Aquaculture 246 (2005) 209 – 225
www.elsevier.com/locate/aqua-online
Incompleteness and statistical uncertainty in
competition/stocking experiments
Marcel Fréchettea,T, Marianne Alunno-Brusciab, Jean-François Dumaisc,
Renée Siroisd, Gaétan Daiglee
a
Institut Maurice-Lamontagne, Ministère des Pêches et des Océans, C.P. 1000, Mont-Joli, QC, Canada G5H 3Z4
b
CREMA, IFREMER, Place du séminaire, B.P. 5, F-17137 L’HOUMEAU, France
c
ISMER, Université du Québec à Rimouski, 310 allée des Ursulines, Rimouski, QC, Canada G5L 3A1
d
Département de mathématiques, dTinformatique et de génie, Université du Québec à Rimouski, 300 allée des Ursulines,
Rimouski, QC, Canada G5L 3A1
e
Département de Mathématiques et de Statistique, Faculté des sciences et de génie, Pavillon Alexandre-Vachon,
Université Laval, Québec, QC, Canada G1K 7P4
Received 28 October 2004; received in revised form 5 January 2005; accepted 11 January 2005
Abstract
In competition experiments, decisions are made not only about experimental conditions such as initial population densities,
of course, but also about population size structure, for instance. Here we use an individual-based simulation model to study the
effect of size-grading of mussels. With low individual variability, predicted yield was lower and less variable, there was no
density-dependent mortality, and optimal stocking density for aquaculture was lower than with high individual variability,
whereby self-thinning occurred and yield was quite variable. Thus, individual variability was a critical factor for estimating
survival effects of overstocking, at the expense of precision of growth estimates. Therefore, competition experiments are
inherently incomplete. We argue that in practice, incompleteness cannot be overcome by using genetic information as a
covariate because evidence from the literature shows that the effect of genetic makeup in competition situations is frequencydependent. Apparently, the only approach presently available to obtain unbiased estimates is to use a size structure similar to
that of the population under study. This contrasts with a literature review of bivalve stocking experiments published in
Aquaculture through the last 30 years which clearly shows that the issue of size structure of test populations has been largely
overlooked. The same principles hold for competition studies in natural settings.
D 2005 Elsevier B.V. All rights reserved.
Keywords: Incompleteness; Intraspecific competition; Self-thinning; Stocking experiments
T Corresponding author. Tel.: +418 775 0625; fax: +418 775 0740.
E-mail address: [email protected] (M. Fréchette).
0044-8486/$ - see front matter D 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.aquaculture.2005.01.015
210
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
1. Introduction
Overstocking has resulted in severe losses of
productivity due to mortality and to growth reductions
in many bivalve aquaculture sites, even among the
most successful (Aoyama, 1989; Héral, 1991). As a
result, assessing the effect of stocking density on yield
has become a central issue in aquaculture management, as well as in agriculture and forestry (Westoby,
1984). In organisms such as suspension feeders whose
food supply depends on natural transport processes,
growth-controlling processes may act at various
scales. At the scale of a site or a whole basin, for
instance, feeding by bivalves and drag exerted on flow
by culture gear may lead to phytoplankton depletion,
with associated spatial gradients in food availability
(Dowd, 1997; Bacher et al., 1998; Grant and Bacher,
2001; Pilditch et al., 2001). The effect of stocking
density at the scale of whole basins is usually studied
using numerical models (Dowd, 1997; Bacher et al.,
1998; Pilditch et al., 2001). At a smaller scale, there
may be food depletion within culture units such as
pearl nets, for instance (Parsons and Dadswell, 1992;
Claereboudt et al., 1994a,b). In contrast to wholebasin situations, stocking effects at the scale of
individual culture units such as ponds, fish pens,
bottom plots, pearl nets or other unit structures are
studied in competition/stocking experiments (e.g.,
Mallet and Carver, 1991; Parsons and Dadswell,
1992; Côté et al., 1993; Hossain et al., 1998; Verhoef
and Austin, 1999; Beal and Kraus, 2002).
In addition to population density itself, however,
size structure within the groups assayed may be a
strong determinant of the outcome of stocking experiments. Size hierarchies are thought to develop initially
because of intrinsic individual variability, but rapidly
may become exacerbated by interindividual interactions. Asymmetric competition (see Weiner, 1990),
whereby smaller individuals suffer disproportionately
more from resource shortage or from agonistic
interactions than larger individuals do, acts to increase
such differences (Schmitt et al., 1987; but see
Baardvik and Jobling, 1990; Adams et al., 2000),
although in many benthic situations spatial effects
may predominate over intrinsic individual variability
as a cause of variability in growth rate (Pfister and
Stevens, 2002). In organisms with well-developed
behavioural interactions, larger individuals are gen-
erally more aggressive and monopolise larger
amounts of resources (usually food) than smaller
individuals (but see Jbrgensen et al., 1993; Jbrgensen
and Jobling, 1993; Ruzzante, 1994 and references
therein; Gardeur et al., 2001). In larval fish, size
hierarchies usually lead to increased cannibalism, the
larger the size differences the stronger the interaction
(Folkvord, 1991; Qin and Fast, 1996; Baras and
d’Almeida, 2001).
