Absolute and relative growth in five Pinna nobilis

Marine Biology 152: 537-548
The original is available at www.springerlink.com
DOI 10.1007/s00227-007-0707-z
Comparison of absolute and relative growth patterns among five Pinna nobilis
populations along the Tunisian coastline: an information theory approach
Lotfi Rabaoui a, Sabiha Tlig Zouari a, Stelios Katsanevakis b* and Oum Kalthoum Ben
Hassine a
a
Research Unit of Biology, Ecology and Parasitology of Aquatic Organisms, Department of Biology,
Faculty of Sciences of Tunis, University campus, El Manar 2092, Tunis, Tunisia
b
Department of Zoology-Marine Biology, Faculty of Biology, University of Athens,
Panepistimioupolis, 15784 Athens, Greece
* corresponding author
Department of Zoology-Marine Biology, Faculty of Biology, University of Athens,
Panepistimioupolis, 15784 Athens, Greece
e-mail: stelios@katsanevakis.com
tel: +30-210-7274634 fax: +30-210-7274608
1
Abstract
The variability in absolute and relative growth of Pinna nobilis along the Tunisian
coastline was investigated. Five populations of P. nobilis were sampled, three from
northern and two from eastern Tunisia. The specimens were aged, and ten
morphometric characters were measured on each individual. To test if differences
existed in absolute and relative growth patterns among the different populations an
information theory approach was followed. For absolute growth, von Bertalanffy,
Gompertz, the logistic and the power models were fitted in combination with three
assumptions regarding inter-population differences in absolute growth patterns: no
differences, differences among all five populations or just between northern and
eastern populations. The assumption of common absolute growth parameters among
all five populations had the greatest support by the data, while the assumption of
different growth patterns among all five populations had no support. Von Bertalanffy
growth model and the power model were both equally supported by the data (while
Gompertz had considerably less support and the logistic model had no support), and
thus it may not be definitely concluded whether P. nobilis grows asymptotically or
not. The P. nobilis populations of the Tunisian coastline had a slow growth and up to
an age of ~7.5 yr their shells were smaller than from all other reported populations in
the Mediterranean. For relative growth, apart from the classical allometric model
Y = aX b , relating the size of a part of the body Y and another reference dimension X,
more complicated models were used in combination with the three abovementioned
assumptions regarding inter-population differences. Those models, of the form
log Y = f (log X ) , either assumed breakpoints in the relative growth trajectories or
non-linearities. For most morphometric characters, the classical allometric model had
no support by the data and more complicated models were necessary. In most cases,
the relative growth patterns differed either among all five populations or between the
northern and eastern population groups. Further investigation is needed to relate the
morphological differences observed among different populations of P. nobilis to
environmental factors.
Introduction
The fan mussel Pinna nobilis is endemic to the Mediterranean Sea. It is one of
the largest bivalves of the world, attaining lengths up to 120 cm (Zavodnik et al.
1991). It is long lived, living up to 20 years according to Butler et al. (1993), while in
Thermaikos Gulf (Greece) an age of 27 years has been reported (Galinou-Mitsoudi et
al. 2006). It has very variable recruitment (Butler et al. 1993), and occurs at depths
between 0.5 and 60 m, mostly in soft-bottom areas overgrown by meadows of the
seagrasses Posidonia oceanica, Cymodocea nodosa, Zostera marina or Zostera noltii
(Zavodnik et al. 1991) but also in bare sandy bottoms (Katsanevakis 2006a, 2007).
The population of P. nobilis has been greatly reduced during the last few
decades as a result of recreational and commercial fishing for food, use of its shell for
decorative purposes, and incidental killing by trawling and anchoring. In the European
Union, it has been listed as an endangered species and is under strict protection
according to the European Council Directive 92/43/EEC. However, it still suffers
illegal fishing (Katsanevakis 2007). Our knowledge on the biology and ecology of the
species is fragmentary and several aspects need further investigation (Butler et al.
1993; Ramos 1998; García-March et al. 2007a). To effectively protect this species
2
there is a pressing need for better information on its biology and the status of all major
local P. nobilis populations. There is no published study on the ecology and status of
the species in the Tunisian coastline as yet.
