Importance of scaling exponents and other parameters in growth mechanism: an analytical approach

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.