In an effort to minimise effects of size variability, it
is customary in some types of cultures to grade
individuals according to size (Bardach et al., 1972;
McVey, 1983a,b; Pillay, 1990). Individual variability
in initial size and growth rate has been a source of
concern also in designing growth and stocking
experiments (e.g., Beal et al., 2001; Gardeur et al.,
2001). To minimise statistical uncertainty in comparisons among densities, it has been attempted to
minimise individual variability by grading individuals
at the beginning of experiments (e.g., Parsons and
Dadswell, 1992; Freites et al., 1995; Román et al.,
1999; Verhoef and Austin, 1999; Alunno-Bruscia et
al., 2000). Consequently, there has been growing
experimental work to assess the effect of size-grading
on yield. The general pattern drawn from these results
appears ambiguous. Some experiments have found
asymmetric competition (e.g., in abalone Haliotis
tuberculata: Mgaya and Mercer, 1995), which supports the usefulness of grading procedures. Lambert
and Dutil (2001) reported lower growth rates in
graded cod (Gadus morhua) groups at low and
intermediate stocking biomass. This is consistent with
expectations based on the allometry of metabolic
requirements. The negative effect of grading
decreased with increasing stocking biomass and
disappeared at the highest biomass tested. Sowka
and Brunkow (1999) reported slower growth in
graded bonytail (Gila elegans) and also found higher
density-dependent mortality in groups with higher
interindividual variability. Other studies, however,
failed to find asymmetric competition and have led
to questioning the usefulness of size-grading (Karplus
et al., 1986; Jobling and Reinsnes, 1987; Kamstra,
1993; Qin et al., 2001). It should be noted, however,
that many of these studies were done only at optimal
stocking density or at biomass-density levels not
conducive to competition. In absence of competition,
it is of course not possible to test whether competition
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
density (N) in what is called a B–N diagram (see
Westoby, 1984). The B–N diagram and related
concepts were developed in plant studies. There is
growing evidence that the B–N diagram may be useful
in animal ecology as well (Latto, 1994; Petraitis, 1995b;
Guiñez and Castilla, 1999). Indeed, B–N diagrams may
provide a more powerful approach to the analysis of
stocking experiments than separate analyses of growth
and mortality (Fréchette et al., 1996).
B–N diagrams may be studied in terms of B–N time
trajectories and B–N curves. A B–N time trajectory is a
curve (N(t), B(t)), t 0VtVt f, representing the time
evolution of the biomass and number of individuals,
starting at an initial point (N(t 0), B(t 0))=(N 0, B 0).
Using different values of N 0 will generate a family of
B–N time trajectories. B–N curves are then obtained
by keeping the time parameter constant: for any given
fixed time tV, a B–N curve is defined by joining those
points from the various time trajectories that correspond to the time value tV, that is all points (N(tV),
B(tV)). The B–N diagram consists of the family of B–N
time trajectories and B–N curves (Fig. 1; alternatively,
one may study m–N curves; m is individual mass;
Petraitis, 1995a). In competition-free situations, individual growth rate is constrained by individual
bioenergetics and background environmental conditions only, with no density-dependent mortality.
t2
Biomass (B)
is asymmetric or symmetric and consequently whether
size-grading might be useful.
Modelling studies of the effect of size variability
on population dynamics suggest that variability in
size promotes long-term stability of populations
(Begon and Wall, 1987; Pacala and Weiner, 1991;
Uchmanski, 1999, 2000). Aquaculture populations
are generally in a different setting than natural
populations. Aquaculture is generally concerned with
short-term, intra-generation productivity. Here we
present a discussion of likely consequences of
grading individuals in bivalve stocking experiments.
To support these speculations and explore some of
their implications, we model the effect of stocking
density (i.e., the effect of initial population density of
groups) and of individual variability on growth and
survival of mussels under the hypothesis of exploitation competition only (i.e., no direct interference as
in some fish, for instance), using an individual-based
model (DeAngelis and Gross, 1992). We also
examine the effect of individual variability on
estimates of optimal stocking density (OSD) in
bivalve aquaculture and on competition mode (i.e.,
symmetric vs. asymmetric). Our simulations are based
on a realistic model of the bioenergetics of densitydependent growth of blue mussels Mytilus edulis
(Fréchette and Bacher, 1998) which has been found to
provide acceptable predictions of individual mussel
growth for the conditions prevailing near the Maurice-Lamontagne Institute, Lower St. Lawrence Estuary (Alunno-Bruscia et al., 2000). The original model
has been modified to further include (1) the effect of
individual variability and (2) criteria for self-thinning
of the mussels. Finally, we review the methodology of
all bivalve stocking experiments published from
Volume 1 through Volume 224 of Aquaculture to
assess experimental practices with respect to size
selection in bivalve stocking experiments.
211
ST
t1
t0
2. The nature and analysis of stocking experiments
Stocking experiments are competition experiments.
In sessile or confined organisms, growth and mortality
are two basic variables from which the effect of
competition is inferred. The case arising, both effects
must be potentially estimated. Competition is best
studied in joint analyses of biomass (B) and population
Population density (N)
Fig. 1. Hypothetical B–N diagram, where yield of groups stocked at
various populations densities is monitored at times t 0, t 1 and t 2. The
plot shows how B–N curves (solid lines) are obtained from B–N
trajectories (dotted lines). Arrows indicate time. ST is the upper
limit to biomass–density combinations set by self-thinning. Densityindependent mortality is assumed to be negligible.
212
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
Assuming negligible natural mortality, time trajectories of the individual groups are vertical (broken lines
in Fig. 1). Since body size does not change with
population density, B–N curves at small t are straight
oblique lines (it is assumed that individual properties
are distributed the same way among the various
density groups). This is the density-independent region
of B–N curves. If population density is sufficiently
high, however, competition may occur as individual
size increases, with its slowing down of individual
growth rate. For those density groups where growth is
retarded by competition with no mortality, this results
in curvilinearity of the B–N curve, as slope decreases
towards 0, exhibiting what has been called the
competition–density (C–D) effect by Shinozaki and
Kira (1956). In extreme situations, the joint effect of
individual size and high population density may result
in mortality as growth proceeds further. In other words,
self-thinning (ST) may occur (Yoda et al., 1963). The
onset of ST is reflected in Fig. 1 by a bending of B–N
time trajectories to the left. As ST becomes fully
developed, however, B–N trajectories exhibit upward
concavity. In summary, B–N diagrams reveal two
patterns: the C–D effect and ST, which reflect the
effects of competition on growth and mortality,
respectively.
