Metabolic rate beyond the 3/4 law

TL;DR

The paper addresses the limitation of fixed-metabolic-scaling exponents in describing ontogeny by introducing a Fibonacci-inspired discrete growth model where mass grows as $M(n)\sim M_0\phi^{n}$ and metabolism follows a stage-dependent law $B(n)=70\,M^{b(n)}$. It derives analytic forms for the exponent, including a refined $b(n)=\dfrac{(n-1)\log\phi-\log\sqrt{5}}{n\log\phi-\log\sqrt{5}}$ and the asymptotic $b_{\text{simp}}(n)=\dfrac{n-1}{n}$, showing $b(n)$ increases with development toward 1. The anchored framework recovers Kleiber's intercept while explaining ontogenetic deviations: early stages exhibit sublinear scaling, shifting toward near-linear scaling as individuals mature, with plausible metabolic rates across mammals. By applying this to nine species and comparing $B(n)$ with the classical $B_K=70\,M^{3/4}$, the work demonstrates a coherent, self-contained conduit between discrete growth, dynamic scaling, and developmental physiology, offering a complementary perspective to the WBE theory.

Abstract

In earlier work, we introduced a discrete Fibonacci-based ontogenetic model in which the metabolic scaling exponent $b(n)$ is treated as a dynamic function of an organism's developmental stage, and we estimated $b(n)$ for selected mammalian species. In the present article, we revisit this framework with a complementary aim. Rather than proposing new parameter estimates or statistical fits, we provide a didactic, step-by-step reconstruction of the derivation that leads from the recursive growth hypothesis to analytical expressions for the stage-dependent exponent $b(n)$. Building directly on these previously obtained exponents, we then incorporate Kleiber's classical result into the model by interpreting the constant $70$ in the law $B \approx 70\,M^{3/4}$ (with $B$ denoting basal metabolic rate and $M$ body mass) as a metabolic "anchoring point". This yields a stage-dependent basal metabolic rate of the form $B(n) = 70\,M^{b(n)}$, which defines an ontogenetic metabolic trajectory linking recursive growth to changes in scaling. We show, at a conceptual level, how this anchored formulation can describe a shift from strongly sublinear behavior at early stages towards an almost linear regime as development proceeds, while still producing basal rates that are compatible, in order of magnitude, with those reported for mammals of different sizes. In this way, the paper offers a self-contained and pedagogical presentation of the model, emphasizing how ontogenetic changes in metabolic rate can be understood through the combined ideas of Fibonacci-like recursion and metabolic anchoring.

Figures (1)

• Figure 1: Metabolic rate $B(n)$ computed from $B(n)=70\,M(n)^{b(n)}$ for $n=1,\dots,10$, assuming $M(n)\propto\phi^n$. Open circles and solid line denote $B_{\text{simp}}(n)$; filled squares and dashed line denote $B_{\text{ref}}(n)$