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\noindent
{\bf 4.0 Overview of modeling efforts - Cisco Werner and Dennis McGillicuddy}

\vskip .2in

A variety of modeling approaches are being used in the Globec
program to study how interactions between zooplankton population
dynamics and physical circulation create space/time variability
in abundance of the target species.  Information on vital rates
is essential to these efforts, although the means by which the
vital rates data are used differs amongst the various applications.
In the so-called ``forward problem,'' a mathematical representation
of the organism's population dynamics is incorporated into a
circulation model.  The coupled model is initialized 
with a set of observations, integrate forward in time and then
compared with the next set of observations.  A successful outcome
results in minor discrepancies between observations and predictions,
and the model solutions thus can be used as a basis for
diagnosis of the processes controlling the observed
behavior.  Unfortunately, satisfactory completion of the forward
problem is not always achievable due to limitations in the models,
the observations, or both.  Adjoint data assimilation methods
provide an alternative approach which is particularly useful in such 
cases.  These techniques can be used to determine the model inputs 
({\it e.g.} parameters, forcing functions, {\it etc.}) that minimize 
the misfit between observations and predictions, thereby producing 
an optimal solution from which the underlying dynamics can be gleaned.
Examples of both forward (Lynch {\it et al.}, 1998; deals with
{\it Calanus finmarchicus}) and inverse (McGillicuddy {\it et al.}, 
1998; deals with {\it Pseudocalanus, spp.}) approaches are described
below.

Individual based models (IBMs) represent yet another approach to
coupled physical--biological problems of this sort.  In this
technique, the equations representing the organism's population 
dynamics are integrated on ensembles of lagrangian particles which 
are moved about by the prescribed circulation.  A major advantage
of this approach is that diagnosis of the simulated physical--biological 
interactions along lagrangian trajectories is relatively straightforward.
However, it is often impractical to compute enough ensembles to
generate eulerian field estimates with robust statistics.  Miller
{\it et al.} (1998) used an IBM for {\it Calanus finmarchicus}
in the Gulf of Maine / Georges Bank region, and results from that
effort are described briefly below.

Linking zooplanktonic trophodynamic models to larval fish models is in its
initial stages.  The full coupling will require 3-D and time-dependent
representation of physical and biological variables, some of which are still
uncertain, such as behavior, prey preference, etc.  However, a simple
conceptual model has been developed for larval fish feeding on stage-structured
prey populations, in an Eulerian framework (Lynch {\it et al.} 2000; see
description below).  The model combines simplified contemporary models of
larval fish trophodynamics, zooplankton population dynamics, and hydrodynamic
turbulence.  The Eulerian view allows instructive maps of larval feeding and
(potential) growth rates for individual prey species, alone or in combination.


\vskip .2in
\noindent
{\bf 4.1 An example of the forward problem:  Biological/physical simulations
of {\it Calanus finmarchicus} population dynamics in the Gulf of
Maine (Lynch {\it et al.}, 1998).}
\vskip .2in

A site-specific, coupled biological/physical model is described for the
dominant winter-spring copepod in the Gulf of Maine (Figure 1).  The 
biological portion describes
temperature- and food-dependent progression through 13 life stages in an
Eulerian (concentration-based) framework.  The population is transported
in a flow field depicting the climatological mean conditions
in 2-month ``seasons''.  Behavioural assumptions account for
depth selection and two limiting cases are contrasted: dispersal
throughout the water column, and aggregation in the surface layer.
Simulations are inspired by MARMAP observations, with an
emphasis on the mid-winter initiation of the annual bloom by diapausing
populations, and their role in supplying reproducing populations to
Georges Bank during spring.

Mathematically, the population dynamics expressed on the right hand
side of the advection-diffusion-reaction equation in Figure 1 
represents a conveyor belt among 13 successive life history stages:

$$ R_i={C_{i-1}\over D_{i-1}}
   -{C_i\over D_{i}}
   -\mu_i C_i $$

\noindent
with $D_{i}$ the stage duration and $\mu_{i}$ the
mortality rate for stage $i$.
Appropriate adjustments at the end points and adult branch
points are implied.  Stages with duration exceeding
1 day (at 5$^\circ C$) are subdivided to control developmental
diffusion which is inherent in the Eulerian approach.