physiological characteristics of the mussels are taken
into account as follows. The volume of water filtered
by a mussel per unit time is assumed to follow a
power law (i.e.: allometry) written as a f m a , where a f
(l day1), is the filtration coefficient. By means of the
energy to mass conversion factor C em (5.1105 g/J)
and the assimilation efficiency e a, this translates into
a rate of mass increase due to phytoplankton
consumption of the form C eme aa f m a q=c f qm a where
c f =C em e aa f. The rate of mass decrease due to
respiratory losses of a mussel is taken to be c r m b
where c r=C ema rT c (J day1), with a r (J day1)
denoting the respiration coefficient and T denoting
temperature (8C). We start with a population of N
mussels, all with the same size and same physiological characteristics except for a r (Koehn and Shumway, 1982; Toro et al., 1996; Bayne et al., 1999). The
value for the ith mussel is denoted a ri , with
corresponding c ri . The a ri are assumed to be normally
distributed. From the above assumptions, it follows
that the evolution equations of the system are
3. Simulation of the effect of restricted individual
variability
Mortality arises under insufficient nutrition. A
criterion to this effect is obtained as follows. The
shell size L of each mussel is made to grow with the
flesh mass at a rate corresponding to the nominal
allometric length–mass relation m=aL b , whenever
dm/dtN0 (a=0.003, b=3, based on suspension-cultured mussels in Îles-de-la-Madeleine, Quebec,
between July 1995 and October 1995; unpublished).
But when dm/dt=0, dL/dt is set to 0. During a
thinning period, L remains constant while m becomes
naL b , with nb1. We consider that death occurs when
n falls below some critical value (de Roos and
Persson, 2001). The period over which the system is
followed is divided into a number of short subintervals over each of which the number of live
individuals is considered constant. Within each such
subinterval, the above differential system is solved
with a standard ode solver (in our case the CVODE
package; Cohen and Hindmarsh, 1994). At the end of
each subinterval, the mortality criterion is applied so
3.1. Yield and mortality
The mathematical model we used to illustrate the
effects of size-grading is as follows (see Appendix A
for parameter values and units). A group of N mussels
resides in a perfectly mixed water reservoir of volume
V, renewed at a rate m (l day1) with incoming water
containing phytoplankton at constant energy density
q in, waste water exiting at the same rate, but with the
ambient phytoplankton density q. In the steady state
and in the absence of mussels, the reservoir is at
constant phytoplankton density q in. Note that if the
mussel-free reservoir originally contained no phytoplankton, the density of phytoplankton (always
assuming mixing in a negligible time) would build
up to q in according to q(t)=q in(1et/s ) where s, the
relaxation time of the reservoir is given by s=V/m. The
dmi =dt ¼ cf qmai cri mbi
dq=dt ¼ ðqin qÞ
s
i ¼ 1; . . . ; N
N
af X
q
ma
V i¼1 i
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
that in the next subinterval the number of live
individuals (and hence the dimension of the differential system) is possibly reduced. We assume that
other sources of mortality are negligible. Therefore
only one competitive mechanism–exploitation competition–is included in the model, although experimental evidence suggests that more than one
mechanism may be acting simultaneously in groups
of competing mussels Mytilus galloprovincialis
(Brichette et al., 2001).
To study the effect of individual variability on
average yield, the model was run with different values
of initial N (10, 20, 40, 60, 80 mussels reservoir1)
until t=365 days. This was repeated for 100 different
sets of a r values (by using different seeds for the
numerically generated a r distribution), both with low
(r=2) and high (r=10) variability. To study the effect of
individual variability on B–N diagrams, the model was
run with different values of initial N (10, 20, 40, 60, 80
mussels/reservoir) with low (r=2) and high (r=10)
variability in a r. We obtained the sets of (N(t), B(t))
points for the B–N curve at t=365 days and at time
t={1, 2, . . ., 365} days for the ST curve. We further
explored the effect of increasing individual variability
by running the model with increasing variability
(r={1, 2, 3,. . ., 15, 20, 30}. Initial stocking densities
were N 0={2, 5, 10, 15, 20, 25, 40, 60, 80} individuals
per reservoir. To study the C–D effect we obtained the
set of (N(t), B(t)) points (i.e. the B–N curve) at t=365
days. To study ST we modelled B(t) and N(t) for the
group with initial density N 0=80 individuals at time
t={10, 20, 30, 40, 60, 100, 180, 365} days. Individuals
with a rb5.0 J day1 were removed to avoid mussels
growing to unrealistic large size.
3.2. Optimal stocking density
In many situations, OSD has been defined as the
maximum population density stocked without affecting individual growth negatively (e.g., Carver and
Mallet, 1990; Grant et al., 1993; Dowd, 1997).
However, aquaculture must be profitable. Therefore
a definition of OSD should incorporate economic
considerations. This cannot be done in the present
situation because the technology and other financial
aspects of operations are not specified. So in order to
reflect the effect of increased investment incurred by
increasing stocking density, we define OSD as the
213
lowest stocking density for which the rate of
increase in yield (with respect to population density)
at harvest time is no longer larger than its value at
initial time. More precisely, this is estimated by
computing the population density at which the
derivative of yield (B) with respect to population
density (N) equals the derivative of initial biomass
with respect to initial population density, i.e. (BB/
BN)z =(BB/BN)0, where the subscripts 0 and z stand
for initial time and harvest time, respectively. We ran
the model to generate B–N time trajectories with all
possible initial population densities (from 1 through
80), and then obtained the B–N curve at t=365. We
repeated the procedure for 100 different seeds
generating 100 different sets of a r values. This was
done with low (r=2) and high (r=10) variability. To
estimate OSD in both cases, we fitted the model
B=gN/(1+N k ), where g and k are parameters) to the
B–N curves, then estimated the initial population
density for which the two derivatives above were
equal (see Appendix B) and tested the estimate of
OSD using a paired t-test.
