The coalescing colony model: mean-field, scaling, and geometry

TL;DR

This paper develops a mean-field and scaling analysis of a coalescing colony model, examining how the primary colony's growth and shape are affected by emission rates, with implications for ecology, tumor growth, and urban development.

Contribution

It introduces a mean-field framework for the coalescing colony model, derives scaling exponents for different emission rate dependencies, and assesses the validity of the circular approximation.

Findings

Mean-field equations derived for primary colony dynamics.

Scaling exponents depend on the emission rate parameter $ heta$.

Circular approximation valid for constant emission rate, breaks down when emission depends on perimeter.

Abstract

We analyze the coalescing model where a `primary' colony grows and randomly emits secondary colonies that spread and eventually coalesce with it. This model describes population proliferation in theoretical ecology, tumor growth and is also of great interest for modeling the development of cities. Assuming the primary colony to be always spherical of radius $r(t)$ and the emission rate proportional to $r(t)^\theta$ where $\theta>0$, we derive the mean-field equations governing the dynamics of the primary colony, calculate the scaling exponents versus $\theta$ and compare our results with numerical simulations. We then critically test the validity of the circular approximation and show that it is sound for a constant emission rate ($\theta=0$). However, when the emission rate is proportional to the perimeter, the circular approximation breaks down and the roughness of the primary colony can not be discarded, thus modifying the scaling exponents.