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

In this work, we improve a previous minimalistic tree-grass savanna model by taking into account water availability, in addition to fire, since both factors are known to be important for shaping savanna physiognomies along a climatic gradient. As in our previous models, we consider two nonlinear functions of grass and tree biomasses to respectively take into account grass-fire feedbacks, and the response of trees to fire of a given intensity. The novelty is that rainfall is taken into account in the tree and grass growth functions and in the biomass carrying capacities. Then, we derive a qualitative analysis of the ODE model, showing existence of equilibria, and studying their stability conditions. We also construct a two dimension bifurcation diagram based on rainfall and fire frequency. This led to summarize different scenarios for the model including multi-stabilities that are proven possible. Next, to bring more realism 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 ecological outcomes that our ODE and IDE models are able to predict. Notably, the expansion of forest into tree-poor physiognomies (grassland and savanna) is systematically predicted when fire return period increases, especially in mesic and humid climatic areas.