3.3. Symmetric vs. asymmetric competition
ST is usually attributed to asymmetric competition, with smaller individuals being suppressed to the
point of dying (White and Harper, 1970). Skewness
in size distribution provides an indication that
competition is asymmetric (Xue and Hagihara,
1999). In cases where group size is small, an
alternative method for examining whether competition is asymmetric is to plot the difference between
growth without competition ( G T) and growth with
competition ( G C), both expressed as a function of a
variable governing competitive ability (in the present
case, it is the respiration parameter a r). Therefore
G T=f(a r). Under the effect of competition, growth is
reduced by a factor u, such that G C=(1u)d f(a r).
Therefore growth retardation attributable to competition may be written as G TG C=f(a r)(1u)d f(a r)
(Fréchette and Despland, 1999). By further dividing
by G T and substituting f(a r) for G T on the right-hand
side, one obtains ( G TG C)/G T=u. Therefore u is an
index of competition strength. The larger the u
values, the stronger the effect of competition on
growth are. Constant values of u are indicative of
symmetrical competition. The relationship between
214
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
individual properties and competition strength can be
assessed by plotting u as a function of a r. In such a
plot, non-zero derivatives are indicative of asymmetric competition. To study the mode of competition, we ran the model for each individual variability
level with flow rate m=432 l day1 and m=432,000 l
day1 for mussels with and without competition,
respectively.
Table 1
Effect of population density and of individual variability on total
yield
(A) Model
Source
Num df
Den df
Population
density
Individual
variability
Interaction
4
990
326.55
b0.0001
1
990
1233.19
b0.0001
4
990
51.58
b0.0001
1
1
1
1
1
4
4
990
990
990
990
990
990
990
20.32
99.24
334.48
668.32
1124.59
534.94
167.31
b0.0001
b0.0001
b0.0001
b0.0001
b0.0001
b0.0001
b0.0001
4. Results
(B) Effect of slices
4.1. Yield and mortality
Population
density
The effect of population density and individual
variability on yield is shown in Fig. 2 (A: total
A
10
20
40
60
80
(A) Two-way heterogeneous-variance ANOVA model with Type 3
tests of fixed effects (SAS, PROC MIXED). (B) Tests of treatment
levels (population density*individual variability; effect of slices).
Total biomass
6
5
4
3
2
1
0
0
20
40
60
80
Initial population density
7
Commercial biomass
P
Individual
variability
2
10
7
F value
B
6
5
4
3
2
1
0
0
20
40
60
80
Initial population density
Fig. 2. Biomass yield (flesh dry mass) for high individual variability
(empty diamonds) and low variability (solid diamonds) groups, as a
function of initial population density (10, 20, 40, 60 and 80
individuals per chamber). (A) Total yield; (B) yield of commercialsize animals.
biomass; B: biomass of commercial-size individuals;
commercial-size individuals are defined as those with
Lz5 cm). Total yield was more variable in high
variability groups than in low variability groups,
irrespective of population density. At low population
density, yield appeared somewhat similar for high
and low interindividual variability groups. At high
population density, however, yield was higher for
high interindividual groups. These trends were tested
using a heterogeneous-variance two-way ANOVA
model (SAS, PROC MIXED; two-way heterogeneous-variance ANOVA model with Type 3 tests of
fixed effects). Differences of least squares means for
high and low variability varied among groups, as
shown by the significant interaction in Table 1A.
The differences between high and low variability
groups, however, were significant for all densities
tested (Table 1B; effect of slices; pb0.0001), yield
being systematically higher in high-variability groups
than in low-variability groups. Commercial yield was
lower than total yield (Fig. 2). The effect of density
and variability treatments, however, was qualitatively
the same for commercial yield as for total yield
(Table 2A,B). In both high and low interindividual
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
(A) Model
Source
Num df
Den df
Population
density
Individual
variability
Interaction
4
990
F value
226.94
P
b0.0001
1
990
1207.43
b0.0001
4
990
154.02
b0.0001
1
1
1
1
1
4
4
990
990
990
990
990
990
990
170.56
536.33
376.54
234.89
41.87
52.71
192.39
b0.0001
b0.0001
b0.0001
b0.0001
b0.0001
b0.0001
b0.0001
(B) Effect of slices
Population
density
Individual
variability
10
20
40
60
80
2
10
(A) Two-way heterogeneous-variance ANOVA model with Type 3
tests of fixed effects (SAS, PROC MIXED). (B) Tests of treatment
levels (population density*individual variability; effect of slices).
variability cases, commercial yield decreased with
increasing stocking density and the differences
between high and low variability groups were
significant for all densities tested (Table 2B; effect
of slices; pb0.0001), commercial yield being systematically higher in high-variability groups than in lowvariability groups.
To assess the effect of interindividual variability
on mortality, we plotted the B–N trajectories and B–
N curves generated by a single run of the model
(Fig. 3). In the low variability groups, growth
proceeded without mortality in all density groups,
as judged from the vertical patterns of the B–N time
trajectories. At the end of the runs, total biomass
increased with population density to a maximum of
about 3.2 g (flesh dry mass) per chamber at N=20
mussels per chamber and increased slightly at higher
population density, as shown by the B–N curve. In
contrast, the B–N trajectories in the high variability
groups were vertical only in the N=10 mussels per
chamber group. At higher initial population density,
mortality occurred and the B–N trajectories were
bent toward the left as growth proceeded. The
strength of this pattern increased with initial population density. As in Fig. 2, maximum final biomass
was higher and more variable in the high variability
group than in the low variability group, except at
low population density (i.e., about 10 mussels/
chamber and less).
