A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve

Abstract

Abstract. Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the AldousShields model [1]. Fix n ≥ 1 and β > 0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate β(n−k)/n, where k is the distance from the node to the root. Denote by Zn(t) the number of nodes with no descendants at time t and let Tn = β −1 n ln(n/ ln 4)+(ln 2)/(2β). We prove that 2 −n Zn(Tn + nτ ), τ ∈ R, converges to the Gompertz curve exp(−(ln 2) e −βτ ). We also prove a central limit theorem for the martingale associated to Zn(t).

Doubly Stochastic Yule Cascades (Part I): The explosion problem in the time-reversible case

“…Recently, such models have also been considered for important cellular biology questions related to ageing and cancer, where generational dependent cell division rates occur and decrease with generations; see e.g. [6,22] and other related references in the medical and biological literature. Remark 1.5.…”