Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

Pattern formation at the ecosystem level is a rapidly growing area of spatial ecology. Theoretical models are a widely used tool for studying e.g. banded vegetation patterns. One important model is the system of advection-diffusion equations proposed by Klausmeier which is a model for vegetation dynamics in semi-deserted areas. It is a generalization of the so-called Gray-Scott system which already exhibits effects similar to Turing patterns.

The underlying mathematics of this model is given by a pair of solutions $(u,v)$ to a partial differential equation system coupled by a nonlinearity. The function $u$ represents the surface water content  and $v$ represents the biomass density of the plants. In order to model the spread of water on a terrain without a specific preference for the direction in which the water flows, the original models were extended by replacing the diffusion operator by a nonlinear porous medium operator, which represents the situation that the ground is  partially filled by interconnected pores conveying fluid under an applied pressure gradient.

We investigate the existence of a pair of nonnegative solutions to the stochastic Klausmeier system with Gaussian multiplicative noise. The proof of existence is based upon a stochastic version of the Schauder-Tychonoff fixed point theorem, for which we shall also provide a proof.

See https://arxiv.org/abs/1912.00996 for the Preprint.