On Phase-Transitions in Spatial Branching Systems With Interaction

We consider a spatial stochastic process with values in (N) S , where S is a countable Abelian group, for example Z d . The process is evolving like a binary branching random walk on Z d , which is supercritical and where in addition at each site after exponential waiting times all particles are killed. This is a population growth model with interaction between population and environment. The resulting process is again critical (mean preserving) for the appropriate choice of parameters and on this case we shall focus here. We call the process the coupled branching process. In a branching random walk on S particles migrate on S independently according to random walks and split or die after exponential rates. Let b be the splitting rate of a particle into two particles, (1 \Gamma p)b the death rate of individual particles and pb the rate of death of a whole colony (site killing), where p 2 [0; 1]. The different families (descendents of one original particle) do not evolve anymore ..