE Contribution No. 298/40. Access to the Bergen Marine Food
Chain Research Infrastructure was supported by the
Improving Human Research Potential Programme
from the European Union, Contract No. HPRI-CT1999-00056.
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