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Modeling of copepods with links to
circulation models
WOLFGANG FENNEL
INSTITUT FÜR OSTSEEFORSCHUNG WARNEMÜNDE AN DER UNIVERSITÄT ROSTOCK, D- WARNEMÜNDE, GERMANY
CORRESPONDING AUTHOR: [email protected]
An important step towards realistic models of the marine ecosystem is the coupling of biological and circulation models. While the modelling of the lower trophic levels has made progress in the last years the
description of stage-resolving zooplankton is still in a preliminary state. The paper presents a zooplankton model which includes the lower trophic levels of the food web and which can be embedded in a circulation model in a consistent manner. The model has two sets of zooplankton state variables, the biomass
and number of individuals of the stages. The model is used to simulate rearing tank experiments under
constant environmental conditions. A link to oceanic conditions, with coupling to the lower levels of the
food web and annual variations of temperature, is studied by a simple box model version. As the ‘modelcopepod’ we choose Pseudocalanus, but the model can be applied to other species in a straightforward way.
I N T RO D U C T I O N
An important step towards a theoretical understanding
and quantitative description of the responses of a marine
ecosystem to physical forcing or nutrient input can be
achieved by realistic models that couple chemical–
biological models and ocean circulation models. This
paper aims at a consistent formulation of a stage resolving
zooplankton model, which can directly be integrated in
physical circulation models.
For the General Circulation Models, (GCMs), the
model equations follow directly from fundamental principles, such as conservation of momentum, energy and
mass.
The model equations are the mathematical formulation
of fundamental laws. The development of the GCMs is
an ongoing process but it can be said, in a somewhat
simplified manner, that problems in running GCMs
concern the parameterizations of sub-grid processes,
treatment of open boundaries, and the availability of
proper data sets for initialization and atmospheric forcing.
The biological models require parameterizations of
sub-grid processes as well as proper initialization and
forcing data, e.g. external nutrient input. Although the
biological processes have to obey the conservation laws of
mass and energy, the model equations do not follow from
these laws, as occurs in physics. Thus, it is a theoretical
challenge to find the right mathematical formulations
which govern the chemical–biological dynamics.
Model equations describe the change of state variables
© Oxford University Press 2001
in time and space driven by physical, chemical and biological processes. The state variables must be well-defined
quantities, such as biomass per unit volume or numbers of
individuals per unit volume, which in principle can be
measured. The processes that drive the changes of the
state variables must be formulated mathematically in a
consistent manner by translating observations into formulas.
Many marine ecosystem models are focused on the
dynamics of nutrients and phytoplankton, while zooplankton grazing is often accounted for implicitly as mortality of phytoplankton [see (Fransz and Verhagen, 1985;
Stigebrandt and Wulff, 1987; Pinazo et al., 1996;
Humborg et al., 1999]. Other approaches consider the
zooplankton biomass as one integrated state variable,
which applies grazing pressure on the phytoplankton. The
feedback to the lower trophic levels is established by lossrates, e.g. respiration, excretion and mortality, which
transfer material to detritus, which in turn is recycled to
the nutrient pool [see (Wroblewski, 1977; Aksnes and Lie,
1990; Fasham et al., 1990; Broekhuizen et al., 1995; Fennel
and Neumann, 1996; Neumann, 2000].
These models were applied to study fluxes of matter
among the state variables to understand and quantify
carbon fluxes in the ocean, to study eutrophication in
coastal seas, or to look at the mesoscale distribution of
nutrients and plankton in response to the circulation patterns. The obvious success of these models indicates that
simplification by parameterizations of unresolved processes must be possible to a certain extent.
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Nevertheless, there are many problems which require a
more detailed description of the variations of zooplankton including growth, development and reproduction as
well as the feedback to the lower and higher levels of the
food web. For example, studies of the recruitment success
involve state-resolving descriptions of zooplankton in
order to address size-selective feeding of larvae and juvenile and adult fish.
Models of zooplankton biomass including several
species and stages have been developed by Vinogradov et
al. (Vinogradov et al., 1972). Stage-resolving population
models were used by Wroblewski (Wroblewski, 1982) and
by Lynch et al. (Lynch et al., 1998). A link of population
models with individual growth has been proposed (Carlotti and Sciandra, 1989) and individual-based modelling
of population dynamics has been considered (Batchelder
and Miller, 1989; Miller and Tande, 1993). An overview
of existing zooplankton models was recently given by Carlotti et al. (Carlotti et al., 2000).
