The coalescent tree of a Markov branching process with generalised logistic growth

TL;DR

This paper analyzes the coalescent tree structure of a broad class of density-dependent branching processes, including logistic and Gompertz growth, showing convergence to a universal limiting tree as population parameters grow large.

Contribution

It introduces a unified framework for the coalescent trees of various density-dependent growth models and proves their convergence to a universal limiting tree.

Findings

Coalescent trees converge to a universal limit as population size and sampling scale increase.

The model generalizes classical exponential, logistic, and Gompertz growth processes.

The limiting tree is robust across different density-dependent growth dynamics.

Abstract

We consider a class of density-dependent branching processes which generalises exponential, logistic and Gompertz growth. A population begins with a single individual, grows exponentially initially, and then growth may slow down as the population size moves towards a carrying capacity. At a time while the population is still growing superlinearly, a fixed number of individuals are sampled and their coalescent tree is drawn. Taking the sampling time and carrying capacity simultaneously to infinity, we prove convergence of the coalescent tree to a limiting tree which is in a sense universal over our class of models.