Understanding the determinants of the emergence of boundaries between savannah and forest and their dependence on resource levels, precipitation and climatic effects is crucial in the context of current climate change. In this presentation, I will present our recent work on the mathematical analysis of equations that govern the interactions between savannah and forest. Starting from the simplest models based on differential equations, I will show the complexity of the possible behaviors that can emerge. In order to understand the spatio-temporal dynamics of these phenomena, I will present a detailed stochastic model and the theory allowing to establish macroscopic equations describing this interaction. I will conclude with our current work describing the dynamics that emerge in heterogeneous domains, modeling for example gradients of precipitation or natural resources, and which pose new mathematical questions that are still very little explored. This work was done in collaboration with Simon Levin of Princeton University, with the assistance of Denis Patterson (Durham) and Carla Staver (Yale).