A new look at duality for the symbiotic branching model

TL;DR

This paper revisits the duality in the symbiotic branching model, providing a new interpretation that extends to the infinite branching rate limit, revealing large-scale behavior and explicit limits under certain conditions.

Contribution

It introduces a novel interpretation of the duality, enabling analysis of the model as the branching rate approaches infinity, which was not previously accessible.

Findings

Extended duality to infinite branching rate limit

Explicit characterization of the limit system as annihilating Brownian motions

Identified the limit behavior under perfectly negatively correlated noises

Abstract

The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate $\gamma$ is sent to infinity. This limit is particularly interesting since it captures the large scale behaviour of the system. As an application of the duality, we can explicitly identify the $\gamma = \infty$ limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.