S R Tega

I V Yatat-Djeumen

J J Tewa

P Couteron

Spatio-temporal modelling of tree-grass dynamics in humid savannas: interplay between nonlocal competition and nonlocal facilitation

Keywords: Tree, Grass, Nonlocal interaction, Facilitation, Competition, Reaction-diffusion equations, Vegetation patterns

For about twenty years, the question about the essential factors promoting the long-lasting coexistence of trees and grasses in humid savannas is at the center of several mathematical works, by the construction of deterministic and/or stochastic mathematical models. A closely related topic is coexistence of open savanna and forest patches at landscape scales, which raises the challenge of accounting for contrasted spatial patterns under similar climate conditions through fire mediated tree-grass interaction models. In this work, we propose and study a deterministic spatio-temporal fire-mediated tree-grass interactions model. The model is based on two nonlocal reaction-diffusion equations with kernels of intra and inter-specific interactions, corresponding to woody and grassy biomasses. A novelty in this paper is the consideration of a kernel-based nonlocal facilitation of trees by other trees to promote growth of seedlings/shrubs and, indirectly, limit fire propagation and its impact. We also take into account a kernel-based nonlocal competition of trees on grasses for light availability and nutrients. A qualitative analysis of the model is carried out and it reveals several ecological thresholds that shape the overall dynamics of the system. Depending on these thresholds, monostability of the forest, grassland or savanna space-homogeneous stationary state and multistabilities (i.e. involving more that one space-homogeneous stationary state) are proven possible. Thanks to the nonlocal biomasses interactions, our model accounts for the occurrence of space inhomogeneous solutions, including a possibly periodic spatial structuring sometimes observed in the humid savanna zone. Specifically, linear stability analyses, performed in the vicinity of space-homogeneous stationary states, provides conditions for the appearance of space inhomogeneous solutions including spatially periodic or aperiodic ones. Finally, numerical simulations are presented to illustrate our theoretical results. Notably, we verify that the computed spatial wavelengths were in good agreement with the predictions from the theoretical analysis.

Introduction

Savannas are complex ecosystems characterized by the co-occurrence of trees and grasses without one lifeforms excluding the other (Higgins and Bond [START_REF] Higgins | Fire, resprouting and variability: a recipe for grass-tree coexistence in savanna[END_REF]). They are also defined as a biome that corresponds to warm mean annual temperatures (> 20 • C) and a broad range of intermediate mean annual rainfall (Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF], Sarmiento [START_REF] Sarmiento | The Ecology of Neotropical Savannas[END_REF]). Covering ca. 12% of the global land surface (February and Higgins [4]), savannas occupy in Africa, ca. 50% of the land area.

Within specific stretches of the rainfall gradient, vegetation may sometimes exhibit plausibly self-organized physiognomies also termed as patchy vegetation or vegetation mosaics. Indeed, as pointed out by Yatat Djeumen et al. [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF], there are several empirical evidences that highlight the existence of vegetation mosaics. Patches of vegetation display dense clusters of shrubs, grasses or trees and can be interpreted as regular spot structures or localized structures (Tlidi et al. [START_REF] Tlidi | Extended patchy ecosystems may increase their total biomass through self-replication[END_REF]). These mosaics involve either bare soil ("desert") versus vegetation (herbaceous or woody) in arid, semi-arid regions (Lefever and Lejeune [START_REF] Lefever | On the origin of tiger bush[END_REF]; Lefever et al. [START_REF] Lefever | Deeply gapped vegetations patterns: on crown-roots allometry, critically and desertifiation[END_REF]; Lefever and Turner [START_REF] Lefever | A quantitative theory of vegetation patterns based on plant structure and the non-local F-KPP equation[END_REF]; Couteron and Lejeune [START_REF] Couteron | Periodic spot patterns in semi-arid vegetation explained by a propagation-inhibition model[END_REF]; Couteron et al. [START_REF] Couteron | Plant clonat morphologies and spatial patterns as self-organized responses to ressource-limited environments[END_REF]; HilleRisLambers et al. [START_REF] Hillerislambers | Vegetation pattern formation in semi-arid grazing systems[END_REF]; Rietkerk et al. [START_REF] Rietkerk | Self organization of vegetation in arid vegetation[END_REF]; Gilad et al. [START_REF] Gilad | A mathematical model of plants as ecosystem engineers[END_REF]; Pueyo et al. [START_REF] Pueyo | Dispersal strategies and spatial organization of vegetation in arid ecosystems[END_REF][START_REF] Pueyo | The role of reproduction plant traits and biotic interactions in the dynamics of semi-arid plant communities[END_REF]; Deblauwe et al. [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF] ), or grasslands/savannas versus forests in temperate as well as humid tropical regions (Youta Happi [START_REF] Happi | Arbres contre graminées: la lente invasion de la savane par la forêt au Centre-Cameroun[END_REF]; Hirota et al. [START_REF] Hirota | Global resilience of tropical forest and savanna to critical transitions[END_REF]; Jeffery et al. [START_REF] Jeffery | Fire management in a changing landscape: a case study from lope national park, gabon[END_REF]; Xu et al. [START_REF] Xu | Temperate forest and open landscapes are distinct alternative states as reflected in canopy height and tree cover[END_REF]; Stall et al. [START_REF] Stall | Bistability, spatial interaction and distribution of tropical forest and savannas[END_REF] and references therein; see also figure 1). Empirical evidences suggest that vegetation mosaics in humid regions barely feature periodic patterns. Most often, they are aperiodic but, with quite sharp boundaries like isolated groves or savanna patches encircled by forests.

Observation of these mosaics further motivated several researches that aimed to study and understand how these patterns may arise and the modalities of transitions between vegetation states that could substantiate or not the theory of abrupt shifts or catastrophic transitions in vegetation ecology (see for instance Scheffer et al. [START_REF] Scheffer | Catastrophic regime shifts in ecosystems[END_REF][START_REF] Scheffer | Generic indicators of ecological resilience inferring the chance of a critical transition[END_REF]; Scheffer and Carpenter [START_REF] Scheffer | Catastrophic regime shifts in ecosystems: linking theory to observation[END_REF]; Staver et al. [START_REF] Ac | Tree cover in sub-saharan africa: rainfall and fire constrain forest and savanna as alternative stable state[END_REF]; Favier et al. [START_REF] Favier | Abrupt shifts in african savanna tree cover along a climatic gradient[END_REF] for more details). It is well-known that at biome scale, vegetation cover displays complex interactions with climate. For instance, any shift from savanna to forest vegetation not only means increase in vegetation biomass and carbon sequestration but also may translate into changes in the regional patterns of rainfall (Oliveras and Malhi [START_REF] Oliveras | Many shades of green: the dynamic tropical forest-savannah transition zones[END_REF]). Therefore, being able to predict or understand the process that shapes savanna dynamics and possible transitions within vegetation patterns can help to figure out global distribution of savannas, orient their evolution in the face of recurring climatic changes in Africa (Dohn et al. [START_REF] Dohn | Spatial vegetation patterns and neighborhood competition among woody plants in a east african savanna[END_REF]) and sustainably manage the natural resources provided by savanna ecosystems.

To understand such self-organized vegetation formations and associated dynamics along the rainfall gradient, theoretical approaches are required. Mathematical modelling is a useful tool to describe dynamics of complex systems and has been used since decades in various fields that include finance, biology, epidemiology, agronomy, ecology. Despite field observations that point out spatial patterns of vegetation or vegetation mosaics (see e.g. figure 1), how tree-grass interactions proceed in space and make vegetation propagate has insufficiently been taken into account in the study of savanna dynamics, in contrast to the insights provided by modelling regarding bare soil-vegetation mosaics in drylands. Indeed, tree-grass interactions in savanna ecosystems (fire-prone or not) have been very often modelled through frameworks that implicitly acknowledge space (see the review of Yatat Djeumen et al. [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF]). According to Borgogno et al. [START_REF] Borgogno | Mathematical models of vegetation pattern formation in ecohydrology[END_REF], the modelling of spatial mechanisms of tree-grass interactions includes discrete kernel-based and partial differential equations (PDE) frameworks. Discrete kernel-based frameworks include cellular automaton (CA) models. CA models have been use in ecology, to explain formation of patterns in fire-prone savannas (Accatino et al. [START_REF] Accatino | Tree, grass and fire in humid savannasthe importance of life historical traits and spatial process[END_REF]), in arid and semi-arid savannas (Borgogno et al. [START_REF] Borgogno | Mathematical models of vegetation pattern formation in ecohydrology[END_REF], Feagin et al. [START_REF] Feagin | Individual versus community level process and pattern formation on a model of stand dune plant succesion[END_REF]). Accatino et al. [START_REF] Accatino | Tree, grass and fire in humid savannasthe importance of life historical traits and spatial process[END_REF] developed a CA model to investigate how trees can invade the grass stratum in humid savannas despite repeated fires. Their results show that trees can invade the grass stratum and finally suppress fire spread because one of the following occurs: (a) trees may frequently resprout and form a population that persists despite repeated effective fires; (b) trees may be fire-resistant; (c) if trees are fire-vulnerable they may cluster and grow in density until grass growth is suppressed and fire prevented. One should note that, only (c) may require spatially-explicit modelling of tree-grass interactions. However, they also show that fire may be effective in preventing the initiation of the invasion process in the grass stratum. But once the invasion process has begun, fire alone is not able to reverse it because of the combinated strategies employed by trees i.e. resprouting, fire resistance or clumping (see also Yatat Djeumen et al. [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF]).

However, since CA models are simulation-based and generally involve a fairly large number of parameters, it is not easy/possible to assess how model parameter variations may influence the model outcomes. In many cases, it is not easy to use mathematical analysis to thoroughly understand the behavior and properties of CA models (Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF]). Therefore, for the specific case of fire-prone savannas, it is desirable to provide insights into the dynamical properties of extensive savanna-forest areas for which data are scarce but that however need decisions in aspects such as fire management, grazing rules, or wood harvest. Spatially-explicit mathematical models that may allow mathematical tractability are thus desirable and rely on PDE frameworks.

Most of the works done using PDE, were carried out in the arid or semi-arid environmental context, using a reaction-diffusion-advection system (emphasizing the dynamics of vegetation and water) or using an integro-differential equation expressing kernel-based modelling of interactions between plants (see the review of Borgogno et al. [START_REF] Borgogno | Mathematical models of vegetation pattern formation in ecohydrology[END_REF]). The goal of that type of modelling is to understand the mechanisms that produce spatial patterns in arid and semi-arid savannas. In reaction-diffusion-advection systems, authors attribute pattern formation to positive feedback between vegetation (trees and grasses) and water availability (Klausmeier [START_REF] Ca | Regular and irregular patterns in semiarid vegetation[END_REF], Gilad et al. [START_REF] Gilad | A mathematical model of plants as ecosystem engineers[END_REF], Meron et al. [START_REF] Meron | Vegetation patterns along a rainfall gradient[END_REF], Sherratt [START_REF] Sherratt | When does colonisation of a semi-arid hillslope generate vegetation patterns?[END_REF]). Two main processes are identified as responsible for this positive feedback. The first one is the flow and infiltration of surface water into vegetated areas and the second feedback process is water up-take by the plant roots that is longer for larger plants (Meron et al. [START_REF] Meron | Vegetation patterns along a rainfall gradient[END_REF]). Such feedback is central to another framework to address vegetation patterns in arid and semi-arid savannas and that is entirely based on kernels that express nonlocal interactions between plants. Two types of non-local mechanisms received a particular attention: facilitative interactions between plants, that promote water infiltration and reduce evapotranspiration, and competitive interactions among them for water and nutrients. It is now acknowledged that pattern formation in arid systems can be explained by a combination of long distance competition and short distance facilitation (Lefever and Lejeune [START_REF] Lefever | On the origin of tiger bush[END_REF], Lejeune et al. [START_REF] Lejeune | Localized vegetation patches: A self-organized responses to resource scarcity[END_REF], Lefever et al. [START_REF] Lefever | Deeply gapped vegetations patterns: on crown-roots allometry, critically and desertifiation[END_REF], Lefever and Turner [START_REF] Lefever | A quantitative theory of vegetation patterns based on plant structure and the non-local F-KPP equation[END_REF], Couteron et al. [START_REF] Couteron | Plant clonat morphologies and spatial patterns as self-organized responses to ressource-limited environments[END_REF]). A common point of these two classes of studies is the view that the pattern formation phenomenon is a symmetry-breaking process that induces instability in an uniform vegetation state. Only a few mathematically tractable and space-explicit tree-grass interactions models have been designed for humid environments. For instance, Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF] studied a PDEbased model where dynamics of a forest-grassland pattern were studied by the mean of a bistable travelling wave. Notably, they showed that depending on the fire frequency, forest could either invade grassland (i.e. forest encroachment) or recede. Goel et al. [START_REF] Goel | Dispersal increases the resilience of tropical savanna and forest distributions[END_REF] examined, using a reaction-diffusion model, the contribution of dispersal to determining savanna and forest distributions. Their reaction-diffusion model considered a one-variable (scalar) equation describing the dynamics of tree cover and took into account fire and mean annual rainfall. Their 2D reaction-diffusion model was able to reproduce the spatial aggregation of biomes with a stable savanna-forest boundary.

In the same vein, Wuyts et al. [START_REF] Wuyts | Tropical tree cover in a heterogeneous environment: A reaction-diffusion model[END_REF] proposed a reaction-diffusion model of Amazonian tree cover. Their model was able to reproduce some observations of spatial distribution of forest versus savanna. However, as pointed out in Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF], modelling biomasses, instead of covers like in [START_REF] Goel | Dispersal increases the resilience of tropical savanna and forest distributions[END_REF][START_REF] Wuyts | Tropical tree cover in a heterogeneous environment: A reaction-diffusion model[END_REF], helps to take into account the fact that plant types are not mutually exclusive at a given point in space since field studies suggested that grass often develops under scattered tree crowns (see Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF] and references therein). Moreover, [START_REF] Goel | Dispersal increases the resilience of tropical savanna and forest distributions[END_REF][START_REF] Wuyts | Tropical tree cover in a heterogeneous environment: A reaction-diffusion model[END_REF] emphasized the effect of precipitation on possible vegetation transitions while Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] suggested that, interplay between fire and water availability may give more realistic scenarios of vegetation distribution or transitions. Recently, Patterson et al. [START_REF] Patterson | Probabilistic foundations of spatial meanfield models in ecology and applications[END_REF] proposed to bridge the gap between ecological models with macroscopic viewpoints (deterministic models) and microscopic descriptions of stochastic transitions (stochastic models). They studied a spatial extension of the tropical cover model of Staver and Levin [START_REF] Staver | Integrating theoretical climate and fire effects on savanna and forest systems[END_REF], characterized by nonlocal interactions describing the evolution of the probability for a patch of landscape to be in a given state (to be understood as, small spatial areas of the typical size of a single tree, allowing growth of new trees). From an ecological stand point, the analysis of their model enabled a more thorough understanding of the determinant of forest-savanna boundary, particularly in the presence of precipitation, resources limitation and climate changes. Notwithstanding notable exceptions, like Patterson et al. [START_REF] Patterson | Probabilistic foundations of spatial meanfield models in ecology and applications[END_REF], a common point of some of these models is that authors mainly relied on numerical simulations to render some spatial structures and relate them to processes. However, due to the absence of qualitative analyses, it is quite difficult to assess how model outcomes respond to model parameter variations.

In the context of humid savannas, patterns approaching regularity are fairly scarce, but not absent (see Figure 1 pannel (a) and also Lejeune et al. [36] or Tlidi et al. [START_REF] Tlidi | Extended patchy ecosystems may increase their total biomass through self-replication[END_REF]). Another class of patterns is made of clearly aperiodic groves in the context of a mosaic that often corresponds to savannas transiting to forests (e.g. see Figure 1 pannel (b)).

