State Dependent Multitype Spatial Branching Processes and their Longtime Behavior

Abstract

The paper focuses on spatial multitype branching systems with spatial components (colonies) indexed by a countable group, for example $Z^d$ or the hierarchical group. As type space we allow continua and describe populations in one colony as measures on the type space. The spatial components of the system interact via migration. Instead of the classical independence assumption on the evolution of different families of the branching population, we introduce interaction between the families through a state dependent branching rate of individuals and in addition state dependent mean offspring of individuals. However for most results we consider the critical case in this work. The systems considered arise as diffusion limits of critical multiple type branching random walks on a countable group with interaction between individual families induced by a branching rate and offspring mean for a single particle, which depends on the total population at the site at which the particle in question is located.

The main purpose of this paper is to construct the measure valued diffusions in question, characterize them via well-posed martingale problems and finally determine their longtime behavior, which includes some new features. Furthermore we determine the dynamics of two functionals of the system, namely the process of total masses at the sites and the relative weights of the different types in the colonies as system of interacting diffusions respectively time-inhomogeneous Fleming-Viot processes. This requires a detailed analysis of path properties of the total mass processes.

In addition to the above mentioned systems of interacting measure valued processes we construct the corresponding historical processes via well-posed martingale problems. Historical processes include information on the family structure, that is, the varying degrees of relationship between individuals.

Ergodic theorems are proved in the critical case for both the process and the historical process as well as the corresponding total mass and relative weights functionals. The longtime behavior differs qualitatively in the cases in which the symmetrized motion is recurrent respectively transient. We see local extinction in one case and honest equilibria in the other.

This whole program requires the development of some new techniques, which should be of interest in a wider context. Such tools are dual processes in randomly fluctuating medium with singularities and coupling for systems with multi-dimensional components.

The results above are the basis for the analysis of the large space-time scale behavior of such branching systems with interaction carried out in a forthcoming paper. In particular we study there the universality properties of the longtime behavior and of the family (or genealogical) structure, when viewed on large space and time scales.