Individual growth in aquatic invertebrates is a process rooted in physiological
processes and is the net result of two opposing processes, catabolism and anabolism
(Bertalanffy 1938). For population analysis, a mathematical expression of the mean
individual body growth is needed, relating the size of the species to its age. Several
models have been proposed to estimate the mean individual growth in a population,
some of these are based on purely empirical relationships, while others have a
theoretical basis and are arrived at by differential equations that link the anabolic and
catabolic processes. The most studied and commonly applied model among all the
length-age models is the von Bertalanffy growth model (VBGM) (Bertalanffy 1938).
However, the practise of a priori using VBGM has often been criticized (e.g.
Katsanevakis 2006b), and for many aquatic species other models like Gompertz
(Gompertz 1825) or the logistic model (Ricker 1975) better describe absolute growth.
All the above models assume asymptotic growth but some aquatic invertebrates, like
cephalopods, seem not to grow asymptotically (e.g. Jackson and Choat 1992) and
non-asymptotic models, like the power model, have been proposed in these cases.
Growth is often accompanied by changes in proportion as well as in size, the
phenomenon of relative or allometric growth. The allometric equation (Huxley 1932)
is the most extensively used model for relative growth during ontogeny; the
relationship between the size of a part of the body Y and another part X (taken as a
reference dimension) has the form Y = aX b , where the exponent b is a measure of the
difference in the growth rates of the two parts of the body. However, the classic
allometric equation frequently fails to adequately fit the data and more complex
models of the form log Y = f (log X ) should be used (Katsanevakis et al. 2007a). The
reason might be either the existence of non-linearity, i.e. f is non-linear, or the
existence of breakpoints, i.e. f and/or its first derivative f ′ are not continuous
functions. The existence of breakpoints in allometric data has been recognized since
the allometric equation was first proposed (Huxley 1932). Such changes in the growth
trajectories of morphological characters during ontogeny are a potentially useful
source of information as they may be caused by marked events in the life history of
the species or fast ecological change, and should not be overlooked.
Model selection based on information theory is a relatively new paradigm in the
biological sciences and is quite different from the usual methods based on null
hypothesis testing. Information theory has been increasingly proposed as a better and
advantageous alternative for model selection (Burnham and Anderson 2002), e.g. in
studies of fish growth (Katsanevakis 2006b), allometric growth of marine species
(Katsanevakis et al. 2007a; Protopapas et al. 2007), and aquatic respiration
(Katsanevakis et al. 2007b). The information theory approach was recommended as a
more accurate, robust and enlightening way to study relative growth of marine species
(Katsanevakis et al. 2007a). It was demonstrated that the use of the classical
allometric model when it is not supported by the data, might lead to characteristic
pitfalls, data misinterpretation, and loss of valuable biological information.
According to the information theory approach, data analysis is taken to mean the
integrated process of a priori specification of a set of candidate models (based on the
science of the problem), model selection based on the principle of parsimony, and the
estimation of parameters and their precision. The principle of parsimony implies the
selection of a model with the smallest possible number of parameters for adequate
representation of the data, a bias versus variance tradeoff. When the data support
3
evidence of more than one model, model averaging the predicted response variable
across models is advantageous in reaching a robust inference that is not conditional on
a single model. Rather than estimating parameters from only the ‘best’ model,
parameter estimation can be made from several or even all the models considered.
This procedure is termed multi-model inference (MMI) and has several theoretical
and practical advantages (Burnham and Anderson 2002).
In the present study, the absolute and relative growth of five populations of P.
nobilis in the Tunisian coastal waters was studied based on an information theory
approach. Several models were defined a priori based on biological assumptions and
it was investigated whether important differences in absolute or relative growth
patterns existed among the five P. nobilis populations.
Materials and Methods
Study area – Sampling
Thirty specimens of P. nobilis were sampled from each of five populations,
three from the north coast (Bizerte Lagoon and Gulf of Tunis) and two from the east
coast (Monastir Bay) of Tunisia (Fig. 1). All samples were randomly taken with free
diving from shallow areas at depths between 1.5 and 3 m. All the habitats of the five
populations were seagrass meadows: Cymodocea nodosa in N1, N2, N3, and E1, and
Posidonia oceanica in E2. On each specimen ten morphometric characters were
measured as in Fig. 2. Six of the morphometric characters (Lp, La, Wp, Wa, L1, L2) were
measured both in the right and left valve and the average value was used in the
analysis. Age was determined by counting the number of adductor-muscle scar rings
on the shells. Because the first year’s muscle-scar ring is either absent or
inconspicuous (Richardson et al. 1999), the age was estimated as the number of rings
plus 1.