Our objective in this paper is therefore to build a mathematically tractable space-explicit PDElike model in order to study dynamics of spatial structuring of vegetation in wet savanna zones (Figure 1, pannel (b)). Tractability is an important property because it allows an efficient exploration of all parts of the parameter space ensuring that interesting situations, notably linked to multistability, are not missed as it might happen if only relying on computer simulations like in CA-based models (Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF]). Another aim is to identify key mechanisms and bifurcation parameters that may shape possible transitions of vegetation physiognomy and trigger spatial pattern emergence in wet savannas. Therefore, based on a mathematical model, we aim to give new insights for the development of relevant management plans of forest-savanna mosaics. Our model is based on, and therefore extend, the recent ODE model of Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF]. Indeed, based on a minimalistic (in terms of state variables and parameters) ODE model, Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] analysed fire-mediated tree-grass interactions and obtained a stability map within the fire vs. mean annual rainfall parameters space. They delineated regions of monostabilities (i.e. where desert, forest, grassland or savanna is stable), regions of multistabilities involving forest, grassland and savanna as well as multistabilities involving several savanna states. In addition, for all levels of rainfall, decreasing woody biomass with increasing fire frequency was verified contrary to almost all recent works of the same complexity or less (e.g. Accatino et al. [START_REF] Accatino | Tree and grass co-existence in savanna:interactions of rain and fire[END_REF]). Our model takes into account the fire resistance strategy of trees, and the main processes present in Yatat's model, such as the grass-fire feed-back and decreasing fire impact with woody biomass. In addition, we incorporate nonlocal interaction terms of intra and interspecific competition. In fact, intraspecific competition influences the growth of species (either trees or grasses) and ultimately changes the dynamics of the entire population (Kothari et al. [START_REF] Kothari | Intraspecific competition in grassland species[END_REF]) and interspecific competition (i.e. asymmetric competition of trees on grasses) leads to a reduction in grass cover and therefore a reduction in the spread and intensity of fires. Though this paper puts emphasis on the conditions for stable, spatially regular patterns, it opens prospects for studying transient and metastable patterns. The rest of the paper is organized as follows: section 2 presents the construction of the model, section 3 deals with the theoretical analyses of the model including the existence and uniqueness of solutions and linear stability analysis of homogeneous stationary solutions. Section 4 deals with numerical illustrations of theoretical results.

Model construction

Our model is based on Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] where authors considered two state variables, G(t) and T (t) that stand for the grassy biomass and the woody biomass at time t, respectively (G in t.ha -1 and T in t.ha -1 ). In Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF], the following hypotheses are done:

• Trees and grasses biomasses have a logistic growth.

• Grass biomass mortality or suppression may result from natural mortality, external factors (grazing, termites, human actions, etc), interactions with tree biomass and fire.

• Tree biomass mortality may result from natural mortality, external factors (browsers, human actions, etc) or is fire-induced. In fact, fire momentum is an increasing nonlinear function of G, while its impact on woody vegetation is a decreasing nonlinear function of woody biomass T .

Starting from these assumptions, we incorporated a spatial component on state variables. Precisely, G(x, t) and T (x, t) denote the normalized densities (by grass and tree carrying capacities K G and K T , in t.ha -1 ) of biomass of grass and tree, respectively, at a spatial point x and at a time t. Then, 0 ≤ T (x, t) ≤ 1 and 0 ≤ G(x, t) ≤ 1. We consider the following assumptions:

• Tree and grass biomasses, have a logistic growth but with an intraspecific competition which takes place in a nonlocal way, through the respective root systems of the two lifeforms. In fact, a tree (respectively grass) located at a point x, can consume resources (water, nutriment) at a point y where, another tree (respectively grass) is located or where its roots are present. Then,

γ T T (x, t) 1 - +∞ -∞ ϕ M 2 (x -y)T (y, t)dy and γ G G(x, t) 1 - +∞ -∞ ϕ M 1 (x -y)G(y, t)dy
(1) describe the logistic growth with intraspecific competition where, for i = 1, 2, the kernel ϕ M i (xy) represents, the level of consumption of resources in the area [-M i ; M i ] of the space domain, γ G (respectively, γ T ) denotes the intrinsic growth rate of grasses (respectively, trees).

• According to Craine and Dybzinski [START_REF] Craine | Mechanisms of plant competition for nutriments, water and ligth[END_REF], trees facilitate the germination and the recruitment of new trees by improving the conditions under or around the canopy. In fact, sapling establishment for example depends on tree cover, not just because of seed production but also by local facilitation of seedlings and saplings by other trees via hydrological facilitation and shading (Li. et al. [45]). Then, we assume that there is a factor of cooperation Ω between trees that promotes regrowth and growth of young trees, helping them to reach a fire and/or browser non-vulnerability height. Hence the γ T coefficient of exponential growth in equation ( 1) is substituted by γ T (1 + ΩT ).

• Trees negatively impact the dynamics of grass biomass in a nonlocal way. Indeed, a tree located at a point y can, either by its root system or by the shade created by its crown, reduces the density of grasses located at a point x by reducing the resources (light availability, nutrients) in x. Then, the term

γ T G G(x, t) +∞ -∞ ϕ M 2 (x -y)T (y, t)dy (2)
describes this nonlocal interspecific impact where γ T G = K T η T G and η T G is the tree-grass interaction parameter in ha.t -1 .yr -1 . The consequence here is the reduction of the grass continuum on the ground, which will reduce the spread of fire. This term will depress grass biomass growth.

• The function describing the impact of fires, ω(G), on tree biomass depends on G. Indeed, in savanna ecology it is widely admitted that dried-up grass biomass is the main factor controlling both fire intensity and spreading capacity. For simplicity, we combined these two properties of fire in a single (fire momentum), increasing function of grass-biomass, expressing that when the average herbaceous biomass is in its highest range, fires simultaneously display the highest intensity and affect all the landscape. Conversely, low grass biomass due to aridity, grazing or tree competition, will make fires of low intensity and/or unable to reach all locations in a given year thereby decreasing the actual average frequency (see for instance Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF]). Following Tchuinte et al. [START_REF] Tchuinte | A minimalistic model of tree-grass interactions using impulsive differential equations and non-linear feedback functions of grass biomass onto fire-induced tree mortality[END_REF], Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF][START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF], we consider a Holling Type III function

ω(G) = G 2 G 2 + g 2 0 , (3)
where g 0 = µ K G and µ is the grass biomass at which fires reach its half maximal momentum.

• We consider a function of fire-induced tree mortality that decreases with the cumulated woody biomass around any space point x. If trees are numerous and/or tall, then their mortality due to fire will be reduced. Indeed, tree parts above the flame zone are immune to topkills. This function is therefore, a decreasing function of tree biomass. In analogy with the work of Martinez-Garcia et al. [START_REF] Martinez-Garcia | Spatial patterns in mesic savannas: The local facilitation limit and the role of demographic stochasticity[END_REF], we consider a function of the form :

V T (x) = exp -p +∞ -∞ ϕ M 2 (x -y)T (y, t)dy , (4)
where p = K T δ and δ is a parameter proportional to the inverse of biomass destroyed at intermediate level of mortality, in t -1 .ha, see also Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] for a nonspatial version of V T .

• We also assume, according to Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF], that grass biomass and tree biomass, display local isotropic biomass diffusion in space with the coefficient D G and D T respectively, that are modelled with Laplace operators. Here, as a first approximation, we consider local diffusion of biomasses and neglect the long-range seed dispersal.

All this leads to the following model:

                       ∂G ∂t = D G ∂ 2 G ∂x 2 + γ G G 1 - +∞ -∞ ϕ M 1 (x -y)G(y, t)dy -δ G G -γ T G G +∞ -∞ ϕ M 2 (x -y)T (y, t)dy -λ f G f G, ∂T ∂t = D T ∂ 2 T ∂x 2 + γ T T (1 + ΩT ) 1 - +∞ -∞ ϕ M 2 (x -y)T (y, t)dy -δ T T -λ f T f ω(G) exp -p +∞ -∞ ϕ M 2 (x -y)T (y, t)dy T, (5)
where x ∈ K = (-l, l) and t > 0. Parameters are defined in table 1 bellow. The initial data are

0 ≤ T (x, 0) = T 0 (x) and 0 ≤ G(x, 0) = G 0 (x), (6)
where G 0 (x) and T 0 (x) are bounded and sufficiently smooth functions. In addition, we also consider homogeneous Neumann boundary condition:

∂T (x, t) ∂x = ∂G(x, t) ∂x = 0 at x = -l and x = l, l > 0. ( 7
)
We assume that the kernels ϕ M i , (i = 1, 2) are nonnegative even functions with compact support in the interval [-M i , M i ]. Then, for 0 ≤ M i ≤ l, we consider the step function kernels:

ϕ M i (x) =    1 2M i , |x| ≤ M i , 0 , |x| > M i , i = 1, 2,
with ϕ 0 a Dirac function and

+∞ -∞
ϕ M i (y)dy = 1. For the chosen kernel function ϕ M i , the strength of nonlocal interaction is the same with the range [x -M i , x + M i ]. However, other forms of kernels have been considered in the literature dedicated to pattern formation, notably Gaussian kernels and Laplace kernels (see for instance Lefever and Lejeune [START_REF] Lefever | On the origin of tiger bush[END_REF], Lefever et al. [START_REF] Lefever | Deeply gapped vegetations patterns: on crown-roots allometry, critically and desertifiation[END_REF], Lefever and Turner [START_REF] Lefever | A quantitative theory of vegetation patterns based on plant structure and the non-local F-KPP equation[END_REF]).

The choice of the step function kernels in this work was mainly motivated by the type of nonlinearities in our model and model's mathematical analysis. Indeed, we found that Gaussian and Laplace kernels are not able to induce patterns with our model (see also Remark 4, page 15 or Remark 6, page 16). Following Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF], the f (in yr -1 ) parameter is taken as constant multiplier of ω(G) in system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF], and we interpret it as a man-induced 'targeted' fire frequency (as for instance in a fire management plan), which will not automatically translate everywhere into actual frequency of fires of notable intensity (because of ω(G)). With this interpretation, the actual fire regime may substantially differ from the targeted one, as frequently observed in the field (see for instance Diouf et al. [START_REF] Diouf | Relationships between fire history, edaphic factor and woody vegetation structure and composition in a semi-arid savanna landscape (Niger, West Africa)[END_REF] in southern Niger). We therefore distinguish fire frequency from fire intensity because grass biomass controls fire spread (see e.g. Govender et al. [START_REF] Govender | The effect of fire season, fire frequency, rainfall and management on fire intensity in savanna vegetation in South Africa[END_REF], McNaughton [START_REF] Mcnaughton | The propagation of disturbance in savannas through food webs[END_REF], Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF] and references therein).

Mathematical analysis

3.1. Existence and uniqueness of solutions of system ( 5)-( 7)

Let K = [-l, l] be the closure of K, and for any τ > 0, we set:

D τ = K × (0, τ ], D τ = K × [0, τ ], S τ = ∂K × (0, τ ]. (8)
Denote by C α (D τ ) the set of Hölder continuous functions in D τ with the exponent α ∈ (0; 1), and C(D τ ), the set of continuous functions in D τ . Denote also by C 2,1 (D τ ) the set of functions that are twice continuously differentiable in x and once continuously differentiable in t. For simplicity, throughout this paper, we denote:

f 1 (G, T ) = γ G G (1 -ϕ M 1 * G) -δ G G -γ T G G (ϕ M 2 * T ) -λ f G f G, f 2 (G, T ) = γ T T (1 + ΩT ) (1 -ϕ M 2 * T ) -δ T T -λ f T f ω(G) exp (-pϕ M 2 * T ) T, (9)
with

(ϕ M 1 * G) (x) = +∞ -∞ ϕ M 1 (x -y)G(y, t)dy and (ϕ M 2 * T ) (x) = +∞ -∞
ϕ M 2 (xy)T (y, t)dy [START_REF] Couteron | Periodic spot patterns in semi-arid vegetation explained by a propagation-inhibition model[END_REF] where ϕ M i is a spatial kernel function satisfying:

+∞ -∞ ϕ M i (y)dy = 1, i = 1, 2. ( 11
)
Definition 1. (Tian et al. [START_REF] Tian | Nonlocal interactions driven pattern formation in a preypredator model[END_REF]) A pair of nonnegative functions Ũ = ( G, T ) ′ and Û = ( Ĝ, T )

′ ∈ C(D τ ) ∩ C 2,1 (D τ ) is called upper and lower solutions of (5) if Ũ ≥ Û and if ∂ G ∂t -D G ∆ G ≥ γ G G 1 -ϕ M 1 * G -δ G G -γ T G G ϕ M 2 * T -λ f G f G, in D τ ∂ T ∂t -D T ∆ T ≥ γ T T (1 + Ω T ) 1 -ϕ M 2 * T -δ T T -λ f T f ω( Ĝ) exp -pϕ M 2 * T T , in D τ ∂ Ĝ ∂t -D G ∆ Ĝ ≤ γ G Ĝ 1 -ϕ M 1 * Ĝ -δ G Ĝ -γ T G Ĝ ϕ M 2 * T -λ f G f Ĝ, in D τ ∂ T ∂t -D T ∆ T ≤ γ T T (1 + Ω T ) 1 -ϕ M 2 * T -δ T T -λ f T f ω( G) exp -pϕ M 2 * T T , in D τ (12) ∂ Ĝ ∂t , ∂ T ∂t ≤ 0, ∂ G ∂t , ∂ T ∂t ≥ 0 on S τ . G(x, 0) ≥ G(x, 0), T (x, 0) ≥ T (x, 0), Ĝ(x, 0) ≤ G(x, 0) T (x, 0) ≤ T (x, 0) x ∈ K.
The ordering relation Ũ ≥ Û means that G ≥ Ĝ and T ≥ T for (x, t) ∈ D τ . For a given pair of ordered upper and lower solutions Ũ and Û, we set:

⟨ Û, Ũ⟩ = U = (G, T ) ′ ∈ C(D τ ) : Û ≤ U ≤ Ũ . ( 13
)
Let us consider the following thresholds:

           R G = γ G δ G + f λ f G , R G,0 = γ G δ G , R T,0 = γ T δ T . ( 14
)
Our model is designed for humid savannas where we assume that rainfall is sufficient to ensure that R G,0 > 1, and R T,0 > 1.

Hence, in the rest of the paper, we assume that (15) holds true.

Theorem 1 (Existence and uniqueness of global solution). Assume that the following three conditions are valid.

• R G > 1,

• the initial functions G(x, 0) and

T (x, 0) ∈ C α (D τ ) ∩ C(D τ ) and • 0 ≤ (G 0 (x), T 0 (x)) ′ ≤ 1.
Then, the nonlocal reaction-diffusion system (5)-( 7) admits a unique global solution U * (x, t) = (G * (x, t), T * (x, t)) ′ for (x, t) ′ ∈ K × (0, +∞) and

0 ≤ G * (x, t) ≤ W 1 , 0 ≤ T * (x, t) ≤ W 2 , (16)
where

W 1 = max sup K G(0, x), 1 - 1 R G , W 2 = max sup K T (0, x), 1 - 1 R T,0 , if Ω = 0, W 2 = max            sup K T (0, x), (1 -Ω) 2 + 4Ω 1 - 1 R T,0 -(1 -Ω) 2Ω            , if Ω > 0.
Proof. See Appendix A, page 35.

Space homogeneous steady states and linear stability analysis

Our aim in this section is to derive a condition on spatial convolution such that an equilibrium or space homogeneous steady state is locally stable in the case M 1 = M 2 = 0 but unstable for some M i > 0, i = 1, 2.

The local case:

M 1 = M 2 = 0
Due to the fact that the local spatio-temporal model (LSTM) associated to the system (5) is quasi-monotone decreasing (Smith [START_REF] Smith | Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems[END_REF]), we have the two following consequences. First, the LSTM can not lead to pattern formation (see e.g Kishimoto and Weinberger [START_REF] Kishimoto | The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domain[END_REF], Banerjee and Volpert [START_REF] Banerjee | Spatio-temporal pattern formation in rosenzweig-macarthur model : Effect of nonlocal interactions[END_REF], [START_REF] Banerjee | Prey-predator model with a nonlocal bistable dynamics of prey[END_REF]) and second, the linear stability analysis of homogeneous steady states associated to LSTM is the same as for the space-implicit model i.e., the ODE model associated to system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF]. The space-implicit ODE model corresponding to system (5) is:

     dG dt = γ G G(1 -G) -δ G G -γ T G T G -λ f G f G, dT dt = γ T T (1 + ΩT ) (1 -T ) -δ T T -λ f T f ω(G) exp(-pT )T, (17)
with non negative initial data.

In this part, we are interested in the behavior of steady states of system [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF], notably in the characterization of their stability properties. Recall that steady states of system [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF] are also space homogeneous steady states of system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF]. Steady states of ( 17) are solutions of system [START_REF] Happi | Arbres contre graminées: la lente invasion de la savane par la forêt au Centre-Cameroun[END_REF]:

γ G G(1 -G) -δ G G -γ T G T G -λ f G f G = 0, γ T T (1 + ΩT )(1 -T ) -δ T T -λ f T f ω(G) exp(-pT )T = 0. ( 18
)
Recall that we assumed that ( 15) is valid meaning that, the desert (the state with absence of vegetation) can not be stable. The following result is valid. Proposition 1. (Steady states of system ( 17))

The system (5) admits three homogeneous steady states.

a) a desert steady state E 0 = (0, 0) ′ . b) a forest steady state such that:

* When Ω > 0, then E T 2 = (0, T 2 ) ′ =       0, (1 -Ω) 2 + 4Ω 1 - 1 R T,0 -(1 -Ω) 2Ω       ′ is
the forest steady state. This is the case of tree-tree facilitation. c) a grassland steady state:

E Ge = (G e , 0) ′ = 1 - 1 R G , 0 ′ . Remark 1.
It is straightforward to observe that E T 2 is an increasing function of Ω.

We are now interested in the coexistence steady state (savanna steady state); set:

a = - λ f G f + δ G γ T G , b = γ G γ T G , θ = 2(a + b)bΩγ T + γ T (1 -Ω)b, α = Ωγ T b 2 , q = (γ T -δ T ) + γ T (Ω -1)(a + b) -Ωγ T (a + b) 2 , m = λ f T f exp (-p(a + b)) , θ * = 24α + mpb (pb) 2 + 6(pb) + 6 exp(pb) 6 ,
and

R T = γ T δ T + λ f T f ω(G e ) , R F,f = γ G δ G + λ f G f + γ T G T i , R 1 Ω = γ T (1 -Ω) pλ f T f ω(G e ) . (19)
Proposition 2. (Savanna steady state)

• case I: f = 0. If R F,f =0 > 1, then we have a unique savanna steady state E s = (G * , T * ) ′ such that G * = 1 - 1 R F,f =0 and T * = T i , i = 1, 2. ( 20
)
• case II: f > 0 and γ T G = 0. The savanna steady state E * = (G * , T * ) ′ verifies:

G * = G e Ωγ T (T * -T 2 )(T * -T 2-) + λ f T f ω(G e ) exp(-pT * ) = 0 ( 21
)
where

T 2-= - (1 -Ω) + (1 -Ω) 2 + 4Ω(1 -δ T γ T ) 2Ω
. Hence: * if R 1 Ω > 1, then there may exist 0 or 1 savanna steady state. * if R 1 Ω < 1, then there may exist 0, 1 or 2 savanna steady states.

• case III: f > 0 and γ T G ̸ = 0.

The savanna steady state E s = (G * , T * ) ′ must satisfy these two relations:

-α(G * ) 4 + θ(G * ) 3 -m exp(pbG * )(G * ) 2 + (q -αg 2 0 )(G * ) 2 + θg 2 0 G * + qg 2 0 = 0, (22)
and

T * = (a + b) -bG * . ( 23
)
Moreover G * must satisfy the inequality

max G e - γ T G γ G ; 0 < G * < G e . (24)
We can therefore summarize the maximum number of savanna steady states according to the following cases:

• Case 1: θ < mpb Condition q < m + αg 2 0 q > m + αg 2 0
Maximal number of savanna steady states 2 3 Proof. See Appendix B, page [START_REF] Wuyts | Tropical tree cover in a heterogeneous environment: A reaction-diffusion model[END_REF].

Remark 2. Let us set R 0 G = 2γ G 2λ f G f + 2δ G + γ T G , R T G = 2γ T G 2λ f G f + 2δ G + γ T G and R * = γ T 1 + Ω R 0 G R T G pλ f T f exp (-p(a + b)) .
Now, we want to characterize local stability property of previous steady states. System ( 17) is a planar, competitive and dissipative system. Hence, based on Smith [52, Theorem 2.2, page 35], we deduce that solutions of system [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF] will always converge toward an equilibrium point. That is, no stable limit cycles may exist for system [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF]. Proposition 3. (Stability properties of trivial and semi trivial steady states). The following results are valid for system [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF].

(a) The desert steady state E 0 = (0, 0) ′ is always unstable.

(b) If R F < 1 then the forest steady state E T i is locally asymptotically stable (LAS). (c) If R T < 1 then the grassland steady state E Ge = (G e ; 0) ′ is LAS.
Now we deal with conditions of stability of a savanna steady state when its exists. Set:

R * 1 = γ T [(1 -Ω) + 2ΩT * ] pλ f T f ω(G * ) exp(-pT * ) , R * 2 = γ T G ω ′ (G * ) pγ G ω(G * ) . ( 25
)
Proposition 4. (Stability condition of a savanna steady state). The stability conditions of a coexistence steady state, of system [START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF], when it exists are given by the following cases:

• case 1:Assume f = 0, then (G * , T * ) ′ is LAS. • case 2: Assume f > 0. If: R * 1 -R * 2 > 1, (26)
then (G * , T * ) ′ is LAS.

Proof. See Appendix C, page 56.

Remark 3. (i) R T,0 = γ T δ T
denotes the primary production of tree biomass relative to tree biomass loss due to human activities and herbivory.

(ii) R G,0 = γ G δ G
is the primary production of grass biomass relative to grass biomass loss due to human activities and herbivory.

(iii) R G = γ G δ G + f λ f G
denotes the primary production of grass biomass relative to grass biomass loss due to grazing or human action and additional fire induced biomass loss.

(iv) R T = γ T δ G + λ f T f ω(G e )
is the primary production of tree biomass relative to fire-induced biomass loss at the grassland equilibrium and the additional loss due to herbivory (grazing) or human action.

(v) R F,f = γ G δ G + f λ f G + γ T G T i
represents the primary production of grass biomass, relative to grass biomass loss induced by fire, herbivory (grazing) or human action and additional grass suppression due to tree competition, at the closed forest equilibrium.

The nonlocal case (M

1 or M 2 > 0)
Our aim now is to derive a condition on spatial convolution such that a steady state (G s , T s ) ′ ∈ {(G e , 0) ′ ; (0, T 2 ) ′ ; (G * , T * ) ′ } is locally asymptotically stable in the case M 1 = M 2 = 0, but unstable for some M i > 0, i = 1, 2. In fact, the spatial patterns appearing in the nonlocal savanna model ( 5) can be studied by performing a linear stability analysis of the stationary homogeneous solution of (5) given by the system [START_REF] Happi | Arbres contre graminées: la lente invasion de la savane par la forêt au Centre-Cameroun[END_REF]. Linearizing the integro-differential system ( 5) around (G s ; T s ) ′ , leads to the following results: Proposition 5. (linearized system) Set: g(x, t) = G(x, t) -G s and h(x, t) = T (x, t) -T s two perturbations around a non trivial homogeneous steady state. The system obtained after linearization is:

                           ∂g ∂t = D G ∂ 2 g ∂x 2 + [γ G (1 -G s ) -δ G -γ T G T s -λ f G f ] g -γ G G s +∞ -∞ ϕ M 1 (x -y)g(y, t)dy -γ T G G s +∞ -∞ ϕ M 2 (x -y)h(y, t)dy, ∂h ∂t = D T ∂ 2 h ∂x 2 + [(γ T (1 + ΩT s )(1 -T s ) -δ T -λ f T f ω(G s ) exp(-pT s )) + γ T ΩT s (1 -T s )] h + (pλ f T f ω(G s ) exp(-pT s )T s -γ T T s (1 + ΩT s )) +∞ -∞ ϕ M 2 (x -y)h(y, t)dy -λ f T f ω ′ (G s ) exp(-pT s )T s g. (27)
Proof. See Appendix D, page 57.

Now we are in position to study linear stability around all non trivial homogeneous steady state.

Linear stability analysis around the grassland homogeneous steady state

E G = (G e , 0) ′ . Set : b 11 = γ G G e , b 12 = γ T G G e , b 22 = γ T -δ T -λ f T f ω(G e ). (28)
The following results hold: Proposition 6. (Linearized system around the grassland homogeneous steady state) Let g(x, t) = G(x, t) -G e and h(x, t) = T (x, t) be two perturbations around the grassland homogeneous steady state. The system obtained after linearization is:

       ∂g ∂t = D G ∂ 2 g ∂x 2 -b 11 +∞ -∞ ϕ M 1 (x -y)g(y, t)dy -b 12 +∞ -∞ ϕ M 2 (x -y)h(y, t)dy, ∂h ∂t = D T ∂ 2 h ∂x 2 + b 22 h. (29)
By considering the eigenvalue problem of the system (29) where λ is the eigenvalue and taking the Fourier transform of this eigenvalue problem, we obtain the following system:

λg(k) = -D G k 2 g(k) -b 11 ϕ M 1 (k)g(k) -b 12 ϕ M 2 (k)h(k), λh(k) = -D T k 2 h(k) + b 22 h(k), ( 30
)
where k is the wavenumber

(k ∈ R) with ϕ M i (k) = sin kM i kM i , where, g(k), h(k) and ϕ M i (k) are
the Fourier transforms of the functions g(x, t), p(x, t) and ϕ M i (x), respectively. Therefore, the system in ( 30) can be written in the matrix form:

λ   g(k) h(k)   =   -D G k 2 -b 11 ϕ M 1 (k) -b 12 ϕ M 2 (k) 0 -D T k 2 + b 22     g(k) h(k)   . (31)
Let us consider:

M =   -D G k 2 -b 11 ϕ M 1 (k) -b 12 ϕ M 2 (k) 0 -D T k 2 + b 22   , (32)
T r(M ) = -(D G + D T ) k 2 -b 11 ϕ M 1 (k) + b 22 , (33)
and

Det(M ) = D G D T k 4 + b 11 D T ϕ M 1 (k) -b 22 D G k 2 -b 11 b 22 ϕ M 1 (k). (34)
Therefore, the grassland homogeneous steady state is locally asymptotic stable if:

T r(M ) < 0, (35)
and

Det(M ) > 0. ( 36
)
If [START_REF] Lejeune | Localized vegetation patches: A self-organized responses to resource scarcity[END_REF] is not satisfied then we have an inhomogeneous solution call pattern (deriving from a Turing bifurcation).

We are now in position to find Turing bifurcation threshold around the grassland homogeneous steady state. Because of the form of ϕ M 1 (k), we set z = kM 1 and denote, for simplicity, ϕ M 1 (k) by ϕ 1 (z).

Theorem 2. (Stability of the Grassland homogeneous steady state)

If R T < 1 and ϕ 1 (z) > 0 for all z, then the grassland homogeneous steady state is locally asymptotically stable for system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF].

Proof. Assume that R T < 1. Then, b 22 < 0 thanks to the stability conditions of the grassland steady state in the space-implicit model (see for instance proposition 3, page 12). Therefore, if ϕ 1 (z) > 0, then T r(M ) < 0 and Det(M ) > 0.

Remark 4. The previous theorem ensures that for this model, the choice of Gaussian kernels can not lead to pattern formation around the grassland homogeneous steady state. More generally, due to the type of nonlinearities involved in our model, the class of kernel-functions called "positivedefinite functions" and characterized by a positive Fourier transform (see also Bochner [56] and Tzanakis [START_REF] Tzanakis | A note on the fourier transform of a positive-definite function[END_REF]) are such that the empirically evidenced vegetation patterns are not reachable with the model. Then, ϕ 1 (z) < 0 is a necessary condition for spatial Turing instability around the grassland homogeneous steady state and this could happen if ϕ 1 has discontinuities.

Theorem 3. (Grassland homogeneous steady state instability) Assume that R T < 1 and we have a range of positive values of z such that:

ϕ 1 (z) < 0 ( 37
)
holds ; If there exists a critical point M T 1 such that :

M 1 > M T 1 ⇒ 1 (M 1 ) 2 ≤ S 1 (z 1 ), (38)
where

S 1 (z) = -ϕ 1 (z) z 2 b 11 D G , (39)
and z 1 is the value of z such that S 1 (z) takes it global maximum, then the homogeneous grassland steady state is unstable. Furthermore, system (5) undergoes Turing bifurcation at

M 1 = M T 1 .
Proof. See Appendix E page 58.

Remark 5. The space period σ G of the spatial structure is given by: σ G = 2πM 1 z 1 (see also Genieys et al. [58, page 71]) where z 1 is given in the previous theorem.

Linear stability analysis around the forest homogeneous steady state

E T = (0, T i ) ′ ,i = 1, 2.. Set: m 11 = -γ G + (δ G + λ f G f ) + γ T G T i , m * 22 = γ T ΩT i (1 -T i ), m * * 22 = γ T T i (1 + ΩT i ).
Proposition 7. (linearized system around the forest homogeneous steady state) Set: g(x, t) = G(x, t) and h(x, t) = T (x, t) -T i two perturbations around the forest homogeneous steady state. The system obtained after linearization is:

       ∂g ∂t = D G ∂ 2 g ∂x 2 -m 11 g, ∂h ∂t = D T ∂ 2 h ∂x 2 + m * 22 h -m * * 22 +∞ -∞ ϕ M 2 (x -y)h(y, t)dy. (40)
By considering the eigenvalue problem of the system [START_REF] Patterson | Probabilistic foundations of spatial meanfield models in ecology and applications[END_REF] and in the same way like in proposition 6, we obtain the following theorem: for all z, then the forest homogeneous steady state is locally asymptotically stable for system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF], where ϕ 2 (z) denotes the Fourier transform of ϕ M 2 .

Proof. The proof is done like for theorem 2, page 14. Hence, we omitted it.

Remark 6. In the case of no tree-tree facilitation, by the previous theorem, the use of Gaussian kernels can not lead to inhomogeneous patterned solution in the vicinity of forest homogeneous steady state because with Ω = 0, m * 22 = 0 and then, the condition (with Gaussian kernels) of local stability of the forest homogeneous steady state is always verified.

Theorem 5. (Forest homogeneous steady state instability)

Assume that R F,f < 1 and we have a range of positive values of z such that:

ϕ 2 (z) < m * 22 m * * 22 ( 41
)
holds. If there exist a critical value M T 2 > 0 such that:

M 2 > M T 2 ⇒ 1 (M 2 ) 2 ≤ S 2 (z 2 ), ( 42
)
where

S 2 (z) = - ϕ 2 (z) z 2 m * * 22 D T + 1 z 2 m * 22 D T , (43)
and z 2 is the value of z such that S 2 (z) takes a global maximum. Then, the forest homogeneous steady state is unstable and system (5) undergoes a Turing bifurcation at

M 2 = M T 2 .
Proof. The proof is similar to the proof of the theorem 3 in Appendix E, page 58. Therefore, it is omitted.

Remark 7. The space period of the spatial structures σ T observed in this case is given by σ T = 2πM 2 z 2 (see also Genieys et al. [START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF]), where z 2 is given in the previous theorem.

Linear stability analysis around the savanna homogeneous steady state E * = (G * , T * ). Set:

a 11 = -γ G G * , a 12 = -γ T G G * , a 21 = -λ f T f ω ′ (G * ) exp(-pT * )T * , a 22 = -γ T (1 -Ω)T * + 2Ω(T * ) 2 + pλ f T f ω(G * ) exp(-pT * )T * , c = γ T ΩT * (1 -T * ).
Proposition 8. (Linearized system around the savanna coexistence state) Let g(x, t) = G(x, t) -G * and h(x, t) = T (x, t) -T * be two perturbations around the savanna homogeneous steady state. The system obtained after linearization is:

       ∂g ∂t = D G ∂ 2 g ∂x 2 + a 11 +∞ -∞ ϕ M 1 (x -y)g(y, t)dy + a 12 +∞ -∞ ϕ M 2 (x -y)h(y, t)dy, ∂h ∂t = D T ∂ 2 h ∂x 2 + (a 22 -c) +∞ -∞ ϕ M 2 (x -y)h(y, t)dy + ch + a 21 g. (44)
Then, by considering the eigenvalue problem of the system [START_REF] Craine | Mechanisms of plant competition for nutriments, water and ligth[END_REF] where λ is the eigenvalue and taking the Fourier transform of this eigenvalue problem, we obtain the following system:

λg(k) = -D G k 2 g(k) + a 11 ϕ M 1 (k)g(k) + a 12 ϕ M 2 (k)h(k), λh(k) = -D T k 2 h(k) + ch(k) + (a 22 -c)ϕ M 2 (k)h(k) + a 21 g(k), ( 45
)
where k is the wavenumber

(k ∈ R) with ϕ M i (k) = sin kM i kM i , i = 1, 2 where, g(k), h(k) and ϕ M i (k)
are the Fourier transforms of the functions g(x, t), p(x, t) and ϕ M i (x).