Fig. 4 shows the effect of individual variability
on the envelope of the B–N diagram (bounded by
the B–N curve and the ST curve) for a selected
subset of the r values tested, using the same a r
values as in Fig. 3. With low variability, maximum
biomass was about 3.8 g dry flesh mass at N=60.
The C–D region is smooth and the ST region is non
existent because of the absence of mortality. As
individual variability increases, so do maximum
biomass and roughness of the C–D region. In
addition, ST appears. At variability above r=10,
maximum biomass tends to decrease and the C–D
region is increasingly rough. With high individual
variability (e.g., r=30), maximum biomass is low,
the C–D region gets smoother and the ST region is
restricted to a limited part of the range of
population densities as nearly half the animals died
before the first sampling date.
4.2. Optimal stocking density
Mean OSD for total yield was 19.7 mussels per
chamber and 42.5 mussels per chamber for the low
(r=2) and high (r=10) individual variability groups,
respectively (Fig. 5). Variances were heterogeneous
(folded F method, s 2=0.2376, s 10=1.8961; F=63.68;
6
5
Total biomass
Table 2
Effect of population density and of individual variability on
commercial yield
215
4
3
2
1
0
0
20
40
60
80
Population density
Fig. 3. B–N time trajectories for mussel groups with low (triangles)
and high (squares) interindividual variability. Dotted lines indicate
B–N curves obtained for the various groups at the end of the
simulations.
216
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
s=2
Biomass
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
Biomass
40
60
80
0
20
40
6
5
5
4
4
3
3
2
2
1
1
60
80
s=20
7
6
0
0
20
40
60
80
s=10
7
Biomass
20
s=5
7
0
s=15
7
0
20
40
6
5
5
4
4
3
3
2
2
1
1
80
s=30
7
6
60
0
0
0
20
40
60
80
Population density
0
20
40
60
80
Population density
Fig. 4. Envelopes of the B–N curves as a function of interindividual variability, as measured by the standard error of the respiration
parameter a r.
df=99, 99; pb0.0001), therefore the test was made
using Satterthwaite’s method. The estimates of OSD
for low and high variability were significantly
different (t=11.95; df=102; pb0.0001). Mean
OSD for commercial yield was 1.0 mussel per
chamber and 2.8 mussels per chamber for the low
(r=2) and high (r=10) individual variability groups,
respectively. Variances were heterogeneous (folded F
method, s 2=0.0058, s 10=0.1590; F=752.35; df=99,
99; pb0.0001), therefore the test was made using
Satterthwaite’s method. The estimates of OSD for
low and high variability were significantly different
(t=10.81; df=99.3; pb0.0001).
4.3. Symmetric versus asymmetric competition
Variations of parameter u are shown in Fig. 6
(model parameters were the same as in Fig. 3). Two
patterns emerged. First, for a given a r, u values
increased with group size, indicating without surprise
that competition increased with population density.
Second, u values increased with increasing a r in all
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
Biomass
5
total
4
3
t0
2
comm
1
0
0
20
40
60
80
Initial population density
Fig. 5. Mussel yield (B–N curves) as a function of initial population
density. The curves are the mean of 100 runs of the model with
different sets of a r values. Solid curves and dashed curves stand for
the low and high variability groups, respectively. The oblique
straight line labelled t 0 is the B–N curve (total biomass) at the
beginning of the simulation. Total biomass and commercial biomass
are the two upper and lower curves, respectively. In the present case,
OSD for total yield is roughly N=19.7 for low variability groups and
N=42.5 for high variability groups. For commercial yield, OSD is
N=1.0 and N=2.8 for the low and high variability groups,
respectively.
groups. Therefore competition was asymmetric in all
cases. The absence of high a r values in the N=80,
high-variability group, indicates that slow-growing
animals had been removed by competition. Therefore
asymmetric competition affected both growth and
survivorship.
5. Literature review
To ascertain the extent to which size structurerelated biases are widespread in experimental procedures, we reviewed the methodology of all bivalve
stocking experiments published from Vol. 1 through
Vol. 224 of Aquaculture. Two of us made the review
independently. We examined all experiments that
included some form of study of stocking or densitydependent effects with individual size, individual
mass, biomass, population density or production as
response variables. We examined first, whether the
papers indicated that there had been size grading,
second, whether the spat used in the experiments
actually had been obtained by size-grading a larger set
of available spat, and third, whether size structure at
the beginning of stocking experiments was the same
as commercial size structure at the onset of culture, as
judged either from explicit information or from
unambiguous implicit considerations. Therefore, personal knowledge was left out. As some papers report
more than one experiment, we report the results in
terms of number of experiments, not number of
papers.
We found 31 studies of OSD of cultured bivalves
(Table 3). These include sessile epifauna (e.g.,
mussels), mobile epifauna (e.g., scallops) and endofauna (e.g., clams). A total of 41 experiments were
reported. Out of these 41 experiments, 17 gave
explicit information about size selection of the spat.
The spat was size-graded in 15 of these 17 experiments. None of the 17 accounts reported whether
experimental and commercial spat populations had
similar size structure. Regarding the 24 experiments
where information about spat selection was not given,
3 experiments were reported with sufficient methodological detail to conclude that there was no sizerelated bias in the experiment. It could not be judged
whether experimental and commercial size structures
were similar in the remaining 21 experiments. Overall,
commercial and experimental size structures were
deemed as similar in 3 of the 41 experiments found. In
the remaining studies, initial size structures were
deemed as different, either because it was indicated so
or because no information showed that similarity was
1.05
Competition strength
6
217
1.00
0.95
0.90
0.85
0.80
0.75
0
10
20
30
Respiration parameter (ar)
Fig. 6. Plot of the index of competition strength (u) as a function of
the respiration coefficient of individual mussels (a r). Empty
symbols and solid symbols are for the low variability and high
variability groups, respectively. Diamonds, squares and triangles are
for the low (N=20), intermediate (N=40) and high (N=80)
population density groups, respectively.
