In this talk, we focus on the numerical solution of a reaction-diffusion vegetation model that has been introduced in 2019 to investigate the coexistence of two different plant species in arid environments (such as the African savannah), where there is scarce presence of water. This model is characterized by high stiffness, metastability, positivity and oscillating behavior in space. Our goal lies in the construction of efficient problem-oriented numerical schemes, which are able to preserve the a-priori known properties of the exact solution for large values of the discretization step-sizes. For the numerical solution, we propose discretizations based on suitably modified non-standard finite differences (Mickens, 2020) combined with the TASE preconditioners (Calvo et al., 2021) and the exponential fitting technique (Ixaru, Vanden Berghe, 2004), proving that the methods thus derived are able to preserve the known properties of the model. Finally, to increase the order of consistency of these schemes over time, we propose a generalization of the TASE-preconditioned methods by exploiting connections with the W-methods. Numerical tests confirm the efficiency of the proposed techniques.
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