Directed percolation and the extinction transition on a diffusive substrate

The extinction transition on a 1D heterogeneous substrate with diffusive correlations is studied. Diffusively correlated heterogeneity is shown to affect the location of the transition point, as the reactants adapt to the fluctuating environment. At the transition point the density decays like t−0.159, as in directed percolation. However, the scaling function describing the behavior away from the transition and other critical exponents shows significant deviations from the known DP behavior. It is suggested, thus, that the off-transition behavior of the system is governed by local adaptation to favored regions.