Spine for interacting populations and sampling

TL;DR

This paper introduces a spine construction method for Markov jump processes in structured populations, enabling detailed analysis of population evolution, competition, and ancestral lineages through a probabilistic change of measure.

Contribution

It develops a novel spine construction for Markov jump processes that incorporates population state information, facilitating analysis of population dynamics and ancestral lineages.

Findings

Derived the diagram phase of a growth fragmentation model with competition

Analyzed the growth of size in birth-death processes with multiple births

Described ancestral lineages in multitype populations

Abstract

We consider Markov jump processes describing structured populations with interactions via density dependance. We propose a Markov construction with a distinguished individual which allows to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of type space of individuals to include the state of the population. The jump rates outside the spine are also modified. We apply this approach to some issues concerning evolution of populations and competition. For single type populations, we derive the diagram phase of a growth fragmentation model with competition and the growth of the size of birth and death processes with multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.