In this paper we propose a consistent stage-resolving
description of zooplankton in a chemical–biological
model which can be embedded in an ocean circulation
model, where the feedback to lower trophic levels is
included. The explicit description of this feedback is
important in the study of, for example, effects of food
quality and it provides a formal check of the model performance by checking the conservation of mass.
The paper is structured as follows. In the next section a
brief outline of existing population models, which are
coupled to GCMs, is given. An alternative stage-resolving
biomass model is then described. This model is then used
for simulations of rearing tank experiments and more
complex box models and discussions and conclusions are
presented in the final section.
P O P U L AT I O N M O D E L S O F
ZOOPLANKTON
We will briefly review two models which are coupled, or
intended to be coupled, with circulation models. For the
integration of a model into a GCM it is important to keep
the number of state variables, and hence the number of
evolution equations, small and to avoid explicit memory
terms as in delayed equations, because the number of
stored time steps must be small. However, we will not
discuss types of models where copepods are just drifting
particles (strongly reduced biology) or vector-population
models (complex biology), which are difficult to link with
GCMs.
Stage-resolving population models
Basically the dynamics of zooplankton is governed by
both universal and species-specific aspects. For a general
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discussion we look first at the universal aspects, such as
growth, development and reproduction. For simplicity we
look at a model of only four stages of a ‘model-copepod’.
Thus we merge several stages into some state variables
and formulate dynamic equations for the numbers of individuals per unit volume of eggs (Ne), nauplii (Nn), copepodites (Nc) and adults, (Na). We assume that the numbers
of individuals per unit volume are high enough that the
state variables behave like continuous functions and,
hence, the dynamics can be expressed by differential
equations. Then the population model, expressed by a set
of equations of the state variables of the ‘model-copepod’
can be written as:
d N =Q -T N - N
e
en
e
e
e
dt e
(1)
d N =T N - N -T N
en
e
n
n
nc
n
dt n
d N =T N - Z -T N
nc
n
c c
ca
c
dt c
d N =T N - N
ca
a
a
a
dt a
(2)
(3)
(4)
The change of the state variables is biologically driven by
a source-term, Qe, describing the numbers of new eggs per
day, by transfer rates, Ti,i+1, for the stage development,
and mortalities, µi. The transfer rates, Ti,i+1 may be prescribed by inverse stage duration as given by Belehradek
expressions. Such an approach was, for example, used to
couple zooplankton dynamics and physical processes
(Wroblewski, 1982; Gupta et al., 1994; Lynch et al., 1998).
Both the egg rate, Qe, and the mortalities, µi, have to be
prescribed in a plausible way.
We note that the time differentiation in equations (1) to
(4) is an Eulerian one and stands symbolically for an
advection–diffusion equation, which defines the interface
of the biological model and the circulation model, i.e.
d = 2 + v $ d - A
dt 2t
where v is the advection and A the diffusion coefficient.
This type of model can easily be integrated into circulation models but the dynamic linkage to the lower parts
of the food web is missing.
The dynamics of the total number of individuals, Ntot =
∑iNi, follows by adding the equations (1) to (4). Assuming
the same mortality for all stages we find
d N = Q - N
e
tot
dt tot
The total number is basically controlled by the egg production as source term and by mortality as loss term, while
the transfer terms cancel. Since the number of individuals is not subject to a conservation law as, e.g. the

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conservation of mass, there is no constraint which can be
used to check the model consistency in a simple way.
Population models linked to individual
growth
A model which integrates the mean individual properties
and the population dynamics was developed (Carlotti and
Sciandra, 1989). This theory is based on the equations for
the numbers of individuals per unit volume, similar to equations (1) and (4) in conjunction with the evolution of the
mean individual mass, m which obeys equations of the type
d m (t ) = (g (t ) - l (t )) mp ,
i
i
i
dt i
(5)
where gi prescribes growth through ingestion and li losses
through egestion and excretion of stage I, and p is an allometric exponent. The transfer and mortality rates are controlled by the metabolism of the ‘mean individual’. As
their central hypothesis Carlotti and Sciandra assume that
the transfer from one stage to the next one is controlled by
the molting mass (Carlotti and Sciandra, 1989). A transfer to the next stage occurs only if the critical molting
mass, Xi, is approached. The transfer among the stages is
not prescribed by data but is controlled by growth, which
is computed by the model. The transfer is not continuous
but occurs only if the corresponding molting mass is
reached. The full set of model equations is rather complex
and involves a large number of control parameters [for a
detailed description see (Carlotti and Sciandra, 1989;
Carlotti and Nival, 1992].
