Metabolic rate beyond the 3/4 law

TL;DR

This paper revisits a Fibonacci-based ontogenetic model of metabolic scaling, providing a detailed derivation of stage-dependent exponents and incorporating a metabolic anchoring point to describe developmental shifts in metabolic rates.

Contribution

It offers a clear, pedagogical derivation of a stage-dependent metabolic rate model that links recursive growth to metabolic scaling, extending previous work with a new anchoring interpretation.

Findings

The model describes a shift from sublinear to near-linear metabolic scaling during development.

Stage-dependent basal metabolic rates align with empirical data for mammals.

The anchored formulation captures developmental changes in metabolic rates across stages.

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.