Absolute growth
A set of twelve candidate models was used to model absolute growth of the five
P. nobilis populations (gi, i = 1 to 12), i.e. the relationship between L and age t (Table
1). In models g1 to g4 it was assumed that there was no difference in absolute growth
among the five populations, i.e. all five populations had common growth parameters.
In models g5 to g8 it was assumed that the growth parameters differed between the
three northern (N1, N2, and N3) and the two eastern (E1 and E2) populations. In
models g9 to g12 it was assumed that all five populations had different growth patterns.
The von Bertalanffy growth model L(t ) = L∞ 1 − e − k1 (t −t1 ) (Bertalanffy 1938) was used
in models g1, g5, and g9, the Gompertz growth model L(t ) = L∞ exp − e − k2 (t −t2 )
(
)
(
(
)
)
− k3 ( t −t3 ) −1
(Gompertz 1825) in g2, g6, and g10, the logistic model L(t ) = L∞ 1 + e
(Ricker
c2
1975) in g3, g7, and g11, and the power model L(t ) = c0 + c1 ⋅ t in g4, g8, and g12
(Table 1); L∞ (asymptotic length), ki, ti, and ci are estimable regression parameters.
Details on the underlying principles, definition of parameters appearing in the
equations, and mathematical description of the corresponding curves are given in
detail in Katsanevakis (2006b).
4
Relative growth
The allometric growth of W, D, T, Wp, Wa, Lp, La, L1, and L2 in relation to L was
investigated. Twelve candidate models (fi, i = 1 to 12) for the relationship
ln Y = f (ln L ) were fitted to the ln-transformed data, where Y is any of the measured
morphometric characters. In models f1 to f4 it was assumed that there was no
difference in relative growth among the five populations, in models f5 to f8 that the
relative growth parameters differed between the three northern and the two eastern
populations, and in models f9 to f12 that all five populations had different relative
growth parameters. The linear model (L), ln Y = a1 + b1 ln L , was used in f1, f5, and f9,
the quadratic model (Q), ln Y = a1 + b1 ln L + b2 (ln L) 2 , in f2, f6, and f10, the brokena + b ln L, L ≤ B
stick (BS), ln Y =  1 1
, in f3, f7, and f11, and the twoa1 + (b1 − b2 ) ln B + b2 ln L, L > B
a + b ln L, L ≤ B
, in f4, f8, and f12 (Table 2).
segment model (TS), ln Y =  1 1
a2 + b2 ln L, L > B
In the current context, the allometric exponent b was generalized to mean the
first derivative of f with respect to lnL, i.e. b = f ′(ln L) , according to Katsanevakis et
al. (2007a). The L model is the classical allometric equation, assuming that allometry
does not change as body size increases (b = b1 = constant). The Q model assumes that
a non-linearity exists in the relationship of lnY and lnL and that b changes
continuously with increasing body size (b = b1 + 2b2lnL). The BS and TS models
assume a marked morphological change at a specific size of L = B; the BS represents
two straight line segments with different slope that intersect at L = B, while the TS
represents two straight line segments that do not intersect; thus, there is a point of
discontinuity at L = B, and the slope of the two segments (i.e., b) may or may not be
equal.
Model fitting – Model Selection - MMI
The candidate models gi for absolute growth were fitted with non-linear least
squares with iterations by means of Marquardt’s algorithm, assuming additive error
structure. The L model was fitted with simple linear regression, while polynomial
regression was used for the Q model. To fit the BS and TS models, the breakpoint B
was allowed to vary between the minimum and maximum value of L with a
sufficiently small step. For each value of the breakpoint, two separate lines were fitted
with linear regression to the data before and after the breakpoint (independent lines in
the case of TS or connected lines at the breakpoint in the case of BS) and the
corresponding residual sum of squares (RSS) was calculated as the sum of the two
RSS for the two lines; this was done automatically in MsExcel by what-if analysis
(one variable data table). The value of the breakpoint that gave the minimum RSS was
found and the corresponding model parameters were estimated.
The small-sample, bias-corrected form AICc (Hurvich and Tsai 1989) of the
AIC (Akaike 1973; Burnham and Anderson 2002) was used for model selection.
