MARINE ECOLOGY PROGRESS SERIES

Vol. 238: 109–114, 2002 Published August 9

Mar Ecol Prog Ser

Modeling growth of different developmental stages

in bivalves
H. Jörg Urban*
Section for Comparative Ecosystem Research, Alfred Wegener Institute for Polar and Marine Research, Postfach 12 01 61,
27515 Bremerhaven, Germany

ABSTRACT: The 5 most common growth models (Gompertz, Special von Bertalanffy, Richards,
Logistic and Generalized von Bertalanffy) were fitted to growth data comprising different develop-
mental stages (i.e. free-swimming larvae, plantigrades, post-larvae and juveniles/adults) of the
Caribbean pearl oyster Pinctada imbricata. As criteria for determining the most suitable model, the
mean deviation of the data from the estimated growth curve (= mean square error, MSE), the devia-
tion of the estimated asymptotic length of the observed maximum length obtained from a natural
population, and the plot of the residuals against estimates were used. The Special von Bertalanffy
and Richards models produce pure asymptotic curves and are therefore unable to simulate the expo-
nential growth of the first 2 developmental stages (free-swimming larvae and plantigrades). Further-
more they tend to overestimate the asymptotic length. Gompertz, Logistic and the Generalized von
Bertalanffy model, in contrast, produce sigmoid curves and thus are much more capable of modeling
growth of the combined developmental stages, thereby, however, underestimating asymptotic
length. Of all models, the Generalized von Bertalanffy model yields the best fit which is explained
and discussed as a special mathematical property of its surface factor ‘D’.

KEY WORDS: Growth models · Modeling · Development · Bivalves · Pinctada imbricata

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INTRODUCTION developed by Pauly (1979). Accordingly, growth is

the net result of anabolism and catabolism. Sub-
Growth, defined as the increase in length or stances necessary for anabolism enter through a sur-
weight per time unit, is a significant aspect in the face which increases according to Eq. (1), and catab-
study of the dynamics of marine species. Regarding olism corresponds to weight, which increases
bivalve growth, a good overview of published results according to Eq. (2).
is given in Vakily (1992) and Brey (1995). Both
authors used large databases compiled from litera- S = pLa (1)
ture to compare the growth of different bivalve pop-
W = qLb (2)
ulations. To quantify growth, the general approach is
to fit mathematical models to data, such as those of with S = surface, L = length, W = weight, p, q = con-
Gompertz (1825) and Richards (1959). The most com- stants, and a, b = exponents.
mon used model is the von Bertalanffy growth model The original function defined by von Bertalanffy
(VBGF), among other reasons, because it incorpo- (1938) is a special case as surface and weight increase
rates an interesting physiological concept which was isometrically (i.e. exponent a of Eq. (1) = 2 and expo-
first introduced by von Bertalanffy (1938) and further nent b of Eq. (2) = 3. Thus the original VBGF (von
Bertalanffy 1938) is called ‘Special VBGF’, compared
to the ‘Generalized VBGF’ which is without these con-
*E-mail: jurban@awi-bremerhaven.de straints (Pauly 1979).

© Inter-Research 2002 · www.int-res.com

110 Mar Ecol Prog Ser 238: 109–114, 2002

Table 1. Pinctada imbricata. Growth parameters of different models with growth data of all developmental stages. n = 268;
L∞: asymptptoc length; K: growth factor; t0: theoretical age at length 0; ti : age at inflexion point (yr); MSE: mean square error;
VBGF: von Bertalanffy growth function; Deviation of L∞ from Lmax: Dev. = (L∞ − Lmax )2 ; Lmax = 84 mm)

Model L∞ K –t0 or ti D MSE r2 Deviation of

(mm) (yr–1) (yr–1) L∞ from Lmax

Gompertz 53.9 3.340 –0.414 5.860 0.981 30.1

Special VBGF 162.7 0.365 –0.047 5.776 0.981 78.7
Richards 165.6 0.365 –0.001 1.018 5.776 0.981 81.6
Logistic 48.6 6.106 –0.503 8.280 0.973 35.4
Generalized VBGF 65.7 1.767 –0.000 1.765 4.799 0.984 18.3

