Spatially explicit modelling of tree–grass interactions in fire-prone savannas: A partial differential equations framework

Yatat, Valaire; Couteron, Pierre; Dumont, Yves

doi:10.1016/j.ecocom.2017.06.004

Cited by 18 publications

(36 citation statements)

References 85 publications

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“…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 ).…”

Section: Application To Fire-prone Grasslandssupporting

confidence: 48%

“…Existence, unicity, positivity and boundedness of solutions of problem (4)-(2)-(3) follow directly from Proposition 1, Lemma 2 and Proposition 2 of Yatat et al (2017b) [46]. Similarly, existence and unicity of solutions of problem (5)- (2)- (3) hold according to [Rogovchenko (1997a) [28] [47]).…”

Section: Models Formulationmentioning

confidence: 74%

“…Existence of vegetation mosaics involving grasslands/savannas and forests in humid tropical regions is supported by several empirical evidences (see Yatat et al (2017b) [46] for more details). Moreover, in such regions fire is a major disturbance and its frequency is not expected to be greater than two (Bond and Keeley (2005) [6]).…”

Section: Numerical Simulationsmentioning

confidence: 75%

“…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.…”

Section: Application To Fire-prone Grasslandsmentioning

confidence: 99%

“…Impulsive reaction-diffusion equations can be used to model ecological systems dynamics such as fire-prone savannas. Readers are referred to Yatat et al (2017b) [46] for travelling waves in savanna reaction-diffusion models with time-continuous fire events. To the best of authors' knowledge, existence of travelling wave solutions for system of impulsive reaction-diffusion equations is not well documented except in Huang et al (2017) [14].…”

Section: Introductionmentioning

confidence: 99%

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FKPP equation with impulses on unbounded domain

Yatat

Dumont

2017

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This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reactiondiffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

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Section: Application To Fire-prone Grasslandssupporting

confidence: 48%

Section: Models Formulationmentioning

confidence: 74%

Section: Numerical Simulationsmentioning

confidence: 75%

Section: Application To Fire-prone Grasslandsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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FKPP equation with impulses on unbounded domain

Yatat

Dumont

2017

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Impulsive Fire Disturbance in a Savanna Model: Tree–Grass Coexistence States, Multiple Stable System States, and Resilience

Hoyer‐Leitzel

Iams

2021

Bull Math Biol

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Savanna ecosystems are shaped by the frequency and intensity of regular fires. We model savannas via an ordinary differential equation (ODE) encoding a one-sided inhibitory Lotka–Volterra interaction between trees and grass. By applying fire as a discrete disturbance, we create an impulsive dynamical system that allows us to identify the impact of variation in fire frequency and intensity. The model exhibits three different bistability regimes: between savanna and grassland; two savanna states; and savanna and woodland. The impulsive model reveals rich bifurcation structures in response to changes in fire intensity and frequency—structures that are largely invisible to analogous ODE models with continuous fire. In addition, by using the amount of grass as an example of a socially valued function of the system state, we examine the resilience of the social value to different disturbance regimes. We find that large transitions (“tipping”) in the valued quantity can be triggered by small changes in disturbance regime.

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

Banasiak

Dumont

Djeumen

2020

Differ Equ Dyn Syst

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Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction-diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction-diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

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Spatially explicit modelling of tree–grass interactions in fire-prone savannas: A partial differential equations framework

Cited by 18 publications

References 85 publications

FKPP equation with impulses on unbounded domain

FKPP equation with impulses on unbounded domain

Impulsive Fire Disturbance in a Savanna Model: Tree–Grass Coexistence States, Multiple Stable System States, and Resilience

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

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