HOW DOES ESTIMATION OF ENVIRONMENTAL
CARRYING CAPACITY FOR BIVALVE CULTURE
DEPEND UPON SPATIAL AND TEMPORAL
SCALES?
Pedro Duarte1, Anthony J. S. Hawkins 2 and António Pereira1
1CEMAS
– University Fernando Pessoa, Praça 9 de Abril, 349, 4249-004 Porto, Portugal,
Marine Laboratory, Prospect Place, The Hoe, Plymouth PL1 3DH, United
2Plymouth
Kingdom
Abstract: The simplest computational approach for estimating environmental carrying
capacity (CC) for bivalve suspension-feeders is to compare the combined rate
of filtration with rates of processes that contribute to food renewal. More
realistic approaches are based on mathematical models that take into account
complex sets of feedbacks, both positive and negative, whereby cultured
organisms interact with ecosystem processes. Each of these methods requires
spatial and temporal integrations. Yet densities of cultured animals and rates of
ecological processes vary in space and time. We illustrate strong dependencies
of estimated CC on the spatial and temporal scales chosen for associated
integrations. Where food availability is the primary limitation upon CC, low
resolution models may lead to overestimates of CC, when the potential for error
increases in positive relation with the spatial scale resolved by a model.
Considering both spatial and temporal integrations, we recommend a procedure
to help evaluate the maximum appropriate scale for the situation at hand,
thereby avoiding bias in estimates of CC stemming from any “dilution” of
bivalve densities.
Keywords: Spatial scales, carrying capacity, modelling
INTRODUCTION
Suspension-feeding bivalves have a remarkable capacity to filter
the water column. Nevertheless, growth may be limited both by the
quantity and composition of available food items, which may include
bacteria, phytoplankton and detritus (Bayne 1992, Grant et al. 1992,
Hawkins et al. 1998). Food availability is therefore a critical element in
estimating environmental carrying capacity for bivalve culture (CC),
when particle supply rate is primarily dependent upon both water
residence time (RT) and primary production rate, as reflected by cell
doubling time (PT) (Dame and Prins 1998).
The rate at which food is supplied to suspension-feeding bivalves
can potentially limit CC at different geographic scales. Among others,
2
these include the scale of the cultivation unit as blocks of ropes or lantern
nets, as seston-depleted water flows from one cultivation unit to another,
and the ecosystem scale. CC also depends on temporal variations in food
availability, which may vary over both short (i.e. tidal) and long (i.e.
seasonal) time scales (Pilditch et al. 2001). Estimates of CC must
therefore integrate over different spatial and temporal scales.
CC has been defined as the maximum standing stock that may be
kept within a particular ecosystem to maximise production without
negatively affecting growth rate (Carver and Mallet 1990). More recently,
CC has been described as the standing stock at which the annual
production of the marketable cohort is maximised (Bacher et al. 1998), or
the total bivalve biomass supported by a given ecosystem as a function of
particle supply rate and the time taken by bivalves to clear water of all
available particles (CT) (Dame and Prins 1998). These and other
definitions are generally based upon food limitation alone, despite
variable and significant interrelated effects of food availability, water
quality and other environmental factors upon bivalve survival and
population dynamics. Indeed, considering that ecosystems have multiple
functions, with a need for integrated management, ecologists are
increasingly challenged to model the various interactions between and
among species, including with their environment, on a large scale. These
interactions underlie more holistic definitions of CC, such as “the amount
of change that a process or variable may suffer within a particular
ecosystem, without driving the structure and function of the ecosystem
beyond certain acceptable limits” (Duarte et al. 2003). Certainly, there are
examples where CCs for bivalve cultivation have been exceeded by
unsustainable practices. These include the bay of Marénnes-Óleron,
France, where growth in the oyster Crassostrea gigas has reduced
significantly with increased stock densities over the years (Raillard and
Ménesguen 1994). In addition, growth in the mussel Mytilus edulis has
been diminished by increased standing stocks in the Oosterschelde
estuary, Netherlands (Smaal et al. 2001).
