Critical Multitype Branching Systems: Extinction Results

TL;DR

This paper investigates extinction conditions for a critical multitype branching system with type-dependent movement, lifetime, and offspring distributions, revealing complex interactions affecting local extinction.

Contribution

It provides new extinction theorems for multitype branching systems with heavy-tailed and finite-mean lifetimes, incorporating mobility and interaction effects.

Findings

Extinction occurs under finite mean lifetime conditions.

Heavy-tailed lifetime distributions influence extinction behavior.

Interaction between mobility and longevity determines local extinction in complex cases.

Abstract

We consider a critical branching particle system in $\R^d$, composed of individuals of a finite number of types $i\in\{1,...,K\}$. Each individual of type $i$ moves independently according to a symmetric $\alpha_i$-stable motion. We assume that the particle lifetimes and offspring distributions are type-dependent. Under the usual independence assumptions in branching systems, we prove extinction theorems in the following cases: (1) all the particle lifetimes have finite mean, or (2) there is a type whose lifetime distribution has heavy tail, and the other lifetimes have finite mean. We get a more complex dynamics by assuming in case (2) that the most mobile particle type corresponds to a finite-mean lifetime: in this case, local extinction of the population is determined by an interaction of the parameters (offspring variability, mobility, longevity) of the long-living type and those of the most mobile type. The proofs are based on a precise analysis of the occupation times of a related Markov renewal process, which is of independent interest.