Metabolic Scaling, Growth and Fibonacci-Based Models
Research on body mass, metabolism, ontogenetic growth and Fibonacci dynamics.
This research line investigates how body mass, metabolism and organismal growth can be related through mathematical models, with particular emphasis on the metabolic scaling exponent and a developmental variable associated with the golden ratio.
Comparison between the empirical range of the metabolic scaling exponent, the approximation
\( b(n) = \frac{n-1}{n} \), the refined expression and the classical WBE value
\( b = 0.75 \).
Figure included as a visual highlight of the article Metabolic Scaling From Fibonacci Dynamics.
Ontogenetic Growth: A Fibonacci-Based Ontogenetic Discretization of Body-Mass Growth Trajectories
In this work, the Fibonacci-based idea is extended from metabolic scaling to
body-mass growth trajectories. Biological growth remains a continuous
process; the discretization is applied to the ontogenetic coordinate
used to represent development, rather than directly to body mass.
The continuous description begins with the relation between body mass and the ontogenetic coordinate:
\[m(t)=m_0\varphi^{n(t)}\]
One possible saturating dynamics for this coordinate is
The formulation starts from a continuous coordinate \(n(t)\) and a discretized version
divided into small substages. When the resolution \(\Delta n\) is small, the discretized
trajectory approaches the continuous curve. The visible steps should therefore be interpreted
as a finite-resolution representation of developmental progression, not as real biological
jumps in body mass.
Main result: the Fibonacci-based formulation reproduced the general shape of
the growth trajectories for guinea pig, guppy, hen and cow. In the quantitative comparison,
the Fibonacci formulation was favored in two of the four cases (Guppy and Hen), whereas the
WBE model was favored for Guinea pig and Cow. Even in those cases, the Fibonacci curves
closely followed the empirical growth patterns.
The GIFs below illustrate, across organisms with very different body-mass and developmental
time scales, how the Fibonacci-based discretized description can follow the continuous growth
trajectory as the ontogenetic resolution is refined.
Guinea pig. Comparison between observed data, the continuous reference curve and the Fibonacci-based ontogenetic discretization.Guppy. A small-body-mass example showing the discretized representation approaching the continuous trajectory.Hen. The discretization represents progression through ontogenetic substages without implying real jumps in body mass.Cow. The same formalism is applied to a trajectory with much larger body mass and growth time.
The aim is not to replace classical growth models universally, but to provide an alternative
ontogenetic coordinate for organizing and analyzing growth trajectories across biological systems.
For sufficiently large values of \(n\), Fibonacci numbers satisfy approximately
\[F_n\approx\frac{\varphi^n}{\sqrt{5}}\]
The golden ratio therefore provides a simple mathematical structure for constructing a sequence of scales and testing quantitative relationships between metabolism and growth.
