A note on the von Bertalanffy growth function concerning the allocation of surplus energy to reproduction.
|Publication:||Name: Fishery Bulletin Publisher: National Marine Fisheries Service Audience: Academic Format: Magazine/Journal Subject: Zoology and wildlife conservation Copyright: COPYRIGHT 2012 National Marine Fisheries Service ISSN: 0090-0656|
|Issue:||Date: April, 2012 Source Volume: 110 Source Issue: 2|
|Topic:||Event Code: 310 Science & research|
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 example, 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 reproductive 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. Additional information on reproduction
will improve the reliability of model estimates.
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 environmental fluctuations have been examined in many studies, we propose that intrinsic physiological dynamics 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 reproduction 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; Pecquerie et al., 2009).
The effects of reproductive energy should also be important for practical curve fitting studies. A biphasic growth model derived by connecting two independent VBGFs at an arbitrary age is often employed for curve fitting (Soriano et al., 1992; Porch et al., 2002; Araya and Cubillos, 2006; Quince et al., 2008a, 2008b; A15s et al., 2010; Tribuzio et al., 2010). Although a biphasic VBGF is one approach used to account for inflections in growth and is similar to the higher-parameter model (Schnute and Richards, 1990), results of model selection based on the Akaike information criterion (AIC; Akaike, 1973) often indicate that the biphasic VBGF is a more suitable model than the original monophasic VBGF (Porch et al., 2002; Araya and Cubillos, 2006; Tribuzio et al., 2010). The better fit implies that the delay in growth due to a reallocation of energy may be detected as a change in the growth trajectory.
Day and Taylor (1997) and Czarnoleski and Kozlowski (1998) identified the lack of an explicit formula for the reproductive process in the VBGF. Although the biphasic 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 selection. An overview of changes in growth-curve shapes is subsequently shown. Finally, we briefly discuss the features of our model.
We start with the general form of VBGF given by
dw / dt = [hw.sup.2/3] - kw, (1)
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
dw/dt + c df/dt = [hw.sup.2/3] -kw. (2)
Two newly introduced terms, f and c, denote the cumulative 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). Equation 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=[beta][l.sup.3], where l is body length and [beta] is a constant proportionality coefficient. Dividing dw/dt=3 [beta][l.sup.2] dl/ dt by each side of Equation 2 and substituting w = [beta][l.sup.3] yields the following equation:
dl/dt = (dw/dt) [(dw/dt + c df/dt).sup.-1] K ([l.sub.[infinity]] - l), (3)
where K=k/3 and [l.sub.[infinity]] (=[hk.sup.-1] [[beta].sup.-1/3]) is the asymptotic length. Let p be the ratio (0 [less than or equal to] p [less than or equal to] 1) of energy invested to reproduction against total surplus energy such that
p = (c df/dt) [(dw/dt + c df/dt).sup.-1]. (4)
Hence, Equation 3 becomes
dl/dt = (1-p)K([l.sub.[infinity]] - l). (5)
Equation 5 comprehensively describes two types of life history strategies, which can be generally classified as determinate and indeterminate growth (Lincoln et al., 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 mathematical treatment is easy when p [equivalent to] p(t). Given that l=0 at t=[t.sub.0] (the initial condition), the general form of the growth function is given as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)
It should be noted that p(t) can take an arbitrary functional form with 0 [less than or equal to] p (t) [less than or equal to] <1.
Among the various possible forms of p(t), we propose the following two models that are relatively easy to derive. The first is a model where p(t) exhibits a discontinuous change in age at maturity [t.sub.m], such that p(t)=0 (t<[t.sub.m]) and p(t)=v (t [greater than or equal to] [t.sub.m]), where 0 [less than or equal to] v [less than or equal to] 1. In this case, T(t) is defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
Equation 7 represents the time delay to attain a certain body size in t [greater than or equal to] [t.sub.m] due to change in energy allocation. Consequently, the growth curve becomes biphasic, combining two independent VBGFs.
The alternative model assumes that p(t) changes continuously throughout an individual's lifetime. In particular, an S-type curve that has an inflection point around t=[t.sub.m] is suitable for describing a change in p(t) due to sexual maturation. Let p(t) be p(t) = v/(1+exp(a(t-[t.sub.m]))) as a general logistic curve such that the analytical solution for T(t) is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)
where v 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[right arrow][infinity]. By inspection, Equation 7 is a special case of Equation 8.