This model is particularly successful in simulating
experiments in rearing tanks under well-defined conditions. Attempts to couple this type of model to a onedimensional water column model were presented within
an Eulerian approach (Carlotti and Radach, 1996) and
with a Lagrangian ensemble theory (Carlotti and Wolf,
1998). However, the mean individual mass, which is
defined as biomass divided by the number of individuals,
obeys an evolution equation (5) only for coherent cohorts.
Hence the mean individual mass is not in general a good
choice of a state variable and a consistent integration of
this model into a GCM model could be very difficult.
A S TAG E - R E S O LV I N G B I O M A S S
MODEL
As an alternative way to achieve a consistent model
description for copepods we propose a stage-resolving
biomass model. The model employs the concept of critical molting masses of Carlotti and Sciandra (Carlotti and
Sciandra 1989).
We include five stages of the model-copepod: eggs,
nauplii, copepodites 1 and 2, and adults. We merge the
nauplii stages into one stage variable ‘model-nauplii’. Similarly we merge the copepodites I to III, and IV to V into
the state variables ‘model-copepodites 1 and 2’, respectively.
The corresponding biomass-variables per unit of
volume are Ze, Zn, Zc1, Zc2 and Za. The corresponding
numbers of individuals per unit of volume are Ne, Nn, Nc1,
Nc2 and Na.
All these state variables are related to a population
density function, (m, t), which describes the distribution
of individuals as a function of mass, m, and time, t. Thus
(m, t)dm is the number of individuals in the interval (m, m
+ dm) at time t. The total zooplankton biomass and the
total number of individuals are
Xa
Z tot =
#m (m) mdm
(6)
e
and
Xa
#m (m) mdm
N tot =
(7)
e
where me is the mass of eggs and Xa is the maximum mass
of matured adults. For a certain stage, i, where m is confined to the interval Xi-1 ≤ m ≤ Xi, the biomass, Zi, and
number of individuals, Ni, are
Xi
Zi=
#X
(m) mdm
(8)
(m) dm
(9)
i - 1
and
Xi
Ni=
#X
i - 1
Owing to the growth of the individuals the distribution
density (m, t) may propagate along the m-axis with the
‘propagation speed’, dm , which is controlled by growth
dt
(grazing minus losses) according to
d m (t ) = (g (t ) - l (t )) m
i
i
i
dt i
(10)
This equation corresponds to equation (5). However, we
have avoided a broken allometric exponent, i.e. p = 1, and
choose instead stage-dependent rates, which are described
below, see equation (25).
An important observable quantity is the stage duration
time, Di, which can be calculated by integration of equation (10) with the initial condition m = Xi – 1 for t = 0
t
m i (t) = X i - 1 exp ( # dt [g i (t) - l i (t)])
0
or, for t = Di,

Di
m (D i ) = X i = X i - 1 exp ( # dt [g i (t) - l i (t)])
0
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d N = - N -
n
n
nc
dt n en
d N = - N -
nc
c
c
c c
dt c
d N = - N -
c c
c
c
c a
dt c
d N = - N
c a
a
a
dt a
Here, Xi is the molting mass, of the stage under consideration, while Xi–1 is the molting mass of the preceding
stage. Thus, the stage duration is implicitly given by
1
1
#0
Di
X
dt [g i (t) - l i (t)) = ln ( X i )
i-1
2
For strictly constant conditions as, for example, in a
rearing experiment we find
X
D i = g 1- l ln ( X i )
i-1
i
i
(11)
Due to the mortality, µ, the magnitude of the distribution
density (m, t ) will decrease with time. Thus the total
change in time is
d =- dt
(12)
Xi
#X
[
i - 1
1
1
1
2
1
2
1
2
2
2
2
d (m)
m + (m) dm ] dm
dt
dt
(13)
= (g i - l i - i ) Z i
#X
i - 1
g i (P ) = i (1 - exp (- I i P 2 )) f (Z i /N i , X i )
d (m)
dm
dt
(14)
= i N i
If the transfer among the stages is included, then the
equation set for the stage-dependent biomass is explicitly
d Z =T Z -T Z - Z
ae a
en e
e e
dt e
(16)
1
d Z = T Z + (g - - l ) Z - T Z (17)
nc
n
c
c
c
c
c c
c
dt c
1
1
1
1
1
1
2
1
d Z = T Z + (g - - l ) Z - T Z (18)
c c
c
c
c
c
c
c a c
dt c
2
1
2
1
2
2
2
2
2
d Z = T Z + (g - - l ) Z - T Z
c a c
a
a
a
a
ae a
dt a
2
(24)
2
2
(19)
The dynamics of the biological stage variables is controlled by the following process-rates: transfer rate to the
next stage, Ti,i+1 grazing rate, gi, loss rate, li, mortality rate
µi where i = (e, n, c1, c2, a). The rates are explained in detail
below. The corresponding equations for the number of
individuals are similar to equations (1) to (4),
d N = - N -
ae
e
e
en
dt e
(25)
where P is the food concentration, i.e. phytoplankton, the
Ii values are stage-dependent Ivlev-constants, and f is a
Fermi function, which is explained below. The maximum
grazing rate, i, depends on the temperature. We choose
an Eppley factor,
i = b i exp (aT)
(15)
d Z = T Z + (g - - l ) Z - T Z
en e
n
n
n
n
nc
n
dt n
1
(23)
The grazing rates prescribe the amount of ingested food
per day in relation to the biomass. Thus, in general, the
lower stages have higher grazing rates than the higher
stages. Only for declining resources it is assumed that the
higher stages have an advantage due to higher mobility to
capture food. These features are formulated in terms of a
modified Ivlev formula,
2
dN i
=
dt
(22)
The death rates, µi, are the same as in the equation set (15)
to (19). The transfer rates, i,i+1 are closely related to the
Ti,i+1.