218
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
Table 3
Criteria for rating the papers listed in Appendix C and number of experiments of each category
1) Information given?
2) Size selection?
3) Science and industry size structures similar?
Y=17
Y=15
Y=0
N=15
N=24
N=2
Y=0
N=2 Y=3
N=21
The papers were published in Vol. 1 through Vol. 224 of Aquaculture. All papers which tested density effects either on individual size,
individual mass, total biomass, production or population density were included in the analysis. Because some papers report more than one
experiment, the numbers account for the various experiments, not the number of papers. The papers were examined according to the following
criteria: first, whether the papers indicated that there had been size grading (Y=yes, N=no), second, whether the spat used in the experiments
actually had been obtained by size-grading available spat, and third, whether size structure at the beginning of stocking experiments was similar
to commercial size structure at the onset of culture, as judged either from explicit information or from unambiguous implicit considerations; in
cases where indications were insufficient to allow firm conclusions, the answer was negative. Only bivalve papers are presented here. Other
groups will be considered elsewhere.
intended. There were some discrepancies in our initial
ratings of a few experiments, which were overcome
easily after discussing the grey zones in the accounts
of experimental methodology. Nonetheless, even after
discussion, two experiments (from a single paper) got
different ratings. We rallied to a common standpoint
about these experiments and present only a single set
of results. The references of these studies are given in
Appendix C.
6. Discussion
Reduced growth and increased mortality are two
basic consequences of competition. Therefore both
should be adequately measured in competition experiments. In B–N diagrams, reduced growth and mortality are ascertained from the C–D effect and ST,
respectively. In ideal situations, assuming negligible
measurement error, both patterns should potentially be
estimated with high accuracy. Our analysis, however,
shows that this was not the case. Estimates of the C–D
effect and of ST varied with the amount of individual
variability in bioenergetic efficiency (Figs. 2 and 3;
Sowka and Brunkow, 1999 found quite similar results
in cage-cultured bonytail G. elegans which were not
provided supplemental particulate food). Furthermore,
the ability to estimate competition effects on growth
on the one hand, and the ability to estimate effects on
mortality on the other hand were mutually exclusive.
Individual variability was required for estimating
survival effects, at the expense of precision on growth
effects (Figs. 2 and 4). Therefore B–N diagrams as
depicted in Fig. 1 are incomplete accounts of
competition/stocking experiments.
The concept of incompleteness refers to the
impossibility of specifying all aspects of a system
without invoking additional axioms or variables (see
Barrow, 1998). Including properties of individuals in
B–N diagrams should allow better accounts of
stocking experiments. This may be achieved by
using variables XV, XU. . . as covariates to remove
the uncertainty on individual growth performance.
An obvious candidate for variables XV, XU. . . is the
genetic makeup of individuals. Interindividual variability in metabolic efficiency has been related to
genetic variability (Koehn and Shumway, 1982; Toro
et al., 1996; Bayne et al., 1999). Here we assume that
epigenetic variability has negligible effects, although
it is not always the case, as found in studies of
developmental stability (see Polak, 2003) and of
variability of clonal animals (e.g., Archer et al.,
2003a,b). The use of covariates, however, implies
assumptions about the relationship between the
variables involved. In the present case, removing
incompleteness by using the genetic makeup of the
individuals as a covariate of a r would require that the
relationship between genetic makeup and a r be
known not only for density-independent situations,
but also for competing individuals possibly undergoing ST. Therefore all interactions between more or
less genetically different individuals, some of them
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
dying in the process, would have to be specified.
Between-genotype interactions do occur in ST, as
shown by the switch from spatially autocorrelated to
spatially random genetic structure in plants undergoing ST (Chung et al., 2003 and references therein).
A similar situation was found in a mussel experiment
where various family genotypes were grown in
different combinations (Brichette et al., 2001).
Growth of the various combinations of families
could not be predicted from their average performance (but see Blanc and Poisson, 2003 for a counter
example). Such results imply that the genetic outcome of ST is frequency-dependent. It is dependent
on initial proportions of the n genotypes present in an
experiment. Therefore, the use of genetic properties
as a covariate would require that not only main
effects be known, but also that all their possible
interactions, which amounts to 2n 1 possibilities, be
tested at all possible population density combinations. It may be attempted to move away from
incompleteness by actually running such multi-group
experiments. Considering the effect of two groups
only, a full-factorial design implies an experiment
three times as large as a single-group experiment. For
three-group or four-group populations, the experiment becomes respectively 7-fold and 15-fold larger.
In addition, with variations in the relative proportions
of groups, the experiment would be inflated even
more. This may be quite manageable for small n and
small-sized species not requiring complex culture
equipment. For fish and other species requiring large
or complex culture gear, however, such a task is
clearly out of practical range.
The present problem of initial size variability is in
some way analogous to that of comparing production
of multi-species groups to that of monospecific
cultures. Experimental strategies such as the replacement series or surface analysis may alleviate the
burden of work in such experiments, as compared to
the full-factorial design discussed above (Jolliffe,
2000). Indices are also available for such studies
(Garnier et al., 1997). The problem of multispecific
production may be seen as even simpler than that of
initial variability because individuals from different
species presumably can be assigned to their group
without error. The situation is different in the problem
of initial variability, where presumably assessing
individual differences usually can be achieved only
219
at the expense of intrusive or destructive genetic
analyses (note that in some cases, however, genetic
information may be gained using non-destructive
methods such as hemolymph sampling, Yanick and
Heath, 2000; ecogenomics may be a useful approach,
but then it would be limited to hatchery-produced
spat, Dicke et al., 2004). Based on the foregoing
arguments, we believe that incompleteness is pervasive in competition experiments. In the present state of
(our) knowledge, it further seems that it is inescapable
since it is a property of the experiment itself. In
addition, it should be pointed out that results of model
runs (not shown) using isometric (as opposed to
allometric) relationships between bioenergetic rates
and body mass were similar to those in Fig. 3,
although in this case mortality did occur in both
groups.
