Continuous-state branching processes, extremal processes and super-individuals

TL;DR

This paper characterizes the long-term behavior of continuous-state branching processes using extremal processes and subordinators, highlighting the role of super-individuals that dominate population dynamics.

Contribution

It introduces the concept of super-individuals as record individuals influencing growth and extinction in continuous-state branching processes, linking extremal processes to population behavior.

Findings

Extremal processes describe long-term behaviors of branching flows.

Super-individuals correspond to Poisson process records impacting growth.

Super-individuals can cause infinite growth or slow extinction.

Abstract

The long-term behaviors of flows of continuous-state branching processes are characterized through subordinators and extremal processes. The extremal processes arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to $+\infty$ in the supercritical case, and slow down the rate of extinction in the subcritical one.