Abstract
Vegetation biome encompasses in wet ecosystems, self organized physiognomies that traduce complex dynamical processes leading to homogeneous distributions of forests, grasslands and savannas, to heterogeneous distributions of trees and grasses. Spatio-temporal patterns of vegetation, are characteristic feature of wetland ecosystems occurring in all continents. The development of a better understanding of their spatial dynamics, is an issue of considerable ecological and social economical importance, by regulating global climate and provides materials need for human. Mathematical modelling is a useful tool to describe dynamics of complex systems and, several mathematical models have been devoted to the study of tree-grass dynamics in savanna ecosystems, but with a scarcely attention, of spatial mechanism of tree and grass interactions that translated in space. This work dedicated to the modelling and the analyse via partial differential equations on tree and grass dynamics in humid savannas is divided in two main parts. In the first part, we propose and analyse a spatio-temporal model of tree-grass interactions in humid savanna. This first model, is based on two nonlocal reaction-diffusion equations with kernels of intra and inter specific competition and also a kernel indirectly acting as facilitation in term of reduction of fire effect on tree mortality; all off these in the reaction part of the model. The diffusion part is modelled via the Laplace operators considered in a one spatial domain. A qualitative analyse of this model reveals several ecological thresholds that shape the overall dynamics of the model. Thanks to linear stability analysis, the model account for the occurrence of space inhomogeneous solutions. All of these lead us to conclude that, the interplay between nonlocal competition and nonlocal facilitation, can explain the spatial periodic structuring sometimes observed in humid savannas. In the second part of this work, we consider nonlocal seed dispersal, as to describe the propagation in space of both tree and grass biomass. We therefore replaced, the Laplace operators by integral operators, and, we focus on the existence of travelling wave connecting the grassland homogeneous steady state to the forest homogeneous steady state of the model. A qualitative analyse of this reaction dispersion model, leads to the characterisation by a mathematical expression depending on several parameters of the model, of the minimal wave speed that controls the forest encroachment into the grassland. We therefore found that, the length of tree seed dispersal and the fire frequency can control the wave propagation.