Proposition 9. (Characteristic equation)

The Characteristic equation of system (45) is:

λ 2 -T r(k, M 1 , M 2 )λ + Det(k, M 1 , M 2 ) = 0, ( 46
)
where:

T r(k, M 1 , M 2 ) = -(D G + D T )k 2 + a 11 ϕ M 1 (k) + a 22 ϕ M 2 (k) + (1 -ϕ M 2 (k))c, (47)
and

Det(k, M 1 , M 2 ) = D G D T k 4 -a 22 D G ϕ M 2 (k) + a 11 D T ϕ M 1 (k) + cD G (1 -ϕ M 2 (k)) k 2 + a 11 (a 22 -c)ϕ M 1 (k)ϕ M 2 (k) + ca 11 ϕ M 1 (k) -a 12 a 21 ϕ M 2 (k). ( 48
)
From the characteristic equation ( 46), we can write the stability conditions of the savanna homogeneous steady state (G * , T * ) as follows:

T r(k, M 1 , M 2 ) < 0, (49)
and

Det(k, M 1 , M 2 ) > 0. ( 50
)
To determine the stability boundary, we need to determine the thresholds for k, M 1 , and M 2 such that only one of the eigenvalue of the characteristic equation ( 46) crosses the origin from the left to the right and other eigenvalues have negative real parts. If [START_REF] Govender | The effect of fire season, fire frequency, rainfall and management on fire intensity in savanna vegetation in South Africa[END_REF] holds and ( 50) is not satisfied, then there is a real eigenvalue crossing the origin. Initially (k = M 1 = M 2 = 0), ( 49) and ( 50) hold. So we find the thresholds k T , M T 1 and M T 2 so that (50) is not satisfied (it is call Turing Bifurcation). Therefore, we find the value of parameters for which Det(k, M 1 , M 2 ) is non-negative for all values of k, M 1 and M 2 and equals to zero at the points of its minima. Then, these thresholds correspond to the minima of the stability boundary region and satisfy:

Det(k, M 1 , M 2 ) = 0, ∂Det(k, M 1 , M 2 ) ∂M 1 = 0, ∂Det(k, M 1 , M 2 ) ∂M 2 = 0, ∂Det(k, M 1 , M 2 ) ∂k = 0. ( 51
)
With the given conditions in [START_REF] Tian | Nonlocal interactions driven pattern formation in a preypredator model[END_REF] we deduce the following result: Theorem 6. (Stationary pattern condition around the savanna homogeneous steady state) Consider z 1 and z 2 two positive solutions of the equation tan(z) = z (z 1 < z 2 ) such that: µ j = sin z j z j < 0, j = 1, 2. Then, suppose that:

R * 1 -R * 2 > 1 and a 11 (c -a 22 )µ 1 µ 2 ca 11 µ 1 -a 12 a 21 µ 2 < 1. ( 52
)
Assume also that:

M j > M T j := z j D G D T (a 11 a 22 -ca 11 ) µ 1 µ 2 + ca 11 µ 1 -a 12 a 21 µ 2 1/4 , j = 1, 2, (53)
and

k > k T := (a 11 a 22 -ca 11 ) µ 1 µ 2 + ca 11 µ 1 -a 12 a 21 µ 2 D G D T 1/4 . ( 54
)
Then we have the appearance of periodic solutions in space in the neighborhood of savanna homogeneous steady state.

Proof. See Appendix F, page 58.

Because of the difficulty of interpretation of the second condition in [START_REF] Smith | Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems[END_REF], we find a sufficient condition to the previous one. Set:

R * 3 = γ T (1 + ΩT * ) pλ f T f ω(G * ) exp(-pT * )
.

It is straightforward to observe that R 1 -R 2 < R 3 and that R 3 > 1 implies that the second condition of ( 52) is valid. Therefore, the following result holds true:

Theorem 7. (Sufficient condition) If R * 1 -R * 2 > 1, then the conclusion of Theorem 6 is valid. Remark 8.
1. R * 3 is the primary production of tree biomass and additional production of tree biomass due to tree-tree facilitation relative to fire induced tree biomass loss. 2. Condition (53) gives the range beyond which the nonlocal interactions are sufficient for the coexistence of both tree and grass inhomogeneous solutions in the same space domain. 3. Due to the implicit nature of the equation [START_REF] Diouf | Relationships between fire history, edaphic factor and woody vegetation structure and composition in a semi-arid savanna landscape (Niger, West Africa)[END_REF], it is difficult to provide explicit expression of Turing bifurcation threshold analytically and hence we have describe one way in previous theorem to determine a triplet (k T , M T 1 , M T 2 ) as a suitable choice of M 1 and M 2 to obtain stationary Turing Pattern (see also Banerjee and Volpert [START_REF] Banerjee | Prey-predator model with a nonlocal consumption of prey[END_REF]). However the space period of spatial structure is σ = 2π k max where k max is the most unstable mode, that could be computed numerically.

Numerical illustration

Our model is designed for humid savannas. Then, carrying capacities considered, before the normalization of biomasses are K G = 17t.ha -1 and K T = 340t.ha -1 . These values were obtained from Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] considering that the mean annual rainfall W is equal to 1500 mm.yr -1 . We also assume that D G = 0.1 and D T = 1 (see e.g. Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF]). The finite difference method was used to discretize the spatial part and on the other hand, the non standard finite difference method (Anguelov et al. [START_REF] Anguelov | On nonstandard finite difference schemes in biosciences[END_REF]) was used to discretize the temporal part of the problem given by the system (5) (see for instance Appendix G). Our numerical illustrations in this paper are suitable for a 9 hectare (ha) savanna square domain (for instance, Martinez-Garcia et al. [START_REF] Martinez-Garcia | Spatial patterns in mesic savannas: The local facilitation limit and the role of demographic stochasticity[END_REF] considered for example a square patch of savanna of 1 ha). Due to the fact that we have restricted the mathematical analysis to a domain of dimension 1, numerical illustrations are carried out in the space interval [0; 300]. The unit of space considered is meter (m) and unit of time is year (yr). Parameter values (see Table 4, page 18) used for model [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF] are based on (Yatat Djeumen et al. [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF][START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF], Accatino et al. [START_REF] Accatino | Tree and grass co-existence in savanna:interactions of rain and fire[END_REF]). Only Ω, D G and D T are assumed. We first illustrate a bifurcation diagram, for the space-implicit model related to system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF], with respect to variations of the fire frequency f and γ T G , the parameter that shapes the competition of tree on grass. From figure 2, one deduces that, at least for parameters considered in table 4, stable forest state is easily found, but, for low values of tree-grass competition, savannas are present. We also notice that when we approach the annual fire regime and proceed beyond we manage to recover the grassland state as part of a bistability situation. Then, the increase of the tree-grass competition parameter γ T G , leads to the transition from savanna to forest or grassland-savanna to forest-grassland. In fact, in humid zone, the vegetation is intrinsically dominated by trees, that exert competition pressure on grass biomass, such that grass may be easily suppressed. The increase of fire frequency leads to the reduction of tree biomass but thanks to tree-tree cooperation, trees can perpetuate. Therefore depending on the tree-grass competition parameter, the system switches are either savanna to grassland-savanna or forest to forest-grassland. Now we want to illustrate the spatial structuring of trees and grasses in the various cases displayed on the previous bifurcation diagram (see figure 2 in page 19).

Parameters D G D T γ G δ G γ T δ T λ f G λ f T p g 0 Ω Values 0.
4.1. Case of forest monostability (f = 0.9 and γ T G = 5.1) With the choice of parameters in table 4, the homogeneous forest steady state E T 2 = (0, 0.9477) ′ is locally asymptotic stable in absence of nonlocal interactions.

Based on theorem 5, figure 3 depicts that the forest homogeneous steady state is unstable for those M 2 such that the curve of S 2 (z) intersects with the line 1 (M 2 ) 2 . For those values, we see that the minimum M 2 required for the emergence of the Turing instability verified approximately 1 (M T 2 ) 2 = 0.0798 (see the red dashed line in figure 3), then M T 2 = 3.54m. Therefore, we choose M 1 = 0.5m and M 2 = 20m and we consider the initial data as a random perturbation of the forest homogeneous steady state (0, T 2 ) ′ : We observe from Figure 4 that the solution of system (5) converges toward a space inhomogeneous forest solution, thanks to a Turing bifurcation.

G(x, 0) = 0 + ϵ 1 , T (x, 0) = T 2 + ϵ 2 with 0 ≤ ϵ i ≤ 10 -3
The key thresholds in that situation are R F (the primary production of grass biomass, relative to grass biomass loss induced by fire, herbivory (grazing) or human action and additional grass suppression due to tree competition, at the closed forest equilibrium) and M 2 the range of nonlocal competition of trees on grasses. Using a periodogram, we can numerically determine the number of patches in our inhomogeneous solution and we can therefore compute the associated spatial wavelength. From figure 5, we have 10 patches in the spatial profile of forest distribution (see also panel (d) in figure 4). Therefore, the numerical wavelength is σ T = 300 10 = 30m. However, from the linear stability analysis and the parameter values considered in this case, the theoretical wavelength is σ T = 31.4m which is quite close of the numerical space period. We also found that for increasing values of M 2 , the space period (wavelength) σ T increases. Figure 6 shows that the numerical wavelength of tree distribution is σ T = 300 7 = 42.8m with M 2 = 30m and σ T = 300 5 = 60m with M 2 = 40m. By linear stability analysis, the space period is σ T = 47.1m for M 2 = 30m, σ T = 62.8m for M 2 = 40m 4.2. Case of savanna monostability (f = 0.9 and γ T G = 1.7)

We find that the savanna steady state E * = (0.1345, 0.9453) ′ is locally asymptotically stable in the absence of nonlocal interactions. Moreover, the minimal positive solution of the equation tan(z) = z is z 1 = 4.49. We take z 2 = 10.9 which is also solution of tan(z) = z. From these two values, we find µ 1 = -0.22, µ 2 = -0.09 and we get the Turing bifurcation condition: M 1 > 5.07m and M 2 > 12.32m. For illustration we choose M 1 = 5.5m, M 2 = 15m and we consider the initial data as a random perturbation of the savanna equilibrium (G * , T * ). We observe from figure 7 that, solutions of system (5) converge toward a space inhomogeneous tree-grass coexistence solution thanks, to a Turing bifurcation. In the same way as before, we are interested in the wavelength resulting from this inhomogeneous solution. From figure 9, k max = 0.28 and then σ = 22.43m.

The value of M 1 used previously could be questioned and seen too large. However, note that the Turing condition that we obtained is only a sufficient condition. Therefore, it may be possible that outside of these values, we can have a change of sign of Det(k, M 1 , M 2 ) which leads to a Turing bifurcation. To illustrate that point, we consider M 1 = 0.5m and M 2 = 25m and we draw Det(k, M 1 , M 2 ). From figure 10, we observe that it is possible to have a Turing bifurcation with M 1 = 0.5m and M 2 = 25m due to the change of sign of Det(k, M 1 , M 2 ). For these values of M 1 and M 2 we can thus illustrate the inhomogeneous solution obtained (see figure 11). The graph of periodogram is illustrated in figure 12. In this case, in absence of nonlocal interactions we have a bistability situation with two homogeneous steady states: a grassland steady state E G and a forest steady state E T 2 . We may observe the spatial structuring of the two state variables in two cases: first around the grassland homogeneous steady state and second around the forest homogeneous steady state.

Around the grassland homogeneous steady state

In this section we will consider D G = 0.01 and, for an easy display of figures, we reduce the size of the domain to 100m, with E G = (0.7089; 0) ′ . Based on theorem 3, figure 13 illustrates that grassland homogeneous steady state is unstable for values of M 1 such that the curve of S 1 (z) intersects with the line 1 (M 1 ) 2 . The minimal value of M 1 such that the grassland equilibrium is unstable verified 1 (M T 1 ) 2 = 2.26 (then M T 1 = 0.6647m) and we choose for illustration around the grassland equilibrium M 1 = 1.5m and M 2 = 20m. Figure 14 suggests the existence of a metastable tree-grass pattern. In fact, from panel (a) one could believe that we have an inhomogeneous solution of coexistence of the two species; but when we increase the simulation time, we observe that we are moving rather towards the inhomogeneous forest solution. So in this case we have the coexistence of unstable grassland inhomogenous solution and stable tree inhomogeneous solution. This type of solution is called a metastable state (see also Eigentler and Sheratt [START_REF] Eigentler | Metastability as a coexistence mechanisms in a model for dryland vegetation pattern[END_REF]). However, if we stop at a final time equal to 1000, we observe that the grassy biomass benefits from the space freed by the trees. We can further illustrate it with figure 15 for M 1 = 3m and M 2 = 20m.

Around the forest homogeneous steady state

The forest homogeneous steady state is E T 2 = (0, 0.9477) ′ . As previously, to find the Turing bifurcation threshold M T 2 , we need to draw the curve of S 2 (z). Then by figure 17 the nonlocal system (5) converges toward a forest inhomogeneous stable solution, and numerical space period is σ T = 33.33m while theoretically, the space period is σ T = 31.42m. Considering parameter values in table 4, the savanna homogeneous steady state E * and the grassland homogeneous steady state E G are both locally asymptotically stable for the space implicit model related to system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF]. In this section, the space domain is [0, 100] and D G = 0.01

Around the savanna homogeneous steady state

Around the savanna homogeneous steady state E * = (0.1136, 0.9455) ′ the Turing bifurcation condition are M 1 > 2.97m and M 2 > 7.21m. To illustrate the appearance of inhomogeneous solution, we choose M 1 = 3m and M 2 = 20m. Therefore, we have figure 18. The system converges toward a space inhomogeneous tree-grass coexistence solution (savanna) thanks, to a Turing bifurcation. We also observe that in figure 18, we have grass localized solution in space and regular tree spots.

Around the grassland homogeneous steady state

The grassland homogeneous steady state is E G = (0.7089, 0) ′ and is the same as before (see section 4.3.1). The Turing bifurcation threshold is the same as before. We choose M 1 = 1.5m, M 2 = 20m for illustration. Figure 19 illustrates the spatial distribution of the inhomogeneous tree-grass (i.e. savanna) solution. Figure 19 shows a high density under the trees which is due to the range of interactions between the grasses which is quite low. However, if we push this range to M 1 = 3m, we obtain the following figure 20 which is similar to the structure obtained around the savanna homogeneous steady state.

Discussion

We analyzed an integro-differential reaction-diffusion fire-mediated tree-grass interactions model, allowing to reach spatial patterns (namely, regular spotted pattern ) sometimes observed in humid savannas. Starting from the parsimonious 2-dimensional ODE-based model of grassy and woody biomasses fire-mediated interactions studied in Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF], we introduced local biomass propagation through Laplace operators, like in Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF], as well as nonlocal interaction terms. Hence, our model improves and extends previous ODE models (e.g. Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF][START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF]) by explicitly taking into account spatial components and nonlocal terms of tree-grass interactions. We showed that the combination of the nonlocal tree-tree facilitation and the nonlocal tree-tree, grass-grass and tree-grass competition, may induce spatial patterns.

In fact, nonlocal interactions break up the homogeneous distribution of tree and grass biomass resulting in the emergence of a regular spotted pattern (see for instance Tian et al. [START_REF] Tian | Nonlocal interactions driven pattern formation in a preypredator model[END_REF]). Then, novelties in this paper include the consideration of nonlocal interaction terms (both facilitation and competition) on both trees and grasses dynamics. Indeed, in the absence of nonlocal terms, our model is unable to produce spatial patterns since the associated reaction-diffusion model is monotone decreasing (see also Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF]). A key technical point is the requirement on kernels that must be constant functions with compact supports. Indeed, we show that Gaussian or Laplace-like kernels are not able to produce patterns in our model. According to Martinez-Garcia et al. [START_REF] Martinez-Garcia | Spatial patterns in mesic savannas: The local facilitation limit and the role of demographic stochasticity[END_REF], kernels whose Fourier transforms take negative values for some wavenumber values, will lead to clustering in some specific models with short range facilitation.

On the other hand, it is now acknowledged that fire is one of the key factors that shape the physiognomy of savanna vegetation, in general, and particularly, in humid savannas where rainfall is sufficient to promote very high grass biomass production which in turn constitutes the principal fuel for fires. However, as a response to the negative impact of fires, trees have developed 'defence' or resilience mechanisms in order to limit or to reduce the fire-induced tree mortality. Indeed, tree-tree facilitation or cooperation promotes germination of tree's seeds, the recruitment of new trees by improving the conditions under canopy (shading, litter and nutriments, enhanced water infiltration). We modelled this effect thanks to the Ω parameter that was added to the reference ODE model of Yatat Djeumen et al. [START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] to make the unsaturated logistic growth a non linear function of trees biomass (T ). By enhancing woody biomass growth, tree-tree facilitation indirectly reduces the grass layer or favours an heterogeneous spatial distribution of the grass layer which reduces fire intensity along with the potential of fire to spread all over the landscape.

