  We propose a model to study the spatiotemporal dynamics of biocrust and
vegetation cover on sand dunes. 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. 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. Such irregular dynamics
can be associated with the temporal variations of dune cover. In addition,
similar behavior can be generated by other models that exhibit temporal
oscillations and bistability.
