Non-Hermitian delocalization and the extinction transition

Logistic growth on a static heterogenous substrate is studied both above and below the drift-induced delocalization transition. Using stochastic, agent-based simulations the delocalization of the highest eigenfunction is connected with the large $N$ limit of the stochastic theory, as the localization length of the deterministic theory controls the divergence of the spatial correlation length at the transition. Any finite colony made of discrete agents is washed away from a heterogeneity with compact support in the presence of strong wind, thus the transition belongs to the directed percolation universality class. Some of the difficulties in the analysis of the extinction transition in the presence of a localized active state are discussed.