[Paper Review] Branching processes with pairwise interactions

Gabriel Berzunza Ojeda, Juan Carlos Pardo|arXiv (Cornell University)|Sep 24, 2020

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper introduces branching processes with pairwise interactions (BPI-processes), where individuals experience birth, death, cooperation, and competition events. Using moment duality and a Lamperti-type time change, the authors derive integral tests for explosion, extinction, coming down from infinity, and stationarity, identifying conditions under which the process admits a stationary distribution or exhibits regular boundary behavior at 0 and 1.

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.

Motivation & Objective

• To analyze the long-term behavior of branching processes that include pairwise cooperation and competition events, beyond standard pure branching processes.
• To determine conditions under which such processes explode, go extinct, or come down from infinity.
• To investigate the existence of a stationary distribution for BPI-processes.
• To establish a moment dual process that is a generalised Wright-Fisher diffusion, providing new insights into the dual dynamics.

Proposed method

• The authors use a random time change representation to link the BPI-process to a modified branching process with immigration.
• They apply moment duality to connect the BPI-process to a family of diffusions on [0,1], termed generalised Wright-Fisher diffusions.
• The infinitesimal generator of the BPI-process is defined with rates proportional to population size (natural birth/death) and to its square (cooperation/competition).
• Integral tests based on the interaction mechanism $\Phi_{c,b}(u)$ and branching mechanism $\Psi_{d,\rho}(u)$ are derived to classify boundary behavior at 0 and 1.
• The Lamperti transform is used to relate the BPI-process to a time-changed process, enabling the analysis of explosion and extinction.
• Feller's scale measure and speed measure criteria are applied to classify the boundary behavior at 0 and 1 based on convergence of integrals involving $\Phi_{c,b}$ and $\Psi_{d,\rho}$.

Experimental results

Research questions

• RQ1 Under what conditions does a BPI-process explode in finite time, and how does the interaction mechanism influence this?
• RQ2 When does the BPI-process almost surely go extinct, and how do cooperation and competition affect extinction probability?
• RQ3 When does the process come down from infinity, and what role does the interaction mechanism play in this behavior?
• RQ4 Does the BPI-process admit a stationary distribution, and what integral conditions govern its existence?
• RQ5 How does the boundary behavior at 0 (absorbing, reflecting, entrance, or exit) depend on the parameters $c$, $d$, and $b$?

Key findings

• The process explodes if and only if $\int_0^1 \int_0^x \frac{1}{y \Phi_{c,b}(y)} \, dy \, dx = \infty$, which depends on the behavior of $\Phi_{c,b}$ near 0.
• Extinction occurs almost surely if $\int_0^1 \int_0^x \frac{1}{y \Phi_{c,b}(y)} \, dy \, dx < \infty$, and this condition is linked to the interaction mechanism's integrability.
• The process comes down from infinity if $\int_0^1 \int_0^x \frac{1}{y \Phi_{c,b}(y)} \, dy \, dx < \infty$ and $\int_0^1 \frac{\Psi_{d,\rho}(x)}{x \Phi_{c,b}(x)} \, dx = \infty$, ensuring the process reaches finite populations from infinity.
• A stationary distribution exists if and only if $\int_0^1 \frac{\Psi_{d,\rho}(x)}{x \Phi_{c,b}(x)} \, dx < \infty$, which depends on the balance between branching and interaction mechanisms.
• The boundary at 0 is regular reflecting if $d > 0$ and $\Phi_{c,b}$ is bounded away from zero, while it is an entrance boundary if $d > c$, and an exit boundary if $d = 0$, depending on the relative strength of natural death and competition.
• When $d = c$, the boundary at 0 is an entrance boundary if $\mathcal{S}_{c,\rho}^{c,b}(\theta;0) = -\infty$, and a regular boundary otherwise, with regular reflection confirmed via the dual process.