Desertification could be described catastrophic responses of an ecosystem to environmental perturbations including internal and outgoing pressures (Fig 1). Desertification is a catastrophic and irreversible with collapse in vegetation patterns at a fast time scale, for instance as a consequence of a slow decrease of yearly rainfall, or through an increased grazing pressure (Fig 1). Arid and semiarid ecosystems have been characterized by regular patterns (periodic); we can find a predictable sequence of self organized patterns based on "Turing instability". Initially stable steady state of a dynamical system can become unstable if we additionally consider diffusion in the system. It is surprising and unexpected phenomenon because diffusion usually makes things more smooth and uniform. Loss of stability due to diffusion is called Turing instability, after Alan Turing who proposed this idea in his work in 1952. Turing instability can occur in reaction diffusion systems with self organization and pattern dynamics which can be considered a sign to forecast system behaviors. In areas with no preferred orientation of the pattern, the form of patterns shifts from gaps to labyrinths and to spots as the system becomes more degraded. In this article, has been explained desertification process into mathematical terms and relation of pattern formation with desertification and ecosystem collapse. It will be shown that the process of desertification poses fundamental mathematical questions and we are able to forecast this process by study of pattern shifts base on PDEs models. So in this paper has been introduced differential equations to estimate vegetation pattern dynamics. The results showed that vegetation patterns collapse into the desert state like Turing bifurcations. Also pattern dynamics is a sign for estimation equilibrium state of ecosystem to forecast desertification. The finding indicated that spotted vegetation patterns can be an early warning signal of imminent desertification in drylands. Also the process of desertification can be modeled by systems of reaction diffusion equations. PDEs/ODEs models introduced in this article show that as a critical transition to a barren state is neared, the vegetation becomes characterized by regular patterns because of Turing instability. These patterns change implying that this may be interpreted as early warning signal for a catastrophic bifurcation of desertification. The results recommend applications of the pattern formation approach to water limited landscapes to predict the possible emergence of spatial heterogeneity as a self-organization phenomenon. Also this article shows that the predicted spatial patterns can be periodic (spots, stripes and gaps), irregular with a characteristic length scale, or scale free. Results indicated that the pattern formation approach provides clear criteria for the realizations of these different pattern types in terms of environmental hazards considering land degradation and desertification. In addition to early warning signals for desertification, there are particular spatial patterns that can arise before a critical transition.

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