Vegetation and biocrust play crucial roles in dune stability and mobility, and their interaction can lead to bistability, temporal oscillations, and hysteresis. We studied a two-dimensional (2D) mathematical model of vegetation and biogenic crust cover dynamics on sand dunes. Under a certain parameter range, the space-independent version of the model exhibited the bi-stability of an oscillatory state and a steady state, and we studied the 2D dynamics of the model under these parameters. The patterns developed by the 2D model showed a high degree of spatial heterogeneity and complexity depending on the initial conditions and on the state type across the front. The results suggest that spatial heterogeneity and complexity can evolve from the intrinsic dynamics between vegetation and biocrust, even without natural geodiversity and spatiotemporal climate fluctuations. In the real world, these two types of intrinsic and extrinsic heterogeneity processes interact such that it is difficult to distinguish between them.