Abstract: Multiple ecosystems (grassland, desert and shrubs) always coexist in ecologically fragile areas. Influence by external environmental conditions and human activities, one ecosystem in ecologically fragile areas would be changed into another, called abrupt change. It is important to analyze the stability of each ecosystem in ecologically fragile areas under the condition of multiple ecosystems coexistence for understanding ecosystem changes. In this article, we reviewed several mathematical models that can describe the dynamic system and stability of multiple ecosystems in ecologically fragile areas. The dynamic system described by differential equations included the uniform and non-uniform ecosystems, as well as multi-ecosystems with the existence of forests, grasses, and deserts. There were three types of methods for analyzing the stability of the ecosystem. One type was the analysis method of ecosystem stability caused by the initial perturbation that could represent human activities, and the other type was the analysis method of ecosystem stability caused by the model perturbation that can represent the changes of environmental factors (precipitation, temperature, etc.), and the third type was the analysis method of ecosystem stability caused by the combination of initial perturbation and model perturbation. Finally, we introduced how to analyze the stabilities of ecosystem in ecologically fragile areas with those methods. These mathematical methods not only help us understand the stability of the ecosystems in the ecologically fragile area, but also provide theoretical guidance for the abrupt change of the ecological system, future prediction, and management.

Key words: ecologically vulnerable area, alternative stable states, abrupt, stability analysis method, nonlinear optimization method