Metabolic rate beyond the 3/4 law

Abstract: This work presents a discrete theoretical model in which basal metabolic rate (B) is described as a dynamic function of an organism's ontogenetic stage (n). Instead of treating (B) only as a static function of body mass (M), we adopt the form (B(n) = B_0 \, M^{\,b(n)}), in which the effective scaling exponent (b(n)) varies systematically throughout development. In contrast to classical approaches, such as Kleiber's empirical law ((B \propto M^{3/4})) and the continuous fractal model of West--Brown--Enquist (WBE), which assume a constant exponent, the present framework emphasizes how the metabolic scaling relationship itself can evolve over the life cycle of a single individual. The model is inspired by a Fibonacci-based description of growth in discrete stages, leading to analytic expressions for (b(n)) that connect ontogenetic progression to changes in the scaling between metabolism and mass. In this setting, Kleiber's constant (B_0 \approx 70) kcal/day is reinterpreted as a \emph{metabolic anchoring point}, linking the classical law (B \approx 70\,M^{3/4}) to a developmentally explicit formulation. We show that the resulting trajectory (B(n)) captures, at a conceptual level, how metabolic scaling can shift from strongly sublinear behavior at early stages towards an almost linear regime as (n) increases, and that the predicted basal rates remain compatible, in order of magnitude, with values reported for mammals of different sizes. In this way, the work offers a unified framework that connects the evolution of (B(n)) across ontogeny to the recursive organization of biological growth.