Global attractors for a vegetation model

Abstract

In this article, a rigorous mathematical treatment of the dryland vegetation model introduced by Gilad et al. (Phys. Rev. Lett. 98(9) (2004), 098105-1-098105-4, J. Theoret. Biol. 244 (2007), 680-691) is presented. We prove the existence and uniqueness of solutions in (L 1 (Ω)) 3 and the existence of global attractors in L 1 (Ω; D), where D is an invariant region for the system. A key step is the regularization of the model by adding eΔ to the diffusion term and by approximating the initial data U0 by a sequence {U0,n} of smooth functions in (L 1 (Ω)) 3 . The various a priori estimates and the maximum principle permit the passage to the limit as e → 0a ndn →∞ , proving the existence and uniqueness of solutions U in the specified space. Also, we deduce from the a priori estimates that the solution meets the necessary hypotheses (see Theorem 1.1 in Chapter 1 of Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997) and hence, we obtain the existence of global attractors.