2k (k + 1)
Specifically,
AICc = AIC +
,
where
for
least
squares
n − k −1
RSS


AIC = n log(2π
) + 1 + 2k , RSS is the residual sum of squares, n the number of
n


5
observations, and k is the number of regression parameters plus 1. The model with the
smallest AICc value (AICc,min) was selected as the ‘best’ among the models tested.
The AICc differences, ∆ i = AICc ,i − AICc ,min were computed over all candidate
models gi. According to Burnham and Anderson (2002), models with ∆i > 10 have
essentially no support and might be omitted from further consideration, models with
∆i < 2 have substantial support, while there is considerably less support for models
with 4 < ∆i < 7. To quantify the plausibility of each model, given the data and the set
of five models, the ‘Akaike weight’ wi of each model was calculated, where
exp(−0.5∆ i )
wi = 5
. The ‘Akaike weight’ is considered as the weight of evidence in
∑ exp(−0.5∆ k )
k =1
favor of model i being the actual best model of the available set of models (Akaike
1983; Buckland et al. 1997; Burnham and Anderson 2002). Akaike weights may be
interpreted as a posterior probability distribution over the model set. To obtain more
robust inferences, the final results were based on model-averaging the response
variable using Akaike weights, rather than simply on the ‘best’ model (Burnham and
Anderson 2002).
Results
Absolute growth
For each candidate model, RSS, AICc, ∆i, and wi were calculated (Table 3).
Models g1 and g4 were both substantially supported by the data, while all other models
had considerably less or no support. These best models were:
g1 : L(t ) = 104.3 ⋅ (1 − e −0.0526( t+ 0.286) ) (t in yr, L in cm)
g 4 : L(t ) = 10.31⋅ t 0.671 − 4.90 (t in yr, L in cm)
The models that assumed different absolute growth parameters among all five
populations (g9 to g12) had a sum of Akaike weights of only 0.1%, thus this
assumption had no support by the data. The models that assumed differences in
growth parameters between the northern and eastern population groups (g5 to g8) had
a sum of Akaike weights of 16.0%, while models assuming that all five populations
had common growth parameters had a sum of Akaike weights of 84.0% (Table 3).
Hence, the data indicated that the assumption of common growth parameters among
the five populations is the most plausible but without conclusively rejecting the
assumption of a different growth pattern between northern and eastern populations.
The VBGM assumes asymptotic growth, while the power model assumes nonasymptotic growth. The models assuming non-asymptotic growth (g4, g8, and g12) had
a sum of Akaike weights of 49.3%, while the models that assumed asymptotic growth
(the rest of the models) had a sum of Akaike weights of 50.7%. Hence, from the
dataset of the present study, it may not be concluded whether the growth of P. nobilis
may be considered asymptotic or non-asymptotic. The size-at-age raw data and the
average model are shown in Fig. 3.
Relative growth
6
For each morphometric character and for each model, RSS, AICc, ∆i, and wi
were calculated (Table 4). The assumption that all populations had a common relative
growth pattern was substantially supported by the data for Wp (f3 and f4 were the two
best models with a sum of Akaike weights of 99.1%) and also had some support for
W, without being the best alternative (f2 that assumed a common relative growth
pattern had a wi of 20.3%, while f6 and f7 that assumed different patterns among
northern and eastern population groups had a sum of wi of 73.1%). The assumption of
different patterns among all five populations was the best alternative in four out of
nine cases (for D, T, La, and L2), while the assumption of different relative growth
patterns between the northern and eastern population groups was the best alternative
in other four out of nine cases (for W, Wa, L1, and Lp). Models assuming a linear
relationship between the ln-transformed morphometric variables had substantial
support by the data in only two cases (La and L2), while in all other cases more
complicated models were supported. The raw data and the average models for all
morphometric characters are shown in Fig. 4.
The relative growth of W was positive allometric (b > 1) up to a size (L) of ~18
cm and then became negative allometric (b < 1) with continuously decreasing b (Fig.
5). Thus, P. nobilis shell widens up to an age of ~3.5 yr (Fig. 3) and then for the rest
of its life its shape becomes more and more elongated. For the relative growth of T, b
was always <1 (Fig. 5); hence, the shell of P. nobilis becomes relatively thinner with
age. The largest variability in relative growth patterns was found for D and T (Figs 4,
5). The adductor muscle scars seemed to grow in a similar pattern to the outer shell.
The relative growth of the vertical dimensions (Lp and La) was close to isometry, i.e.
growing analogously to L, while the horizontal dimensions (Wp and Wa) grew initially
isometrically or with positive allometry but then their relative growth slowed down or
even diminished (Figs 4, 5).
Discussion
The usual approach when studying absolute growth in marine species is to a
priori adopt the VBGM, which is not a good practise for inference and robust
predictions (Katsanevakis 2006b). To our knowledge, all other studies on P. nobilis
growth (e.g. Moreteau and Vicente 1982; Richardson et al. 1999; Šiletić and Peharda
2003; Richardson et al. 2004; Galinou-Mitsoudi et al. 2006; García-March et al.
2007a) a priori used VBGM and did not check other non-asymptotic models or
whether growth of P. nobilis may be considered non-asymptotic. This may have
implications in the accuracy and precision of the estimated growth parameters (e.g.
asymptotic length), especially when based on a bad model. Furthermore, growth
parameters of VBGM are quite imprecise when estimated based on a dataset without
available large sizes close to the asymptotic length, which is sometimes the case in
reported results. For these reasons it is better to compare the curves of the reported
growth models within the range of sizes encountered in each dataset than to compare
the estimated parameters.
Comparing absolute growth of different P. nobilis populations (Fig. 6), large
variability in reported patterns is observed. Some P. nobilis populations seem to reach
an asymptote in their growth at quite small sizes and early in their life, while others
grow much larger becoming more than double in size. Even among individuals of the
same population, depth-related differences in absolute growth were found (Fig. 6;
García-March et al. 2007a). The P. nobilis populations of the Tunisian coast had a
7
slow growth and up to an age of ~7.5 yr their shells were smaller than all other
reported populations (Fig. 6). The oldest individual found in this study was 15 yr old
and at that age absolute growth seemed not to be close to a size plateau. At this age
the size of P. nobilis in the Tunisian coast was larger than the asymptotic length of
many other populations (Fig. 6). If based on VBGM (with the reservations stated
before), the asymptotic length of the Tunisian populations ( L∞ = 104.3 ) is the highest
among all reported values (Fig. 6).
When the classical allometric model is a priori ‘picked’ to study allometric
growth instead of adopting an information-theory approach, false conclusions may be
reached by ‘smoothing’ the true pattern, a large part of information could be lost, and
there is a serious risk is to judge the type of allometry wrongly, e.g. concluding
positive allometry when there is actually negative or vice versa (Katsanevakis et al.
2007a; Protopapas et al. 2007). For example (Katsanevakis et al. 2007a), Pinna
nobilis in Lake Vouliagmeni (Greece) exhibited a marked change in the relative
growth of width in relation to length; initially there was strong positive allometry
which after a length of ~20 cm became strongly negative (Fig. 7). The classical
allometric model ‘smoothed’ this picture and derived a constant allometric exponent
with a 95% confidence interval between 0.90 and 1.03, supporting isometric growth
during ontogeny and thus reaching a quite different conclusion.
Unfortunately, apart from the abovementioned study, information theory and
multi-model inference has not been used in the study of relative growth in other P.
nobilis populations and so there are no much relevant data to be compared with those
of this study. Katsanevakis et al. (2007a) reported only the relative growth of W in
relation to L and found a similar pattern to that found in the present study, i.e. a shift
from positive to negative allometry at a similar size (Fig. 7).
Several other methods are frequently used to detect morphological
differentiation among different populations of a species (for a review see Cadrin
2000). To detect such differences in morphology, it is critical to separate differences
in body size from differences in size-corrected morphology. Most studies of among
population variation in morphology either use analysis of covariance (ANCOVA)
with a univariate measure of body size as the covariate or compare residuals from
ordinary least squares regression of each morphological character against body size or
the first principal component of the pooled data (shearing). Both approaches have
been criticized and McCoy et al. (2006) proposed a better alternative based on
common principal components analysis combined with Burnaby’s back-projection
method (Burnaby 1966). A normalization technique to scale morphometric data to a
common size, adjusting their shape according to allometry has also been proposed
(Thorpe 1975, 1976; Lleonart et al. 2000). However, all the abovementioned
approaches assume the validity of Huxley’s classical allometric equation, i.e. linearity
between the log-transformed morphometric characters. Principal components analysis
(PCA) inefficiently summarizes non-linear patterns (Hopkins 1966; Somers 1986) and
the results of principal-component methods should be interpreted with caution
(Hopkins 1966; Somers 1986; Houle et al. 2002; McCoy et al. 2006). It has been
shown that in many cases the classical allometric equation is inappropriate and more
complicated models should be used (Hall et al. 2006; Katsanevakis et al. 2007a;
Protopapas et al. 2007; present study), thus this basic underlying assumption of the
above commonly used approaches is often invalid. The multi-model informationtheory approach used in the present context does not make such an assumption and is
thus a good alternative when the classical allometric equation is not supported by the
data.
