Generalized contact process on random environments

Spreading from a seed is studied by Monte Carlo simulation on a square lattice with two types of sites affecting the rates of birth and death. These systems exhibit a critical transition between survival and extinction. For time-dependent background, this transition is equivalent to those found in homogeneous systems (i.e., to directed percolation). For frozen backgrounds, the appearance of the Griffiths phase prevents the accurate analysis of this transition. For long times in the subcritical region, the spreading remains localized in compact (rather than ramified) patches, and the average number of occupied sites increases logarithmically in the surviving trials.