  The transition from localized to systemic spreading of bacteria, viruses and
other agents is a fundamental problem that spans medicine, ecology, biology and
agriculture science. We have conducted experiments and simulations in a simple
one-dimensional system to determine the spreading of bacterial populations that
occurs for an inhomogeneous environment under the influence of external
convection. Our system consists of a long channel with growth inhibited by
uniform UV illumination except in a small ``oasis'', which is shielded from the
UV light. To mimic blood flow or other flow past a localized infection, the
oasis is moved with a constant velocity through the UV-illuminated ``desert''.
The experiments are modeled with a convective reaction-diffusion equation. In
both the experiment and model, localized or extinct populations are found to
develop, depending on conditions, from an initially localized population. The
model also yields states where the population grows everywhere. Further, the
model reveals that the transitions between localized, extended, and extinct
states are continuous and non-hysteretic. However, it does not capture the
oscillations of the localized population that are observed in the experiment.
