223 
A note on the von Bertalanffy 
growth function concerning 
the allocation of surplus energy 
to reproduction 
Shuhei Ohnishi (contact author ) 1 
Takashi Yamakawa 2 
Hiroshi Okamura 3 
Tatsuro Akamine 4 
Email address for contact author: ohnishi @scc.u-tokai.ac. ip 
1 School of Marine Science and Technology 
Tokai University 
Shimizu, Shizuoka 424 8610, Japan 
2 Graduate School of Agricultural and Life Sciences 
The University of Tokyo 
Bunkyo, Tokyo 1 13-8657, Japan 
3 National Research Institute of Far Seas Fisheries 
Fisheries Research Agency 
Yokohama, Kanagawa 236-8648, Japan 
4 National Research Institute of Fisheries Science 
Fisheries Research Agency 
Yokohama, Kanagawa 236-8648, Japan 
Abstract— We propose an extended 
form of the von Bertalanffy growth 
function (VBGF), where the allocation 
of surplus energy to reproduction is 
considered. Any function can be used 
in our model to describe the ratio of 
energy allocation for reproduction to 
that for somatic growth. As an exam- 
ple, two models for energy allocation 
were derived: a step-function and a 
logistic function. The extended model 
can jointly describe growth in adult 
and juvenile stages. The change in 
growth rate between the two stages 
can be either gradual or steep; the 
latter gives a biphasic VBGF. The 
results of curve fitting indicated 
that a consideration of reproduc- 
tive energy is meaningful for model 
extension. By controlling parameter 
values, our comprehensive model gives 
various growth curve shapes ranging 
from indeterminate to determinate 
growth. An increase in the number of 
parameters is unavoidable in practical 
applications of this new model. Addi- 
tional information on reproduction 
will improve the reliability of model 
estimates. 
Manuscript submitted 17 February 2011. 
Manuscript accepted 5 December 2011. 
Fish. Bull. 110:223-229(2012). 
The views and opinions expressed 
or implied in this article are those of the 
author (or authors) and do not necessarily 
reflect the position of the National Marine 
Fisheries Service, NOAA. 
The von Bertalanffy growth function 
(VBGF) has been used to analyze 
somatic growth data in a wide range 
of studies. It is now commonly put into 
practice to partially reparameterize 
the VBGF parameters to avoid their 
covariation and to ensure statistical 
accuracy (Quinn and Deriso, 1999). 
Although variations in the growth 
rate influenced by extrinsic envi- 
ronmental fluctuations have been 
examined in many studies, we pro- 
pose that intrinsic physiological dy- 
namics are also of great importance. 
The interaction between growth rate 
and sexual maturation has often 
been debated in life history studies 
(Roff, 1984; Beverton, 1992; Jensen, 
1996). Mathematical treatments for 
representing the switch in energy 
allocation between growth and re- 
production have been introduced to 
discuss optimal life-history strategies 
(Roff, 1983; Kozlowski, 1992, 1996; 
Kozlowski and Teriokhin, 1999). 
When the energy budget of fishes 
has been quantified, dynamics of the 
energy distribution between growth 
and reproduction have often been 
considered in simulated scenarios 
(Jorgensen and Fiksen, 2006; Pec- 
querie et ah, 2009). 
The effects of reproductive energy 
should also be important for practi- 
cal curve fitting studies. A biphasic 
growth model derived by connect- 
ing two independent VBGFs at an 
arbitrary age is often employed for 
curve fitting (Soriano et ah, 1992; 
Porch et ah, 2002; Araya and Cu- 
billos, 2006; Quince et ah, 2008a, 
2008b; Alos et ah, 2010; Tribuzio et 
ah, 2010). Although a biphasic VBGF 
is one approach used to account for 
inflections in growth and is similar 
to the higher-parameter model (Sch- 
nute and Richards, 1990), results of 
model selection based on the Akaike 
information criterion (AIC; Akaike, 
1973) often indicate that the bipha- 
sic VBGF is a more suitable model 
than the original monophasic VBGF 
(Porch et ah, 2002; Araya and Cubil- 
los, 2006; Tribuzio et ah, 2010). The 
better fit implies that the delay in 
growth due to a reallocation of en- 
ergy may be detected as a change in 
the growth trajectory. 
