On a nonlocal system for vegetation in drylands

Matthieu Alfaro (1) , Hirofumi Izuhara (2) , Masayasu Mimura (3)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 University of Miyazaki
3 MIMS - Meiji Institute for Advanced Studies of Mathematical Sciences

Several mathematical models are proposed to understand spatial patchy vegetation patterns arising in drylands. In this paper, we consider the system with nonlocal dispersal of plants (through a redistribution kernel for seeds) proposed by Pueyo et al. (Oikos 117:1522-1532, 2008) as a model for vegetation in water-limited ecosystems. It consists in two reaction diffusion equations for surface water and soil water, combined with an integro-differential equation for plants. For this system, under suitable assumptions, we prove well-posedness using the Schauder fixed point theorem. In addition, we consider the stationary problem from the viewpoint of vegetated pattern formation, and show a transition of vegetation patterns when parameter values (rainfall, seed dispersal range, seed germination rate) in the system vary. The influence of the shape of the redistribution kernel is also discussed.

Domaines: Mathématiques [math]

Matthieu Alfaro, Hirofumi Izuhara, Masayasu Mimura. On a nonlocal system for vegetation in drylands. Journal of Mathematical Biology, 2018, 77 (6-7), pp.1761-1793. ⟨10.1007/s00285-018-1215-0⟩. ⟨hal-01982126⟩