8
For comparison, a correlation-based PCA (i.e. on normalised data) of the pooled
log-transformed morphometric data of the five populations was conducted and the
graph of the first two principal components is given in Fig. 8. The first two
components explained 95.2% of the variability in the data. The ten morphometric
characters were almost equally weighted in the first principal component (PC1) with
weights varying between 0.30 and 0.33, while PC2 was bipolar with unequal
weightings that varied between –0.40 and 0.67. PC1 may be considered to represent
overall size, while PC2 may be interpreted as a ‘shape component’ (Somers 1986;
Cadrin 2000). PC2 effectively separated the five populations (ANOVA, p < 0.0001)
and a Student-Neuman-Keuls multiple-range test separated three homogenous groups:
N1+N2, N3, E1+E2. This separation matches with the geographical separation of the
five populations (Fig. 1). These results generally agree with those obtained with the
information-theory approach but they should be viewed with caution as explained
above.
It seems that in both absolute and relative growth of P. nobilis there is large
variability in the patterns among different populations (Figs 4, 5, 6, 7). Great
differences in growth patterns have also been reported between individuals of the
same population that are subject to different environmental conditions (García-March
et al. 2007a). Food availability, temperature, upwelling intensity, sediment type, and
hydrodynamics are among the factors that have been reported to affect the absolute
growth of several bivalve species (e.g. Newell and Hidu 1982; Steffani and Branch
2003; Ackerman and Nishizaki 2004; Philips 2005). Specifically for P. nobilis, severe
sediment disturbance and high hydrodynamic stress have been proposed as causes for
reduced growth (García-March et al. 2007b). Effect of depth, hydrodynamics,
predators, temperature, and sediment type on relative growth and shell morphology
has been found before for many bivalves (e.g. Seed 1980; Newell and Hidu 1982;
Hinch and Bailey 1988; Akester and Martel 2000; Steffani and Branch 2003). Further
studies are needed to relate the observed variability in absolute and relative growth of
P. nobilis to specific factors and uncover the ways in which the environment may
affect its growth pattern.
Acknowledgements
We wish to acknowledge the suggestions and comments of two anonymous reviewers,
which helped to improve the quality of the manuscript. This survey was conducted in
compliance with the current laws of Tunisia.
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11
Table 1: The set of candidate models used for the analysis of absolute growth. k is the
total number of estimable parameters (regression parameters plus 1).
Model
g1
Description
Bertalanffy,common parameters for all 5 populations
k
4
g2
Gompertz, common parameters for all 5 populations
4
g3
Logistic, common parameters for all 5 populations
4
g4
Power, common parameters for all 5 populations
4
g5
Bertalanffy, different parameters for north and east population groups
7
g6
Gompertz, different parameters for north and east population groups
7
g7
Logistic, different parameters for north and east population groups
7
g8
Power, different parameters for north and east population groups
7
g9
Bertalanffy, different parameters for each of the five populations
16
g 10
Gompertz, different parameters for each of the five populations
16
g 11
Logistic, different parameters for each of the five populations
16
g 12
Power, different parameters for each of the five populations
16
Table 2: The set of candidate models used for the analysis of relative growth. k is the
total number of estimable parameters (regression parameters plus 1).
Model
f1
Description
L model,common parameters for all 5 populations
k
3
f2
Q model, common parameters for all 5 populations
4
f3
BS model, common parameters for all 5 populations
5
f4
TS model, common parameters for all 5 populations
6
f5
L model, different parameters for north and east population groups
5
f6
Q model, different parameters for north and east population groups
7
f7
BS model, different parameters for north and east population groups
9
f8
TS model, different parameters for north and east population groups
11
f9
L model, different parameters for each of the five populations
11
f 10
Q model, different parameters for each of the five populations
16
f 11
BS model, different parameters for each of the five populations
21
f 12
TS model, different parameters for each of the five populations
26
12
Table 3: Values of residual sum of squares (RSS), the small-sample bias-corrected
form of Akaike information criterion (AICc), AICc differences (∆i) and of the ‘Akaike
weights’ wi for the twelve models gi of absolute growth. The models with ∆i < 4 are
given in bold.
