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Mathematical models of sand dune dynamics have considered different types of sand dune cover. However, despite the important role of psammophilous plants (plants that flourish in moving-sand environments) in dune dynamics, the incorporation of their effects into mathematical models of sand dunes remains a challenging task. Here we propose a nonlinear physical model for the role of psammophilous plants in the stabilization and destabilization of sand dunes. There are two main mechanisms by which the wind affects these plants: (i) sand drift results in the burial and exposure of plants, a process that is known to result in an enhanced growth rate, and (ii) strong winds remove shoots and rhizomes and seed them in nearby locations, enhancing their growth rate. Our model describes the temporal evolution of the fractions of surface cover of regular vegetation, biogenic soil crust, and psammophilous plants. The latter reach their optimal growth under either (i) specific sand drift or (ii) specific wind power. The model exhibits complex bifurcation diagrams and dynamics, which explain observed phenomena, and it predicts new dune stabilization scenarios. Depending on the climatological conditions, it is possible to obtain one, two, or, predicted here for the first time, three stable dune states. Our model shows that the development of the different cover types depends on the precipitation rate and the wind power and that the psammophilous plants are not always the first to grow and stabilize the dunes.