RESEARCH NOTE
The self-thinning rule applied to cultured populations in aggregate
growth matrices
Ramón Filgueira, Laura G. Peteiro, Uxı́o Labarta
and Marı́a José Fernández-Reiriz
C.S.I.C.—Instituto de Investigaciones Marinas, C/Eduardo Cabello 6, 36208 Vigo, Spain
The increase in size of individuals in a population implies an
increment of their space and food requirements. These factors
when limiting can cause intraspecific competition, resulting in
a decrease of the growth rate and an increase in the mortality
rate (Alunno-Bruscia et al., 2000). The self-thinning process
describes the negative relationship between body size and
population density when growth results in mortality through
intraspecific competition (Westoby, 1984). Self-thinning can be
studied by means of successive sampling of the growth of
even-aged individuals that are maintained at different densities
(see Fig. 1 in Alunno-Bruscia et al., 2000). The following
equation describes the self-thinning rule (Westoby, 1984):
B ¼ kN 1=2 ;
ð1Þ
where B is the total biomass, k the intercept of the relationship
and N the population density. The last expression is obtained
from the mean morphometric relationships between the substratum area occupied by the individual (S), length of the individual (l) and its weight (m). Assuming isometric growth (S a l 2
and m a l 3) and an inverse relationship between density and
area occupied (N a S 21), the following equation is obtained:
m ¼ kN 3=2 ;
ð2Þ
which given the relationship between population biomass and
individual average weight (B ¼ Nm) can be expressed as
Equation 1.
Although this theory is based on studies focused on plants, it
can be applied to both sessile and mobile animal populations
(Hughes & Griffiths, 1988; Steingrı́msson & Grant, 1999).
Damuth (1981) observed in primary consumers an exponent of
24/3 between average weight and density and suggested that
this can be related to the metabolic requirements of individuals. In this way, Begon, Firbank & Wall (1986) stated a
theoretical relationship between the self-thinning process and
the population metabolic rate (MR), which is related to the
individual average weight (MR a Nm 3/4) and environmental
conditions (MR a F, where F is the total amount of food consumed by the population). Assuming a constant consumption
of food by the population (Fcte), in stable environmental conditions the following relationship can be obtained:
m ¼ kN 4=3 ;
ð3Þ
which given the relationship between population biomass and
individual average weight (B ¼ Nm) can be expressed as:
B ¼ kN 1=3 :
ð4Þ
Correspondence: M.J. Fernández-Reiriz; e-mail: [email protected]
Fréchette & Lefaivre (1990) established a method to distinguish between both causes of competition – space or food –
in the relationship between the maximum biomass that can be
supported by a population and its density (self-thinning). The
method is based on the equations described above, ascribing to
the space-limiting factor (spatial self-thinning, SST) the
relation m ¼ kN 23/2 (or B ¼ kN 21/2) and to the food-limiting
factor (food self-thinning, FST) the relation m ¼ kN 24/3 (or
B ¼ kN 21/3).
In the present study we analysed a dataset of dry weights
and densities of Mytilus galloprovincialis cultivated in the Rı́a de
Ares-Betanzos (NW Spain). The data came from the regular
sampling carried out by our research group for various other
studies. At each sampling, two replicates of 50 cm of rope were
scraped free of mussels. The mussels were counted, weighted,
measured and classified into length classes. The data were
taken during the years 2004 (n ¼ 111), 2005 (n ¼ 149) and at
the beginning of 2006 (n ¼ 8). The use of data sampled across
years allowed us to consider the assumption of constant energy
flow less likely violated because fluctuations in food abundance
over years should only cause increased variance around the
self-thinning line (Steingrı́msson & Grant, 1999). Mussels were
cultivated on ropes at high densities, which presented an ideal
situation to study the self-thinning processes because this
method involves mortality by intraspecific competition. The
diameters of 25 commercial ropes were measured to calculate
the available area for mussel attachment. Ropes were hung
from rafts following the technology used commonly in the
Galician Rı́as, which includes three phases: seed settlement
(initial), early thinning-out and thinning-out. In the two last
phases, the mussels are detached from the ropes and replaced
in culture at lower densities. The reduction of the density
caused by this manipulation alters the natural process of selfthinning and invokes mitigation in the mortality by intraspecific competition. The subsequent inclusion of values in
which the mortality by intraspecific competition has not yet
begun will flatten the estimated slope of the self-thinning fit
(Zhang et al., 2005). Several studies have suggested corrective
techniques that include the removal of these data points
(Westoby, 1984) or the use of different regression models that
allow a good fit without the omission of data points (see review
by Zhang et al., 2005). These last authors compared several
regression models [quantile regression, deterministic frontier
function and stochastic frontier function (SFF)] that have the
potential to establish an upper limiting boundary line above
all plots for the maximum size –density relationship, without
subjectively selecting a subset of data points based on predefined criteria. Zhang et al. (2005) concluded that the SFF
was the most suitable model given that this method can easily
yield statistical inference on the model coefficients.
