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A subcritical pattern-forming system with nonlinear advection in a boundeddomain is recast as a slow-fast system in space and studied using a combinationof geometric singular perturbation theory and numerical continuation. Two typesof solutions describing the possible location of stationary fronts areidentified, whose origin is traced to the onset of convective and absoluteinstability when the system is unbounded. The former are present only fornonzero upstream boundary conditions and provide a quantitative understandingof noise-sustained structures in systems of this type. The latter correspond tothe onset of a global mode and are present even with zero upstream boundarycondition. The role of canard trajectories in the nonlinear transition betweenthese states is clarified and the stability properties of the resulting spatialstructures are determined. Front location in the convective regime is highlysensitive to the upstream boundary condition and its dependence on thisboundary condition is studied using a combination of numerical continuation andMonte Carlo simulations of the partial differential equation. Statisticalproperties of the system subjected to random or stochastic boundary conditionsat the inlet are interpreted using the deterministic slow-fastspatial-dynamical system.

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system