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In this work, we improve a previous minimalistic tree-grass savanna model bytaking into account water availability, in addition to fire, since both factorsare known to be important for shaping savanna physiognomies along a climaticgradient. As in our previous models, we consider two nonlinear functions ofgrass and tree biomasses to respectively take into account grass-firefeedbacks, and the response of trees to fire of a given intensity. The noveltyis that rainfall is taken into account in the tree and grass growth functionsand in the biomass carrying capacities. Then, we derive a qualitative analysisof the ODE model, showing existence of equilibria, and studying their stabilityconditions. We also construct a two dimension bifurcation diagram based onrainfall and fire frequency. This led to summarize different scenarios for themodel including multi-stabilities that are proven possible. Next, to bring morerealism in the model, pulsed fire events are modelled as part of an IDE(Impulsive differential Equations) system analogous to the ODE system.Numerical simulations are provided and we discuss some important ecologicaloutcomes that our ODE and IDE models are able to predict. Notably, theexpansion of forest into tree-poor physiognomies (grassland and savanna) issystematically predicted when fire return period increases, especially in mesicand humid climatic areas.

Effect of rainfall and fire frequency on tree--grass dynamics: Capturing the forest--savanna distributions along biogeographic gradients