Ohmshi et al.: The von Bertalanffy growth function concerning the allocation of surplus energy to reproduction 
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Table 1 
Parameter estimates for two types of von Bertalanffy growth functions (VBGFs). Both types of VBGF have three common 
parameters: asymptotic length (/„), growth coefficient (K), and initial condition of age (t 0 ). Additional parameters, namely age 
at maturity (t m ), the upper limit of the allocation rate in reproductive energy (u), and rapidity of maturation (a), were used in 
the extended model. Values within parentheses show the square root of the variance of the estimates derived from the matrix 
inverse of the Hessian matrix. The 4AIC shows the relative difference of the Akaike information criterion (AIC) value compared 
with the minimum AIC. 
Type of VBGF 
L 
K 
^0 
V 
a 
AIC 
4AIC 
Conventional model 
260.72 
0.34 
-0.45 
— 
— 
— 
26881.3 
20.8 
(1.51) 
(0.01) 
(0.04) 
Extended model 
463.01 
0.15 
-0.77 
3.41 
0.79 
1.01 
26860.5 
0 
(49.08) 
(0.01) 
(0.14) 
(0.35) 
(0.05) 
(0.21) 
the boundary of mature 
“size.” In most 
cases, 
however, 
Results 
it is not easy to obtain an explicit solution, as shown in 
Equation 6, owing to the complexity of the integrand in 
Equation 9. 
Allocated reproductive energy can be derived as fol- 
lows by rearranging Equation 4 with the condition 
p^pit) as 
(i-PMEf-PWf. 
Substituting dwldt=3fH 2 dlldt and Equation 5 into this 
equation, one obtains 
f = 3 -frW(L-D, 
f = ^~ J'p(s )/ 2 {L-l)ds. 
( 10 ) 
( 11 ) 
Equation 10 represents the instantaneous reproduc- 
tive energy at age t. Equation 11 shows the cumulative 
investment of reproductive energy until age t. Thus, 
changes in two quantities (body size in Eq. 6 and energy 
investment in Eq. 10) are treated in an extended VBGF. 
We fitted the growth curve in Equations 6 and 8 to 
individual measurements in length-at-age as L l (i = 1,..., 
AD, where N is the total number of samples. Param- 
eters were estimated by minimizing the residual sum of 
squares ofS=X, iL-lf 2 . The numerical optimization for 
S was accomplished by using the quasi-Newton method 
(BFGS algorithm) in “optim( )” with R statistical soft- 
ware (R Development Core Team, 2011). The comparison 
between this model and the original monophasic VBGF 
was based on the AIC value as follows: AIC=ATogS+20, 
where 8 is the number of free parameters. 
We used measurement data on willowy flounder 
( Tanakius kitaharai) males collected by bottom-trawl 
surveys in the coastal area of Fukushima Prefecture, 
Japan, from 2004 to 2006. The sample size was n=2169. 
Age ranged from 1.38 to 14.30 years and length ranged 
from 113 to 298 mm (standard length). Otoliths were 
used to determine yearly ages and dates of birth were 
assigned as January 1st. 
Results for curve fitting and model selection are sum- 
marized in Table 1 and Figure 1. As shown in Table 1, 
the AIC difference (AAIC=20.8) between the two types of 
VBGF suggests that the trajectory given by the extended 
model more appropriately described lifetime growth. 
This result implies that a consideration of reproductive 
energy can be meaningful for model extensions. The 
variance of two common parameters (i.e., and t 0 ) in the 
extended model was larger than that in the conventional 
VBGF (Table 1). 
Twelve types of energy allocation schedules, pit), and 
the corresponding somatic growth (in length /) based 
on different combinations of parameter values in Equa- 
tion 8 (u = Q.4, 0.6, 0.8, 1.0 and a- 1, 3, 100) are shown 
in Figure 2. The behavior of df/dt and f describing the 
energy investment in reproduction is shown in Figure 3. 
When v = 0, the growth curve is identical to the origi- 
nal VBGF (Fig. 2). Although somatic growth curves 
generated by lower v values (i.e., v = 0.4, Fig. 2, A-C) do 
not differ substantially from the original VBGF, there 
are distinctive differences for shapes with higher v 
values (i.e., u = 0.8, 1.0, Fig. 2, G-L). In these cases, the 
somatic growth of the adult and juvenile stages can be 
clearly distinguished. Gradual but steady growth after 
maturation is typical with indeterminate growth (Fig. 
2, A-I). We can see a continuous phase shift of inde- 
terminate growth in Figure 2. When u = 1.0, the growth 
rate after maturation converges to zero because most 
surplus energy is devoted to reproduction, generating 
more determinate growth (Fig. 2, J-L). 
The variation in a leads to a difference in the degree 
of continuity of growth rate during the sexual matura- 
tion transition period (Fig. 2, A, D, G, and J vs. Fig. 
2, C, F, I, and L). The curves given by sufficiently high 
a (a = 100) represent biphasic VBGF resulting from an 
abrupt change in growth rate around age t m (Fig. 2, C, 
F, I, and L). 
In Figure 3, an apex can be found on the convex 
shape of df/dt, and the height and degree of curvature 
changes according to the values of v and a. An increase 
in the value of v raises the reproductive investment df! 