On the above basis, approaches used to define CC may be divided
into two main categories: traditional budget calculations and ecological
models. Budget calculations consider the time taken for phytoplankton
renewal, calculated from PT, the time for water renewal, expressed as RT,
and the time taken for bivalves to filter all water within a particular area
(CT) (Dame and Prins 1998). These comparisons may be performed over
daily, seasonal or other intervals to determine the biomass of bivalve that
can be sustained in a given ecosystem. Ecological models, on the other
hand, may be divided in box models, local depletion models and
integrated physical-biogeochemical models, all based on established
interrelations between environmental variables, biogeochemical processes
and animal or plant physiology (e.g. Bacher 1989, Jørgensen et al. 1991,
Bacher et al. 1998, Hawkins et al. 2002; Duarte et al. 2003). Such models
divide ecosystems into distinct state variables (e.g. bivalve biomass,
phytoplankton biomass). Flows of energy or material between state
variables are quantified as biological fluxes (e.g. grazing), mediated by
external forcing functions (e.g. light intensity). Fluxes are normally
3
represented by a series of differential equations that define internal
processes. To account for spatial heterogeneity, the ecosystem may be
divided in boxes. The size of each box determines the spatial resolution of
the model. Typically, box size in coastal ecosystem models has a scale of
hundreds to thousands of meters. For a description of the general structure
of an ecosystem box model with bivalve suspension-feeders, see Herman
(1993) and Dowd (1997).
The main difference between box models and coupled physicalbiogeochemical models is that in the latter, physical and biogeochemical
processes are computed simultaneously over the same temporal and
spatial frameworks. Coupled physical-biogeochemical models calculate
the velocity field with the equations of motion and the equation of
continuity (Knauss 1997), solving the transport equation for all pelagic
variables as follows:
dS   uS    vS    wS 
 2S
 2S
 2S



 Ax
 Ay
 Az
 Sources  Sinks
2
2
dt
x
y
z
z2
x
y
(1)
where u, v and w = current speeds in x, y and z directions (m s-1); A = the
coefficient of eddy diffusivity (m2 s-1); S = a conservative (Sources and
Sinks are null) or a non-conservative variable in the respective
concentration units. Whilst coupled physical-biogeochemical models are a
more accurate representation of natural systems than box models, their
main drawback is the required computing time, which especially
complicates calibration and validation.
Local depletion models are applied to reduced spatial scales,
usually being restricted to a single cultivation unit, towards being a
practical tool for local farmers or managers. This unit may be divided in
several cells, but with no feedback to the ecosystem. Instead, models are
forced solely by seston availability and current velocities at the
boundaries, solving the transport equation using those boundary
conditions for local sources and sinks. By these means, outputs may be
used to predict effects of bivalve filtration, including the geometry of
cultivation structures, on seston supply and bivalve growth downstream
(e.g. Pilditch et al. 2001, Bacher et al. 2003).
Calculated for the whole system, or in models with low spatial
resolution, average bivalve density may be lower than at the scale of
cultivation units; because bivalves are not normally spread evenly
throughout the ecosystem. This distribution factor would not matter in
estimations of CC, if seston renewal was sufficiently fast that bivalve
growth was not limited in densely cultivated areas. Otherwise, irrespective
of whether using box or coupled physical-biogeochemical models, unless
appropriate scales are chosen to account for variable bivalve densities,
predicted CC may exceed the true potential.
Our objectives are to help answer the following questions:
1) How does CC depend on the spatial resolution of the computing
method?
4
2) How should one evaluate the right spatial and temporal scales to
compute CC?
METHODS AND CONCEPTS
In the present work, some theoretical considerations are
developed in order to estimate appropriate spatial and temporal scales in
CC estimates. Model simulations and comparisons of results obtained
with different distributions and densities are also carried out to obtain
some empirical insight into the dependence of CC estimates on the spatial
resolution of the computing methods. Both approaches are applied to
Sungo Bay (People’s Republic of China), based upon our previous
modelling there (Duarte et al. 2003).
