A Geometrical Class of Population Growth Models, across Scales and Population Structures

A new geometrical framework has been developed to describe population growth processes where the rate of growth is exponentially dependent on the number of active components. This framework allows for the separation of the fraction of active components from the size of the accessible component space, leading to distinct growth laws that emerge from the same underlying construction.

One example of a fixed-size component space is bacterial population growth, which is primarily controlled by activation and deactivation. On the other hand, human population growth and speciation involve an expanding component space where interactions between individuals facilitate further growth. This observation leads to the formulation of a Replicative geometrical law, expressed as dX/dt = X exp({beta} X^{gamma}), where the exponent characterizes the scaling of the effective growth-supporting component space.

This law has been tested against historical human-population regimes and data on the development of eukaryotic lineages over time.

The same framework also applies to innovation indicators, such as patents and arXiv manuscript submissions, where the existing item itself does not reproduce. In these cases, a Direct Geometrical production law without the prefactor X can be derived from the same construction. Therefore, this geometrical framework offers a unified description of both fixed and expanding component spaces, while simultaneously distinguishing between Replicative and Direct Geometrical growth.