  We consider an infinite time horizon spatially distributed optimal harvesting
problem for a vegetation and soil water reaction diffusion system, with
rainfall as the main external parameter. By Pontryagin's maximum principle we
derive the associated four component canonical system, and numerically analyze
this and hence the optimal control problem in two steps. First we numerically
compute a rather rich bifurcation structure of flat (spatially homogeneous) and
patterned canonical steady states (FCSS and PCSS, respectively), in 1D and 2D.
Then we compute time dependent solutions of the canonical system that connect
to some FCSS or PCSS. The method is efficient in dealing with non-unique
canonical steady states, and thus also with multiple local maxima of the
objective function. It turns out that over wide parameter regimes the FCSS,
i.e., spatially uniform harvesting, are not optimal. Instead, controlling the
system to a PCSS yields a higher profit. Moreover, compared to (a simple model
of) private optimization, the social control gives a higher yield, and
vegetation survives for much lower rainfall. In addition, the computation of
the optimal (social) control gives an optimal tax to incorporate into the
private optimization.