However, all growth models are normally used to larvae reared in the laboratory, (2) mobile plantigrades
describe the growth of adults, thus limiting their reared in the laboratory, (3) sessile post-larvae grown
applicability to the complete life history of bivalves, as in culture systems in the field and (4) juveniles/adults
different developmental stages may display different cultured in the field.
growth patterns. Most marine bivalves have an initial
free-swimming larval stage followed by a post-larval
stage. These 2 stages grow linearly or exponentially MATERIALS AND METHODS
(e.g. Beaumont & Budd 1982, Rose & Baker 1994),
whereas adult growth — in the case of growth in Growth data of different developmental stages. Lar-
length — is asymptotical. Bayne (1965), examining the val growth in the laboratory: Adult individuals of the
growth of Mytilus edulis, showed linear growth for pearl oyster Pinctada imbricata were collected from
larvae and asymptotic growth for adults. Both devel- Cabo de Vela (12° 10’ N, 72° 20’ W) and left for 4 wk
opmental stages were analyzed independently. to adapt to laboratory conditions using ultraviolet-
Based on this consideration the present study in- irradiated, 45 µm filtered sea water kept between 23 to
tends to evaluate the suitability of different historical 25°C and a salinity of 34, and fed on a 1:1 (by volume)
growth models to explain the growth of all develop- mixture of Chaetoceros gracilis and Isochrysis galvana
mental stages in bivalves. Firstly, the growth rates of supplied by a continuous dropping system of 1 drop s–1.
different developmental stages of a marine bivalve During a spawning trial the temperature was gradually
were compared. All obtained growth data were then increased by 1°C every 30 min up to 34°C. Fertilization
fitted to the different models. Growth data of the was initiated by mixing eggs and sperm at 1:10 (vol-
Caribbean pearl oyster Pinctada imbricata covered the ume-ratio). Larvae (3 replicates) were transferred to
following developmental stages: (1) free-swimming 500 l tanks filled with ultraviolet-irradiated, 1 µm fil-
tered seawater kept at 27°C, stocked
at an initial density of 6 larvae ml–1
and fed on I. galvana and C. gracilis
(up until Day 15: 50 000 cell ml–1, after
Day 15: 100 000 cell ml–1), changing
the water every 2 d. From Day 15 on-
wards a propylene net material was
placed in the tanks during larval set-
tlement. Larval growth was studied for
37 d by measuring the anterior-poste-
rior length of approximately 10 to 15
larvae every 5 d with a microscope.
Post-larval growth under natural
conditions: From Day 47 to Day 88,
growth of the post-larvae obtained
from the experiment described above
continued under natural conditions.
Fifty post-larvae (3 replicates, mean ±
Fig. 1. Pinctada imbricata. Mean daily growth rates of different developmental
stages. The equation shown is the polynomial of the 4th order fitted to the data. SD) measuring 2.9 ± 0.675 SD mm shell
GR: growth rate length, chosen randomly from the lar-
 Urban: Modeling growth in bivalves 111

