May 2024 Spine for interacting populations and sampling
Vincent Bansaye
Author Affiliations +
Bernoulli 30(2): 1555-1585 (May 2024). DOI: 10.3150/23-BEJ1645


We consider some Markov jump processes which model structured populations with interactions via density dependence. We propose a Markov construction involving a distinguished individual (spine) which allows us to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of the type space of individuals to include the state of the population. The jump rates off the spine individual can also be modified. We exploit this approach to study issues concerning population dynamics. For single type populations, we derive the phase diagram of a growth fragmentation model with competition as well as the growth of the size of transient birth and death processes which permit multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.

Funding Statement

This work was partially funded by the Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MNHN-F.X and ANR ABIM 16-CE40-0001.


The idea of this project started during IX Escuela de Probabilidad y Procesos Estocásticos, in Mexico in 2018. The author is grateful to the organizers for the invitation. The author is also grateful to Simon Harris, Andreas Kyprianou and Bastien Mallein for stimulating discussions on this topic. Finally, the author thanks Madeleine Kubasch for corrections and the referees and the editor for their suggestions and comments.


Download Citation

Vincent Bansaye. "Spine for interacting populations and sampling." Bernoulli 30 (2) 1555 - 1585, May 2024.


Received: 1 July 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699564
Digital Object Identifier: 10.3150/23-BEJ1645

Keywords: interactions , jump Markov process , Martingales , populations , Positive semigroup , Random tree , Spine


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Vol.30 • No. 2 • May 2024
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