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Effect of grazing on plant diversity was studied based in 92 sites of the semiarid Magellan Steppe, South Patagonia, Argentina. 85 species were registered, 96% of them perennials and mostly native and endemic, with only 7.6% exotics. Cover of tussock grass Festuca gracillima was used as a Proxy variable for grazing intensity. Maximum likelihood analysis of diversity vs. tussock cover selected a quadratic function with a maximum at intermediate grazing intensities, closer to predictions of equilibrium models (MSL) for sub humid environments with a long history of grazing than those for semi arid-short evolutionary grazing history. State and Transition (ST) models indicate that the most diverse areas have probably lost tussock cover by intense use in their grazing history but are at present under lower grazing pressure, as this induces gain of diversity of short grasses without reestablishing tussock cover. A new model is proposed combining previous MSL and ST models. It shows a smaller increase in diversity with intermediate/high grazing, loops in the trajectory and multiple alternate states under low grazing depending on the management history. It may be an example of non-linear models that are supposed to be common in semiarid areas but have few published examples. (C) 2015 Elsevier Ltd. All rights reserved.