Mixed state branching evolution for cell division models

TL;DR

This paper establishes a scaling limit theorem for two-type interacting Galton-Watson processes, leading to mixed state branching processes modeling cell division with parasites, including conditions for extinction and ergodicity.

Contribution

It introduces a new class of mixed state branching processes derived from two-type Galton-Watson models with interaction, with proofs of extinction and ergodicity conditions.

Findings

Derived a scaling limit theorem for interacting two-type processes

Characterized conditions for extinction with probability one

Proved exponential ergodicity in total variation distance

Abstract

We prove a scaling limit theorem for two-type Galton-Waston branching processes with interaction. The limit theorem gives rise to a class of mixed state branching processes with interaction using to simulate the evolution for cell division affected by parasites. Such process can also be obtained by the pathwise unique solution to a stochastic equation system. Moreover, we present sufficient conditions for extinction with probability one and the exponential ergodicity in the total variation distance of such process.