  We consider the Lefever-Lejeune nonlinear lattice, a spatially discrete
propagation-inhibition model describing the growth of vegetation densities in
dry-lands. We analytically identify parametric regimes distinguishing between
decay (associated with spatial extinction of vegetation patches) and
potentially non-trivial time-asymptotics. To gain insight on the convergence
dynamics, a stability analysis of spatially uniform states is performed,
revealing the existence of a threshold for the discretization parameter which
depends on the lattice parameters, below which their destabilization occurs and
spatially non-uniform equilibrium states may emerge. Direct numerical
simulations justified that the analytical stability criteria and parametric
thresholds effectively describe the above transition dynamics and revealed the
rich structure of the equilibrium set. Connections with the continuous sibling
Lefever-Lejeune partial differential equation are also discussed.