Based on parameter values used for the bifurcation diagram (see figure 2, page 19), we explore and illustrate in the different regions of the bifurcation diagram, the spatial structuring of trees and grasses resulting from nonlocal interaction terms. We obtained broadly four types of inhomogeneous solutions: first, what we call forest inhomogeneous solution (obtained around the monostable forest space-homogeneous equilibrium) which are characterized by an absence of grass biomass and regular tree spots in the space domain. Second, the savanna inhomogeneous solutions which featured both tree and grass spots. Third, the coexistence of "localized" grass pattern and regular tree spots and, finally, the presence of metastable patterns obtained in the conditions of the forest-grassland bistable state. In each of these cases we were able to characterize a minimal range of nonlocal interactions for the appearance of spatial structures. In the case of the forest inhomogeneous state, we note that the grass biomass does not take advantage of the space between the ligneous plants, where it is absent. This may result from the fact that grassland space-homogeneous equilibrium is unstable and also from the strong pressure (competition) led by trees on grasses. We also observe the presence of extinction zones were none of the two life forms establish (see for example figure 4, page 20). In the case of the savanna inhomogeneous solution, we consider an initial distribution of the vegetation around the monostable savanna equilibrium. We find that the ligneous plants are in phase with the grass biomass. Likewise in this case, grasses do not take advantage of the space between the trees and exclusion zones are also created (see figure 7, page 21). On the other hand, we notice that the savanna inhomogeneous state is favored by the high level of woody biomass due to the fact that R * 1 -R * 2 is an increasing function of T * . In fact, one of the necessary conditions for the existence of savanna inhomogeneous solution is R * 1 -R * 2 > 1. We also notice the appearance of metastable structures when the initial setting is the forest-grassland bistability. Precisely, we considered vegetation initial distribution around the stable grassland homogeneous steady state while parameter values ensured that the forest homogeneous state is also stable. Therefore, for a substantial time of simulation up to an order of 10 3 years, we can see that the grass biomass takes advantage of the space between the trees (see figure 15, page 25).

Here, nonlocal competition between the grass tuft is responsible for this configuration. However, when the final simulation time is high (> 10 3 years), the previous tree and grass spots configuration is no longer observed. Instead, we find a regular structure of tree spots (see panel (d) figure 14, page 24). In this case, coexistence of tree and grass spots appears as a long transient phase to a tree spots pattern, which seems qualitatively compatible with the type of pattern illustrated in Fig. 1a). According to Eigentler and Sheratt [START_REF] Eigentler | Metastability as a coexistence mechanisms in a model for dryland vegetation pattern[END_REF], metatastable pattern is an unstable pattern whose instability is caused by a very small unstable eigenvalue. In case of savanna-grassland bistability, we numerically observe another type of structure that we assimilate to a coexistence of localized grass inhomogeneous solution and regular tree spots (see figure 20, page 27). In fact, Vanag and Epstein [START_REF] Vanag | Localized patterns in reaction-diffusion systems[END_REF] suggested that if the system is in the spatial bistability domain, then we must apply a perturbation of appropriate shape and sufficient amplitude in order to cause a transition to possibly localized inhomogeneous patterns. The necessary and sufficient condition for localized patches is the coexistence of homogeneous cover and periodic pattern (Tlidi et al. [START_REF] Tlidi | Extended patchy ecosystems may increase their total biomass through self-replication[END_REF], Koga and Kuramoto [START_REF] Koga | Localized patterns in reaction-diffusion systems[END_REF]). In this case, localized inhomogeneous solutions can be interpreted as a nonlinear front between spatially periodic tree distribution and aperiodic grass distribution.

Another line of discussion relies on the size of tree patches observed numerically (i.e. σ T ) and its comparison with the size (width) of the tree nonlocal interaction kernel (M 2 ) and the value of the cooperation factor Ω. Our illustration around, the forest and savanna homogeneous steady states showed that σ T ≈ 1.5M 2 . Note that, M 2 is to be related to the lateral extend of tree roots or tree canopy. In all cases, where we obtained regular spots, we find that the size of vegetation patches goes above 20m. In Lejeune et al. [START_REF] Lejeune | Localized vegetation patches: A self-organized responses to resource scarcity[END_REF] for example, the size of vegetation patches in Marahoué National Park in Ivory Coast, ranges from 10m to 20m. The value of Ω used in our work was chosen for illustrative purposes. Nevertheless, within the framework of this paper, we noticed that Ω plays a role on the kinetics of our structures. In fact, for low values of Ω (Ω < 1), the structures take longer time to set up, while the reverse occurs with Ω at large values. Lefever et al. [START_REF] Lefever | Deeply gapped vegetations patterns: on crown-roots allometry, critically and desertifiation[END_REF] gave a range of value for Ω in the case of arid vegetation. Finally in this paper, first, we choose to work in first approximation with local operator for spatial propagation (Laplace operators). This choice allows us, from a mathematical point of view, to find a good characterization of the ranges of nonlocal interactions enabling the appearance of structures. Without these local operators it would become difficult to find a mathematical characterization of spatial ranges of nonlocal interactions that can be easily manipulated numerically. Secondly and as a perspective of this work, it is necessary to improve our numerical schemes, where for which during the simulations the densities sometimes exceed the carrying capacities. This ambiguity has also been observed in other models with similar structures of equations, notably in Banerjee and Volpert [START_REF] Banerjee | Prey-predator model with a nonlocal consumption of prey[END_REF] and Genieys et al. [START_REF] Genieys | Pattern and waves for a model in population dynamics with nonlocal consumption of resources[END_REF]. It is also necessary to emphasize on the mathematical conditions allowing this model to exhibit localized structures and metastable patterns, that we observed numerically, and that may be of substantial relevance to account for field observations.

Conclusion

In this work, we developed and studied a spatio-temporal tree-grass fire-mediated interactions model allowing to illustrate the spatial structuring of vegetation in the wet savanna zone, where regular spotted patterns (tree groves) have been casually reported in presence of high grass production and frequent fires. To achieve this aim, we extended previous temporal models studied in Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF][START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF] into integro-differential reaction-diffusion systems. We explore in this model, the combination of nonlocal facilitation and nonlocal competition for the emergence of inhomogeneous solutions. In this context, we integrated kernel functions describing the area of influence of tree and grass roots and the extent of tree canopy-induced shadow effect. Both are modeled like in Martinez-Garcia et al. [START_REF] Martinez-Garcia | Spatial patterns in mesic savannas: The local facilitation limit and the role of demographic stochasticity[END_REF], Banerjee and Volpert [START_REF] Banerjee | Spatio-temporal pattern formation in rosenzweig-macarthur model : Effect of nonlocal interactions[END_REF][START_REF] Banerjee | Prey-predator model with a nonlocal consumption of prey[END_REF], Banerjee and Zhang [START_REF] Banerjee | Stabilizing role of nonlocal interaction on spatio-temporal pattern formation[END_REF] by a constant function of finite range. Accordingly, one of the major key in this paper is the simultaneous presence of nonlocal tree-tree facilitation along with nonlocal tree-tree, tree-grass and grass-grass competition. In fact, the associated model that results from the present contribution, takes into account the tree-tree cooperation mechanisms modelled by the parameter Ω which is not considered in most of the works dedicated to tree-grass interactions in fire-prone savannas, specifically in Yatat Djeumen et al. [START_REF] Yatat Djeumen | Spatially explicit modelling of tree-grass interactions in fire prone savannas: a partial differential equations framework[END_REF][START_REF] Yatat Djeumen | A minimalistic model of vegetation physiognomies in the savanna biome[END_REF]. Thanks to the stability analysis, we found conditions of existence of patterned inhomogeneous solutions around space-homogeneous steady states of our system. From a mathematical point of view, our work summarizes all the methods generally used to capture inhomogeneous solutions in nonlocal reaction-diffusion systems, and it appeared necessary to include nonlocal terms as to induce the symmetry breaking instability leading to the patterns. The sequences of patterns observed in this paper consist of regular spot vegetation (tree and grass spots noticed around the forest and grassland homogeneous steady state), "localized" grass structures and metastable pattern. In all cases where we obtained regular spotted patterns, wavelength is an increasing function of the range of tree competitive or tree canopy influence, M 2 . As a first approximation, we assumed that both grass and tree biomasses have local propagation through Laplace operators which is in line with rendering clonal propagation. But in reality, wind or even animals may also favor plant propagation through propagule dipersion. Thus Pueyo et al. [START_REF] Pueyo | Dispersal strategies and spatial organization of vegetation in arid ecosystems[END_REF] suggested that it is more reasonable to use nonlocal terms to describe plant dispersal, than diffusion terms. Hence, a line of improvement of the current work could rely on the consideration of nonlocal dispersion terms. Another important objective is to consider the same problem in a two-dimensional spatial domain as to reach more realistic prospects on the patterning processes addressed in the present paper. In so doing, we may expect to obtain very interesting multi-scale vegetation patterns.

k 1 = (γ G + δ G + λ f G f ) + (γ G + γ T G )∥G 1 ∥ 0 + γ G ∥G 2 ∥ 0 + γ T G ∥T 2 ∥ 0 , k 2 = (γ T + δ T ) + γ T (1 + Ω)(∥T 1 ∥ 0 + ∥T 2 ∥ 0 ) + Ωγ T ∥T 1 ∥ 2 0 + ∥T 2 ∥ 2 0 (∥T 1 ∥ 0 + ∥T 2 ∥ 0 ) +λ f T (1 + (θ 1 + θ 2 )∥T 2 ∥ 0 )
where θ 1 and θ 2 are respectively the Lipschitz constants of the function ω(G) and exp(-pT ) and

∥G 1 ∥ 0 = sup Dτ |G 1 |, ∥T 1 ∥ 0 = sup Dτ |T 1 | then, f = (f 1 ; f 2 ) defined in (9) is Lipschitz continuous with respect to ⟨ Û, Ũ⟩.
In addition, we define the following operators:

L 1 G = ∂G ∂t -D G ∆G + k 1 G, L 2 T = ∂T ∂t -D T ∆G + k 2 T, F 1 (G, T ) = k 1 G + f 1 (G, T ), F 2 (G, T ) = k 2 T + f 2 (G, T ). (A.2)
Then the system (5) can be reformulated as follows:

                       L 1 G = F 1 (G, T ) in D τ , L 2 G = F 2 (G, T ) in D τ , ∂G ∂x = ∂T ∂x = 0 on S τ , G(x, 0) = G 10 (x), T (x, 0) = T 20 (x) in K. (A.3)
Now we are in position to show that the system (A.3) has a unique global solution. To this aim, we construct a sequence U (m) ≡ G (m) , T (m) according to the following iteration process:

                       L 1 G (m) = F 1 (G (m-1) , T (m-1) ) in D τ , L 2 T (m) = F 2 (G (m-1) , T (m-1) ) in D τ , ∂G (m) ∂x = ∂T (m) ∂x = 0 on S τ , G (m) (x, 0) = G 10 (x), T (m) (x, 0) = T 20 (x) in K (A.4) with U (0) ∈ C α (D τ ) ∩ C(D τ ).
To show the convergence of the sequence U (m) , set:

w 1 = e -γt G, w 2 = e -γt T, (A.5)
where γ is a positive constant. The system (A.3) is equivalent to the following system:

             L i w i + γw i = H i (w 1 , w 2 ) for i = 1, 2 in D τ , ∂w ∂x = ∂w 2 ∂x = 0 on S τ w 1 (x, 0) = w 10 (x), w 2 (x, 0) = w 20 (x) in K (A.6)
where,

w 10 (x) = e -γt G 10 (x), w 20 (x) = e -γt T 20 (x), H 1 (w 1 , w 2 ) = k 1 w 1 + γ G w 1 (1 -e γt ϕ M 1 * w 1 ) -δ G w 1 -γ T G w 1 e γt ϕ M 2 * w 2 -λ f G f w 1 , H 2 (w 1 , w 2 ) = k 2 w 2 + γ T w 2 (1 + Ωe γt w 2 )(1 -e γt ϕ M 2 * w 2 ) -δ T w 2 -λ f T f ω(w 1 ) exp(-pe γt ϕ M 2 * w 2 )w 2 (A.7)
with ω(w 1 ) = w 2 1 w 2 1 + (g 0 e -γt ) 2 . According to (A.6), we can construct sequences w (m) via the following iteration process:

                 L i w (m) i + γw (m) i = H i (w (m-1) 1 , w (m-1) 2
)

for i = 1, 2 in D τ , ∂w (m) 1 ∂x = ∂w (m) 2 ∂x = 0 on S τ , w (m) 1 (x, 0) = w 10 (x), w (m) 2 (x, 0) = w 20 (x) in K. (A.8)
In term of the integral representation theory for linear parabolic boundary-value problems, the sequence w (m) can be expressed as:

w (m) i (x, t) = t 0 dτ K Γ i (x, t, ξ, τ )(H i (w (m-1) ))(ξ, τ )dξ + t 0 dτ ∂K Γ i (x, t, ξ, τ )(ψ i (w (m-1) ))(ξ, τ )dξ + K Γ i (x, t, ξ, τ )w i0 (ξ)dξ (A.9)
The second equation of system (B.1) give :

T * = T i , i = 1, 2 (depending on the values of Ω).

The first system of (B.1) leads to

G * = 1 - δ G + γ T G T i γ G , = 1 - 1 R F,f =0 .
If f > 0 and γ T G = 0, then the savanna equilibrium (G * ; T * ) ′ , satisfies:

G * = G e , Ωγ T (T * -T 2 )(T * -T 2-) + λ f T f ω(G e ) exp(-pT * ) = 0. (B.2) Let us set J(T ) = Ωγ T (T -T 2 )(T -T 2-) + λ f T f ω(G e ) exp(-pT ), then: lim T →0 J(T ) = δ T + λ f T f ω(G e )(1 -R T ). (B.3)
We have also the first derivative of J:

J ′ (T ) = Ωγ T [2T -T 2 -T 2-] -pλ f T f ω(G e ) exp(-pT ), lim T →0 J ′ (T ) = pλ f T f ω(G e )[R 1 Ω -1], lim T →1 J ′ (T ) = pλ f T f ω(G e ) exp(-p)[R 2 Ω - 1], (B.4)
where -p) .

R 2 Ω = γ T (1 + Ω) pλ f T f ω(G e ) exp(
The second derivative:

J"(T ) = 2Ωγ T + p 2 λ f T f ω(G e ) exp(-pT ) > 0. (B.5) Therefore, J ′ is increasing on [0; 1]. (I) if R 1 Ω > 1, then J ′ (T ) > 0 on [0; 1], and J is increasing on [0; 1]; (a) if R T < 1, then J(T ) > 0 on [0; 1]. (b) if R T > 1,
then there exists at most one savanna steady state.

