This talk discusses two efficient parallel domain decomposition algorithms for fast solving the fully-mixed Stokes-Darcy-type models, which can decouple such systems into some more minor sub-physics problems naturally. The equivalence between the modified weak form and the original weak form of the fully mixed Stokes-Darcy coupling system is proved, which solves the rationality of decoupling space matching of multi-physical models. For the random Stokes-Darcy model, we utilize the Monte Carlo method for the coupled model with random inputs to derive some deterministic Stokes-Darcy numerical models and use the idea of the ensemble to realize the fast computation of multiple problems. One remarkable feature of the algorithm is that multiple linear systems share a common coefficient matrix in each deterministic numerical model, which significantly reduces the computational cost and achieves comparable accuracy with the traditional methods. Mesh-independent convergence rates of the algorithms are rigorously derived by choosing suitable Robin parameters. We also derive and analyze the optimized Robin parameters to accelerate the convergence of the proposed algorithms. In addition, the flow-vegetation interaction problem is currently receiving wide attention, which can also be seen as a multi-domain, multi-physics model. Therefore, at the end of this discussion, we have provided some ideas on the flow-vegetation interaction problem.