Average developent times $D_i$ are sensitive to both temperature
and food availability.  Generally, increasing temperature $T$
decreases the stage durations, $D_i$. These are traditionally related
through Belehradek equations of the form $D_i (T) = a_i (T - \beta)^\alpha$,
where $\beta$ and $\alpha$ are constants and $a_i$ is a stage
varying parameter. Parameters used in this model for Egg through C5
are based on a recent rearing experiment by
Campbell {\em et al.} and appear in
Table 1.
The estimate for
immature females is based on Plourde and Runge (1993).
Stage durations of males and mature females are treated as
infinite.

Food effects are less precisely known.  The stage duration
equation above holds for maximal development rate assuming
non-limiting food concentrations ($\ge 400 \mu g C/l$).
We assume that Eggs, N1 and N2
do not feed (Marshall and Orr, 1955; Carlotti and Radach, 1996).
For the other stages, we expect that
stage durations will increase as food concentrations are reduced.
Campbell {\em et al.} (1998)
indicate an increase by almost a factor of 1.5 for food levels of 50
$\mu g C/l$ and a factor of 2 for food levels at $25 \mu g C/l$.
Durbin et al (1995) observed stage durations of 24.8 days for C4's
at 11$^{\circ}$ in the field, under food limited conditions.

Completion of the final stages of oogenesis is highly dependent upon
external food supply (Plourde and Runge 1993; Runge and
Plourde 1996; Hirche 1996a; Niehoff
and Hirche 1996). Thus
the stage duration of the immature female is lengthened as food
abundance is reduced. The literature suggests that development in the
field can be
delayed by as much as 1 month (Hirche 1996a) when food is scarce.
This could have a significant effect on the
first generation as it would delay its initiation and displace
it ``downstream'' in the advective system.

As a point of departure, food effects on development
are modeled by decreasing development rate
($1\over D_i (T)$) linearly toward zero when food is below a satiation
value. (Equivalently, development time varies inversely with food
availability, with the Belehradek function representing
the lower
limit.)  The baseline parameters used are summarized in Table 1.

Spawning is an intermittent process.  A mature female releases
a clutch of eggs
over a short period of time (minutes) with longer time intervals
(of order 1 day)
between clutches (Hirche, 1996a).
Generally the
interclutch interval $d$ is considered to be sensitive to variations in
the environment, while by comparison the clutch size is relatively
insensitive.
Averaging over many individuals, we model this as a continuous
process with constant clutch size of 50 eggs per female:

$$ R_e= {50 C_F \over d}
   -{C_e\over D_{e}}
   -\mu_e C_e $$

Both laboratory and field experiments have
found temperature and egg production to be positively correlated
(Hirche 1990; Plourde and Runge 1993; Hirche and Kwasniewski 1997;
Runge and Plourde 1996; Hirche et al. 1997).
Miller {\em et al.} (1997) fit the Hirche {\em et al.}
data for clutch interval variations
with temperature using the Belehradek function $d = 218 * ( T +9.11)^{-2.05}$,
and we use this relation in table \ref{tab:rates}.
This relation holds for superabundant food
levels ($>400 \mu g C / l$ in Hirche, 1990).  As food
concentration is reduced, egg production decreases with a few days time
lag before the effect is seen (Runge and Plourde 1996; Hirche
1990).  In
laboratory experiments at lower levels ($<300 \mu g C/l$),
Hirche and Kwasniewski (1997) observed
both a tendency for increased interclutch interval
and decreased clutch size.  Plourde and Runge (1993)
also saw pronounced
reduction in egg production in the field for ambient levels $<1.25\mu
g Chl/l$.   We approximate this effect in the same way as development
rate:  decreasing the egg production rate linearly to zero when food
is below a satiation value ($150 \mu g C / l$).
Like the food effect on female development, this is a critical
parameterization relative to the initiation of Generation 1.

Stage-dependent mortality is represented.  Generally, it is assumed
highest for earlier life-stages stages (Aksnes and Blindheim 1996).
Male mortality is assumed higher than female
mortality (a factor of 5 herein), reflecting observations of
relative abundance in the field.
Predation is likely to be the dominant source of mortality (Davis
1984a,c; Madin et al 1996; Sullivan and Meise 1996) at least over Georges
Bank.  It is not easily quantified.  Predation risk
from
visual predators would decrease with depth
(Aksnes and Blindheim 1996);
and diel migration may be triggered by the presence of predators
(Durbin et al., 1995).
At this time, the details of predator abundance and
distribution, and the behavioural response to same,  must await a
fuller understanding.