(II) if R 1 Ω < 1, then lim T →0 J ′ (T ) < 0 and due to J ′ increasing, we have:

(a) if R 2 Ω < 1, then J ′ (T ) < 0 on [0; 1] and J is decreasing on that interval. Then (a 1 ) if R T > 1, then J(T ) < 0 on [0; 1]. (a 2 ) if R T < 1, we have at most one savanna steady state. (b) if R 2 Ω > 1, then by the intermediate value theorem, there exist T 0 ∈ [0; 1] such that J ′ (T 0 ) = 0. Then: (b 1 ) if J(T 0 ) > 0, then J(T ) > 0 on [0; 1]. (b 2 ) if J(T 0 ) < 0, we have at most two savanna steady states (G e , T * i ) ′ ,i = 1, 2 where T * 1 ∈ [0; T 0 ] and T * 2 ∈ [T 0 ; 1]
If f > 0 and γ T G ̸ = 0, savanna equilibrium is a solution of the system:

γ G (1 -G) -δ G -γ T G T -λ f G f = 0, γ T (1 + ΩT )(1 -T ) -δ T -λ f T f ω(G) exp(-pT ) = 0, (B.6)
the first equation of system (B.6) gives:

T = - δ G + λ f G f γ T G + γ G γ T G (1 -G) set: a = - δ G + λ f G f γ T G and b = γ G γ T G then T = (a + b) -bG. (B.7)
Using the fact that G, T ∈]0; 1], (B.7) gives that:

a -1 b + 1 < G < a b + 1. Note that a -1 b + 1 = (1 - 1 R G ) - γ T G γ G and a b + 1 = 1 - 1 R G . Therefore, because R G > 1 then G e - γ T G γ G < G < G e . (B.8)
The second equation of system (B.6) gives :

λ f T f ω(G) exp(-pT ) = (γ T -δ T ) + γ T (Ω -1)T -γ T ΩT 2 . (B.9)
Substituting (B.7) in (B.9) we obtain first:

(γ T -δ T ) + γ T (Ω -1)T -γ T ΩT 2 = (γ T -δ T ) + γ T (Ω -1)(a + b) -γ T Ω(a + b) 2 + (2(a + b)bγ T Ω -bγ T (Ω -1)) G -γ T Ωb 2 G 2 then, λ f T f ω(G) exp(-pT ) = (γ T -δ T ) + γ T (Ω -1)(a + b) -γ T Ω(a + b) 2 + (2(a + b)bγ T Ω -bγ T (Ω -1)) G -γ T Ωb 2 G 2 . (B.10) Set: q = (γ T -δ T ) + γ T (Ω -1)(a + b) -γ T Ω(a + b) 2 , θ = 2(a + b)bγ T Ω -bγ T (Ω -1), α = γ T Ωb 2 .
Then, we obtain in (B.10)

λ f T f ω(G) exp(-pT ) = q + θG -αG 2 . (B.11)
Substituting (B.7) in (B.11), we obtain:

λ f T f exp (-p(a + b)) exp(pbG)G 2 = qg 2 0 + θg 2 0 G + (q -αg 2 0 )G 2 + θG 3 -αG 4 . (B.12) Set m = λ f T f exp (-p(a + b)) .
Hence,

-αG 4 + θG 3 -m exp(pbG)G 2 + (q -αg 2 0 )G 2 + θg 2 0 G + qg 2 0 = 0. (B.13)
Define the function f by:

f (G) = -αG 4 + θG 3 -m exp(pbG)G 2 + (q -αg 2 0 )G 2 + θg 2 0 G + qg 2 0 (B.14)
and fine the roots of f in the interval [0

; 1].    lim G→0 f (G) = qg 2 0 , lim G→+∞ f (G) = -∞, lim G→1 f (G) = (θ -α + q)(g 2 0 + 1) -m exp(pb). (B.15)
The first derivative of f is :

f ′ (G) = -4αG 3 + 3θG 2 -mpb exp(pbG)G 2 -2m exp(pbG)G + 2(q -αg 2 0 )G + θg 2 0 (B.16) and    lim G→0 f ′ (G) = θg 2 0 , lim G→+∞ f ′ (G) = -∞, lim G→1 f ′ (G) = -4α + 3θ + 2(q -αg 2 0 ) + θg 2 0 -m exp(pb)[pb + 2].
(B.17

)
The second derivative of f is given by:

f ′′ (G) = -12αG 2 +6θG-m(pb) 2 exp(pbG)G 2 -4mpb exp(pbG)G-2m exp(pbG)+2(q-αg 2 0 ) (B.18)
and:

   lim G→0 f ′′ (G) = 2 q -(m + αg 2 0 ) , lim G→+∞ f ′′ (G) = -∞, lim G→1 f ′′ (G) = 6θ + 2(q -αg 2 0 ) -12α -m exp(pb) (pb) 2 + 4pb + 2 . (B.19)
The third derivative of f is given by: [START_REF] Jeffery | Fire management in a changing landscape: a case study from lope national park, gabon[END_REF] and:

f ′′′ (G) = -24αG + 6θ -m(pb) 3 exp(pbG)G 2 -6m(pb) 2 exp(pbG)G -6m(pb) exp(pbG) (B.
   lim G→0 f ′′′ (G) = 6(θ -mpb), lim G→+∞ f ′′′ (G) = -∞, lim G→1 f ′′′ (G) = 6θ -mpb (pb) 2 + 6pb + 6 exp(pb) + 24α . (B.21)
The fourth derivative of f is given by: m exp(pb) m exp(pb)

f ′′′′ (G) = -m(pb) 4 exp(pbG)G 2 -8m(pb) 3 exp(pbG)G -6m(pb)
g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. B. if q < m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 1 ∈ [0; 1] such that f (G * 1 ) = 0. (I.1.2) If θ > 0 then f ′ (G)
g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (b) If q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 2 ∈ [0; 1] such that f (G * 2 ) = 0. (I.1.2.2) If q < 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (0) ∈ [0; 1] such that f ′ (G (0) ) = 0 on and f ′ (G) > 0 on [0; G (0) ] and f ′ (G) < 0 on [G (0) ; 1]. (I.1.2.2.1) If f (G (0) ) < 0, then f (G) < 0 on [0; 1]. (I.1.2.2.2) If f (G (0) ) > 0, then:
(a) If q > 0 and q > m exp(pb)

g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1].
(b) If q < 0 and q > m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 3 ∈ [0; G (0)
] is the unique root of f . (c) If q > 0 and q < m exp(pb) m exp(pb)

g 2 0 + 1 + α -θ, then G * 4 ∈ [G (0) ; 1] is the unique root of f on [0; 1]. (d) If q < 0 and q < m exp(pb) g 2 0 + 1 +α-θ, then ∃G * 3 ∈ [0; G (0) ] and G * 4 ∈ [G (0) ; 1] such that f (G * 3 ) = f (G * 4 ) = 0. (I.2) If q > m + αg 2 0 then f ′′ (G)
g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (b) If q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 5 ∈ [0; 1] such that f (G * 5 ) = 0. (I.2.1.2) If θ < 0, then because f ′ is strictly increasing on [0; 1] we have: (I.2.1.2.1) If q < 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then f ′ (G) < 0 on [0; 1]. Then, f strictly decreases on [0; 1]. (a) If q < 0 then, f (G) < 0 on [0; 1]. (b) If q > 0 then, (b.1) If q > m exp(pb) g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. (b.2) If q < m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 6 ∈ [0; 1] such that f (G * 6 ) = 0. (I.2.1.2.2) If q > 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (00) ∈ [0; 1] such that f ′ (G (00) ) = 0 and then f ′ (G) < 0 on [0; G (00) ] and f ′ (G) > 0 on [G (00) ; 1]. (a) If f (G (00) ) > 0 then f (G) > 0 on [0; 1]. (b) If f (G (00)
) < 0 then, (b.1) If q < 0 and q < m exp(pb)

g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (b.
2) If q > 0 and q < m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 7 ∈ [0; G (00) ] such that f (G * 7 ) = 0. (b.
3) If q < 0 and q > m exp(pb) 00) ; 1] such that f (G * 8 ) = 0. (b.4) If q > 0 and q > m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 8 ∈ [G (
g 2 0 + 1 + α -θ, then G * 7 ∈ [0; G (00) ] and G * 8 ∈
[G (00) ; 1] are the two roots of f . m exp(pb)

(I.2.2) If q < 6α -3θ + αg 2 0 + 1 2 m (pb) 2 + 4(pb) + 2 exp(pb), then ∃ G (000) ∈ [0; 1] such that f ′′ (G (000) ) = 0. Then, f ′′ (G) > 0 on [0; G (000) ] and f ′′ (G) < 0 on [G (000) ; 1]. (I.2.2.1) If f ′ (G (000) ) < 0, then f ′ (G) < 0 on [0; 1] and f strictly decreases on that interval. (a) If q < 0 then f (G) < 0 on [0; 1]. (b) If q > 0, then: (b.1) If q > m exp(pb) g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. (b.2) If q < m exp(pb) g 2 0 + 1 + α -θ, then, ∃G * 9 ∈ [0; 1] such that f (G * 9 ) = 0. (I.2.2.2) If f ′ (G ( 000
g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (b.2) If q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 10 ∈ [0; 1] such that f (G * 10 ) = 0. (I.2.2.2.2) If θ < 0 and q > 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (0000) ∈ [0; G (000) ] such that f ′ (G (0000) ) = 0. Therefore f ′ (G) < 0 on [0; G (0000) ] and f ′ (G) > 0 on [G (0000) ; 1]. (a) If f (G (0000) ) > 0, then f (G) > 0 on [0; 1]. (b) If f (G (0000) ) < 0, then: (b.1) If q < 0 and q < m exp(pb) g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (b.
2) If q > 0 and q < m exp(pb) 0000) ; 1] such that f (G * 12 ) = 0. (b.4) If q > 0 and q > m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 11 ∈ [0; G (0000) ] such that f (G * 11 ) = 0. (b.3) If q < 0 and q > m exp(pb) g 2 0 + 1 + α -θ then ∃G * 12 ∈ [G (
g 2 0 + 1 +α-θ then G * 11 ∈ [0; G (0000) ] and G * 12 ∈
[G (0000) ; 1] are the two roots of f . (I.2.2.2.3) If θ > 0 and q < 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (00000) ∈ [G (000) ; 1] such that f ′ (G (00000) ) = 0. Therefore, f ′ (G) > 0 on [0; G (00000) ] and f ′ (G) < 0 on [G (00000) ; 1]. (a) If f (G (00000) < 0, then f (G) < 0 on [0; 1]. (b) If f (G (00000) > 0, then: (b.1) If q > 0 and q > m exp(pb)

g 2 0 + 1 + α -θ then f (G) > 0 on [0; 1]. (b.
2) If q < 0 and q > m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 13 ∈ [0; G (00000) ] such that f (G * 13 ) = 0. (b.
3) If q > 0 and q < m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 14 ∈ [G (00000) ; 1] such that f (G * 14 ) = 0. (b.4) If q < 0 and q < m exp(pb) g 2 0 + 1 +α-θ, then G * 13 ∈ [0; G (00000)
] and G * 14 ∈

[G (00000) ; 1] are the two roots of f . (I.2.2.2.4) If θ < 0 and q < 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (0000) ∈ [0; G (000) ] and

G (00000) ∈ [G (000) ; 1] such that f ′ (G (0000) ) = f ′ (G (00000) ) = 0. Therefore f ′ (G) < 0 on [0; G (0000) ] ∪ [G (00000) ; 1] and f ′ (G) > 0 on [G (0000) ; G (00000) ]. (a) If f (G (0000) ) > 0, then (a.1) If q > m exp(pb) g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. (a.2) If q < m exp(pb) g 2 0 + 1 + α -θ, G * 14 ∈ [G (00000) ; 1] is the unique root of f . (b) If f (G ( 0000
) ) < 0 and f (G (00000) ) < 0 then (b.1) q < 0 on then f (G) < 0 on [0; 1]. (b.2) q > 0 then ∃G * 11 ∈ [0; G (0000) ] is the unique root of f . (c) If f (G (0000) ) < 0 and f (G (00000) ) > 0 then ∃G * 15 ∈ [G (0000) ; G (00000) ] such that f (G * 13 ) = 0. (c.1) If q < 0 and q > m exp(pb)

g 2 0 + 1 + α -θ then G * 15 is the unique root of f in the interval [0; 1].
(c.2) If q < 0 and q < m exp(pb) g 2 0 + 1 + αθ, then with G * 15 we have also

G * 14 ∈ [G (00000) ; 1] roots of f . Therefore,f (G * 15 ) = f (G * 14 ) = 0. (c.3) If q > 0 and q > m exp(pb) g 2 0 + 1 + α -θ,then with G * 15 we have also G * 11 ∈ [0; G (0000) ] roots of f . Therefore,f (G * 15 ) = f (G * 11 ) = 0. (c.4) If q > 0 and q < m exp(pb) g 2 0 + 1 + α -θ,
(a) If q > 0, then f (G) > 0 on [0; 1]. (b) If q < 0, then: (b.1) If q < m exp(pb) g 2 0 + 1 + α -θ, then f (G) < 0, on [0; 1]. (b.2) If q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 17 ∈ [0; 1] such that f (G * 17 ) = 0. (II.1.1.2) If θ < 0, then : (II.1.1.2.1) If q < 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then f ′ (G) < 0 on [0; 1] and then f decrease strictly on [0; 1]. (a) If q < 0 then f (G) < 0 on [0; 1]. (b) If q > 0 then: (b.1) If q > m exp(pb) g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. (b.2) If q < m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 18 ∈ [0; 1] such that f (G * 18 ) = 0. (II.1.1.2.2) If q > 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (0) ∈ [0; 1] such that f ′ (G (0) ) = 0. Therefore, f ′ (G) < 0 on [0; G (0) ] and f ′ (G) > 0 on [G (0) ; 1]. (a) If f (G (0) ) > 0 then f (G) > 0 on [0; 1]. (b) If f (G (0) ) < 0, then: (b.1) If q < 0 and q < m exp(pb) g 2 0 + 1 + α -θ then f (G) < 0 on [0; 1]. (b.
2) If q > 0 and q < m exp(pb)

g 2 0 + 1 + α -θ, then G * 19 ∈ [0; G (0) ] such that f (G * 19 ) = 0. (b.3) If q < 0 and q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 20 ∈ [G (0) ; 1] such that f (G * 20 ) = 0. (b.4) If q > 0 and q > m exp(pb) g 2 0 + 1 +α-θ, then G * 19 ∈ [0; G (0)
] and G * 20 ∈ [G (0) ; 1] are the two roots of f . (II.1.2) If q < m + αg 2 0 , then we have the following situations:

(II.1.2.1) If q < 6α -3θ + αg 2 0 + 1 2 m exp(pb) (pb) 2 + pb + 2 , then f ′′ (G) < 0 on [0; 1]. Then f ′ decrease strictly on [0; 1]. (II.1.2.1.1) If θ < 0, then f ′ (G) < 0 on [0; 1] and f is therefore decreasing on [0; 1]. (a) If q < 0, then f (G) < 0 on [0; 1]. (b) If q > 0, then: (b.1) If q > m exp(pb) g 2 0 + 1 + α -θ,then f (G) < 0 on [0; 1]. (b.2) If q < m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 21 ∈ [0; 1] such that f (G * 21 ) = 0. (II.1.2.1.
2) If θ > 0, then, we have the following situations:

(a) If q > 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] ,then f ′ (G) > 0 on [0; 1] and f increase strictly on [0; 1]. (a.1) If q > 0, then f (G) > 0 on [0; 1]. (a.2) If q < 0, then: (a.2.1) If q < m exp(pb) g 2 0 + 1 + α -θ,then f (G) < 0 on [0; 1]. (a.2.2) If q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 22 ∈ [0; 1] such that f (G * 22 ) = 0. (b) If q < 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] . We use the fact that f ′ is decreasing (strictly) on [0; 1]. By the intermediate values theorem: ∃G (00) ∈ [0; 1] such that f ′ (G (00) ) = 0. Therefore, f ′ (G) > 0 on [0; G (00) ] and f ′ (G) < 0 on [G (00) ; 1]. (b.1) If f (G (00) ) < 0, then f (G) < 0 on [0; 1]. (b.2) If f (G (00) ) > 0, then: (b.2.1) If q > 0 and q > m exp(pb) g 2 0 + 1 + α -θ then f (G) > 0 on [0; 1]. (b.2.2) If q < 0 and q > m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 23 ∈ [0; G (00) ] such that f (G * 23 ) = 0. (b.2.3) If q > 0 and q < m exp(pb) g 2 0 + 1 + α -θ, then G * 24 ∈ [G (00) ; 1] such that f (G * 24 ) = 0. (b.2.4) If q < 0 and q < m exp(pb) g 2 0 + 1 +α-θ, then G * 23 ∈ [0; G (00) ] and G * 24 ∈ [G (00) ; 1] are the two roots of f . (II.1.2.2) If q > 6α -3θ + αg 2 0 + 1 2 m exp(pb) (pb) 2 + pb + 2 , because f ′′ increase strictly on [0; 1], by the intermediate values theorem ∃G (000) ∈ [0; 1] such that f ′′ (G (000) ) = 0. So, f ′′ (G) < 0 on [0; G (000) ] and there f ′′ (G) > 0 on [G (000) ; 1]. (II.1.2.2.1) If f ′ (G (000) ) > 0, then f ′ (G) > 0 on [0; 1], therefore f is increasing on that interval. (a) If q > 0, then f (G) > 0 on [0; 1]. (b) If q < 0, then (b.1) If q < m exp(pb) g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (b.2) If q > m exp(pb) g 2 0 + 1 +α-θ, then ∃G * 25 ∈ [0; 1] such that f (G * 25 ) = 0. (II.1.2.2.2) If f ′ (G (000) ) < 0, then: (a) If θ < 0 and q < 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then f ′ (G) < 0 on [0; 1]. Therefore f strictly decreases on [0; 1]. (a.1) If q < 0, then f (G) < 0 on [0; 1]. (a.2) If q > 0, then: (a.2.1) If q > m exp(pb) g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. (a.2.2) If q < m exp(pb) g 2 0 + 1 + α -θ, then ∃G * 26 ∈ [0; 1] such that f (G * 26 ) = 0. (b) If θ > 0 and q < 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (0000) ∈ [0; G (000) ] such that f ′ (G (0000) ) = 0. Therefore f ′ (G) > 0 on [0; G (0000) ] and f ′ (G) < 0 on [G (0000) ; 1]. (b.1) If f (G (0000) ) < 0, then f (G) < 0 on [0; 1]. (b.2) If f (G (0000) ) > 0, then:
(b.2.1) If q > 0, and q > m exp(pb)

