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

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Emerging Trends in Applied Mathematics and Mechanics Perpignan, May 30 - June 3, 2016

[MS 9: Mathematical Modelling, Analysis and Simulations in Biology]

A Mathematical Model of Tree-Grass interactions using Impulsive Differential Equations and non-linear feedback functions of grass biomass onto fire-induced

tree mortality

A. Tchuint´e Tamen1,3, Y. Dumont2, S. Bowong1,3, J.J. Tewa1,3, P. Couteron4 1 LIRIMA, GRIMCAPE, University of Yaounde 1, Cameroon

2 CIRAD, Umr AMAP, Montpellier, France

3 IRD, UMI 209, UMMISCO, IRD France Nord, Bondy, France 4 IRD, Umr AMAP, Montpellier, France

a.tchuinte@uy1.uninet.cm

Abstract: Explanations accounting for tree-grass interactions in savanna ecosystems are of-ten based on either facilitation/competition for resources, or differential sensitivity to fires. We propose a tree-grass model which accounts both phenomena (i.e. competition and fires). In the literature, several models used positive feedback between grass biomass (fuel load) and fire intensity, however, they did not incorporate nonlinear response of fire frequency/intensity to tree biomass. Our model explicitly considers the impact of pulse fire on tree biomass by mean of two non-linear impact functions including the response function of woody biomass to fire intensity. By considering two impact functions, our system yields complex dynamics, allowing for various possibilities of bistability and periodic equilibria in the ecosystem. We investigate local and global properties of equilibria. We highlight three parameters of bifur-cations: the fire period, the tree-grass competition parameter, and the fire intensity. Using an appropriate nonstandard numerical scheme, we provide numerical simulations to discuss some ecologically interesting cases our model is able to exhibit along a rainfall gradient. In order to represent spatial patterns observed in different environmental regions, a preliminary work that explicitly represent the spatial interactions among trees and grasses is introduced.

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