Bifurcation Analysis in Multi-Species Reaction–Convection–Diffusion Systems Under Nonlinear Boundary Constraints

In this paper, we establish a bifurcation framework for a multi-species reaction–diffusion–advection system with nonlinear boundary constraints, revealing two fundamental pattern-generation mechanisms: (i) flow-velocity-triggered symmetry breaking and (ii) boundary-mediated Turing instability. Through Crandall–Rabinowitz theory combined with spectral graph analysis, we prove that advection-diffusion coupling governs pattern selection, while Lyapunov–Schmidt reduction derives universal equations for nascent patterns. Crucially, nonlinear boundary interactions are shown to induce hybrid patterns unobtainable in classical boundary-value systems. Stability analysis demonstrates non-monotonic dependence on advection strength and boundary sensitivity parameters, resolving paradoxical stabilization phenomena in ecological flows. These results provide a mathematical foundation for predicting spatial segregation patterns in advection-dominated ecosystems.