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

Abstract

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.

On the degradation of forest ecosystems by extreme events: Statistical Model Checking of a hybrid model

“…We emphasize that these degraded forest dynamics cannot be described by a single deterministic model, and thus appear as the result of the antagonism between the deterministic and the probabilistic processes which constitute our hybrid model. On this point, our statements corroborate the results obtained with other approaches in the aforementioned papers [4], [18] and [28]. Therefore, the degradation pattern of the forest-climate ecosystem is described through an innovative and explanatory approach.…”

“…A great deal of work has already been done in this regard. For instance, the impacts of drought-driven wildfires have been studied for the Central Amazonian forest in [37]; the role of forest ecosystems in the fluxes of carbon have been analyzed in [5], [11] or [51]; hydrological services of the forests are investigated in [9] or in [10]; interactions with wildlife and biodiversity are studied in [33]; original models coupling the dynamics of trees and grass have also been analyzed in [4], [15], [24] or in [42], and it has been proved that their spatial structure well reproduces ecologically relevant patterns. Finally, the dynamics of transition from forest to savanna have been investigated for instance in [18], [28], [49].…”

A minimalistic model of vegetation physiognomies in the savanna biome

“…A former version of the present model has already inspired a spatially explicit model featuring local propagation of grass and tree biomass (e.g. clonal reproduction) through diffusion operators, taking into account continuous fire (Yatat et al [2018a]), or impulsive periodic fire (Yatat and Dumont [2018], Banasiak et al [2019]). Several studies (e.g.…”