The practical consequences of incompleteness
vary depending on experimenters’ decisions. It is
clear that maximum yield varies with interindividual
variability (Fig. 2). If interindividual variability is
too low, experiments will fail to reveal actual
mortality patterns (Fig. 3). Furthermore, the level
of interindividual variability affects OSD estimates,
which were lower with low (r=2) interindividual
variability, compared with those with high (r=10)
individual variability (Fig. 5). Maximum yield and
OSD estimates are basic aspects of biological advice
to aquaculturists. According to our literature review,
the issue of size structure has been overlooked in a
large proportion (nearly 3/5) of bivalve stocking
experiments (Table 1). Furthermore, inadequate
population size structure is possibly a serious
concern in roughly 9 out of 10 experiments. The
data in Table 1 were obtained from a rather rigid
analysis. For instance, based on a strict application
of the questionnaire summarised in Table 1, Experiment A in Fréchette et al. (1996) was attributed the
ratings Yes, Yes and No to the first, second and third
questions, respectively, because the paper stated that
the spat was obtained using a declumper/grader, but
there was no mention that experimental and commercial spat size structures had actually been tested
and no differences found. Based on personal knowledge, however, the paper should have earned the
ratings Yes, Yes and Yes. Other cases of this sort
probably occurred. Therefore our analysis of the
literature may be biased toward the rating bNoQ to
220
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
the third question to some extent, although the issue
of failing to provide information about initial size
structure unquestionably remains.
A further aspect related to the question of spat-size
selection is that given unlimited spat supply, aquaculturists would of course select high performers, in
an attempt to boost maximum production. Simulations
not shown here suggest that maximum yield would
indeed increase accordingly. Therefore we are not
suggesting that variability in spat be increased with
the purpose of increasing yield. Maximum biomass
was higher at r=10 than at r=2 only because higher
variability allowed to include better performers than
the baverageQ individual obtained with low variability.
Increasing interindividual variability to higher levels
(e.g., r=30) resulted in depressed yield because of
high mortality which caused total production to be
limited by the reduced number of survivors. In
addition, our analysis shows that for the time being,
the consequences of incompleteness of competition/
stocking experiments cannot be alleviated by the use
of covariates. It appears, therefore, that the only
practical way to avoid biased advice is to use
experimental sizes similar to those of cultured
populations. A related point is the necessity to repeat
stocking experiments if actual size structure of spat
changes through time. This would occur if spat
availability varied from year to year and growers
adjusted the size structure of commercial spat to take
advantage of the abundance of preferred sizes.
Although these considerations may sound as truisms,
they were overlooked in a large proportion of the
studies we reviewed (Table 1). A similar situation is
encountered in studies of competition in non-cultured
bivalve populations. In some cases, experimenters
maintained size structure in their plots as similar as
possible to that of the natural population (e.g.,
Peterson, 1982). In other cases, however, it was
preferred to select sizes in order to limit individual
variability in an effort to reduce measurement error on
growth (Alunno-Bruscia et al., 2000; Beal et al., 2001;
Petraitis, 2002). Therefore the issue of incompleteness
and its practical consequences is not restricted to
aquaculture research.
Intraspecific competition in bivalves has been
reported to be asymmetric in many instances (Bertness and Grosholz, 1985; Montaudouin and Bachelet, 1996; Fréchette et al., 2000; Brichette et al.,
2001). Asymmetric competition in bivalves has been
thought to require some form of interference
(Peterson, 1982; Fréchette and Bourget, 1985).
Brichette et al. (2001) performed an extensive study
of mussel growth in a design that allowed quantifying the relative importance of symmetric and
asymmetric modes of competition. They found a
mixture of both effects, which might be attributed to
the presence of more than one competition mechanism (in mussels, possibly impaired shell gaping and
food regulation: Fréchette and Despland, 1999). Joint
asymmetric and symmetric components of competition have also been found in plants and interpreted
as reflecting the action of more than one competition
mechanism (Thomas and Weiner, 1989). Our simulations, however, suggest that competition may have
both asymmetric and symmetric components,
although only a single competition mechanism is
present, i.e., food regulation. Apparently symmetric
effects were brought about by differences in group
size while asymmetric effects were caused by
variability in bioenergetic efficiency.
Other simulation results (not shown here) predict
that B–N trajectories of identical plants or animals
would behave in an all-or-nothing fashion, with all
individuals surviving or all dying simultaneously.
These results are quite similar to those obtained
experimentally by Puntieri (1993, Fig. 6) with
populations of the plant Galium aparine which did
not exhibit size hierarchy. In these G. aparine
populations, B–N time trajectories increased vertically
to a point where they bifurcated abruptly toward the
origin of the B–N space instead of moving toward a
classical ST curve. This reveals that large proportions
of the populations died at once, instead of progressively as found in classical ST. Therefore, our model
results were consistent with both typical and atypical
ST patterns. This suggests that the concept of
incompleteness of competition experiments is amenable to experimental testing.
Acknowledgements
We thank P. Petraitis, F. Guichard and J.-M.
Sévigny for critical discussions. This study was
supported by the authors’ institutions and Région
Poitou-Charente, France, and IFREMER.
