  We argue that the spatial discretization of the strongly nonlinear
Lefever-Lejeune partial differential equation defines a nonlinear lattice that
is physically relevant in the context of the nonlinear physics of ecosystems,
modelling the dynamics of vegetation densities in dry lands. We study the
system in the lattice $\mathbb{Z}^2$, which is especially relevant because of
its natural dimension for the emergence of pattern formation. Theoretical
results identify parametric regimes for the system that distinguish between
extinction and potential convergence to non-trivial states. Importantly, we
analytically identify conditions for Turing instability, detecting thresholds
on the discretization parameter for the manifestation of this mechanism.
Numerical simulations reveal the sharpness of the analytical conditions for
instability and illustrate the rich potential for pattern formation even in the
strongly discrete regime, emphasizing the importance of the interplay between
higher dimensionality and discreteness.
