A new deterministic model for the transmission dynamics of the lowly- and highly-pathogenic avian influenza (LPAI and HPAI) strains is designed and rigorously analyzed. The model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. It is shown that the re-infection of birds infected with the LPAI strain causes the backward bifurcation phenomenon. In the absence of such re-infection, the disease-free equilibrium of the model is globally-asymptotically stable when the associated reproduction number is less than unity. Using non-linear Lyapunov functions of Goh-Volterra type, the LPAI-only and HPAI-only boundary equilibria of the model are shown to be globally-asymptotically stable when they exist. A special case of the model is shown to have a continuum of co-existence equilibria whenever the associated reproduction numbers of the two strains are equal and exceed unity. Furthermore, numerical simulations of the model suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity.