The infinite rate symbiotic branching model: from discrete to continuous space

TL;DR

This paper demonstrates that the infinite rate symbiotic branching model on discrete space converges to its continuous counterpart under diffusive rescaling, linking discrete and continuous models and analyzing interface behavior.

Contribution

It establishes the convergence of the discrete infinite rate model to the continuum model through diffusive rescaling, bridging a significant gap in the understanding of these systems.

Findings

Discrete model converges to continuum model under rescaling

Initial ordering of types is preserved in the limit

Interface between types reduces to a single point

Abstract

The symbiotic branching model describes a spatial population consisting of two types that are allowed to migrate in space and branch locally only if both types are present. We continue our investigation of the large scale behaviour of the system started in Blath, Hammer and Ortgiese (2016), where we showed that the continuum system converges after diffusive rescaling. Inspired by a scaling property of the continuum model, a series of earlier works initiated by Klenke and Mytnik (2010, 2012) studied the model on a discrete space, but with infinite branching rate. In this paper, we bridge the gap between the two models by showing that by diffusively rescaling this discrete space infinite rate model, we obtain the continuum model from Blath, Hammer and Ortgiese (2016). As an application of this convergence result, we show that if we start the infinite rate system from complementary Heaviside initial conditions, the initial ordering of types is preserved in the limit and that the interface between the types consists of a single point.