Summary: General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated as nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of J. Pejsachowicz
and P. J. Rabier
[J. Anal. Math. 76, 289–319 (1998; Zbl 0932.47046
)] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.