val experiments, were held in suspended boxes (plastic tal stages exhibit different growth rates, each repre-
boxes, 40 length × 60 width × 15 height [cm], covered by sented by an ellipsoid in Fig. 1. From free-swimming
propylene nets of 0.7 cm mesh size) at 5 m depth in larvae to plantigrades and post-larvae, both growth
the sea situated in the near vicinity of the laboratory rates and variability increase reaching a maximum at
(11° 20’ N, 74° 10’ W). The length on the anterior- about 90 d. Growth rate of juveniles decreases almost
posterior axis of 30 individuals was measured every 5 d asymptotically with the exception of a peak at 300 d.
using vernier calipers. A polynomial of the 4th order was fitted to these data
Growth of juveniles/adults in suspended boxes: accounting for 50% of the variability of growth rates
Juveniles were obtained as natural spat from collectors (r2 = 0.504).
(see Urban 2000a) with an initial shell length of 12.8 ± The growth parameters estimated with the 5 differ-
1.881 mm. Growth of individuals from 3 replicates was ent models are summarized in Table 1. To evaluate the
monitored between September 1997 and June 1998 in fit of the data, the residuals (of years) were plotted
suspended boxes (plastic boxes, 40 length × 60 width × against estimated shell length (Fig. 2a). The corre-
15 height [cm], covered by propylene nets of 0.7 cm sponding growth curves were plotted over the raw
mesh size) at a density of 30% (= percentage of the data (shell length against years, Fig. 2b).
available area covered). Boxes were connected to ver-
tical lines on a bottom long line kept in the water col-
umn at least 5 m above the bottom and 5 m below the DISCUSSION
surface. Long line systems were located at Gayraca
Bay in the Tayrona Park (11° 20’ N, 74° 10’ W, see Fig. 1 shows a remarkable growth increase from Day
Urban 2000b). Anterior-posterior shell length of 30 in- 23 onwards. According to the morphological develop-
dividuals was measured monthly (vernier calipers). ment, the free-swimming larvae developed into planti-
Culture systems were cleaned at the same time. grades after 23 d, having acquired branchial filaments
Analysis of growth data. As an initial step, based on some days before. Thus the growth difference between
means of all individuals of 3 replicates of each measur- these 2 stages can be explained by the fact that indi-
ing date, mean growth rates of shells [mm d–1] were viduals with a fully functional feeding apparatus were
calculated. In a second step, different growth models able to feed on the available phytoplankton at a much
(Gompertz Eq. 3, Special VBGF Eq. 4, Richards Eq. 5, higher rate than the previous larval stage.
Logistic Eq. 6 and Generalized VBGF Eq. 7) were fitted Growth rates of juveniles reached a small peak at
to length-at-age data of all developmental stages in 300 d (Fig. 1) although according to the growth model
order to estimate the parameters using an iterative one would expect an asymptotic decrease. Urban
non-linear least-square method (Simplex algorithm, (2000b) related the growth rates of this data set (only
Press et al. 1986). juveniles) to different abiotic factors. One month
[ − K (t −t 0 ) ] before growth increase, an increase of particulate
Gompertz: Lt = L∞ee (3)
organic matter (POM) was observed in the water. In
the study area the POM cycle depends on a seasonal
Special VBGF: Lt = L∞ (1 − e− K (t −t 0 ) ) (4)
pattern of the rainy season (POM increase owing to
run-off induced by rainfall). Thus, an oscillating
Lt = L∞ [1 − De[ − K (t −t i )] ]
1/ D
Richards: (5) growth pattern is indicated, although, due to little or
no seasonality in the tropics, non-oscillating growth is
L∞
Lt = usually assumed.
Logistic: (1 + e− K (t −t 0 ) ) (6) Based on the graphical results in Fig. 2a and the esti-
mated parameters, and referring to the statistics given
Generalized VBGF: Lt = L∞ (1 − e− K (t −t 0 ) )D (7) in Table 1, 3 criteria were used to evaluate the 5 mod-
els: (1) the plot of residuals, (2) the mean deviation of
where L∞ = asymptotic length (mm), K and DRichards = the data from the estimated curve (given by the mean
growth parameters (yr–1), DGen VBGF = surface factor, t = square error, MSE), and (3) the deviation of the esti-
age (yr), t0 = theoretical age at zero length, and ti = age mated L∞ from the maximum length of the natural
at the inflexion point (yr). population.
The latter can be explained as follows: Urban (2000a)
studied the population dynamics of this species from
RESULTS Cabo de la Vela. From pooled monthly samples over
1 yr of more than 1000 specimens, a maximum length
Mean daily growth rates for all developmental of Lmax = 84 mm was obtained. It can be assumed that
stages are plotted in Fig. 1. Clearly the 4 developmen- this Lmax is close to the true biological L∞. However, it is
112 Mar Ecol Prog Ser 238: 109–114, 2002

Fig. 2. Pinctada imbricata. (a) Plot of residuals of estimated shell lengths for 5 growth models of different developmental stages.
(b) Shell length (mm) plotted against age (yr)
 Urban: Modeling growth in bivalves 113

