A Geometrical Class of Population Growth Models, across Scales and Population Structures
- 1University of Exeter
Abstract
We develop a general geometrical framework for growth processes in which the growth rate depends exponentially on the number of active growth-supporting components. The framework separates the fraction of active components from the size of the accessible component space, allowing qualitatively different growth laws to arise from the same underlying construction. Bacterial population growth provides an example of a fixed-size component space controlled primarily by activation and deactivation, whereas examples such as human population growth or speciation provide an expanding component space in which interactions between individuals promote further growth. This motivates a Replicative geometrical law, dX/dt= X exp({beta} X^{gamma}), whose exponent characterizes the scaling of the effective growth-supporting component space and is tested against historical human-population regimes and eukaryotic lineage-through-time data. For innovation indicators such as patents and arXiv manuscript submissions, where an existing item need not itself reproduce, the same construction motivates a Direct Geometrical production law without the prefactor X. The framework therefore provides a common description of fixed and expanding component spaces while distinguishing Replicative from Direct Geometrical growth.
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BibTeX
@misc{tunstall2026geometrical,
title = {{A Geometrical Class of Population Growth Models, across Scales and Population Structures}},
author = {T. Tunstall and A. Zaccone},
howpublished = {bioRxiv},
year = {2026},
month = oct,
doi = {10.64898/2026.10.09.757927},
url = {https://www.biorxiv.org/content/10.64898/2026.10.09.757927v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - A Geometrical Class of Population Growth Models, across Scales and Population Structures
AU - T. Tunstall
AU - A. Zaccone
PY - 2026
DA - 2026/10/09
PB - bioRxiv
DO - 10.64898/2026.10.09.757927
UR - https://www.biorxiv.org/content/10.64898/2026.10.09.757927v1
AB - We develop a general geometrical framework for growth processes in which the growth rate depends exponentially on the number of active growth-supporting components. The framework separates the fraction of active components from the size of the accessible component space, allowing qualitatively different growth laws to arise from the same underlying construction. Bacterial population growth provides an example of a fixed-size component space controlled primarily by activation and deactivation, whereas examples such as human population growth or speciation provide an expanding component space in which interactions between individuals promote further growth. This motivates a Replicative geometrical law, dX/dt= X exp({beta} X^{gamma}), whose exponent characterizes the scaling of the effective growth-supporting component space and is tested against historical human-population regimes and eukaryotic lineage-through-time data. For innovation indicators such as patents and arXiv manuscript submissions, where an existing item need not itself reproduce, the same construction motivates a Direct Geometrical production law without the prefactor X. The framework therefore provides a common description of fixed and expanding component spaces while distinguishing Replicative from Direct Geometrical growth.
ER -