Model
g1
g2
RSS
754.3
779.2
AICc
676.2
681.1
∆i
0.2
5.1
wi
38.2%
3.3%
g3
822.0
753.2
738.2
753.7
689.1
676.0
679.5
682.6
13.1
0.0
3.5
6.6
0.1%
42.3%
7.4%
1.6%
g8
g9
786.6
738.9
694.8
689.0
679.6
691.7
13.0
3.6
15.7
0.1%
6.9%
0.0%
g 10
693.1
691.3
15.3
0.0%
g 11
704.1
693.7
17.7
0.0%
g 12
692.8
691.3
15.3
0.0%
g4
g5
g6
g7
13
Table 4: Values of the small-sample bias-corrected form of Akaike information
criterion (AICc), AICc differences (∆i) and of the ‘Akaike weights’ wi for the twelve
models fi of the measured morphometric characters. For each character, values of wi
corresponding to models with ∆i < 4 are given in bold.
Model
W=f(L)
D=f(L)
T=f(L)
W p =f(L)
W a =f(L)
AICc
L 1 =f(L)
L p =f(L)
L a =f(L)
L 2 =f(L)
f1
-295.9
-62.2
-290.8
-234.7
-256.0
-366.1
-394.2
-314.1
-233.9
f2
-353.9
-60.1
-289.4
-245.3
-258.9
-369.6
-401.7
-312.3
-232.3
f3
-344.7
-86.3
-288.0
-256.9
-263.1
-369.1
-400.5
-313.4
-231.6
f4
-342.8
-100.1
-288.8
-256.2
-264.0
-372.2
-402.0
-314.4
-234.7
f5
-314.9
-109.0
-301.1
-232.5
-262.8
-394.6
-430.5
-323.7
-257.4
f6
-355.9
-105.8
-303.5
-240.3
-271.2
-397.0
-439.6
-339.4
-259.3
f7
-353.5
-148.2
-304.9
-251.6
-281.1
-395.6
-438.2
-343.4
-261.3
f8
-351.1
-154.6
-306.0
-250.0
-281.6
-399.1
-446.2
-353.7
-274.1
f9
-326.5
-141.5
-315.1
-230.9
-258.3
-393.0
-441.1
-372.4
-286.0
f 10
-343.8
-133.2
-325.4
-240.4
-272.3
-393.2
-436.0
-364.6
-285.1
f 11
-347.8
-160.2
-312.6
-238.2
-266.9
-387.8
-424.6
-353.7
-273.4
f 12
-343.3
-163.3
-324.8
-251.6
-265.2
-385.6
-428.6
-364.0
-278.1
∆i
f1
60.0
101.1
34.6
22.2
25.7
33.0
52.0
58.3
52.1
f2
2.0
103.2
36.0
11.6
22.7
29.5
44.6
60.1
53.7
f3
11.2
77.0
37.4
0.0
18.5
30.0
45.7
59.0
54.4
f4
13.1
63.2
36.6
0.7
17.6
26.8
44.2
58.0
51.3
f5
41.0
54.4
24.3
24.4
18.8
4.5
15.7
48.7
28.6
f6
0.0
57.5
21.9
16.6
10.4
2.0
6.6
33.0
26.7
f7
2.4
15.1
20.5
5.3
0.5
3.5
8.0
29.0
24.7
f8
4.8
8.7
19.4
6.9
0.0
0.0
0.0
18.7
11.9
f9
29.4
21.8
10.3
26.0
23.4
6.1
5.1
0.0
0.0
f 10
12.1
30.1
0.0
16.5
9.3
5.9
10.2
7.8
0.9
f 11
8.1
3.1
12.8
18.7
14.8
11.3
21.7
18.7
12.6
f 12
12.6
0.0
0.6
5.3
16.4
13.5
17.7
8.4
7.9
wi
f1
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
f2
20.3%
0.0%
0.0%
0.2%
0.0%
0.0%
0.0%
0.0%
0.0%
f3
0.2%
0.0%
0.0%
53.6%
0.0%
0.0%
0.0%
0.0%
0.0%
f4
0.1%
0.0%
0.0%
37.0%
0.0%
0.0%
0.0%
0.0%
0.0%
f5
0.0%
0.0%
0.0%
0.0%
0.0%
6.1%
0.0%
0.0%
0.0%
f6
55.9%
0.0%
0.0%
0.0%
0.3%
20.7%
3.2%
0.0%
0.0%
f7
17.2%
0.0%
0.0%
3.7%
43.2%
9.9%
1.6%
0.0%
0.0%
0.0%
1.7%
55.9%
57.2%
87.8%
0.0%
0.2%
f8
5.2%
1.1%
f9
0.0%
0.0%
0.3%
0.0%
0.0%
2.7%
6.9%
96.6%
59.9%
f 10
0.1%
0.0%
56.7%
0.0%
0.5%
3.1%
0.5%
1.9%
38.7%
f 11
1.0%
17.0%
0.1%
0.0%
0.0%
0.2%
0.0%
0.0%
0.1%
f 12
0.1%
81.9%
42.9%
3.8%
0.0%
0.1%
0.0%
1.5%
1.2%
14
Figure captions
Fig. 1: The location of the five populations of the present study. N1: Echaâra, N2:
Njila, N3: Sidi Rais, E1: Stah Jaber, E2: Téboulba.
Fig. 2: The morphometric characters measured in P. nobilis specimens. L: length, W:
width (maximum dorso-ventral length), T: thickness, D: distance from the top of the
shell hinge to the top of the valves, Lp, La: lenghts of the posterior and anterior
adductor muscle scars respectively, Wp, Wa: maximum width of the posterior and
anterior adductor muscle scars respectively, L1, L2: lengths from the top of the
posterior and anterior adductor muscle scars respectively to the top of the shell.
Fig. 3: The size-at-age raw data and the average model. Differences between North
and East population groups were so slight that were not visible in the graph. The data
were jittered by adding a small random quantity to the horizontal coordinate to
separate overplotted points and have a better visualization of the dataset.
Abbreviations as in Fig. 1.
Fig. 4: (Left panel) The raw data of the measured morphometric characters in relation
to the shell length (L). (Right panel) The average models of relative growth. Ni are the
northern populations and Ei the eastern ones; abbreviations as in Figs 1 and 2.
Symbolization: ◊: N1, : N2, ∆: N3, +: E1, x: E2.
Fig. 5: The generalized allometric exponent b for the relative growth of the measured
pairs of morphometric characters, based on the average models.
Fig. 6: Absolute growth curves of several P. nobilis populations: A1: 13 m depth,
Moraira Bay – Spain (García-March et al. 2007a), A2: 6 m depth, Moraira Bay –
Spain (García-March et al. 2007a), B: Thermaikos Gulf – Greece (Galinou-Mitsoudi
et al. 2006), C: Mljet National Park – Croatia (Šiletić and Peharda 2003), D1:
Aguamarga – Spain (Richardson et al. 1999), D2: Rodalquilar – Spain (Richardson et
al. 1999), D3: Carboneras – Spain (Richardson et al. 1999), E1: Veliko jezero Croatia (Richardson et al. 2004), E2: Malo jezero - Croatia (Richardson et al. 2004),
E3: Mali Ston Bay - Croatia (Richardson et al. 2004), F: National Park of Port Cros –
France (Moreteau and Vicente 1982), PS: present study (model g1). The reported
asymptotic length L∞ in each case is given in parenthesis.
Fig. 7: Comparison of W-L relative growth curves (Top) and generalized allometric
exponents (Bottom) of the P. nobilis populations of the present study and the
population in Lake Vouliagmeni – Greece (Katsanevakis et al. 2007a).
Fig. 8: Scatterplot of the first two principal components of the pooled normalized logtransformed morphometric data of the five P. nobilis populations.
15
Fig. 1
16
Fig. 2
Fig. 3
17
Fig. 4a
18
Fig. 4b
19
Fig. 5
20
Fig. 6
Fig. 7
21
Principal Component 2
1.5
1
N1
0.5
N2
0
N3
E1
-0.5
E2
-1
-1.5
-11
-8
-5
-2
1
4
Principal Component 1
Figure 8
22