For the model simulations we have to specify our modelspecies. As model-copepod we choose in particular Pseudocalanus, which is described in detail in a previously
published review article (Corkett and McLaren, 1978).
and
Xi
(21)
Grazing rates
The dynamic equations for Zi and Ni follow from equations (10) and (12), apart from the transfer terms among
the stages, which are ignored here for simplicity, as
dZ i
=
dt

(26)
with a = 0.063(ºC)–1. The numerical values of the parameters are listed in Table I. The decrease of bi for the
higher stages reflects that higher stages with more biomass
ingest a smaller amount of food, expressed as percentage
of the body mass, than the lower stages, while the increasing Ivlev constants reflect the advantage of higher stages
at low food levels, see Figure 1 (upper panel).
The function f(Zi/Ni,Xi) in equation (25) is a stagedependent filter function, (low-pass filter), which makes
sure that the growth decreases if the mean individual mass
Zi/Ni reach the maximum mass, Xi, of the corresponding
stage and may molt to the next stages. We choose a Fermifunction, which provides a representation of the stepfunction with a smooth transition.
f (Z i /N i , X i ) =
(20)

1
1 + exp c X (Z i /N i - X i ) m
i
(27)
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Table I: Grazing rates
Stage
bi = I (0ºC)d–1
I (10º;C)d–1
I (15ºC)d–1
Nauplii
0.5
0.93
1.3
2.5
Copepodites 1
0.35
0.66
0.9
4.7
Copepodites 2
0.25
0.47
0.64
7
Adults
0.12
0.22
0.31
10.1
I2i 10–3mmolC–2m6
Fig. 1. Top panel, grazing rates as functions of the available food for T = 10ºC. For the lower stages the ratio of ingested food to body mass is
larger than for the higher stages. At food shortage the higher stages are better in catching food due to a higher mobility. Bottom panel, Fermifunctions with properties of low pass and high pass filter f (Zi/Ni, Xi) (solid) and f (< m >I, Zi/Ni) (dash-dot), respectively, to control the development, for i = c2.
The function (27) provides, in a statistical sense, the link of
the individual level to the bulk state variables. The function is activated, i.e. different from unity, if the mean individual mass, Zi/Ni, of a stage approaches the molting mass
Xi, see Figure 1 (lower panel). The mass parameters listed
in Table II were derived from data of the Baltic Sea
(Hernroth, 1985).
Loss rates
The ingested food is partly used for growth and partly for
the metabolism of the animals. We assume (Corkett and
McLaren, 1978) that 35% of the ingestion is lost as egestion, 10% as excretion and 10% by respiration. Moreover,
about 15% of the ingestion is assumed to be needed for
the molting processes. These losses are expressed by the
rates li = 0.7 gi (P,T), for (i = n, c1, c2). For the adults we
assume la = 0.8 ga (P,T).