g 2 0 + 1 + α -θ, then f (G) > 0 on [0; 1]. (b.2.
2) If q < 0, and q > m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 27 ∈ [0; G (0000) ] such that f (G * 27 ) = 0. (b.2.
3) If q > 0, and q < m exp(pb)

g 2 0 + 1 + α -θ, then ∃G * 28 ∈ [G (0000) ; 1] such that f (G * 28 ) = 0. (b.2.4) If q < 0, and q < m exp(pb) g 2 0 + 1 + α -θ, then * 27 ∈ [0; G (0000) ] and G * 28 ∈ [G (0000) ; 1] are the two roots of f . (c) If θ < 0 and q > 1 2 4α -3θ + (2α -θ)g 2 0 + m exp(pb) [pb + 2] , then ∃G (00000) ∈ [G (000) ; 1] such that f ′ (G (00000) ) = 0. Therefore f ′ (G) < 0 on [0; G (00000) ] and f ′ (G) > 0 on [G (00000) ; 1]. (c.1) If f (G (00000) ) > 0, then f (G) > 0 on [0; 1]. (c.2) If f (G (00000) ) < 0:
(c.2.1) If q < 0, and q < m exp(pb)

g 2 0 + 1 + α -θ, then f (G) < 0 on [0; 1]. (c.2.
2) If q > 0, and q < m exp(pb) In the same way we have :

g 2 0 + 1 +α-θ, then ∃G * 29 ∈ [0; G (00000) ] such that f (G * 29 ) = 0. (c.2.3) If q < 0, and q > m exp(pb) g 2 0 + 1 +α-θ, then ∃G * 30 ∈ [0; G (00000) ] such that f (G * 30 ) = 0. (c.
(00000) ∈ [G (000) ; 1] such that f ′ (G (0000) ) = f ′ (G (00000) ) = 0. Therefore f ′ (G) > 0 on [0; G (0000) ] ∪ [G (00000) ; 1] and f ′ (G) > 0 on [G (0000) ; G (0000) ]. (d.1) If f (G (0000) ) < 0,then (d.1.1) if q < m exp(pb) g 2 0 + 1 + α -θ then f (G) < 0 on [0; 1]. (d.1.2) if q > m exp(pb) g 2 0 + 1 + α -θ, then G * 30 is the unique root of f . (d.2) If f (G (0000) ) > 0 and f (G (00000) ) > 0, then: (d.2.1) If q > 0, then f (G) > 0 on [0; 1]. (d.2.2) If q < 0, then G * 27 ∈ [0; G (0000) ] is the unique root of f . (d.3) If f (G (0000) ) > 0 and f (G (00000) ) < 0, then: ∃G * 31 ∈ [G (0000) ; G (00000) ] such that f (G * 31 ) = 0. (d.3.1) If q > 0 and q > m exp(pb) g 2 0 + 1 + α -θ, then with G * 31 , we have also G * 30 ∈ [G (00000) ; 1] such that f (G * 31 ) = f (G * 30 ) = 0. (d.3.2) If q > 0 and q < m exp(pb) g 2 0 + 1 + α -θ, then G * 31 is the unique root of f . Therefore f (G * 31 ) = 0. (d.3.3) If q < 0 and q < m exp(pb) g 2 0 + 1 + α -θ, then with G * 31 , we have also G * 27 ∈ [0; G (0000) ] such that f (G * 31 ) = f (G * 27 ) = 0. (d.3.4) If q < 0 and q > m exp(pb) g 2 0 + 1 +α-θ, then with G * 31 and G * 30 ∈ [G (00000) ; 1], we have also G * 27 ∈ [0; G (0000) ] such that f (G * 29 ) = f (G * 30 ) = f (G * 32 ) = 0 (II.2) If θ < 1
∂g ∂t = D G ∂ 2 g ∂x 2 + γ G (g + G * ) 1 - +∞ -∞ ϕ M 1 (x -y)(g(y, t) + G * )dy -δ G (g + G * ) -λ f G f (g + G * ) -γ T G +∞ -∞
ϕ M 2 (xy)(h(y, t) + T * )dy (g + G * ).

(D.2) Developing the right-hand side of equation (D.2) and neglecting the nonlinear expressions in g we get:

∂g ∂t = D G ∂ 2 g ∂x 2 + [γ G (1 -G s ) -δ G -γ T G T s -λ f G f ] g -γ G G s +∞ -∞
ϕ M 1 (xy)g(y, t)dy

-γ T G G s +∞ -∞
ϕ M 2 (xy)h(y, t)dy.

and

ϕ M i (k) = sin(kM i ) kM i , i = 1, 2.
We are interested by the determination of thresholds k T , M T 1 and M T 2 so that:

Det(k T , M T 1 , M T 2 ) = 0.
These thresholds are solutions of the equations: time-step function 0 < φ(∆t) < 1 such that φ(∆t) = ∆t+O(∆t). The non-standard approximation for the system (5) are given by:

Det(k, M 1 , M 2 ) = 0, ∂Det(k,
                 G i+1 j -G i j φ 1 (∆t) = D G G i j+1 + G i j-1 -2G i j ∆x 2 + (γ G -δ G -λ f G f ) G i j -γ G ϕ M 1 * G i j + γ T G ϕ M 2 * T i j G i+1 j , T i+1 j -T i j φ 2 (∆t) = D T T i j+1 + T i j-1 -2T i j ∆x 2 + γ T -δ T + Ωγ T T i j T i j -γ T ϕ M 2 * T i j + Ωγ T T i j ϕ M 2 * T i j T i+1 j -λ f T f ω(G i j ) exp (-pϕ M 2 * T i j )T i+1 j , (G.1) with            φ 1 (∆t) = e (γ G -δ G -λ f G f )∆t -1 γ G -δ G -λ f G f , φ 2 (∆t) = e (γ T -δ T )∆t -1 γ T -δ T , (G.2) and                        ∆x ≤ min 2D G γ G -δ G -λ f G f ; 2D T γ T -δ T ∆t ≤ min      ln 1 + γ G -δ G -λ f G f 2D G ∆x 2 -(γ G -δ G -λ f G f ) γ G -δ G -λ f G f ; ln 1 + γ T -δ T 2D T ∆x 2 -γ T -δ T γ T -δ T      . (G.3)
Recall that R G > 1 implies that γ Gδ Gλ f G f > 0 and R T,0 > 1 implies that γ Tδ T > 0.

Second, in system (G.1) ϕ M 2 * T i j is an approximation of the convolution term

Figure 1 :

 1 Figure 1: Some vegetation mosaics of trees and grasses in Zambia and in Cameroon.

Theorem 4 .

 4 (Forest homogeneous steady state stability) If R F < 1 and ϕ 2 (z) > m * 22 m * * 22

1 ]Figure 2 :

 12 Figure2: Bifurcation diagram according to variations of γT G and f . The blue triangle corresponds to the savanna monostability, the red square stands for the forest-grassland bistability, the green star denotes the forest monostability and the yellow circle represents the grassland-savanna bistability.

Figure 3 :

 3 Figure 3: Graph of S2 as a function of z. The parameter values are given in table 4. The red dashed line stands for 1 (M2) 2 .

  time of Grass in the non local PDE FINAL TIME GRASS =8000 (a) Grass distribution in space at t = 8000. time of Trees in the non local PDE FINAL TIME TREE =8000 (b) Tree distribution in space at t = 8000.

Figure 4 :

 4 Figure 4: Illustration of Grass and Tree profiles in space.

Figure 5 :

 5 Figure 5: Graph of a periodogram of forest inhomogeneous solution

Figure 6 :

 6 Figure 6: Illustration of Trees distribution profiles in final time and the corresponding periodogram.

  time of Trees in the non local PDE FINAL TIME TREE =8000 (b) Tree distribution at t = 8000.

Figure 7 :

 7 Figure 7: Illustration of Grass and Tree profiles in space at final times.

Figure 8 :Figure 8 Figure 9 :

 889 Figure 8: Periodogram of savana inhomogeneous solution

Figure 10 :

 10 Figure 10: Graph of Det(k, M1; M2) with M1 = 0.5m and M2 = 25m.

12 G

 12 Profil in final time of Grass in the non local PDEFINAL TIME GRASS =12000 (a) Grass distribution at t = 12000. time of Trees in the non local PDE FINAL TIME TREE =12000 (b) Tree distribution at t = 12000.

Figure 11 :

 11 Figure 11: Illustration of Grass and Tree profiles in space at final times.

Figure 12 : 4 . 3 .

 1243 Figure 12: Periodogram of savana inhomogeneous solution

Figure 13 :

 13 Figure 13: Graph of S1 as a function of z with the parameter values given in table 4. The red dashed line stands for 1 (M1) 2 .

  Tree-grass distribution at t = 1000. Tree-grass distribution at t = 1200. Tree-grass distribution at t = 5000.

Figure 14 :

 14 Figure 14: Illustration of Grass and Tree distributions.

Figure 15 :

 15 Figure 15: Tree-grass distribution at t = 1000

Figure 16 :Figure 16

 1616 Figure16: Graph of S2 as a function of z with the parameter values given in table[START_REF] February | The distribution of tree and grass roots in savannas in relation to soil nitrogen and water[END_REF]

Figure 17 :

 17 Figure 17: Illustration of Grass and Tree distributions.

4. 4 .

 4 Case of bistability savanna-grassland (f = 0.98 and γ T G = 1.7)

Figure 18 :

 18 Figure 18: Tree-grass distribution at t = 10000.

Figure 19 :

 19 Figure 19: Tree-grass distributions at t = 10000.

5 G

 5 Tree-grass distribution at t = 10000.Profil in final time of Grass in the non local PDE FINAL TIME GRASS =10000(b) zooming of grass distribution at t = 10000.

Figure 20 :

 20 Figure 20: Illustration of Grass and Tree profiles in space at final times.

  ) we have: (I.1.1) if θ < 0 then f ′ (G) < 0 on [0; +∞[ and then f strictly decreases on [0; +∞[. According to (B.15) (I.1.1.1) If q < 0 then f (G) < 0 on [0; +∞[. (I.1.1.2) If q > 0 then, A. if q >

  has a positive root on [0; +∞[. (I.1.2.1) If q > 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then f ′ (G) > 0 on [0; 1] and f strictly increases on [0; 1]. (I.1.2.1.1) If q > 0 , then f (G) > 0 on [0; 1]. (I.1.2.1.2) If q < 0, then: (a) if q <

+ 1 2 m

 2 has a unique positive root on [0; +∞[. (I.2.1) If q > 6α -3θ + αg 2 0 (pb) 2 + 4(pb) + 2 exp(pb) then f ′′ (G) > 0 on [0; 1] , then f ′ strictly increases on [0; 1]. (I.2.1.1) If θ > 0 then f ′ (G) > 0 on [0; 1]. So, f strictly increases on [0; 1]. (I.2.1.1.1) If q > 0, then f (G) > 0 on [0; 1]. (I.2.1.1.1) If q < 0 , then: (a) If q <

  ) ) > 0, then: (I.2.2.2.1) If θ > 0 and q > 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then f ′ (G) > 0 on [0; 1] and f is increasing on [0; 1]. (a) If q > 0 then f (G) > 0 on [0; 1]. (b) If q < 0, then : (b.1) If q <

then with G * 15 we have also G * 11 ∈

 11 [0; G (0000) ] and G * 14 ∈ [G (00000) ; 1] roots of f . Therefore,f (G * 15 ) = f (G * 11 ) = f (G * 14 ) = 0. (II) we suppose that θ > mpb, because f ′′′ is decreasing on [0; 1], by the intermediate value theorem f ′′′ has a unique positive root on [0; +∞[. (II.1) If θ > 1 6 24α + mpb (pb) 2 + 6(pb) + 6 exp(pb) , then f ′′′ (G) > 0 on [0; 1] and therefore, f ′′ is increasing on [0; 1]. (II.1.1) If q > m + αg 2 0 , then f ′′ (G) > 0 on [0; 1] therefore, f ′ strictly increases on [0; 1]. (II.1.1.1) If θ > 0, then f ′ (G) > 0 on [0; 1] and therefore f strictly increases on [0; 1].

2 . 4 )

 24 If q > 0, and q > m exp(pb)g 2 0 + 1 + αθ, then G * 29 ∈ [0; G (00000)] and G * 30 ∈ [0; G (00000) ] are the two roots of f . (d) If θ > 0 and q > 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then: ∃G (0000) ∈ [0; G (000) ] and G

6 24α+(C. 2 )

