A multitype Markovian branching process with one-type population size dependence
Abstract
Motivated by a within-host framework of immunity-modulated parasitic disease, we formulate a multitype Markovian branching process with reproductive parameters dependent on the number of individuals of a single type only. We impose a soft carrying capacity K with respect to the controlling type, serving as a threshold between subcritical and supercritical dynamics. We prove that extinction occurs almost surely when the offspring batch of each non-controlling type includes an individual of the controlling type with positive probability, with the time to extinction possessing finite moments of all orders. We then construct a sequence of density-dependent processes indexed by K and study scaling limits of the process using the canonical functional law of large numbers (FLLN) and central limit theorem, with a view towards characterising the emergence of metastable behaviour in the vicinity of asymptotically stable equilibria, or stable limit cycles of the FLLN.
Related
- The Q-process for the general mother-dependent multitype branching process
- On conservativity of multitype continuous-state branching processes
- Weighted moments and sharp convergence rates for supercritical multitype branching processes
- Convergence of a Critical Multitype Branching Particle System with Mixed Finite and Infinite Mean Lifetimes
- Limit theorems for supercritical branching processes in Markovian environment with linear fractional offspring distributions
BibTeX
@misc{mehra2026multitype,
title = {{A multitype Markovian branching process with one-type population size dependence}},
author = {Somya Mehra and Peter G. Taylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12567},
eprint = {2609.12567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12567},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A multitype Markovian branching process with one-type population size dependence
AU - Somya Mehra
AU - Peter G. Taylor
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12567
UR - https://arxiv.org/abs/2609.12567
AB - Motivated by a within-host framework of immunity-modulated parasitic disease, we formulate a multitype Markovian branching process with reproductive parameters dependent on the number of individuals of a single type only. We impose a soft carrying capacity K with respect to the controlling type, serving as a threshold between subcritical and supercritical dynamics. We prove that extinction occurs almost surely when the offspring batch of each non-controlling type includes an individual of the controlling type with positive probability, with the time to extinction possessing finite moments of all orders. We then construct a sequence of density-dependent processes indexed by K and study scaling limits of the process using the canonical functional law of large numbers (FLLN) and central limit theorem, with a view towards characterising the emergence of metastable behaviour in the vicinity of asymptotically stable equilibria, or stable limit cycles of the FLLN.
ER -