Temperature and food effects on mortality are not represented
directly.  Hirche(1987) found that the animal has a high tolerance
to temperature variation.
One direct effect of food limitation would be
to increase the susceptibility  to predation  (Kleppel et al. 1996); but we
have insufficient empirical support.
The effects of temperature and food on stage duration, and therefore on
exposure to early stage predation, are captured as indirect effects.
Another direct effect would be that of female diet on viability of eggs
and early nauplii (Laabir et al., 1995; Uye, 1996).
This is not represented; instead it is bundled
into the limitation on egg production, avoiding the production of eggs
which are not viable through stage N2.
We have adopted baseline mortality estimates in Table 1
as lower limits.  Additional detail must be inferred at present.

\begin{table}[p]
\begin{center}
\caption{\label{tab:rates}
  Vital Rates for {\em Calanus finmarchicus} (days),
  at $5^{\circ}$C and $10^{\circ}$C.
  Development times are given by the Belehradek form
  $D(T) = a(T - \beta)^{\alpha}$ with $\alpha= -2.05$
  and $\beta=-9.11^{\circ}C$.}
\vspace{0.25cm}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
%    \makebox[1.0in]{Stage} &
%    \makebox[1.0in]{Belehradek a} &
%    \makebox[1.0in]{D at $5^{\circ}$C)} &
%    \makebox[1.0in]{Satiation, $\mu g C/l$} &
%    \makebox[1.0in]{Mortality Time, $\tau$} \\
%
       & Belehradek a
       & D($5^{\circ}$C)
       & D($10^{\circ}$C)
       & Satiation, $\mu g C/l$
       & Mortality, \%/day \\
 \hline
   Eggs&  595 &  2.62 & 1.41 & - & 20 \\
  \hline
    N1 &  387 &  1.70 & 0.91 &  -  & 15 \\
    N2 &  582 &  2.56 & 1.38 &  -  & 15 \\
    N3 & 1387 &  6.10 & 3.28 & 100 & 15 \\
    N4 &  759 &  3.34 & 1.79 & 100 & 15 \\
    N5 &  716 &  3.15 & 1.69 & 100 & 15 \\
    N6 &  841 &  3.70 & 1.99 & 100 & 15 \\
  \hline
    C1 &  966 &  4.25 & 2.28 & 150 & 5 \\
    C2 & 1137 &  5.00 & 2.69 & 150 & 5 \\
    C3 & 1428 &  6.28 & 3.37 & 150 & 5 \\
    C4 & 2166 &  9.53 & 5.12 & 150 & 5 \\
    C5 & 4083 & 17.97 & 9.65 & 150 & 5 \\
  \hline
    Males                & -    &  -   &  -   & -   & 10\\
    Immature Females     & 1757 & 7.73 & 4.15 & 150 &  2\\
    Mature Females       & -    & -    & -    & -   &  2\\
    Interclutch Interval & 218  & 0.96 & 0.52 & 150 & - \\
  \hline
    Diapause & - & - & - & - & 1  \\
 \hline
 \end{tabular}
\end{center}
\end{table}


Baseline population dynamics are initiated on January 1 by activating
a diapausing population ($G_0$) based on 10-year mean abundance and
distribution  from the MARMAP program (Figure 2).  The abundance and 
distribution
of $G_0$ adults is reproduced with a three-layer model, spatially variable
mortality, an extended period of activation, no food limitation, and a
large, heretofore unobserved source of diapausing C5's near-bottom.

Reproduction modeled in this system shows significant development
of generation 1 ($G_1$) over the Gulf of Maine in February-March,
which is not observed (Figure 3).  Delay of
reproduction over the Gulf, and/or severe early-stage
mortality, is required to conform with the data.  The
space-time pattern of this effect is consistent with the observed
chlorophyll distribution and timing, and reasonable food-limitation
thresholds.  Inclusion of this effect initiates the spring {\em Calanus}
bloom in the correct space-time pattern, with significant cohorts of
$G_0$ females and $G_1$ nauplii over important cod and haddock spawning
grounds on Georges Bank.  The implication is that $G_1$ is locally
spawned in food-rich waters over Georges Bank by
females advected in the food-poor
Gulf of Maine surface layer.



