FKPP equation with impulses on unbounded domain

Abstract

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.

A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial scales in spite of scarce data

“…However, as an improvement of [109], there are also some ongoing works that aim to deal with existence of travelling waves for system of Impulsive Partial Differential Equations (IPDE) that model savanna dynamics (e.g. [111]). This type of modelling is of great interest considering that the forest-savanna ecotone is the most widespread in the tropics and that forests have been encroaching during the last decades in many humid savannas of West and Central Africa, and to a lesser extent in other regions of the world (Oliveras and Malhi (2016) [72]).…”

“…Based on Table IV, page 18, we further consider in our group (Yatat et al (2017) [110], (2018) [109], [111], Tchuinté Tamen (2016) [89], (2017) [90], (2018) [88]) impulsive time-periodic fire events which is an approximation that keeps the potential of analytical investigation as large as possible while modelling discrete fires. An impulsive differential equations system can be used to express fire through impul-sive periodic occurrences (e.g.…”

A minimalistic model of vegetation physiognomies in the savanna biome

“…This is a problem because outcomes spanning limited areas in parameter space (as for the multistable states we evidenced) may be missed by simulations if no qualitative result is available to pinpoint their existence. Another line of thought relies on the modelling of fires as impulsive events (Yatat et al [2017(Yatat et al [ , 2018b, Yatat and Dumont [2018], Tchuinté Tamen et al [2016,2017] and references therein). This leads to the impulsive differential equation (IDE) framework.…”

“…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.…”

Spreading speeds and traveling waves for monotone systems of impulsive reaction-diffusion equations: application to tree-grass interactions in fire-prone savannas

“…Here, for simplicity of the exposition, we will focus on mathematical models depending only on time. We note that there exist several possibilities to model pulse events such as the formalism presented in (Lakshmikantham et al [28], Bainov and Simeonov [7], Dumont and Tchuenche [16], Dufourd and Dumont [15], Tchuinté Tamen et al [49], [50], Yatat [62], Yatat et al [59], [61] and the references therein), or the approach used by Lewis and Li [29], see also Weinberger et al [55], Lewis et al [30], Li et al [31], Vasilyeva et al [52], Fazly et al [18], Huang et al [25], Yatat and Dumont [60] and the references therein.…”