Branching random walk in a random time-independent environment

In a lattice population model, particles move randomly from one site to another as independent random walks, split into two offspring, or die. If duplication and mortality rates are equal and take the same value over all lattice sites, the resulting model is a critical branching random walk (characterized by a mean total number of offspring equal to 1). There exists an asymptotical statistical equilibrium, also called steady state. In contrast, when duplication and mortality rates take independent random values drawn from a common nondegenerate distribution (so that the difference between duplication and mortality rates has nonzero variance), then the steady state no longer exists. Simultaneously, at all lattice sites, if the difference between duplication and mortality rates takes strictly positive values with strictly positive probability, the total number of particles grows exponentially. The lattice Z(d) includes large connected sets where the duplication rate exceeds the mortality rate by a positive constant amount, and these connected sets provide the growth of the total population. This is the supercritical regime of branching processes. On the other hand, if the difference between duplication and mortality rates is almost surely negative or null except when it is almost surely zero, then the total number of particles vanishes asymptotically. The steady state can be reached only if the difference between duplication and mortality rates is almost surely zero.