Spatial and Temporal Scales – Theoretical Considerations
We suggest that one possible way to evaluate appropriate spatial
and temporal scales to compute CC, thereby avoiding potential errors
described above, is to compare CT with PT at different scales ranging
from the whole system to cultivation units. The larger the scale chosen,
the more likely it is that PT will be lower than CT, given “dilution” of
high bivalve densities. This, we define here as Condition 1.
PT < CT
(1)
At lower scales, focussing upon cultivation sites with high bivalve
densities, the opposite may happen. If there is a geographic scale (X)
below which CT is smaller than PT, then bivalves below X remove
phytoplankton cells faster than they divide, and therefore depend upon
food input from adjacent waters. This, we define here as Condition 2.
PT > CT
(2)
At this stage, CT should be compared with water residence time
(RT), which may be estimated as X/v, where v = resultant current speed
(m s-1). If RT is smaller than CT, then X may be an appropriate scale to
use in further computations. If not, then X must be reduced to ensure
that water properties do not change much across the scale considered.
This, we define here as Condition 3.
RT < CT
(3)
If Condition 2 is true and Condition 3 does not hold even for very
small X, predicted bivalve biomass will exceed system CC. If Condition
2 never holds, predicted bivalve biomass will be below CC. If Condition 2
5
holds and Condition 3 is always attained, predicted bivalve biomass may
be above or below CC, depending on whether water renewal decreases or
increases the available seston.
After assessing and choosing an appropriate spatial scale,
temporal scale may be assessed using the numerical Courant condition
(Condition 4) in such a way to avoid all water in any box being cleared
within each model time step for numerical stability:
(4)
where CR = bivalve clearance rate (m3 d-1), Phyt = phytoplankton
biomass (mg m-3) and V = box volume (m3).
Model Simulations
We assess the above a priori expectations on the basis of
empirical evidence for Sungo Bay, an area of 180 km2 of intensive multispecies aquaculture of kelp (Laminaria japonica), oyster (C. gigas) and
scallop (Chlamys farreri) in the People’s Republic of China (Fig. 1).
Firstly, we calculate values of CT, PT and RT at different spatial
scales, ranging from 500 m to 15000 m, to assess how estimated carrying
capacity may depend on the spatial and temporal scales chosen for
associated integrations.
Secondly, we use the two-dimensional vertically integrated,
coupled hydrodynamic-biogeochemical model developed and described
by Duarte et al. (2003) for Sungo Bay. The model has a land and an ocean
boundary, and is based on a finite difference bathymetric staggered grid
(Vreugdenhil 1989) with 1120 cells (32 columns X 35 lines) and a spatial
resolution of 500 m. The model time step is 18 seconds. It is forced by
tidal height at the sea boundary, light intensity, air temperature, wind
speed, cloud cover and boundary conditions for some of the simulated
state variables. It solves the general 2D transport equation (Equation 1)
(Neves 1985, Knauss 1997). The hydrodynamic sub-model solves the
speed components, whereas biogeochemical processes such as primary
productivity and grazing, as well as physical processes such as sediment
deposition and resuspension provide the sources and sinks terms of
Equation 1.
Duarte et al. (2003) used this model to estimate CC for oysters (C.
gigas) and scallops (C. farreri) in Sungo Bay. The model resolved bivalve
density in aquaculture and non-aquaculture areas, with no bay-averaged
values. Scenarios modelled by Duarte et al. (2003) represented past
(Scenario I) and current culture practise (Scenario IIa), with initial stocks
of 2850, 0.6 and 1860 tons DW of kelps, oysters and scallops,
respectively; including hypothetical changes of x 0.5, x 2 and x 3 in
seeding densities of scallop and oyster (Scenarios IIb to d,
6
16 km
N
17.5 km
Sungo
Bay
Kelp culture
Scallop culture
Oyster culture
Fig. 1 – Areas cultivated in Sungo Bay since 1999, including part of the model grid (upper
left corner), for which the spatial step is 500 m (refer Methods and Concepts, Model
simulations).
