  We propose a model equation for the dynamics of tree density in mesic
savannas. It considers long-range competition among trees and the effect of
fire acting as a local facilitation mechanism. Despite short-range facilitation
is taken to the local-range limit, the standard full spectrum of spatial
structures obtained in general vegetation models is recovered. Long-range
competition is thus the key ingredient for the development of patterns. The
long time coexistence between trees and grass, and how fires affect the
survival of trees as well as the maintenance of the patterns is studied. The
influence of demographic noise is analyzed. The stochastic system, under the
parameter constraints typical of mesic savannas, shows irregular patterns
characteristics of realistic situations. The coexistence of trees and grass
still remains at reasonable noise intensities.
