  On the one hand, we investigate the existence and pathwise uniqueness of a
nonnegative martingale solution to the stochastic evolution system of nonlinear
advection-diffusion equations proposed by Klausmeier with Gaussian
multiplicative noise. On the other hand, we present and verify a general
stochastic version of the Schauder-Tychonoff fixed point theorem, as its
application is an essential step for showing existence of the solution to the
stochastic Klausmeier system. The analysis of the system is based both on
variational and semigroup techniques. We also discuss additional regularity
properties of the solution.