In the present study, the mathematical expression that
describes the self-thinning (m ¼ kN g) was calculated by means
of the logarithmic linear transformation (log m ¼ log
k þ g log N) following two regression models: reduced major
Journal of Molluscan Studies (2008) 74: 415–418
# The Author 2008. Published by Oxford University Press on behalf of The Malacological Society of London, all rights reserved.
RESEARCH NOTE
(1999) developed a tridimensional model in which the multilayered structure of an intertidal mussel community is taken
into account. In this model, the individual density (N) is considered inversely proportional to the area projected on the substratum (S) and directly proportional to the number of layers
(L), i.e. NaL S 21. Therefore, the B–N–L tridimensional
model that defines the self-thinning can be expressed in its logarithmic form as (Guiñez & Castilla, 1999):
log B ¼ log k ð1 bÞ log L þ b log Ne :
ð5Þ
Given the direct relationship between the number of layers (L)
and density (N), the B–N–L tridimensional model could be
reduced to a B–N bidimensional model, although this change
alters the exponent of the model and, therefore, it could not be
compared with the theoretical exponents for space- and
food-limiting factors. Guiñez & Castilla (2001) suggested a
new tridimensional model that allows comparison of the
empirical exponent with the theoretical exponents for spaceand food-limiting factors. This model introduces the concept of
effective density (Ne), which is defined as the expected density
if individuals within a sample occurred as a monolayer. The
log L term of the tridimensional model is then removed and
the model can be expressed as:
Figure 1. Dry weight of individual mussels vs culture density, and
linear fits performed by RMA and SFF. ANCOVA following Zar
(1984) was performed to compare the observed exponents with
theoretical values for FST and SST (24/3 and 23/2, respectively).
axis (RMA) and SFF (Zhang et al., 2005). The regression
models were carried out using the statistical package SPSS
14.0 and Frontier 4.1c (Coelli, 1996), respectively. The results
shown in Figure 1 highlight the importance of the regression
model used for fitting the self-thinning equation. The RMA
regression results in a slope of 21.58 + 0.106 (mean + 95%
confidence interval), which is statistically similar to the theoretical slope of the space limitation relationship (SST;
ANCOVA: t ¼ 1.481, P ¼ 0.140, n ¼ 268), whereas the SFF
regression results in a slope of 21.16 + 0.102, which is statistically different from the theoretical values for FST (ANCOVA:
t ¼ 3.213, P , 0.005, n ¼ 268) and space limitation (SST;
ANCOVA: t ¼ 6.476, P , 0.001, n ¼ 268). Although the RMA
has yielded an exponent similar to the theoretical value, the
handling of cultivation ropes, with a consequent reduction in
mussel density, introduces data points for which mortality by
intraspecific competition has not occurred. So, caution must be
taken in interpreting the meaning of the self-thinning exponent
estimated by means of the RMA regression model. Therefore,
the comparison between theoretical values and the selfthinning exponent calculated by means of the SFF (–1.16 +
0.102) indicates that the population’s regulation is caused by
neither space nor food limiting factors.
However, distinguishing between space or food limiting
factors by means of the value of the exponent is a difficult task
because it is possible that both limiting factors exert a densitydependent effect on sessile populations (Fréchette, Aitken &
Pagé, 1992), and the exponent could show deviations from the
theoretical value (Alunno-Bruscia et al., 2000). In an analogous
way, wide variability has been observed in the exponent of the
relationship between MR and weight (Latto, 1994), which
would modify the theoretical value of the FST given the
dependence of MR and self-thinning established by Begon
et al. (1986). In the case of a space factor, the assumptions of
the classic model are: (1) isometric growth, (2) a complete
occupation of the sampled area and (3) an inverse relationship
between the sampled area and the individual density (NaS 21;
Westoby, 1984). However, growth usually is allometric instead
of isometric causing a deviation from the theoretical exponent
(Westoby, 1984). In addition, the multilayered structure
formed by the mussel on the substratum violates the third
assumption and therefore alters the theoretical exponent
because the available area is underestimated (Hughes &
Griffiths, 1988). These deviations have been included in newer
models (Fréchette & Lefaivre, 1990; Guiñez & Castilla, 1999)
of mussel growth that modify classical models to introduce the
effect of allometric growth, multilayered structure and/or the
effect of the roughness of the substratum. Guiñez & Castilla
log m ¼ log k þ g log Ne :
ð6Þ
The application of this model requires knowledge of the area
occupied by each individual. In this way, assuming that the
maximum length of an individual is perpendicular to the substratum, Guiñez & Castilla (1999) estimated the area projected
onto the substratum by multiplying the maximum width by
the maximum height; that is, assuming that the mussel is contained in a parallelepiped. The application of this model to the
present study dataset results in a reduction of the exponent
from 21.16 + 0.102 to 21.62 + 0.024 (Figs 1 and 2A, respectively), which is statistically different from the theoretical values
for FST (24/3; ANCOVA: t ¼ 24.167, P , 0.001, n ¼ 268)
and SST (23/2; ANCOVA: t ¼ 10.000, P , 0.001, n ¼ 268).
