Propagation dynamics of a free boundary problem with advection and preferred density

This paper mainly studies a reaction-diffusion-advection free boundary problem with a preferred density boundary condition, which models the spatial spread of a species. The species tends to maintain a specific density δ > 0 at the spreading fronts, and this condition significantly affects the invasion dynamics. We establish the well-posedness of the solution and rigorously prove a trichotomy for population evolution: spreading occurs when 0 < δ < 1 , transition occurs when δ = 1 , and vanishing happens when δ > 1 . The method used for the case δ = 1 also resolves Conjecture 4 in [19] in the symmetric version. In the spreading case, we precisely obtain for the first time the asymptotic speeds of the left and right spreading fronts and their dependence on the advection strength β . Moreover, we reveal a structural transition in the left propagation pattern from a semi-wave to a traveling wave when β exceeds a threshold β ⁎ . This study systematically elucidates the coupling influence of the preferred density and advection effects on biological invasion dynamics.