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.

[1]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[2]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[3]  Jorge Bustamante,et al.  Bernstein Operators and Their Properties , 2017 .

[4]  John Wakeley,et al.  The many-demes limit for selection and drift in a subdivided population. , 2004, Theoretical population biology.

[5]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[6]  G. M. Lieberman.  SECOND ORDER PARABOLIC DIFFERENTIAL EQUATIONS , 1996 .

[7]  Djalil CHAFAÏ,et al.  Recueil de modèles aléatoires , 2016 .

[8]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[9]  Pierre Priouret,et al.  Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum , 1968 .

[10]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[11]  Stewart N. Ethier,et al.  Diffusion approximations of Markov chains with two time scales and applications to population genetics, II , 1980 .

[12]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[13]  Sylvie Méléard.  Modèles aléatoires en écologie et évolution , 2016 .

[14]  A Stochastic Model for Growing Sandpiles and its Continuum Limit , 1998 .

[15]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[16]  S. Ethier.  A class of degenerate diffusion processes occurring in population genetics , 1976 .