A minimalistic model of tree–grass interactions using impulsive differential equations and non-linear feedback functions of grass biomass onto fire-induced tree mortality

Abstract

Since savannas are important ecosystems around the world, their long term dynamics is an important issue, in particular when perturbations, like fires, occur more or less often. In a previous paper, we developed and studied a tree–grass model that take into account fires as pulse events using impulsive differential equations. In this work, we propose to improve this impulsive model by considering the impact of pulse fire on tree biomass by means of combination of two nonlinear functions of grass and tree biomasses respectively. By considering two impact functions, our model yields more complex dynamics, allowing for the possibility of various bistabilities and periodic solutions, in either grassland or savanna states in the ecosystem. Our mathematical analysis allows extensive and realistic description of savannas ecosystems, than previous modelling approaches. We also highlight several threshold parameters that summarize all possible dynamics, as well as three main parameters of bifurcations in the tree–grass dynamics: the fire period, the tree–grass facilitation/competition parameter, and the fire intensity. Using an appropriate nonstandard numerical scheme, we provide numerical simulations to discuss some ecologically interesting cases that our model is able to exhibit along a rainfall gradient, observable in Central Africa.