Transfer rates
In order to formulate the development we need a prescription of the transfer of one stage to the next. To this
end we adopt the ideas in Carlotti and Sciandra and use
a critical molting mass, Xi, for the nauplii and copepodites
to describe the transfer (Carlotti and Sciandra, 1989). The
stages are characterized by a mass-interval between the
previous and the actual molting masses, see Table II. The
transfer rates, Ti,i+1, are defined as

Ti,i+1 = gif (<m>i, Zi/Ni)
(28)
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Table II: Mass parameters
Egg mass
me = 0.1 µgC
Stage
Molting-mass
Maturation-mass
Mean-mass
Nauplii
Xn = 0.3µgC
–
< m >n = 0.22 µgC
Copepodites 1
Xc1 = 0.8 µgC
–
< m >c1 = 0.6 µgC
Copepodites 2
Xc2 = 2 µgC
–
< m >c2 = 1.6 µgC
Adults
–
Xa = 3 µgC
< m >a = 2.6 µgC
where gi is the grazing rate and f (< m >I, Zi/Ni) is again a
filter function (high-pass filter), which makes sure that the
transfer to the next stage starts not before the mean individual mass, Zi/Ni, exceeds a certain value <m>i. Note the
following property of the Fermi-functions f(x,y) = 1 – f (y,x).
In order to define the parameter < m >i, we choose a massvalues somewhat smaller than the corresponding molting,
see Figure 1 (lower panel).
The corresponding transfer-rates for the number of
individuals follows as
i,i+1 =Ti,i+1Zi/Xi
(29)
We note that the use of the filter functions in the grazingrates, equation (25), and in the transfer-rates, equations
(28) and (29), implies that two sets of state variables, the
biomasses and the number of individuals, are needed.
Reproduction
The rate of reproduction describes the egg production of
the female adults after reaching the maturation mass. This
is a transfer rate from adults to eggs. We assume that half
of the adults are female and 30% of the ingested food is
transferred into egg biomass.
T ae = ( 12 ) 0.3g i
(30)
Then the number of new eggs per day is given by
ea = T ae Z a /m e
(31)
This approach respects that the egg-rate depends on food
availability and temperature, here through the grazing
rates, but ignores the discontinuous release of clutches of
eggs with time intervals of a couple of days [see e.g.
(Hirche et al., 1997)]. The transfer from eggs to nauplii
(hatching) is set by the embryonic duration, about 10 days,
for low temperatures, (Corkett and McLaren, 1978). The
effect of temperature is accounted for by an Eppley factor,
Ten = h (T – T0) exp(a(T – T0)
with h = 0.124, T0 = 2.5ºC and (x) being the step-function (x), = 1 for x > 0 and (x) = 0 for x < 0. Thus only
for temperatures exceeding 2.5ºC do the model eggs hatch
to model-nauplii.
Mortality and overwintering
Since the model food chain is truncated at the level of the
model zooplankton the mortality involves death rates and
predation by planktivorous fish. The choice of the mortality rate is difficult because the basic factors are poorly
known. In the literature, stage-dependent mortalities with
some seasonal variation are often used (Gupta et al., 1994).
The typical orders of magnitude (per day) vary from µ ~
0.01 . . . 0.1d–1. In our model we choose constant mortality rates for eggs and model-nauplii, µe = 0.2d–1 and µn =
0.033d–1. Since constant rates for all stages are certainly an
oversimplification we relate the mortality of the copepodites and adults to a time-dependent function, which is
proportional to a constant minus the normalized length of
the day, d(t), which is shown in Figure 2. Thus µc1 = µc2 =
0.1 (1 – d(t) ) d–1 and µa = 0.05 (1 – d(t) ) d–1. Since d(t) varies
between 0.2…0.8 we have µc1 = µc2 = 0.02…0.08d–1 and
µa = 0.01…0.04d–1. This choice is motivated by the low
abundance in autumn and winter as reported from the
Baltic Monitoring Programme [HELCOM, 1996), P. 96].
It is assumed that only the adult stages overwinter. Due
to mortality the state variables go to zero during the
winter. In order to mimic the overwintering we assume
that the mortality of the adults vanishes below a threshold
of Za ≤ 0.3 mmol C m–3.
Phytoplankton, nutrients and detritus
We include a very simple model of the lower trophic levels
with the state variables: limiting nutrients (NDIN), phytoplankton (P ) and detritus (D).
d N
=- M (N DIN ) P + l PN P + L ZN Z
dt DIN
(33)
+ l DN D + Q ext
(32)

DIN
DIN
DIN
d P = M (N
DIN ) P - G (P) Z - P P - l PN P
dt
DIN
(34)
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Fig. 2. Annual cycle of the temperature and normalized length of the day at 54º N.
d D=L Z-l
ZD
DN
dt
DIN
D + P P - Q
ext
The uptake is controlled by a modified Michaelis–Menten
Formula
2
M (N DIN ) =
G (P) Z = ! g i (P ) Z i ,
(35)
rN DIN
2 .
2
+ N DIN
The role of the daylight is included qualitatively in the
rate r by means of an astronomical standard formula of
the normalized length of the day for temperate latitudes,
= 54ºN, see Figure 2. The effect of temperature is
included by an Eppley factor in r. Moreover, as a crude
reflection of mixed layer formation we set r to zero for
temperatures below 2.5ºC, i.e.
i
where i = (n, c1, c2, a). The rate lDNDIN prescribes the recycling of detritus to nutrients, see Table III.
The mortality rate, µp, prescribes the transfer of phytoplankton to detritus including a crude approach for
sinking, while lPNDIN describes how much phytoplankton
biomass is transferred directly to nutrients through respirational losses.
Finally, the rate Qext prescribes an external input of
nutrients. In Table III the numerical values of the various
parameters are listed in carbon units, where the Redfield
ratio, C/N = 106:16, is assumed.
E X P E R I M E N TA L S I M U L AT I O N S
r = rmax(T – T0)d(t) exp(aT )
The loss rates LZNDIN and LZD describe how much biomass
of the different stages of the zooplankton is transferred to
nutrients, NDIN, and detritus, D.
L ZN Z = !l i Z i ,
DIN
i
L ZD Z = ! i Z i ,
In the preceding section we have constructed the model by
defining state-variables, constructing the dynamic equations and formulation of the processes. The modelcopepod is characterized by tables of mass-intervals and
process-parameters, see Tables I and II. We used values
corresponding to Pseudocalanus, but the changes for other
species and modified sets of state variables can readily be
made.
Rearing tanks
i
while G(P)Z represents the loss of phytoplankton due to
grazing of the different stages of the model-copepod.
We start with a simulation of the development of a
cohort of eggs in a rearing tank with constant food and
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Table III: phytoplankton model
Parameter
Meaning
Numerical value
LDNDIN
recycling D to N
0.01 exp (aT) d–1
µp
mortality
0.02 d–1
lPNDIN
transfer P to N
0.1 M (NDIN)
rmax
maximum uptake
0.5 d–1
2
half-saturation
40 mmol C2m–6
Qext
nutrient-input
0.2 d(t) mmol C m–3 d–1
temperature condition. To this end we integrate the
model equations (15)–(19) and (20)–(24) with the initial
conditions at t0 = 0, Zi = 0 and Ni = 0 for i = (n, c1, c2, a)
and Ze = 0.03mmolCm-3 and Ne = Ze/megg. First we consider
the case of no reproduction. The results are shown in
Figures 3–5. The development of eggs to adults is shown
for the stage-resolving biomass, Figure 3, as well as for the
stage-resolving number of individuals, Figure 4.
Without reproduction (no new eggs) the biomass
accumulates eventually in the adult stage. The control
of the high pass filters in the transfer-rates, equations
(28) and (29), can clearly be recognized in Figures 3 and
4. Owing to the overlapping of the high- and low-pass
filters the transition between the stages is relatively
smooth. The duration times of the stages can be estimated from Figures 3 and 4. A full generation time, i.e.
the development of eggs to mature adults, takes about
30 days.
The development of the i-th stage to the next stage occurs
when the mean individual mass, Zi/Ni, approaches the
molting mass, Xi. The total number of individuals, Ntot,
decreases while the total biomass, Ztot, and the mean individual mass, mmean = Ztot/Ntot, increase, see Figure 5. Thus, in
this case, the evolution of the mean individual mass can be
used to describe the dynamics of the model zooplankton [as
in (Carlotti and Sciandra, 1989; Carlotti and Nival, 1992)].
Next, we repeat the simulation but with reproduction.
Then we find a similar development as in the previous
case until the adults start to lay eggs. After the onset of
reproduction there is a clear increase in the number of
individuals and the biomass of the model zooplankton as
shown in Figures 6–8. Now the time-evolution of the
Fig. 3. Stage resolving simulation of the zooplankton biomass in a rearing tank without reproduction.
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Fig. 4. Stage resolving simulation of the abundance in a rearing tank without reproduction.
Fig. 5. Total zooplankton biomass (top panel), numbers of individuals (middle panel) and mean individual mass (bottom panel), without reproduction.
mean individual mass cannot be used to describe the
system in a non-ambiguous way. For example, the mean
individual mass of the population, Ztot/Ntot, decreases
after the onset of reproduction, because the number of
eggs with small mass increases. The mean individual
mass is no longer related to a more or less coherent cohort
but the mean value of all stages which exists at the same
time.
