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.
Citation: Gauthier Delvoye, Olivier Goubet, Frédéric Paccaut. Comparison principles and applications to mathematical modelling of vegetal meta-communities[J]. Mathematics in Engineering, 2022, 4(5): 1-17. doi: 10.3934/mine.2022035
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
| [1] | D. Bakry, I. Gentil, M. Ledoux, Analysis and geometry of Markov diffusion operators, Springer Science & Business Media, 2013. |
| [2] | O. Blondel, C. Cancès, M. Sasada, M. Simon, Convergence of a degenerate microscopic dynamics to the porous medium equation, arXiv: 1802.05912. |
| [3] | J. M. Bony, P. Courrège, P. Priouret, 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, Ann. de l'Institut Fourier, 18 (1968), 369–521. |
| [4] | P. Brémaud, Markov chains. Gibbs fields, Monte Carlo simulation, and queues, Springer, 1998. |
| [5] | H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2011. |
| [6] | J. Bustamante, Bernstein operators and their properties, Birkauser, 2017. |
| [7] | D. Chafai, F. Malrieu, Recueil de modèles aléatoires, Berlin Heidelberg: Springer-Verlag, 2016. |
| [8] | S. Ethier, A class of degenerate diffusion processes occuring in population genetics, Commun. Pur. Appl. Math., 29 (1976), 483–493. |
| [9] | S. Ethier, T. Kurtz, Markov processes - characterization and convergence, New York: John Wiley & Sons, Inc., 2005. |
| [10] | S. Ethier, T. Nagylaki, Diffusion approximations of Markov chains with two time scales and applications to population genetics, Adv. Appl. Prob., 12 (1980), 14–49. |
| [11] | C. Evans, F. Rezakhanlou, A stochastic model for growing sandpiles and its continuum limit, Commun. Math. Phys., 187 (1998), 325–347. |
| [12] | D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag Berlin Heidenberg, 2001. |
| [13] | O. Kallenberg, Foundations of modern probability, New York: Springer-Verlag, 2002. |
| [14] | O. Ladyzenskaja, V. Solonnikov, N. Ural'ceva, Linear and quasilinear equations of parabolic type, AMS, 1968. |
| [15] | G. Lieberman, Second order parabolic differential equations, Singapore: World scientific publishing, 1996. |
| [16] | S. Méléard, Modèles aléatoires en écologie et évolution, Berlin Heidelberg: Springer-Verlag, 2016. |
| [17] | A. Personne, A. Guillin, F. Jabot, On the Simpson index for the Moran process with random selection and immigration, arXiv: 1809.08890. |
| [18] | D. Stroock, S. Varadhan, Multidimensional diffusion processes, Berlin Heidelberg New York: Springer, 1997. |
| [19] | J. Wakeley, T. Takahashi, The many-demes limit for selection and drift in a subdivided population, Theor. Popul. Biol., 66 (2004), 83–91. |
Gauthier Delvoye, Olivier Goubet, Frédéric Paccaut. Comparison principles and applications to mathematical modelling of vegetal meta-communities[J]. Mathematics in Engineering, 2022, 4(5): 1-17. doi: 10.3934/mine.2022035
DownLoad: