226 
Fishery Bulletin 1 10(2) 
Figure 1 
Two types of growth curve estimated with the conventional von Bertalanffy growth 
function (A) and with the extended von Bertalanffy growth function (B) for individual 
measurement data of the willowy flounder ( Tanakius kitaharai ) male (n = 2169). 
dt and shifts the maximum df/dt to older ages. When 
u = 1.0 (Fig. 3, J-L), df/dt converges to a constant value 
after maturation as a result of determinate growth and 
constant surplus energy, defined by Equation 2. Lower 
a values show a slower initial rise in df/dt around t m 
(Fig. 3, A, D, G, and J), whereas higher a values yield 
a steeper initial rise in df/dt around t m (Fig. 3, C, F, 
I, and L). 
Discussion 
A notable feature of our mode! is that energy allocation 
can be quantified by the arbitrary functional form p(-). 
The introduction of p(-) provides a unified platform to 
treat the trade-off between somatic growth and repro- 
duction. The extended model can jointly describe adult 
and juvenile growth. The change in growth rate between 
the two stages can be either gradual or steep, with the 
latter case showing a biphasic VBGF. By controlling the 
value of p(-), our comprehensive model yields various 
shapes of growth curves that range from indeterminate 
to determinate growth. Therefore, our model can be 
used for life history studies, as well as practical curve 
fitting studies. When allocation dynamics are not fully 
described by a simple model, such as seen in Equation 
8, additional parameters beyond a, u, and t m , or a par- 
ticularly designed form ofp(-) would be useful for further 
model development. 
The extended VBGF in Equation 5 can theoretically 
incorporate an unlimited number of parameters. How- 
ever, an increase in the number of free parameters 
in p(-) will be disadvantageous for model estimation 
because the functional form of p(-) does not directly 
appear in the age-length relationship. Increases in the 
variance of estimates imply instability due to curve 
fitting (Table 1). Therefore, it is necessary to consider 
methods of overcoming the trade-off between an in- 
creased number of parameters and estimation stability. 
Data sets other than those for length-at-age data will 
be useful for estimating the parameters in p(-) because 
the dynamics of p(-) are readily apparent in the behav- 
ior of df/dt (Fig. 3) rather than in length (Fig. 2). We 
expect that the robustness of this estimation will be 
improved by means of a combined likelihood-function 
(Martin and Cook, 1990; Eveson et ah, 2004) described 
by two heterogeneous relationships: length-at-age and 
reproductive energy-at-age. 
Our model development has similarities to that of 
Lester et al. (2004). Both studies explicitly give a 
growth function that can quantify a delay in somatic 
growth due to reproductive energy allocation. Lester 
et al. (2004) initially assumed a linear function of pre- 
mature growth in length and derived the conventional 
VBGF by introducing an intensive energy investment 
at postmature ages. Additionally, Lester et al. (2004) 
assumed that the ratio of gonad to body weight at 
postmature ages is constant. This assumption causes 
the linear function to yield a delay in growth after 
maturation equivalent to that yielded with the VBGF. 
Alternatively, our model derivation started from the 
VBGF. Additional hypotheses for model formulation 
other than ours and those of Lester et al. (2004) are 
possible. Hence, the adequacy of these assumptions for 
model derivation must be evaluated with a wide range 
of practical applications. 