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Fig. 6. Stage resolving simulation of the zooplankton biomass in a rearing tank with reproduction.
Fig. 7. Stage resolving simulation of the abundance in a rearing tank with reproduction.
Simulation of biennial cycles in a box-model
After the preceding simulations under the constant
environmental conditions of rearing tanks we wish now to
approach marine in situ conditions and take the dynamic
links to the lower trophic levels into account. As a first step
to apply the zooplankton-model as formulated in the
equation sets (15)–(19) and (20)–(24) to marine systems we
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Fig. 8. Total zooplankton biomass (top panel), numbers of individuals (middle panel) and mean individual mass (bottom panel), with reproduction.
consider a box-model. The lower part of the food web is
described by the equations (33) to (35). The box can be
thought of as part of the upper mixed layer with no horizontal advection and restricted vertical exchange. The
box could also be considered as a grid box of a circulation
model but with reduced interaction with the neighbouring grid-boxes. We consider a minimum physical control,
i.e. we use the annual cycle of the surface temperature in
the Arkona basin of the Baltic Sea and simulate the seasonal variation of the irradiation by a standard formula of
the normalized length of the day, as shown in Figure 2.
As our standard experiment we look at overwintering
model-copepods that start to lay eggs after the onset of the
vernal phytoplankton bloom. The dynamics of the state
variables is shown in Figure 9–11. The development of
nutrients and phytoplankton is shown in Figure 9. In the
frame of this simple model we control the onset of the
vernal bloom by means of the temperature. We assume
that the development of thermal stratification, which is
not resolved in a box-model, starts if the temperature of
the water exceeds 2.5ºC. The peak of the phytoplankton
bloom is controlled by sinking, which is implicitly included
in the mortality of phytoplankton (transfer to detritus) and
by grazing.
The total biomass of the model-zooplankton shows a
pronounced maximum in the summer. The reduced
grazing pressure in the autumn allows a secondary bloom
of the model-phytoplankton. During the winter the
model-detritus is recycled to model-nutrients.
The stage-resolving dynamics of the model-copepods is
shown in Figures 10 and 11 for the biomasses and the
number of individuals. After the onset of the vernal
bloom the overwintering model-adults start to lay eggs. A
significant increase of the model-zooplankton follows
after the ‘egg-signal’ has ‘propagated’ through the stages
(time-scale about 50 days) and the newly developed adults
start the reproduction. The biomass of the modelzooplankton accumulates in the adult-stage. The highest
number of individuals is found for the model-nauplii.
The effect of ‘bottom up’ control was simulated by
adding nutrients to the box-model. The increase of nutrients had only a small effect on the model phytoplankton
because the excess phytoplankton is rapidly grazed by the
zooplankton. The nutrient signal propagates through the
food web and accumulates finally in the model-adults.
Only the peak of the spring bloom showed an increase in
the second year of the simulation, because the response of
the model zooplankton to the food signal was delayed by
about a generation time. Since the effects were relatively
small we omit the figures of the results.
In our next simulations we study the effects of ‘topdown’ control by means of a temporary increase of mortality, which may be caused by predation. We confine this
experiments to two cases, where the mortality of the
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Fig. 9. Biennial simulation of nutrients and detritus (top panel), phytoplankton and total zooplankton biomass (bottom panel).
Fig. 10. Biennial simulation of the biomass of the different stages.
model-adults and of the model-copepodites 2 is drastically enhanced during the time interval of day 200 to 230
of the biennial simulations, i.e. µa = 1 d–1 in one case and
µc2 = 1 d–1 in the other case. The effects of the predation
event on the adults is shown in Figure 12.
The high mortality rate with a time scale of about a
generation time decreases the adult biomass almost to
zero. Since at the same time the reproduction breaks down
it appears that all stages are affected significantly by the
predation event. After the event new adults can develop
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Fig. 11. Biennial simulation of the number of individuals of the different stages.
Fig. 12. Stage resolving simulation of the biomass for a temporary increased mortality of the model adults.
and start to lay eggs (Figure 12). After the winter season
the system has no memory of this event and develops as
in the standard run (Figure 10).
The effects are less strong if a predation event concerns
stages other than the adults, as for example the modelcopepodites 2. The results of a simulation with temporary
increased mortality, µc2 = 1/d, are shown in Figure 13.
