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

Abstract

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.

Spreading Speeds and Traveling Waves for Monotone Systems of Impulsive Reaction–Diffusion Equations: Application to Tree–Grass Interactions in Fire-prone Savannas

“…(50) and we check if it satisfies Hypotheses 2.1 of Li et al [29]. Recall that n,0 = (u n,0 , v n,0 ) and Q is the time--map solution operator of reaction-diffusion system (47). We consider the order interval…”

“…However, as pointed out in Yatat Djeumen et al [63], it is questionable whether it makes sense to model fire as a permanent forcing that continuously removes a fraction of the fire sensitive biomass. Indeed, since several months and even years can pass between two successive fires, they can be rather considered as instantaneous perturbations of the savanna ecosystem (see also Yatat Djeumen [58], Yatat Djeumen et al [60], Tchuinté et al [47,48]). Several recent papers have proposed to model fires either as stochastic events, while keeping the continuous-time differential equation framework (Baudena et al [7], Beckkage et al [8], Synodinos et al [46]), or by using a time-discrete model (Higgins et al [22], Accatino et al [1,2], Klimasara and Tyran-Kamińska [24]).…”

“…As previously, we deduce from properties (38) that w is a decreasing C 1 (ℝ) function such that For simplicity, we set (u n,0 , q n,0 ) = (u n (x, 0), q n (x, 0)) . We also set n,0 = (u n,0 , q n,0 ) and let Q denote the time--map operator solution of system (47). Then where (u, q) = (H 1 (u, q), H 2 (u, q)).…”

FKPP equation with impulses on unbounded domain

“…Actually, there exist several possibilities including the formalism presented in system (4) (see also Lakshmikantham et al (1989) [18], Bainov and Simeonov (1995) [4], Dumont and Tchuenche (2012) [10], Dufourd and Dumont (2013) [9], Tchuinté Tamen et al (2016) [35], (2017) [36], [47] and references therein) as well as the formalism advocated by Lewis and Li (2012) [19] (see also Vasilyeva et al (2012) [42], Fazly et al (2017) [11], Huang et al (2017) [14] and references therein). For τ-periodic impulsive harvest events, the inter-harvest saison (i.e.…”

“…It decreases with the rate µ (yr −1 ) due to external disturbances including human activities and herbivory. The fire-induced death rate for grass biomass is denoted by η. f (yr −1 ) is the fire frequency (see also Tchuinté Tamen et al (2014) [37], (2016) [35], (2017) [36] for similar assumptions). τ = 1/ f is the length of inter-fire saison, i.e.…”