We propose a model to study at the first time the spatiotemporal dynamics of the coupling between biocrust and vegetation cover on sand dunes; previous studies modeled the temporal dynamics of vegetation-biocrust-sand system while other focused only on the spatiotemporal dynamics of vegetation on sand dunes, excluding the effect of biocrust. The model consists of two coupled partial nonlinear differential equations and includes diffusion and advection terms for modeling the dispersal of vegetation and biocrust and the effect of wind on them. In the absence of spatial variability, the model exhibits self-sustained relaxation oscillations and regimes of bistability-the first state is dominated by biocrust and the second by vegetation. We concentrate on the one-dimensional dynamics of the model and show that the front that connects these two states propagates mainly due to the wind advection. In the oscillatory regime the front propagation is complex and very interesting compared to the non-spatial relaxation oscillations. For low wind DP (drift potential) values, a series of spatially oscillatory domains develops as the front advances downwind. These domains form due to the oscillations of the spatially homogeneous states away from the front. However, for higher DP values, the dynamics is much more complex, becoming very sensitive to the initial conditions and exhibiting an irregular spatial pattern as small domains are created and annihilated during the front advance. The irregular spatiotemporal dynamics reported here seems to be unique, at least in the context of vegetation dynamics and possibly also in context of other dynamical systems.