Mathematical Analysis of a Size Structured Tree-Grass Competition Model for Savanna Ecosystems

Abstract

Abstract-Several continuous-time tree-grass competition models have been developed to study conditions of long-lasting coexistence of trees and grass in savanna ecosystems according to environmental parameters such as climate or fire regime. In those models, fire intensity is a fixed parameter while the relationship between woody plant size and fire-sensitivity is not systematically considered. In this paper, we propose a mathematical model for the tree-grass interaction that takes into account both fire intensity and size-dependent sensitivity. The fire intensity is modeled by an increasing function of grass biomass and fire return time is a function of climate. We carry out a qualitative analysis that highlights ecological thresholds that summarize the dynamics of the system. Finally, we develop a non-standard numerical scheme and present some simulations to illustrate our analytical results.

Tree–grass interaction dynamics and pulsed fires: Mathematical and numerical studies

“…The broad objective of this study is to examine the influence of pulse events with regard to fires impact on the Tree-Grass dynamics along the rainfall gradient in Africa. Tree-grass savanna models can not be studied without the important role of fires (Tilman 1994 [21]; Higgins et al 2000 [22]; Sankaran et al 2004 [23], 2005 [24]; D' Odorico et al 2006 [25]; Accatino et al 2010 [26]; Beckage et al 2011 [27]; Staver et al 2011 [28]; Yatat et al 2014 [29] and Tchuinte et al 2014 [30]). This paper extends our earlier work (Tchuinte et al 2014 [30]) where we consider a continuous tree-grass interaction model that featured a fairly generic family of non-linear functions of grass biomass to model fire intensity and its impact on tree.…”

“…We highlight thresholds that summarize the dynamics of the model and explain the theoretical meaning of these thresholds. Prior to illustrate our theoretical results numerically, in section 4, based on the scheme developed in [29], we develop a reliable nonstandard finite difference method (NSFD) that preserves the qualitative properties of the system (Anguelov et al 2012 [31], 2013 [32], 2014 [33]). Section 6 concludes the paper.…”

FKPP equation with impulses on unbounded domain

“…In this section, we apply results of section 3 in order to study grassland dynamics subjected to fire events. As it was discussed in Yatat et al (2017b) [46], in the contexts where fires are frequent (humid savannas), they generally occur in specific periods in the year (Jeffery et al (2014) [15]), while in protected areas (mesic savannas) fires are often set in the first part of the dry season (Scholes and Walker (1993) [48], (2017) [47], (2017b) [46], we assume that the grass biomass (G, in t.ha −1 ) evolves following a logistic growth where the carrying capacity is K (t.ha −1 ). Grass biomass originates from the existing grass biomass (intrinsic growth) with the rate γ (yr −1 ).…”

“…Modelling of grassland dynamics is usually encompassed in models of tree-grass interactions in savanna ecosystems (Walker et al (1981) [44], Tilman (1994) [ [48], (2017) [47], (2017b) [46] and references therein). See also Tchuinté Tamen et al (2017b) [38] for an overview of spatially-implicit tree-grass interactions models.…”

Modelling and optimization for palm oil plantation management

“…The problem of optimal harvesting time for a biological population consisting of individuals of different sizes was studied in [9], and applied to farming in Mexico. The coexistence of trees and grass in savanna was modelled by size dependent model [10].…”