The solution for Equation 5 is complicated when p [equivalent to] p(l) (or p [equivalent to] p(w)) such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
The explicit solution for l is a biphasic VBGF when p(l) is a step-function that has discontinuous switching at the boundary of mature "size." In most cases, however, 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 follows by rearranging Equation 4 with the condition p [equivalent to] p(t) as
(1 - p(t)) c df/dt = p(t) dw/dt.
Substituting dw/dt=3[beta][l.sup.2]dl/dt and Equation 5 into this equation, one obtains
df/dt = 3[beta]K/c p(t)[l.sup.2] ([l.sub.[infinity]] - l), (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)
Equation 10 represents the instantaneous reproductive 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.sub.i] (i=1, ..., N), where N is the total number of samples. Parameters were estimated by minimizing the residual sum of squares of S=[[SIGMA].sub.i] [([L.sub.i] - [l.sub.i]).sup.2]. The numerical optimization for S was accomplished by using the quasi-Newton method (BFGS algorithm) in "optim( )" with R statistical software (R Development Core Team, 2011). The comparison between this model and the original monophasic VBGF was based on the AIC value as follows: AIC=NlogS+20, where [theta] 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 summarized in Table 1 and Figure 1. As shown in Table 1, the AIC difference ([DELTA]AIC=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., [l.sub.[infinity]] and [t.sub.0]) in the extended model was larger than that in the conventional VBGF (Table 1).
Twelve types of energy allocation schedules, p(t), and the corresponding somatic growth (in length l) based on different combinations of parameter values in Equation 8 (v=0.4, 0.6, 0.8, 1.0 and a=l, 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 original 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., v=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 indeterminate growth in Figure 2. When v=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 maturation 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.sub.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/dt and shifts the maximum df/dt to older ages. When v=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.sub.m] (Fig. 3, A, D, G, and J), whereas higher a values yield a steeper initial rise in df/dt around [t.sub.m] (Fig. 3, C, F, I, and L).
[FIGURE 1 OMITTED]
A notable feature of our model is that energy allocation can be quantified by the arbitrary functional form p(x). The introduction of p(x) provides a unified platform to treat the trade-off between somatic growth and reproduction. 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(x), 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, v, and [t.sub.m], or a particularly designed form of p(x) would be useful for further model development.
The extended VBGF in Equation 5 can theoretically incorporate an unlimited number of parameters. However, an increase in the number of free parameters in p(x) will be disadvantageous for model estimation because the functional form of p(x) 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 increased 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(x) are readily apparent in the behavior 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 al., 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 premature 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.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
The authors would like to thank to Y. Narimatsu of Tohoku National Fisheries Research Institute, Fisheries Research Agency, who provided us with the willowy flounder data for analysis. The authors also express our gratitude to the anonymous reviewers who provided valuable, insightful suggestions for the improvement of the manuscript.
Manuscript submitted 17 February 2011.
Manuscript accepted 5 December 2011.
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Shuhei Ohnishi (contact author) (1)
Takashi Yamakawa (2)
Hiroshi Okamura (3)
Tatsuro Akamine (4)
Email address for contact author: email@example.com
(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 113-8657, 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
Table 1 Parameter estimates for two types of von Bertalanffy growth functions (VBGFs). Both types of VBGF have three common parameters: asymptotic length ([l.sub.[infinity]]), growth coefficient (K), and initial condition of age ([t.sub.0]). Additional parameters, namely age at maturity ([t.sub.m]), the upper limit of the allocation rate in reproductive energy (v), and rapidity of maturation ([alpha]), 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 [DELTA]AIC shows the relative difference of the Akaike information criterion (AIC) value compared with the minimum AIC. [[l.sub. Type of VBGF [infinity]] K [t.sub.0] [t.sub.m] Conventional model 260.72 0.34 -0.45 -- (1.51) (0.01) (0.04) Extended model 463.01 0.15 -0.77 3.41 (49.08) (0.01) (0.14) (0.35) Type of VBGF v [alpha] AIC [DELTA]AIC Conventional model -- -- 26881.3 20.8 Extended model 0.79 1.01 26860.5 0 (0.05) (0.21)
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