Table 1 – Aquaculture scenarios simulated with the model (refer Methods and Concepts)
(Fig. 1).
Culture densities
Past simulations
Current simulations
(Duarte et al. 2003)
Scenario
s
Oysters
(indiv.
m-2)
Scallops
(indiv.
Scenarios
m-2)
Oysters
(indiv.
m-2)
Scallops
(indiv. m-2)
IIa
55
56
IIIa
18.2
19.8
IIb
27.3
28
IIIb
9.1
9.9
IIc
110
112
IIIc
36.5
39.6
IId
165
168
IIId
54.7
59.3
respectively), whilst maintaining seeding periods and spatial distributions
as in Scenario IIa (Fig. 1 and Table 1). Scenarios analysed in the present
study (IIIa to d) maintain the same total numbers of oysters and scallops
as under Duarte et al’s (2003) Scenarios IIa to d, respectively. However,
whereas our past Scenarios IIa to d restricted the spatial distributions of
each cultured species to their normal areas of culture, present Scenarios III
a to d distribute each cultured bivalve species both through their own
normal area of culture and that area used for kelp culture (Fig. 1).
7
Densities were therefore reduced, whilst maintaining the same biomass
throughout the bay (Table 1). Comparing CCs predicted for Scenarios IIIa
to d with those reported by Duarte et al. (2003) for Scenarios IIa to d, we
can assess any effects of local food limitation on CC estimates. A priori,
one would expect that spreading the bivalves over a larger area should
increase estimates of bay-scale CC, thereby providing empirical evidence
for the importance of spatial resolution.
RESULTS AND DISCUSSION
In Sungo Bay, spatially and temporally averaged CT (10 days) is
smaller than RT (20 days) but larger than PT (5 days), suggesting an
unutilized environmental capacity for increased culture of filter-feeding
bivalves (Table 2) (Duarte et al. 2003). Considering issues of scale,
current velocities were in fact slowest within the nearshore regions
Table 2 – Physical and biological characteristics of Sungo Bay (refer Results and
Discussion).
Characteristics
Values
Area (km2)
179.5
Depth (m)
10.0
Volume (106 m3)
1800
Residence time (RT; d)
20.0
Average annual Chl a
1.5
concentration (mg
m-3)
Primary production (106 g C d-1)
26.5
Cell doubling time (PT; d)
5.0
Total biomass
(106 g)
Bivalve clearance time (CT; d)
44000
10.1
of Sungo Bay (Duarte et al. 2003). At the scallop culture site in
northwestern part of the bay, average current velocities were as low as
0.025 m s-1. On the basis of a spatial resolution of 500 m as used in the
ecosystem model of Duarte et al. (2003), it is possible to estimate the
maximum RT as roughly 0.2 days in one cell of the model grid. Assuming
an average clearance rate for a commercial-sized scallop of 5 to 6 cm shell
length as 2.5 l h-1 (Hawkins et al. 2002), a cultivated density of 59 ind. m-2
(Duarte et al. 2003), and an average depth of 6 m, it is possible to estimate
CT for one 500 m x 500 m cell of the model as 1.7 days. It was
impractical within the context of present study to determine effects of
scale on PT, as this would have required revision of hydrodynamic
component for input to our coupled model at each chosen scale.
8
20
18
CT – 56-5.5 ind. m-2
(a)
16
RT, PT
CTFT(days)
RT,and
PT and
(da ys)
14
12
10
Max PT
8
6
RT
Mean PT
4
Min PT
2
0
500
CT – 112-11 ind. m-2
2500
4500
6500
8500
10500
12500
14500
Spatial scale (m)
12
10
(b)
and CT
FT (days)
RT, RT,
PTPTand
(days)
Max PT
8
CT – 55-4.5 ind. m-2
RT
6
Mean PT
4
Min PT
2
0
500
CT – 110-9 ind. m-2
2500
4500
6500
8500
10500
12500
14500
Spatial scale (m)
Fig. 2 – Relationships between water residence time (RT), mean, maximum and minimum
production time (mean PT, max PT and min PT), clearance time (CT, for two ranges of
bivalve densities) and spatial scale for (a) scallops and (b) oysters. Arrows on lower left
corners of both graphs depict spatial scales appropriate for the Sungo model (refer to
Results and Discussion).