Although the biomass of the c2- stage is almost zero during
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Fig. 13. Stage resolving simulation of the biomass for a temporary increased mortality of the model copepodites 2, c2.
the predation event, adults remain on a relatively high
level and maintain the reproduction. Thus, it is obvious
that a predation on the adults has the most severe effects
on the model-zooplankton because then the reproduction
is affected as well.
DISCUSSION AND
CONCLUSIONS
We have proposed a stage-dependent model, that integrates elements of biomass models (Vinogradov et al.,
1972) and stage-dependent population models (Wroblewski, 1982). Moreover, we employed the ideas of Carlotti and Sciandra to prescribe the onset of molting by the
concept of critical molting masses (Carlotti and Sciandra,
1989). Since the molting mass concerns individuals, we
introduced statistical aspects by means of the filter functions (Fermi-function) in order to connect the level of individuals with the state variables.
In order to achieve a consistent formulation, where
both biomass and number of individuals are related to the
same population density function, we avoided an allometric exponent different from unity in the growth equation
of the individual. Mass-dependent growth is implicitly
accounted for by a stage-dependent growth rate.
It was shown that the mean individual mass of the
population, Ztot/Ntot, can be used as a state variable only for
coherent cohorts. If reproduction is involved, then the
dynamics of the mean individual mass is no longer
governed by the growth equation of the individuals.
As ‘model-copepod’ we choose Pseudocalanus, which
have been thoroughly described (Corkett and McLaren,
1978). The signature of the model-copepod is given by
tables which list the parameters involved in the process
formulations.
The process-formulations of growth and development
allow the calculation of the stage duration in the model.
Thus, observations in terms of Belehradek formulas of
stage duration can be used to check the processformulation, but are not being used as prescribed rates.
The duration of stages can be directly determined either
expressed by the mass interval (Xi–1, Xi) and growth and
loss rates as in formula (11) or from the rearing tank simulations, as for example shown in Figure 6.
The food web is truncated at the level of the zooplankton. Thus, the zooplankton mortality includes predation
by planktivorous fish. The inclusion of the lower trophic
levels is useful, for example, to simulate how signals propagate through the food web and to study the effects of food
quality on reproduction.
In order to check the consistency and plausibility of the
model we simulated biennial cycles for several cases
including bottom up and top down effects on the model
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copepods. We considered how a nutrient signal propagates through the food chain and how the population
responded to predation events.
An attractive property of the present model is that it
conserves mass explicitly. Such a conservation law cannot
generally be applied to population models, because there
is no law of conservation of the number of individuals.
Wroblewski assumed in his population model that the
total number of individuals is constant, but this may apply
only for a certain time period in a special case (Wroblewski, 1982).
The present model can be easily embedded into a
GCM by including advection–diffusion equations for the
state variables. An unavoidable disadvantage of the model
is the need for two sets of state variables, biomass and
number of individuals, and hence the need for two equations for each stage. This is due to the linkage of individual, population and bulk biomass level in a statistical
treatment.
An important point is the applicability of the model to
other species. What are the universal and the speciesspecific aspects? In a first approach it can be assumed that
the process-formulations are more or less ‘universal’, i.e.
they apply to several other copepods as well. Speciesspecific properties are mainly the parameters that specify
the processes quantitatively.
An application of this model for another species is
straightforward by using other tables of the speciesspecific parameters. However, it might be necessary to
rearrange the state variables (e.g. merging of stages) and
to adjust the process formulations if specific features are
important as, for example, the differentiation between
body mass and structural mass.
It is generally accepted that models should be as simple
as reasonable and as complex only as necessary. This
implies that the answer to the question ‘which are the right
equations’ depends to a certain extent also on the problem
considered. This in turns implies that it must be possible
to increase or reduce the complexity of a problem within
a certain range, for example by parameterization of unresolved processes or by integration of some state variable
into one, where appropriate.
Since our aim was a consistent model that could be used
directly for coupling into GCMs, we have, for example, to
avoid memory terms (delayed equations), because only a
few time steps can be stored in GCMs. Thus we have to
prescribe the production of eggs by a continuous rate,
which depends on temperature and food availability, but
we cannot resolve the ‘history’ of the female adults.
The next step is an integration of this model into an
existing coupled model, where the bulk-biomass state variable of the zooplankton will be replaced by the state
resolving model. However, the model may also serve as an
easy-to-use workbench model and is available as
MATLAB code on request.
AC K N O W L E D G E M E N T
The author thanks Dr L. Postel and an anonymous referee
for helpful comments and suggestions.
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Received on June 5, 2000; accepted on June 18, 2001
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