An impulsive modelling framework of fire occurrence in a size-structured model of tree-grass interactions for savanna ecosystems

Fires and mean annual rainfall are major factors that regulate woody and grassy biomasses in savanna ecosystems. Within the savanna biome, conditions of long-lasting coexistence of trees and grasses have been often studied using continuous-time modelling of tree-grass competition. In these studies, fire is a time-continuous forcing while the relationship between woody plant size and fire-sensitivity is not systematically considered. In this paper, we propose a new mathematical framework to model tree-grass interactions that takes into account both the impulsive nature of fire occurrence and size-dependent fire sensitivity (via two classes of woody plants). We carry out a qualitative analysis that highlights ecological thresholds and bifurcation parameters that shape the dynamics of the savanna-like systems within the main ecological zones. Through a qualitative analysis, we show that the impulsive modelling of fire occurrences leads to more diverse behaviors including cases of grassland, savanna and forest tristability and a more realistic array of solutions than the analogous time-continuous fire models. Numerical simulations are carried out with respect to the three main ecological contexts (moist, mesic, semi-arid) to illustrate the theoretical results and to support a discussion about the bifurcation parameters and the advantages of the model.