9
However, considering the variability of phytoplankton standing stock and
net primary production obtained in the simulations described in Duarte et
al. (2003) it can be estimated that PT would have varied over a range of 3
to 9 days, with an average of 5 days. Therefore, Conditions 2 and 3 were
true at the spatial scale of 500 m x 500 m cells used by Duarte et al.
(2003). In addition, the Courant Condition 4, with a model time step of 18
s, chosen to ensure hydrodynamic model stability, gives a value of 8 x 1012
(cf.- Methods and Concepts, Spatial and temporal scales – theoretical
considerations, above), which is considerably less than one. Therefore,
both the spatial scale of 500 m and the model time step of 18 s used by
Duarte et al. (2003) seem appropriate for the scallop cultivation site.
Repeating the above calculations of CT and RT for different
spatial scales ranging from 500 m to 15000 m, it is important to note that
bivalve density decreased as
(Fig. 2)
was recalculated by a weighted average of density within the cultivated
areas and density of 0 ind. m-2 outside those areas. This produced a linear
decrease in density from 56 ind. m-2 in culture areas to 5.5 ind. m-2 when
integrated for the whole bay. To assess consequences of increased culture
densities, similar calculations were also carried out with a starting density
of 112 ind m-2. Resulting CT, RT and PT values are illustrated in Fig. 2a,
ould be lower than 1800 m for Conditions 2 and 3 to
hold. Certainly, only by working at this lower spatial resolution within
areas of existing culture, is the potential afforded to establish areas with
greater potential for bivalve production. Similar calculations were carried
out for oysters, and results presented in Fig. 2b. Given that the average
clearance rate of a commercial-sized oyster of 7 cm shell length is 4 l h-1
be less than 900 m. Minimum length scales in areas of bivalve cultivation
in Sungo Bay all exceed a minimum of 1000 m (Fig. 1), and which
scallops and oysters.
The results presented in Table 3 summarise scallop and oyster
productions predicted under scenarios IIa, IIb, IIc, IId, IIIa, IIIb, IIIc and
IIId (cf. – Methods and Concepts). Bivalve growth isolines prior to
harvesting (cf. – Fig. 1) for simulations IIa and IIIa are shown in Fig. 3.
The model predicts that under scenarios IIIa, IIIc and IIId, there is a
considerable increase in scallop production (between one and two orders
of magnitude). It also predicts that compared with a 2x increase in bivalve
density, a 3x increase in density results in a decreased scallop yield.
Oysters, on the other hand, show a decrease in production for scenario IIIa
relative to scenario IIa. This result may be explained by the poorer
performance of oysters that are near the sea boundary (Fig. 3), as a result
of lower seston organic contents (model results not shown). However, as
bivalve density increases in scenarios IIIc and IIId the model predicts an
important increase in production compared with scenarios IIc and IId.
This reflects both intra and interspecific competitions, when oysters outcompete scallops (Duarte et al. 2003), over-riding effects of lower food
10
May
May
Scallops
Scallops
October
Oysters
October
Oysters
February
February
Oysters
Oysters
Fig. 3 - Scenario IIa (on the left) and IIIa (on the right). Growth isolines (cm shell length)
predicted by the model for scallops and for oysters. For the former, results are shown for
May, just before harvest (top two figures). For the latter, results are shown for October and
February, just before the first and second harvest periods, respectively. (refer to text and
Table 3).
11
Table 3 – Harvest (103 T FW) predicted for aquaculture scenarios IIa to IId by Duarte et al.
(2003) and in the present study for aquaculture scenarios IIIa to IIId. Scenarios and results
are given for normal, decreased (1/2) and increased (2 and 3 fold) bivalve loads (refer
Methods and Concepts).