\vskip .2in
\noindent
{\bf 4.2 An adjoint data assimilation approach to
       diagnosis of physical and biological controls on
       {\it Pseudocalanus} spp.
       in the
       Gulf of Maine - Georges Bank region -
McGillicuddy {\it et al.} (1998).}
\vskip .2in

An adjoint data assimilation approach has been used to study the seasonal
variation in climatological abundance of the calanoid copepod
{\it Pseudocalanus} spp. in the Gulf of Maine - Georges Bank region 
(McGillicuddy {\it et al.}, 1998).  The physical basis of the
coupled system is the Dartmouth finite element model, 
which has been used extensively to simulate the circulation in this
region (Lynch {\it et al.}, 1996).  The adjoint method is used to
invert for the population dynamics which are consistent with changes in
{\it Pseudocalanus} spp. abundance and the physical transports
specified by the hydrodynamic model.  An illustrative example of the 
results from McGillicuddy {\it et al.} (1998) is shown in Figure 4.
From the period January-February to March-April, the inferred biological
source term consists of strong growth
(red shading) on the crest of the bank and moderate growth
(yellow shading) in a coastal strip
just offshore of Cape Ann.  In the latter location, physical transport
has little impact, resulting in a modest accumulation of animals in
an area coincident with the patch of local growth.
The situation on Georges Bank is quite different, where
a balance exists between the
advection and source terms.  Flow onto the crest across the northern
flank of the bank imports low concentrations of animals
from the interior of the Gulf of Maine.  This results in a strong
negative contribution from the divergence of the advective flux.
Similar in pattern, of opposite sign, and stronger in magnitude,
the source term overshadows advection such that the tendency is
for concentration on the bank to increase.  Notice that the peak
in the source term is slightly ``upstream'' of the resulting
maximum in animal abundance.  Diffusion contributes very little to the
overall balance of terms in these simulations.

McGillicuddy {\it et al.} (1998) show that
these spatial patterns in net population dynamics vary seasonally
according to the animal distributions, the climatological currents,
and their orientation with respect to each other.  In cases when
the flow is either weak or aligned with gradients in organism abundance,
changes in concentration over time are dominated by local population
dynamics.  In situations where the currents are normal to these gradients,
complex three-way balances arise between the local tendency, advective
transport, and the population dynamics source term.
With some notable exceptions, the inferred population dynamics
appears generally consistent with what is known about ecological controls
such as food limitation and predatory consumption.
These inverse calculations do not necessarily produce an incontrovertible
representation of reality, but they do provide a mathematical formalism
for backing out the ecological variables controlling population
distributions.  This work is at the very beginning of a scientifically
iterative process by which forward and inverse approaches
are used in tandem to systematically refine each other.

\vskip .2in
\noindent
{\bf 4.3 Coupling of an Individual-Based Population Dynamic Model of 
{\it Calanus finmarchicus} to a Circulation
Model for the Georges Bank Region  (Miller {\it et al.}, 1998)}
\vskip .2in

An individual-based life history and population dynamic model for 
the winter-spring dominant copepod of the subarctic North Atlantic, 
{\it Calanus finmarchicus}, is coupled with a regional model of
advection for the Gulf of Maine and Georges Bank.   The population
dynamics model is similar to that used in Lynch {\it et al.} (1998)
described above.
Large numbers of vectors, each representing individual copepods 
with elements for age, stage, ovarian status and other population dynamic
variables, are carried in a computation through hourly time steps. 
Each vector is updated at each time step according to development 
rate and reproductive functions derived from experimental
data. Newly spawned eggs are each assigned new vectors as needed. 
All vectors are subject to random mortality. Thus, both life history 
progression and population dynamics of {\it C. finmarchicus}
are represented for the temperatures in the Gulf of Maine-Georges 
Bank region in the active season. All vectors include elements 
representing depth, latitude and longitude. This allows
coupling of the population dynamics to the tide- and wind-driven 
Dartmouth model of New England regional circulation. Summary data 
from the physical model are used to advance vectors
from resting stock locations in Gulf of Maine basins through 
two generations to sites of readiness for return to rest. 

An example of model output is shown in Figure 5, in which an
ensemble of particles were released from Jordan Basin.  In general,
supply of {\it Calanus} stock to Georges Bank comes from all of 
the Gulf and from the Scotian Shelf.
The top of the bank is stocked from western 
Gulf basins; the Northeast Peak is stocked from Georges Basin 
and the Scotian shelf. All sources contribute to stock that
accumulates in the SCOPEX gyre off the northwest shoulder of 
Georges Bank, explaining the high abundance recurringly seen 
in that region. There is some return of resting stock to Wilkinson
Basin in the western Gulf, but other basins must mostly be 
restocked from upstream sources to the northeast.
An electronic archive of these results is available in animated form at

\noindent
{\tt http://www-nml.dartmouth.edu/Publications/internal\_reports/NML-98-7/}

\vskip .2in
\noindent
{\bf 4.4 An example of an Eulerian-based/diagnostic approach to linking
stage-structured models of zooplanktonic target species and larval fish
growth models: Can Georges Bank Larval Cod Survive on a Calanoid Diet?
-- Lynch {\it et al.) (2000)}.