M. Fréchette et al. / Aquaculture 246 (2005) 209–225
Appendix A. List of model parameters and
variables
Parameter
Symbol Value
Units
Volume of
reservoirs
Water flow rate
Water
temperature
Phytoplankton
energy density
V
10
l
r
T
432
14.0
Initial mussel
soft tissue
mass
Filtering
coefficient
Filtering
exponent
Assimilation
efficiency
Respiration
exponent for
mass
Respiration
exponent for
temperature
Average
respiration
coefficient
Critical thinning
factor
Coefficient of
length–mass
relationship
Exponent of
length–mass
relationship
Population
density
Individual size
(shell length)
Individual mussel
soft tissue mass
m0
q
af
10.4
Jl
1
50.88
0.408
ea
0.85
b
0.844
c
1.358
N0 ¼
ð
g ð1 k Þ 0:058 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ gð1 k Þ 0:0582 þ 0:116ðg 0:029Þ
0:058
Þ
ð1=kÞ
ð2AÞ
(Alunno-Bruscia
et al., 2000)
(Bayne and
Widdows, 1978;
in Thompson,
1984)
l
day1
(Fréchette and
Bacher, 1998)
(Fréchette and
Bacher, 1998)
(Fréchette and
Bacher, 1998)
(Fréchette and
Bacher, 1998)
(Fréchette and
Bacher, 1998)
13.584 J
day1
(Fréchette and
Bacher, 1998)
n
0.66
a
0.003
Alunno-Bruscia,
unpublished
b
3
Alunno-Bruscia,
unpublished
N
and solve Eq. (1A) for N 0 with the condition BB/
BN=0.029. The solution is
N 0 is an estimate of OSD.
Appendix C. List of papers used for the analysis of
published bivalve stocking experiments
0.029 g
a
ar
l day1
8C
Source
221
L
mussels
reservoir1
cm
m
g
Appendix B. Estimation of OSD
To estimate OSD, we use the derivative of B=gN/
(1+N k ), which is
g 1 þ ð1 k ÞN k
BB=BN ¼
ð1AÞ
2
ð1 þ N k Þ
Beal, B.F., Kraus, M.G., 2002. Interactive effects
of initial size, stocking density, and type of predator
deterrent netting on survival and growth of cultured
juveniles of the soft-shell clam, Mya arenaria L., in
eastern Maine. Aquaculture 208, 81–111.
Beal, B.F., Lithgow, C.D., Shaw, D.P., Renshaw,
S., Ouellette, D., 1995. Overwintering hatchery-reared
individuals of the soft-shell clam, Mya arenaria L.: a
field test of site, clam size and intraspecific density.
Aquaculture 130, 145–158.
Dare, P.J., Davies, G., 1975. Experimental suspended culture of mussels (Mytilus edulis L.) in Wales
using spat transplanted from a distant settlement
ground. Aquaculture 6, 257–274.
Ford, S.E., Kraeuter, J.N., Barber, R.D., Mathis,
G., 2002. Aquaculture-associated factors in QPX
disease of hard clams: density and seed source.
Aquaculture 208, 23–38.
Fréchette, M., Bergeron, P., Gagnon, P., 1996. On
the use of self-thinning relationships in stocking
experiments. Aquaculture 145, 91–112.
Fréchette, M., Gaudet, M., Vigneau, S., 2000.
Estimating optimal population density for intermediate culture of scallops in spat collector bags. Aquaculture 183, 105–124.
Fuentes, J., Gregorio, V., Giráldez, R., Molares,
J., 2000. Within-raft variability of the growth rate
of mussels, Mytilus galloprovincialis, cultivated in
the Rı́a de Arousa (NW Spain). Aquaculture 189,
39–52.
González, M.L., López, D.A., Pérez, M.C.,
Riquelme, V.A., Uribe, J.M., Le Pennec, M., 1999.
Growth of the scallop, Argopecten purpuratus (Lamarck, 1819), in southern Chile. Aquaculture 175,
307–316.
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Grice, A.M., Bell, J.D., 1999. Application of
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Hadley, N.H., Manzi, J.J., 1984. Growth of seed
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Holliday, J.E., Maguire, G.B., Nell, J.A., 1991.
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Sydney rock oysters (Saccostrea commercialis).
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Holliday, J.E., Allan, G.L., Nell, J.A., 1993.
Effects of stocking density of juvenile Sydney rock
oysters, Saccostrea commercialis (Iredale & Roughley), in cylinders. Aquaculture 109, 13–26.
Honkoop, P.J.C., Bayne, B.L., 2002. Stocking
density and growth of the Pacific oyster (Crassostrea
gigas) and the Sydney rock oyster (Saccostrea
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213, 171–186.
Jeng, S.S., Tyan, Y.M., 1982. Growth of the hard
clam Meretrix lusoria in Taiwan. Aquaculture 27,
19–28.
Laing, I., Millican, P.F., 1991. Dried-algae diets
and indoor nursery cultivation of Manila clam
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Maeda-Martı́nez, A.N., Reynoso-Granados, T.,
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Maguire, J.A., Burnell, G.M., 2001. The effect of
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carbohydrate content of the adductor muscle in two
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Manzi, J.J., Hadley, N.H., Maddox, M.B., 1986.
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Parsons, J., 1974. Advantages in tray cultivation of
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Parsons, J.G., Dadswell, M.J., 1992. Effect of
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Paul, J.D., Brand, A.R., Hoogesteger, J.N., 1981.
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Román, G., Campos, M.J., Acosta, C.P., Cano, J.,
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Rose, R.A., Baker, S.B., 1994. Larval and spat
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Sukhotin, A.A., Kulakowski, E.E., 1992. Growth
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Taylor, J.J., Rose, R.A., Southgate, P.C., Taylor,
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Toro, J.E., Sanhueza, M.A., Winter, J.E., Senn,
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Wada, K.T., Komaru, A., 1994. Effect of selection
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Walne, P.R., 1976. Experiments on the culture in
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