 62 mpb (pb) 2 + 6(pb) + 6 exp(pb) , then because of the decreasing of f ′′′ on [0; 1] and by using the intermediate values theorem, ∃ G(0) ∈ [0; 1] such that f ′′′ ( G(0) ) = 0. Then f ′′′ (G) > 0 on [0; G(0) ] and f ′′′ (G) < 0 on [ G(0) ; 1]. (II.2.1) If f ′′ ( G(0) ) < 0, then f ′′ (G) < 0 on [0; 1]. Therefore f is decreasing on that interval. (II.2.1.1) If θ < 0, then f ′ (G) < 0 on [0; 1] and f is decreasing on [0; 1].(II.2.1.1.1) If q < 0, then f (G) < 0 on [0; 1]. (II.2.1.1.2) If q > 0, then: (a) If q > m exp(pb) g 2 0 + 1 + αθ, then f (G) > 0 on [0; 1]. (b) If q < m exp(pb) g 2 0 + 1 + αθ, then ∃G * 33 ∈ [0; 1] such that f (G * 33 ) = 0. (II.2.1.2) If θ > 0, then we have the following cases:(II.2.1.2.1) If q > 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then f ′ (G) > 0 on [0; 1]. Therefore f is increasing on [0; 1]. (a) If q > 0, then f (G) > 0 on [0; 1]. (b) If q < 0, then: (b.1) If q < m exp(pb) g 2 0 + 1 + αθ, then f (G) < 0 on [0; 1]. (b.2) If q > m exp(pb) g 2 0 + 1 + αθ, then ∃G * 34 ∈ [0; 1] such that f (G * 34 ) = 0. (II.2.1.2.2) If q < 1 2 4α -3θ + (2αθ)g 2 0 + m exp(pb) [pb + 2] , then ∃ G(1) ∈ [0; 1] such that f ′ ( G(1) ) = 0.Therefore f ′ (G) > 0 on [0; G(1) ] and f ′ (G) > 0 on [ G(1) ; 1]. (a) If f ( G(1) ) < 0, then f (G) < 0 on [0; 1]. (b) If f ( G(1) ) > 0, then: (b.1) If q > 0 and q > m exp(pb) g 2 0 + 1 + αθ, then f (G) > 0 on [0; 1]. (b.2) If q < 0 and q > m exp(pb) g 2 0 + 1 + αθ, then ∃G * 35 ∈ [0; G(1) ] such that f (G * 35 ) = 0. (b.3) If q > 0 and q < m exp(pb) g 2 0 + 1 + αθ, then ∃G * 36 ∈ [ G(1) ; 1] such that f (G * 36 ) = 0. (b.3) If q < 0 and q < m exp(pb) g 2 0 + 1 + αθ, then G * 35 and G * 36 are the two roots on [0; 1] of f . (II.2.2) If f ′′ ( G(0) ) > 0, then: (II.2.2.1) If q > m + αg 2 0 and q > 6α -3θ + αg 2 0 + 1 2 m exp(pb) (pb) 2 + pb + 2 , then f ′′ (G) > 0 on [0; 1].Therefore f ′ is increasing on [0; 1]. (d.4.2) If f ( G9 ) < 0,then with G * 59 and G * 58 , we have also: (d.4.2.1) If q > 0 and q > m exp(pb) g 2 0 + 1 + αθ, G * 59 and G * 58 roots of f . (d.4.2.2) If q < 0 and q > m exp(pb) g 2 0 + 1 + αθ,G * 43 . Therefore G * 59 ,G * 58 ,G * 43 are the three roots of f . (d.4.2.3) If q > 0 and q < m exp(pb) g 2 0 + 1 + αθ, G * 53 . Therefore G * 59 ,G * 58 ,G * 53 are the three roots of f . (d.4.2.4) If q < 0 and q < m exp(pb) g 2 0 + 1 + αθ, G * 43 and G * 53 . Therefore G * 59 ,G * 58 , G * 53 and G * 43 are the fourth roots of f Appendix C. Proof of Proposition 4Set:a 11 = -γ G G * , a 12 = -γ T G G * , a 21 = -λ f T f ω ′ (G * ) exp(-pT * )T * , a 22 = -γ T (1 -Ω)T * + 2Ω(T * ) 2 + pλ f T f ω(G * ) exp(-pT * )T * .For the savanna steady state, we have the Jacobian Matrix:M (G * ; T * ) = a 11 a 12 a 21 a 22 . (C.1)If f = 0, then:a 11 = -γ G G * , a 12 = -γ T G G * , a 21 = 0, a 22 = -γ T (1 -Ω)T * + 2Ω(T * ) 2 = -γ T T * [(1 -Ω) + 2ΩT * ] .Therefore:(a) If Ω = 0, then a 22 < 0. Consequently a 11 < 0 and a 22 < 0. So because a 21 = 0, (G * ; T * ) is LAS.(b) If Ω > 0, thena 22 = -γ T T * (1 -Ω) 2 + 4Ω 1 -δ T γ T < 0, then a 11 < 0 and a 22 < 0. So, (G * ; T * ) is LAS. If f ̸ = 0, T r(M (G * ; T * )) = a 11 + a 22 , Det((G * ; T * )) = a 11 a 22a 12 a 21 . Det(M ) > 0 ⇔ a 11 a 22a 12 a 21 > 0 a 11 a 22a 21 a 12 > 0 ⇔ γ G γ T G * T * [(1 -Ω) + 2ΩT * ]pγ G λ f T f ω(G * ) exp(-pT * )G * T * -γ T G λ f T f ω ′ (G * ) exp(-pT * )G * T * > 0, ⇔ γ T [(1 -Ω) + 2ΩT * ] pλ f T f ω(G * ) exp(-pT * ) -γ T G ω ′ (G * ) pγ G ω(G * ) > 1. 56 Second T r(M (G * , T * )) < 0 ⇔ γ G G * pλ f T f ω(G * exp(-pT * )T * + γ T [(1 -Ω) + 2ΩT * ] pλ f T f ω(G * exp(-pT * ) > 1.But,γ T [(1 -Ω) + 2ΩT * ] pλ f T f ω(G * ) exp(-pT * ) -γ T G ω ′ (G * ) pγ G ω(G * ) > 1 ⇒ γ G G * pλ f T f ω(G * exp(-pT * )T * + γ T [(1 -Ω) + 2ΩT * ] pλ f T f ω(G * exp(-pT * ) > 1. Consequently, if γ T [(1 -Ω) + 2ΩT * ] pλ f T f ω(G * ) exp(-pT * ) -γ T G ω ′ (G * ) pγ G ω(G * ) > 1, then (G * ; T * ) is stable.Appendix D. Proof of Proposition 5We have h= T -T s and g = G -G s , then ∂h ∂t ∂x 2 + γ T (h + T s ) (1 + Ω (h + T s )) 1 -+∞ -∞ ϕ M 2 (xy)(h(t, y) + T s )dy -δ T (h + T s )λ f T f ω(g + G s ) exp -p +∞ -∞ ϕ M 2 (xy)(h(t, y) + T s )dy (h + T s ) (D.1) Developing the right-hand side of equation (D.1) and neglecting the nonlinear expressions in h we get:∂h ∂t = D T ∂ 2 h ∂x 2 + (γ T (1 + ΩT s ) + γ T ΩT s ) (1 -T s ) hδ T hλ f T f ω(G s ) exp(-pT s )h -λ f T f ω ′ (Gs ) exp(-pT s )T s gγ T T s (1 + ΩT s ) +∞ -∞ ϕ M 2 (xy)h(y, t)dy +λ f T f pω(G s ) exp(-pT s )T s +∞ -∞ ϕ M 2 (xy)h(y, t)dy. = D T ∂ 2 h ∂x 2 + [(γ T (1 + ΩT s )(1 -T s )δ Tλ f T f ω(G s ) exp(-pT s )) + γ T ΩT s (1 -T s )] h + (pλ f T f ω(G s ) exp(-pT s )T sγ T T s (1 + ΩT s )) +∞ -∞ ϕ M 2 (xy)h(y, t)dy -λ f T f ω ′ (G s ) exp(-pT s )T s g.

ϕ M 2

 2 (xy)T (y, t)dy, done by the Matlab function "trapz". It is the same for ϕ M 1 * G i j .

Table 1 :

 1 Definition of parameters used in the model.

	Symbols	Description	Units
	γ G δ G	Intrinsic growth of grasses Grass biomass loss due to human activities and herbivory	yr -1 yr -1
	λ f G γ T G γ T δ T	Portion of grass biomass loss due to fire Tree grass interaction parameter Intrinsic growth of trees Tree biomass loss due to human activities	yr -1 yr -1 yr -1
	λ f T	Portion of tree biomass loss due to fire
	p	proportional to the inverse of biomass destroyed at intermediate level of mortality
	Ω	Cooperation factor
	f D G D T	fire frequency Grass biomass diffusion rate Tree biomass diffusion rate	yr -1 ha 2 .yr -1 ha 2 .yr -1
	M 1	Range of grass spatial nonlocal interaction	m
	M 2	Range of tree spatial nonlocal interaction	m

Table 2 :

 2 Maximal number of savanna steady states of system[START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF] with θ < mpb

• Case 2: θ > mpb, Condition θ < θ * θ > θ * Maximal number on savanna steady states 4 3

Table 3 :

 3

Maximal number of savanna steady states of system

[START_REF] Deblauwe | Environnemental modulation of self-organized periodic vegetation in sudan[END_REF]

with θ > mpb

Table 4 :

 4 Parameter values for simulation.

  M 1 , M 2 )Differentiating Det(k, M 1 , M 2 ) with respect to M 1 and M 2 and using the fact that:∂Det(k, M 1 , M 2 ) ∂M 1 = 0 and ∂Det(k, M 1 , M 2 ) ∂M 2 = 0 we obtain: (a 11 a 22ca 11 ) ϕ M 2 (k) + ca 11a 11 D T k 2 a 11 a 22ca 11 11 a 22ca 11 ) ϕ M 1 (k) -D G (a 22c)k 2 + a 12 a 21 a 11 a 22ca 11 M 2 (k) = a 11 D T k 2ca 11 a 11 a 22ca 11 = D T k 2c a 22c or ∂ϕ M 1 ∂M 1 = 0, ϕ M 1 (k) = D G (a 22c)k 2 +a 12 a 21 a 11 a 22ca 11 D T k 2c a 22c and ϕ M 1 (k) = D G (a 22c)k 2 + a 12 a 21 a 11 a 22ca 11 then, Det(k, M 1 , M 2 ) = a 12 a 21 ca 22 D T k 2 + c a 12 a 21 a 22c . Using the fact that ∂Det(k, M 1 , M 2 ) Det(k, M 1 , M 2 ) = a 12 a 21 ca 22 D T k 2 + c a 12 a 21 a 22c and we can not have Turing bifurcation there. Third, if: ϕ M 1 (k) = D G (a 22c)k 2 + a 12 a 21 a 11 a 22ca 11

			∂M 1	= 0,	∂Det(k, M 1 , M 2 ) ∂M 2	= 0	∂Det(k, M 1 , M 2 ) ∂k	= 0. (F.1)
							∂ϕ M 1 ∂M 1	= 0,
	and	(a ∂ϕ M 2 ∂M 2	= 0.
	Then we have:
		          	ϕ or	∂ϕ M 2 ∂M 2	= 0.	(F.2)
	First, if:
		ϕ M 2 (k) =
	obtain k = 0 and then we return to the temporal case.	∂k	= 0, we
	Second, if:
			ϕ M 2 (k) =	D T k 2 -c a 22 -c	and	∂ϕ M 2 ∂M 2	= 0
	then as previously: and	∂ϕ M 2 ∂M 2	= 0
	then we have the same results as before. Finally, if:
			∂ϕ M 1 ∂M 1	= 0 and	∂ϕ M 2 ∂M 2	= 0,
	we obtain
			tan(kM

1 ) = kM 1 and tan(kM 2 ) = kM 2 .

=

3.54m). Hence, for illustration, we choose M 1 = 0.5m and M 2 = 20m.

Acknowledgements

We thank all the four anonymous reviewers and the associated editor for their insightful comments that helped us to improve the presentation of previous drafts of the manuscript.

Appendix A. Proof of Theorem 1.

Before we give the proof of Theorem 1, we first recall some results: Definition 2. A function f i is called Lipschitz continuous with respect to ⟨ Û, Ũ⟩ if there exist a constant k i > 0 for any U = (G 1 , T 1 ) ′ V = (G 2 , T 2 ) ′ ∈ ⟨ Û, Ũ⟩ such that:

Furthermore, if f 1 and f 2 are Lipschitz continuous with respect to ⟨ Û, Ũ⟩, then we call f = (f 1 ; f 2 ) ′ is Lipschitz continuous with respect to ⟨ Û, Ũ⟩. Proposition 10. (Lipschitz condition) If Ûand Ũ are bounded, direct calculations show that there exists constants k 1 and k 2 such that:

where Γ i is the fundamental solution of the Linear parabolic operator L i + γ i and ψ i is the singlelayer potential. We now show that the sequence {w (m) } converges in C(D τ ) to a unique solution of the associated integral in (A.9). Set X = X 1 × X 2 , where :

(A.10) Lemma 1. If w and w ′ ∈ X , then H i (w) ∈ C α (D τ ) ∩ C(D τ ) and:

Proof. First H i (w) ∈ C α (D τ ) ∩ C(D τ ), because the plus, multiplication, spatial convolution and composition do not change the Hölder continuous property of the functions. Secondly,

) + e -γt f i (e γt w 1 , e γt w 2 , e γt ϕ M 1 * w 1 , e γt ϕ M 2 * w 2 )

-f i (e γt w 1 , e γt w 2 , e γt ϕ M 1 * w 1 , e γt ϕ M 2 * w 2 ) ,

Theorem 8. Let ( Ũ, Û) be a pair of coupled upper and lower solutions of system [START_REF] Yatat Djeumen | A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address a broad spatial scales in spite of scarce data[END_REF]. Then the system (5) has a unique solution U * (x, t) and U * ∈ ⟨ Ũ, Û⟩. Moreover, for any

Proof. The proof is based on the contraction mapping theorem in the Banach space C(D τ ). For each i = 1, 2, we define the operators

) by:

where D(A i ) is the domain of A i given by:

) is given by (A.7). In terms of the operators A i and H i , the iteration process in (A.8) can be written as:

) (w

and in vector form it becomes:

From the standard parabolic theorem the inverse operator A -1 exists and possesses the property:

where k 3 = min {k 1 , k 2 }. This implies that (A.15) is equivalent to:

), (w (m-1) ∈ D(A)), (A.17) which can be considered as a compact form for the integral representation (A.9) in the space C α (D τ ) ∩ C(D τ ). In term of lemma (1), there exists a constant k, independent of γ, such that:

Combining (A. [START_REF] Pueyo | The role of reproduction plant traits and biotic interactions in the dynamics of semi-arid plant communities[END_REF]) and (A.18), we have:

Thus, the operator A -1 H possesses a contraction property in X . This ensures that the sequence w (m) converges in C(D τ ). By the equivalence between (A.17) and (A.9) the sequence w

given by (A.9) converges in C(D τ ) to w * i for i = 1, 2. To show that w * is the unique solution of (A.4). Since U (m) = e γt w (m) , the sequence U (m) governed by (A.4) converges to a unique solution U * = e γt w * to the equation (A.3). By the equivalence between (A.3) and ( 5), U * is the unique solution of the system (5).

In theorem [START_REF] Lefever | Deeply gapped vegetations patterns: on crown-roots allometry, critically and desertifiation[END_REF] we prove that to show the existence and the uniqueness of the solution to the system (5), we only need to find a pair of coupled upper and lower solution Ũ and Û which satisfy the Lipschitz condition. If we choose Ũ and Û to be constant vectors c and ĉ, these constant need to satisfy:

We choose ĉ1 = ĉ2 = 0. Then c1 = max sup K G(0, x), 1 - 2) ) < 0, then we have the following cases: (b.2.1) If q < 0 and q < m exp(pb)

2) If q > 0 and q < m exp(pb) 3) ) < 0, then we have the following cases:

2) If q < 0 and q > m exp(pb) 5) ) < 0, then: (c.2.1) If q < 0 and q < m exp(pb)

2) If q > 0 and q < m exp(pb) 4) ) < 0, then we have the following cases: 7) ) < 0, then: (b.2.1) If q < 0 and q < m exp(pb)

2) If q > 0 and q < m exp(pb) 8) ) > 0, then: (c.2.1) If q > 0 and q > m exp(pb)

2) If q < 0 and q > m exp(pb) m exp(pb)

53 as a root of f . (d.3.2) If q < 0 and q > m exp(pb)

is the unique root of f . . (d.3.3) If q > 0 and q > m exp(pb) g 2 0 + 1 + αθ, then with G * 54 we have also G * 50 as a root of f . (d.3.4) If q > 0 and q < m exp(pb)

53 are the three roots on [0; 1] of f .

52

(II.2.2.4) If q < m+αg 2 0 and q < 6α-3θ+αg 2 0 + 1 2 m exp(pb) (pb) 2 + pb + 2 , then: G(3) and G( 6) are the two roots of f ′′ . Therefore, f ′′ (G) > 0 on [ G(3) ; G(6) ] and 8) ) > 0, then: (b.2.1) If q > 0 and q > m exp(pb)

2) If q < 0 and q > m exp(pb) (a.2.1) If q > m exp(pb) 4) ) > 0, then:

(b.2.1) If q > 0 and q > m exp(pb)

2) If q < 0 and q > m exp(pb)

(b.2.4) If q < 0 and q < m exp(pb)

(a.2.1) If q < 0 and q < m exp(pb)

(a.2.2) If q > 0 and q < m exp(pb)

we have also G * 56 roots of f . (b.3.4) If q < 0 and q > m exp(pb)

Therefore, (G e , 0) is unstable. To show that system (5) undergoes spatial Turing bifurcation at M T 1 , we need to verify that spatial Turing bifurcation occurs prior to the temporal Hopf bifurcation (case where T r(M ) = 0 and Det(M ) > 0) as M 1 increases to M T 1 . From the above argument, we only need to show that if [START_REF] Sherratt | When does colonisation of a semi-arid hillslope generate vegetation patterns?[END_REF] fails then [START_REF] Lejeune | Localized vegetation patches: A self-organized responses to resource scarcity[END_REF] must have already failed as M 1 increases. When [START_REF] Sherratt | When does colonisation of a semi-arid hillslope generate vegetation patterns?[END_REF] fails, we have:

Plugging (E.2) into E.1), we see that

Thus, [START_REF] Lejeune | Localized vegetation patches: A self-organized responses to resource scarcity[END_REF] does not hold and this ends the proof.

Appendix F. Proof of Theorem 6

Suppose that R * 1 -R * 2 > 1 and a 11 (ca 22 )µ 1 µ 2 ca 11 µ 1a 12 a 21 µ 2 < 1, we have: Appendix G. Numerical scheme

The numerical scheme for the problem given by system (5) is obtained by using non standard finite method for the discretization of the temporal part of the system and difference finite method for the spatial part. We subdivided the space domain (0, l) in n + 1 intervals such that:

x 0 = 0 < x 1 < x 2 < ... < x n < x n+1 = l, where ∀ j = 1, ..., n ∆x = x j+1x j = l n + 1 and x j = j∆x.

In the same way, we subdivided the time interval such that: t 0 < t 1 < t 2 < ... < t i < t j+1 < ... and t i = i∆t.

We denote by G i j and T i j respectively the value of G and T at the time t i and at the space point x j . Remark first that in non standard method, non linear terms are substituted by a non local approximation. Second, the standard denominator ∆t in each discrete derivative is replaced by a