\vskip .2in

A simple conceptual model was developed for larval fish feeding on
stage-structured prey populations, in an Eulerian framework.  The model
combines simplified contemporary models of larval fish trophodynamics,
zooplankton population dynamics, and hydrodynamic turbulence.  The
Eulerian view allows instructive maps of larval feeding and growth rates
for individual prey species, alone or in combination.

Decadally-averaged MARMAP surveys of {\it Calanus finmarchicus} and {\it
Pseudocalanus} spp. were analyzed for the March-April period.
Quasi-static population dynamics were used to infer the abundance of the
smallest stages from adult female abundance.  Computed growth rates show
that {\it Calanus} alone is insufficient to support the smallest cod
larvae (4 and 6 mm), but provides good growth ($\geq$ 10\% /day) for
large larvae (10, 12 mm); see Figure (6a).  {\it Pseudocalanus} alone
provides generally good growth for all larvae but is mismatched
spatially with observed cod spawning and subsequent larval advection;
see Figure (6b). Both species together provide good growth, matched
spatially with larval cod, for 6 mm and larger larvae; see Figure (6c). 
A dietary supplement beyond these two species is needed for the smallest
(first feeding or yolk-sac) larvae.

The procedure provides a general method for mapping observations of
zooplankton abundance, distribution and reproductive status, and their
relevance to larval fish survival.  Future work is recommended:
continued scrutiny of the turbulent predator/prey interaction is
essential; the break point between growing (6 mm and larger) and
starving (4 mm) larvae in this study is very sensitive to this
parameterization.  The role of {\em Calanus} eggs in larval fish diet
deserves careful evaluation, as they are a major contributor to these
modeled rates.  Relaxation of the various simplificatons to the
population dynamics is an obvious need, and will add considerable
complexity to these simple maps.  The 2-D analyses used herein need
obviously to be generalized to 3-D.

A postscript version of this paper available at

\noindent
{\tt http://www-nml.dartmouth.edu/Publications/external\_publications/PUB-99-1/}


\clearpage
\noindent
{\bf Figure Captions}

\vskip .2in
\noindent
Figure 1 (model.ps): The coupled physical/biological model.

\vskip .2in
\noindent
Figure 2 (g0.ps): Simulations of generation 0.

\vskip .2in
\noindent
Figure 3 (g1.ps): Simulations of generation 1.

\vskip .2in
\noindent
Figure 4 (adjoint.eps):  Bi-monthly climatological {\it Pseudocalanus} spp.
distributions (adults only) derived from the MARMAP data
(number of animals per m$^3$) for (a) January-February, and
(b) March-April.  Panel (c) below shows the source term $R(x,y)$
which results from the population dynamics inversion.  The remaining
panels show the remaining terms in the ADR equation
averaged over the period of integration: advective flux divergence (d),
diffusive flux divergence (e), and overall tendency (f).  Fields in the
bottom four rows have been normalized to the bottom depth, so
the units are
number of animals per m$^4$ per second.  The sign convention is
such that the overall tendency equals the sum of advection,
diffusion and source terms.

\vskip .2in
\noindent
Figure 5 (jord6.ps):  An example of IBM output, in which an ensemble
of particles were released from Jordan basin.

\vskip .2in 
\noindent 
Figure 6a (CalanusPG.ps): Growth rates (\% per day) for larval cod
feeding on {\it Calanus finmarchicus}.  Adult concentrations from
MARMAP; stage-structure from population dynamics; turbulence from the
circulation model.  Grey indicates negative growth; regions of 'good
growth' (10\% /day or greater) are enclosed within the yellow line.

\vskip .2in
\noindent
Figure 6b (PCalG.ps): Growth rates (\% per day) for larval cod feeding
on {\it Pseudocalanus}. Same detail as Figure 6a.

\vskip .2in
\noindent
Figure 6c (CPCalPG.ps): Growth rates (\% per day) for larval cod feeding
on {\it Calanus} and {\it Pseudocalanus combined}.  Same detail as
Figure 6a.

\end{document}
