224 
Fishery Bulletin 1 10(2) 
Day and Taylor (1997) and Czarnot^ski and Kozlowski 
(1998) identified the lack of an explicit formula for the 
reproductive process in the VBGF. Although the bipha- 
sic VBGF is an empirical approach, a deductive model 
that can incorporate both growth and reproduction 
should be developed to help to understand the process of 
energy allocation and to improve curve fit. In this study, 
we begin with an extension of the VBGF with respect 
to a continuous change in energy allocation. We also 
present an application of curve fitting and model selec- 
tion. An overview of changes in growth-curve shapes 
is subsequently shown. Finally, we briefly discuss the 
features of our model. 
Methods 
We start with the general form of VBGF given by 
= hw 2/> - kw, (1) 
dt 
where w, t, h, and k are body weight, age, and coefficients 
of anabolism and catabolism, respectively. The right 
hand side of Equation 1 is the total production rate of 
surplus energy. 
If we consider the reallocation of surplus energy for 
reproduction. Equation 1 can be expanded as 
— + c- = hw 2/3 -kw. (2) 
dt dt 
Two newly introduced terms, f and c, denote the cumula- 
tive energy investment for reproduction until age t and 
the conversion factor of reproductive energy to body 
weight, respectively. Note that f is not equivalent to the 
weight of the gamete (i.e., eggs or spermatozoa). Equa- 
tion 2 is equivalent to the exoskeleton growth model 
(Ohnishi and Akamine, 2006) in that energy allocation 
to activities or appendages unrelated to somatic growth 
are explicitly described. 
Suppose w=fH 3 , where l is body length and /3 is a con- 
stant proportionality coefficient. Dividing dw/dt=3pl 2 dl/ 
dt by each side of Equation 2 and substituting w=f3l 3 
yields the following equation: 
ctt 
dt 
dw df 'l 
+ c — — 
dt dt ) 
K(L-l), 
(3) 
where K=k/3 and l oa (=hk~ 1 j3~ 1/3 ) is the asymptotic length. 
Let p be the ratio (0<p<l) of energy invested to reproduc- 
tion against total surplus energy such that 
P = 
dw +c dl\ l 
dt dt J 
Hence, Equation 3 becomes 
f t =(l- P)K(L-l). 
(4) 
(5) 
Equation 5 comprehensively describes two types of life 
history strategies, which can be generally classified 
as determinate and indeterminate growth (Lincoln et 
ah, 1998). It tends towards determinate growth when 
p is close to 1.0 and otherwise towards indeterminate 
growth. The value of the parameter p increases with 
sexual maturation, and it can be replaced by p(w), p(l), 
or p(t) as a function of size or age. In particular, a math- 
ematical treatment is easy when p^pit). Given that 1=0 
at t=t 0 (the initial condition), the general form of the 
growth function is given as 
l = l„ (l - e KT{I) \ , where T(t) = [ jl-p(s)}o?s. (6) 
It should be noted that pit) can take an arbitrary func- 
tional form with 0<p(t)<l. 
Among the various possible forms of pit), we propose 
the following two models that are relatively easy to 
derive. The first is a model where pit) exhibits a discon- 
tinuous change in age at maturity t m , such that pit )= 0 
it<t m ) and pit)=v it>t rn ), where 0<e<l. In this case, Tit) 
is defined by 
t-h (*<0 
T(t) = j . (7) 
t-t 0 -v(t-t m ) ( t>t m ) 
Equation 7 represents the time delay to attain a cer- 
tain body size in t>t m due to change in energy alloca- 
tion. Consequently, the growth curve becomes biphasic, 
combining two independent VBGFs. 
The alternative model assumes that pit) changes 
continuously throughout an individual’s lifetime. In 
particular, an S-type curve that has an inflection point 
around t=t m is suitable for describing a change in pit) 
due to sexual maturation. Let pit) be p(D = u/(l+exp(- 
ait-t m ))) as a general logistic curve such that the ana- 
lytical solution for Tit) is given by 
T(t) = (l-o)(t-t„)- (g) 
where u and a are the upper limit of the allocation rate 
in reproductive energy and the rapidity of maturation, 
respectively. The logistic function converges to a step- 
function when a —> By inspection, Equation 7 is a 
special case of Equation 8. 
The solution for Equation 5 is complicated when 
pspil) (or p^piw)) such that 
|0£i,< " ,9) 
The explicit solution for / is a biphasic VBGF when pit) 
is a step-function that has discontinuous switching at 