The exponent reduction is caused by the use of effective
density, which corrects the underestimation of the bidimensional model in which the multilayered structure is not considered (Guiñez, Petraitis & Castilla, 2005).
Mussels are gregarious animals that form complex matrices
with several overlapped layers (Fig. 3), in which the individuals that are placed in the external layers are overlapped with
the internal layers, occupying the empty space of the internal
layers. Therefore, calculating the occupied area by means of
the parallelepiped projection, defined as the exclusive individual space occupied per mussel, could overestimate the real
area occupied by the mussels because the overlapping is not
considered. An approach more representative of mussel morphology could yield a more realistic value of the occupied area
per individual. Given the predominant position of the mussel,
with the maximum length of the individual perpendicular to
the substratum, the apical projection onto the substratum
could be a better approach to determine the occupied area. In
this way, the projected area occupied by the population will be
more realistic as the population packing increases. An accurate
tool to determine complex areas is image analysis. In the
present study, apical photographs (Nikon Coolpix 4500) of 50
individuals (from 15 to 90 mm length) were taken to determine the projected area onto the substratum by means of
ImageJ 1.37v. The allometric antero-posterior axis length vs
projected area (projected area ¼ 0.8 1023 length2.22, N ¼
50, r 2 ¼ 0.99, P , 0.001) was applied to the frequency distributions to calculate the occupied area of the population. The
416
RESEARCH NOTE
Figure 2. Dry weight of individual mussels vs culture density, and linear fits performed by RMA, where N is the effective density calculated from
the projected area of the parallelepiped (A), and where N is the effective density calculated from the image analysis (B). In both cases, only RMA
is shown because the SFF estimates identical parameters and the likelihood value is less than that obtained using RMA. ANCOVA following Zar
(1984) was performed to compare the exponents observed in the present study with theoretical values for FST and SST (24/3 and 23/2,
respectively).
ACKNOWLEDGEMENTS
effective density was calculated by dividing the number of individuals by the occupied area of the population. Figure 2B
shows the average weight vs density, expressed as number of
individuals per effective area calculated by means of image
analysis. The observed slope value (21.32 + 0.027) is statistically similar to the theoretical exponent of 24/3 (ANCOVA:
t ¼ 1.111, P ¼ 0.268, n ¼ 268), which would indicate that the
population growth is limited by available food (Fréchette &
Lefaivre, 1990). The application of image analysis techniques
allow the calculation of a more realistic value of occupied area,
which in the particular case of Mytilus galloprovincialis confirms
that in densities commonly used in commercial cultivation on
rafts, self-thinning is probably regulated by the food-limiting
factor.
The arrangement of mussels in complex matrices makes it
difficult to apply classical self-thinning models. The model
suggested by Guiñez & Castilla (2001) introduces the effective
density term that allows comparison of the self-thinning exponent between different populations or species with different
crowding strategies. The use of image analysis techniques to
determine the effective density, as applied in this study of
M. galloprovincialis, could improve the results of the model
because the measured value of the occupied area is more realistic
than the parallelepiped projection. The image analysis technique
used in the present study is easy to apply to other species, and its
application could improve estimation of occupied area. This
variable is essential in obtaining realistic patterns of densitydependence relationships for gregarious populations that are
arranged in complex matrices.
We thank PROINSA mussel farm and their employees,
especially H. Regueiro, M. Garcı́a, C. Brea and
O. Fernández-Rosende for technical assistance. We wish to
thank A. Ayala for helping us with Figure 3.
FUNDING
This study was supported by the contract-project PROINSA,
Code CSIC 20061089, Xunta de Galicia PGIDIT06RMA018E.
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