The dynamics of ecosystems is of crucial importance. Various model-based approaches exist to understand and analyze their internal effects. In this paper, we model the space structure dynamics and ecological dynamics of meta-ecosystems using the formal technique of Graph Transformation (short GT). We build GT models to describe how a meta-ecosystem (modeled as a graph) can evolve over time (modeled by GT rules) and to analyze these GT models with respect to qualitative properties such as the existence of structural stabilities. As a case study, we build three GT models describing the space structure dynamics and ecological dynamics of three different savanna meta-ecosystems. The first GT model considers a savanna meta-ecosystem that is limited in space to two ecosystem patches, whereas the other two GT models consider two savanna meta-ecosystems that are unlimited in the number of ecosystem patches and only differ in one GT rule describing how the space structure of the meta-ecosystem grows. In the first two GT models, the space structure dynamics and ecological dynamics of the meta-ecosystem shows two main structural stabilities: the first one based on grassland-savanna-woodland transitions and the second one based on grassland-desert transitions. The transition between these two structural stabilities is driven by high-intensity fires affecting the tree components. In the third GT model, the GT rule for savanna regeneration induces desertification and therefore a collapse of the meta-ecosystem. We believe that GT models provide a complementary avenue to that of existing approaches to rigorously study ecological phenomena.