Scallops
Oysters
Scenarios
Past
simulations
(Duarte et
al. 2003)
Current
simulations
Past
simulations
(Duarte et
al. 2003)
Current
simulations
Normal
9 (IIa)
18(IIIa)
42 (IIa)
37(IIIa)
1/2X
8 (IIb)
9(IIIb)
26 (IIb)
20(IIIb)
2X
0.6 (IIc)
31(IIIc)
58 (IIc)
61(IIIc)
3X
0.9 (IId)
20(IIId)
25 (IId)
75(IIId)
quality near the sea boundary. Even following a 3x increase in bivalve
density, the model still predicts increased oyster production, including an
increased total production of both scallops and oysters (Table 3).
Average concentrations of chlorophyll, total particulate matter
and particulate organic matter over the whole bay and simulation period
(Jan 99 – August 2000) are compared for simulations IIa – IId and IIIa –
IIId in Table 4. Concentrations for former simulations (IIa to d) are all
lower than those for current simulations (IIIa to d). This result indicates
that spreading bivalve biomass over a larger area, towards the sea
boundary, where RT is lower, results in reduced impacts of bivalve filter
feeding on ecosystem properties, consistent with the higher bivalve
production predicted by simulations IIIa, IIIc and IIId. These results
confirm the a priori hypothesis that spreading bivalves over a larger area
should increase estimates of bay-scale CC (cf. – Methods and Concepts).
In fact, the results shown in Table 3 suggest a significant potential for
increasing scallop and oyster production under scenarios III, whereas the
opposite was predicted under scenarios II, mostly for the scallops.
Table 4 – Annual mean concentrations of chlorophyll, total particulate matter (TPM) and
particulate organic matter (POM) predicted by the model under the 99-00 aquaculture
simulations IIa, IIb, IIc and IId (Duarte et al., 2003) and the current simulations IIIa, IIIb,
IIIc and IIId (cf. – Methods and Concepts, Table 1 and Fig. 1).
IIa
IIIa
IIb
IIIb
IIc
IIIc
IId
IIId
Chl. (g l-1)
1.5
2.1
2.0
2.5
1.0
1.6
0.8
1.2
TPM (mg l-1)
14.9
15.1 15.1
15.1
14.6
14.9
14.5
14.6
2.4
2.6
2.7
2.1
2.4
2.0
2.2
-1
POM (mg l )
2.6
The local depletion model of Bacher et al. (2003) was used to
evaluate bivalve growth and food depletion at different parts of Sungo
12
Bay. Results obtained suggest that scallop growth under current densities
(c.a. 50 ind m-3) is below optimal at inner parts of the bay, where current
velocities and bay-sea exchanges are reduced. The negative effects of
density on scallop growth ranged from 5% in the eastern part of the bay to
more than 30% in the southwestern part of the bay. This is consistent with
the model results of Duarte et al. (2003), and with the larger RT at the
inner parts of the bay predicted by the same authors.
CONCLUSIONS
A general conclusion from the above results is that in estimating
CC, X should be smaller than the length scale of the areas currently used
for suspended bivalve cultivation within Sungo Bay. This would not be
the case if food supply was greater, perhaps associated with faster RT or
PT. Whatever, as high bivalve densities at sites of cultivation are “diluted”
upon integration over larger areas, CT increases geometrically with X.
This is because there is a similar decrease in bivalve density. Assuming
that CT may define an upper threshold for CC, it follows that low
resolution models may lead to overestimates of CC, when the potential for
error increases in positive relation with the spatial scale resolved by a
model. We have demonstrated a strong dependence of estimated CC on
the spatial scales chosen for associated integrations. To evaluate
appropriate scales under different culture situations, we recommend
following the procedures described above, comparing RT, PT and CT at
scales ranging from less than the smallest cultivation units to the scale of
whole system, for an indication of maximum appropriate scale that must
be resolved to avoid bias in estimates of CC stemming from any
“dilution” of bivalve densities (refer Methods and Concepts).
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