Hook(両方のoの頭に¨),M., Li,J., Oba,N. and Snowden,S.(2011): Descriptive and predictive growth curves in energy system analysis. Natural Resources Research, 20(2), 103-116.


 This study reviews a variety of growth curve models and the theoretical frameworks that lay behind them. In many systems, growth patterns are, or must, ultimately be subjected to some form of limitation. A number of curve models have been developed to describe and predict such behaviours. Symmetric growth curves have frequently been used for forecasting fossil fuel production, but others have expressed a need for more flexible and asymmetric models. A number of examples show differences and applications of various growth curve models. It is concluded that these growth curve models can be utilised as forecasting tools, but they do not necessarily provide better predictions than any other method. Consequently, growth curve models and other forecasting methods should be used together to provide a triangulated forecast. Furthermore, the growth curve methodology offers a simple tool for resource management to determine what might happen to future production if resource availability poses a problem. In the light of peak oil and the awareness of natural resources being considered as a basis for the continued well-being of the society and the mankind, resource management should be treated as an important factor in future social planning.

Key words: Growth curve models; curve fitting; logistic model; resource management』

Aim of this study
Theoretical background
 General growth modes
 Unbounded growth
 Bounded growth
 Bell-shaped growth
Examples of growth curves
Applications of growth curves
Concluding discussion

Figure 1. Three different growth modes: Unbounded, bounded and bell-shaped growths behave similarly in the beginning before saturation starts to play a significant role.

Table 1. General properties of selected growth models


Equation for y(t)


Point of inflection

Brody A*(1-b*exp(-kt)) 1 Not defind A
Bertalanffy A*(1-b*exp(-kt))3 3 8/27≒0.30 A
Logistic A*(1+b*exp(-kt))-1 -1 0.5 A
Gompertz A*exp(-b*exp(-kt)) M→∞ e-1≒0.37 A
Richards A*(1±b*exp(-kt))M Variable [(M-1)/M]M A

Table 2. Growth rate properties of selected models


Equation for y(t)

growth rate

growth rate
Brody A*(1-b*exp(-kt)) kA(1-U(t)) k*(u-1-1)
Bertalanffy A*(1-b*exp(-kt))3 3ky(u-1/3-1) 3k(u-1/3-1)
Logistic A*(1+b*exp(-kt))-1 ky(1-u) k(1-u)
Gompertz A*exp(-b*exp(-kt)) ky*ln(u-1) k*ln(u-1)
Richards A*(1±b*exp(-kt))M Mky(u-1/M-1) Mk(u-1/M-1)
For simplicity, u=y/A is used.

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