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Branching processes with pairwise interactions

Gabriel Berzunza Ojeda, Juan Carlos Pardo

Abstract

In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. Here, we provide a series of integral tests that explain how competition and cooperation regulate the long-term behaviour of BPI-processes. In particular, such integral tests describe the events of explosion and extinction and provide conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use a random time change representation in terms of a modified branching process with immigration and moment duality. The moment dual of BPI-processes turns out to be a family of diffusions taking values on $[0,1]$ which are interesting in their own right and that we introduce as generalised Wright-Fisher diffusions.

Branching processes with pairwise interactions

Abstract

In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. Here, we provide a series of integral tests that explain how competition and cooperation regulate the long-term behaviour of BPI-processes. In particular, such integral tests describe the events of explosion and extinction and provide conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use a random time change representation in terms of a modified branching process with immigration and moment duality. The moment dual of BPI-processes turns out to be a family of diffusions taking values on which are interesting in their own right and that we introduce as generalised Wright-Fisher diffusions.

Paper Structure

This paper contains 13 sections, 31 theorems, 154 equations, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Explosion
  3. Extinction
  4. Stationary distribution
  5. Coming-down from infinity
  6. Examples
  7. A modified branching process with migration and the Lamperti transform
  8. Explosion: Proofs of Theorem \ref{['Theo1']}, Theorem \ref{['Theo3']}, Proposition \ref{['Pro5']} and Proposition \ref{['lemma3']}
  9. Duality: Generalisations of Wright-Fisher diffusions
  10. Extinction and stationarity: Proof of Theorem \ref{['lemma4']}, Theorem \ref{['lemma6']}, Lemma \ref{['lemma11']} and Theorem \ref{['Lamtype']}
  11. Coming-down from infinity: Proof of Theorem \ref{['lemma9']}
  12. Appendix
  13. Acknowledgements.

Key Result

Theorem \oldthetheorem

Let $z \in \mathbb{N}$, $\rho>0$, $b\ge0$ and $\varsigma< 0$.

Theorems & Definitions (70)

  • Theorem \oldthetheorem
  • Corollary 1: LB-processes
  • Proposition 1
  • Theorem \oldthetheorem
  • Proposition 2
  • Corollary 2
  • Lemma 1
  • Lemma 2
  • Conjecture 1
  • Theorem \oldthetheorem: Extinction
  • ...and 60 more