Tree-Grass interactions dynamics and Pulse Fires: mathematical and numerical studies

Savannas are dynamical systems where grasses and trees can either dominate or coexist. Fires are known to be central in the functioning of the savanna biome though their characteristics are expected to vary along the rainfall gradients as observed in Sub-Saharan Africa. In this paper, we model the tree-grass dynamics using impulsive differential equations that consider fires as discrete events. This framework allows us to carry out a comprehensive qualitative mathematical analysis that revealed more diverse possible outcomes than the analogous continuous model. We investigated local and global properties of the equilibria and show that various states exist for the physiognomy of vegetation. Though several abrupt shifts between vegetation states appeared determined by fire periodicity, we showed that direct shading of grasses by trees is also an influential process embodied in the model by a competition parameter leading to bifurcations. Relying on a suitable nonstandard finite difference scheme, we carried out numerical simulations in reference to three main climatic zones as observable in Central Africa.Comment: 51 pages, 7 figure

A generic modeling of fire impact in a tree-grass savana model

We propose and study a model for tree-grass interactions in the context of savannas which are subjected to fire pressure. Several theoretical models in the literature which have highlighted the impact of fire on tree-grass interactions did not explicitly deal with the indirect feedback of dry grass biomass onto tree dynamics through fire intensity and frequency. The novelty in our work is to consider a fairly generic modeling of fire impact on woody biomass by means of a family of increasing and bounded functions of grass biomass. The characteristic feature of this family of functions is that, it could include several forms: linear as well as non-linear ones (sigmoidal or not). Since the nonlinear shape brings more diverse results than the previous attempts using a linear function, it could be used to show that several vegetation equilibria exist with some of them showing tree-grass coexistence features.We show that the number of equilibria with both grass and trees depends on the choice of the fire impact function. We also established thresholds defining the stability domains of the equilibria and highlighted some bifurcation parameters to provide numerical simulations complying with the theoretical properties of the model

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

International audienceSince 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 mean 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