well known that the estimated L∞ may differ consider- growth of all developmental stages including the larval
ably from Lmax if growth data do not cover most of the stages, although the fit for the first developmental
species’ growth range (Ralph & Maxwell 1977, Urban stage is not optimal. Thus, from all models compared
& Mercuri 1998). This applies to the present data set the Generalized VBGF is clearly the most suitable.
because growth followed for only 12 mo. The deviation In order to explain these findings some mathematical
of the estimated L∞ from the maximum length of the properties of the models have to be discussed. The
natural population was calculated as: Generalized VBGF used here differs from the Special
VBGF only by the surface factor D, defined as D = b – a
Deviation = (L∞ − Lmax )2
(b, a: exponents of Eqs. 1 & 2). The ‘D’-parameter is
By plotting the residuals against the estimated shell also found in Richards model, and indeed, as stated by
lengths the randomness of their distribution can be Pauly (1981), Richards model is a ‘first version of the
evaluated (Fig. 2a). For the Special VBGF and Generalized VBGF, …however, without setting theo-
Richards, a strong linear trend of the first 2 stages and retical limits to possible values of the exponents relat-
a nonlinear trend of the last stage (juveniles/adults) is ing anabolism and catabolism to weight’. When D = 1,
clearly expressed. In the Logistic and Gompertz, these the Generalized VBGF and Richards are equal to the
trends are similar but less clear, while for the General- Specialized VBGF. A good example is given in Table 1:
ized VBGF they are not pronounced, indicating an D(Richards) is almost 1, so we would expect the parame-
acceptable fit of the data. This leads to the following ters of Richards to be equal or similar to the parameters
ranking of the models (beginning with best fit): Gen- of the Special VBGF. As it turns out, both K-values are
eralized VBGF, Gompertz/Logistic, Special VBGF/ equal and the L∞ values are very similar. However, the
Richards. According to the MSE, the models can be parameters D in the Generalized VBGF and Richards
arranged in the following order (from smallest to high- model have different modeling properties taking into
est MSE): (1) Generalized VBGF, (2) Special VBGF, account that Richards produced an asymptotic curve,
(3) Richards, (4) Gompertz and (5) Logistic. For the while Generalized VBGF resulted in a sigmoid curve.
deviation of L∞ from Lmax the ranking of the models is Pauly (1981) mentions as an ‘interesting property’ of
different: The smallest deviation was also found for the the Generalized VBGF the occurrence of an inflexion
Generalized VBGF (=18.3 mm), followed by the Gom- point when D(Generalized VBGF) > 1 and no inflexion point
pertz (=30.1 mm) and the Logistic (=35.4 mm). High when D = 1. (The original paper says ‘D < 1’, but I
deviations were obtained for the Special VBGF assume this must be a printing error.)
(=78.7 mm) and Richards (=81.6 mm). In other words, when D(Generalized VBGF) ≤ 1 the curve is
So far the following conclusions can be drawn: asymptotic and only when D(Generalized VBGF) > 1 does the
The maximum growth rates at 90 d (=0.25 yr, Fig. 1) curve become increasingly sigmoid. On the other
corresponds to the point of inflexion of the sigmoid hand, D(Richards) merely shifts the entire growth curve on
curves in Fig. 2b. In other words, owing to the maxi- the abscissa without changing its form.
mum growth rates of Fig. 1, growth data of Fig. 2 are In summary, when D(Generalized VBGF) > 1, the General-
difficult to simulate with a simple asymptotic growth ized VBGF has an inflexion point, in other words the
model. Instead, a model capable of explaining all asymptotic curve is changed into a sigmoid curve,
stages of the sigmoid character of the growth data is thus, giving the best fit to the data consisting of all
more suitable. The Special VBGF and Richards pro- developmental stages.
duce pure asymptotic curves, thus they are unable to
simulate the near-exponential larval growth. Further-
more, they tend to overestimate L∞ indicated by the Acknowledgements. Juan Pablo Assmus, Socorro Sánchez
high deviation of L∞ from Lmax. Such high values for the and Yadira Caballero helped in the field and worked on the
estimated L∞ (Special VBGF: 162.7 mm; Richards: samples. Considerations of Dr. T. Brey regarding data analy-
sis were helpful, and several constructive critics of 4 referees
165.6 mm) are completely out of this species’ growth improved this paper. The idea for this paper was born several
range and therefore without biological relevance. The years ago after discussing this subject with Dr. D. Pauly.
Generalized VBGF, Logistic and Gompertz, produce
sigmoid curves and thus are much more capable of
modeling the growth of these bivalves. They tend to LITERATURE CITED
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Editorial responsibility: Otto Kinne (Editor), Submitted: August 21, 2001; Accepted: January 31, 2002
Oldendorf/Luhe, Germany Proofs received from author(s): July 9, 2002