State Dependent Multitype Spatial Branching Processes and their Longtime Behavior
Donald A. Dawson, Carleton University
Andreas Greven, Universitat Erlangen-Nurnberg
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.
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