We investigate the role of migratory birds in the spread of H5N1 avian influenza, focusing on the interaction of a migratory bird species with nonmigratory poultry. The model is of patch type and is derived with the aid of reaction-advection equations for the migratory birds in the air along the flyways. Poultry may reside at some or all of the four patches of the model, which consist of the breeding patch for the migratory birds, their winter feeding patch, and two stopover patches where birds rest and refuel on their migration. Outward and return migratory routes can be different. The equations for the migratory birds contain time delays which represent the flight times for migratory birds along particular sectors. Delays also appear in the model coefficients via quantities which represent flight survival probabilities for the various sectors. We establish results on positivity, boundedness, global asymptotic stability of the disease-free equilibrium, and the persistence of infection. We also discuss extensions of the model to include the seasonality of the migration phenomenon. Numerical simulations support the analytical findings; here we used data on H5N1 infected ducks in the Poyang Lake region of China.