In the present study, the individual growth process of an organism has been shown with the help of a mathematical model. The surplus energy production rate, i.e. intake rate minus metabolic cost, plays a crucial role in controlling the growth rate. Considering the existence of an optimum mass, which maximizes the surplus energy, it has been found that the scaling exponent for the metabolic cost has to be greater than the exponent for the intake rate. On the basis of the consideration that the system always generates some surplus energy, a relationship among the empirical constants has been established. The growth is found to continue with an ever decreasing rate. When the system attains its optimum mass, the growth rate is found to be the maximum. The mass variation with time has been graphically shown using the expression obtained by solving a differential equation involving surplus energy. Using figures, the dependence of mass variation upon various scaling parameters, has been thoroughly discussed. As mass increases, the surplus energy production rate per unit mass is found to decrease and this may be the probable reason for the smaller number of organisms with larger mass. As the scaling exponent regarding intake increases, the maximum attainable mass increases along with an increase in the time required for its attainment.

The growth process of a living organism is studied with the help of a mathematical model where a part of the surplus power is assumed to be used for growth. In the present study, the basic mathematical framework of the growth process is based on a pioneering theory proposed by von Bertalanffy and his work is the main intellectual driving force behind the present analysis. Considering the existence of an optimum size for which the surplus power becomes maximum, it has been found that the scaling exponent for the intake rate must be smaller than the exponent for the metabolic cost. A relationship among the empirical constants in allometric scaling has also been established on the basis of the fact that an organism never ceases to generate surplus energy. The growth process is found to continue forever, although with a decreasing rate. Beyond the optimum point the percentage of shortfall in energy has been calculated and its dependence on scaling exponents has been determined. The dependence of optimum mass on the empirical constants has been shown graphically. The functional dependence of mass variation on time has been obtained by solving a differential equation based on the concept of surplus energy. The dependence of the growth process on scaling exponent and empirical constants has been shown graphically.