Life and death near a windy oasis

We propose a simple experiment to study delocalization and extinction in inhomogeneous biological systems. The nonlinear steady state for, say, a bacteria colony living on and near a patch of nutrient or favorable illumination ("oasis") in the presence of a drift term ("wind") is computed. The bacteria, described by a simple generalization of the Fisher equation, diffuse, divide A --> A + A, die A --> 0, and annihilate A + A --> 0. At high wind velocities all bacteria are blown into an unfavorable region ("desert"), and the colony dies out. At low velocity a steady state concentration survives near the oasis. In between these two regimes there is a critical velocity at which bacteria first survive. If the "desert" supports a small nonzero population, this extinction transition is replaced by a delocalization transition with increasing velocity. Predictions for the behavior as a function of wind velocity are made for one and two dimensions.