Simple mathematical models can exhibit rich and complex behaviors. Prototypical examples of these drawn from biology and other disciplines have provided insights that extend well beyond the situations that inspired them. Here, we explore a set of simple, yet realistic, models for savanna-forest vegetation dynamics based on minimal ecological assumptions. These models are aimed at understanding how vegetation interacts with both climate (a primary global determinant of vegetation structure) and feedbacks with chronic disturbances from fire. The model includes three plant functional types-grasses, savanna trees, and forest trees. Grass and (when they allow grass to persist in their subcanopy) savanna trees promote the spread of fires, which in turn, demographically limit trees. The model exhibits a spectacular range of behaviors. In addition to bistability, analysis reveals (i) that diverse cyclic behaviors (including limit and homo- and heteroclinic cycles) occur for broad ranges of parameter space, (ii) that large shifts in landscape structure can result from endogenous dynamics and not just from external drivers or from noise, and (iii) that introducing noise into this system induces resonant and inverse resonant phenomena, some of which have never been previously observed in ecological models. Ecologically, these results raise questions about how to evaluate complicated dynamics with data. Mathematically, they lead to classes of behaviors that are likely to occur in other models with similar structure.
Keywords: bifurcations; critical transitions; ecological dynamics; savanna; stochastic resonance.