Universal curve, biological time, and dynamically varying scaling exponent in growth law

A general quantitative model for ontogenetic growth of organisms was derived by West et al. based on fundamental principles for the allocation of metabolic energy in which a universal parameterless growth curve was established in terms of the biomass ratio and a dimensionless biological time. This model was then extended by Guiot et al. to account for the growth of tumors in vivo   in which the fractional scaling exponent pp becomes a dynamic parameter depending on time tt. In this paper, we present a method that may be used effectively to construct a generalized universal growth curve for such a growth law model with an arbitrary dynamically varying fractional exponent. As by-products of this method, we show that the assumption that a biological time variable flows forward in the universal curve allows us to predict the behavior of pp with regard to the developmental stages of organisms characterized by body mass which is consistent with the findings of Guiot et al. based on biomedical data, and that, the universal curve is independent of the properties of pp as a function of tt. We also consider the situation when time increases discretely and the fractional scaling exponent is periodic. We observe the familiar, but superimposed, period-doubling and transition-to-chaos phenomenon.