Comparison principles and applications to mathematical modelling of vegetal meta-communities

This article partakes of the PEGASE project the goal of which is a better understanding of the mechanisms explaining the behaviour of species living in a network of forest patches linked by ecological corridors (hedges for instance). Actually we plan to study the effect of the fragmentation of the habitat on biodiversity. A simple neutral model for the evolution of abundances in a vegetal metacommunity is introduced. Migration between the communities is explicitely modelized in a deterministic way, while the reproduction process is dealt with using Wright-Fisher models, independently within each community. The large population limit of the model is considered. The hydrodynamic limit of this split-step method is proved to be the solution of a partial differential equation with a deterministic part coming from the migration process and a diffusion part due to the Wright-Fisher process. Finally, the diversity of the metacommunity is adressed through one of its indicators, the mean extinction time of a species. At the limit, using classical comparison principles, the exchange process between the communities is proved to slow down extinction. This shows that the existence of corridors seems to be good for the biodiversity.