Multi-type spatial branching models for local self-regulation I: Construction and an exponential duality

TL;DR

This paper develops a spatial multi-type branching model with migration and state-dependent reproduction, analyzing its diffusion limit, proving existence and uniqueness of solutions, and establishing a new exponential duality.

Contribution

It introduces a new exponential duality for spatial multi-type branching models and analyzes the diffusion limit, extending previous models with a focus on local self-regulation.

Findings

Existence of the infinite particle model established.

Uniqueness of solutions proved in the exchangeable case.

A new exponential duality for the model is developed.

Abstract

We consider a spatial multi-type branching model in which individuals migrate in geographic space according to random walks and reproduce according to a state-dependent branching mechanism which can be sub-, super- or critical depending on the local intensity of individuals of the different types. The model is a Lotka-Volterra type model with a spatial component and is related to two models studied in \cite{BlathEtheridgeMeredith2007} as well as to earlier work in \cite{Etheridge2004} and in \cite{NeuhauserPacala1999}. Our main focus is on the diffusion limit of small mass, locally many individuals and rapid reproduction. This system differs from spatial critical branching systems since it is not density preserving and the densities for large times do not depend on the initial distribution but mainly on the carrying capacities. We prove existence of the infinite particle model and the system of interacting diffusions as solutions of martingale problems or systems of stochastic equations. In the exchangeable case in which the parameters are not type dependent we show uniqueness of the solutions. For that purpose we establish a new exponential duality.