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Savannas are characterized by the coexistence of two contrasting plant life‐forms: woody and herbaceous vegetation. During the last decades, there has been a global trend of an increase in woody cover and the spread of shrubs and trees into areas that were previously dominated by grasses. This process, termed bush encroachment, is associated with severe losses of ecosystem functions and typically difficult to reverse. It is assumed to be an example of a critical transition between two alternative stable states. Overgrazing due to unsustainable rangeland management has been identified as one of the main causes of this transition, as it can trigger several self‐reinforcing feedback loops. However, the dynamic role of grazing within such feedback loops has received less attention. We used a set of coupled differential equations to describe the competition between shrubs and grasses, as well as plant biomass consumption via grazing and browsing. Grazers were assumed to receive a certain level of care from farmers, so that grazer densities emerge dynamically from the combined effect of vegetation abundance and farmer support. We quantified all self‐reinforcing and self‐dampening feedback loops at play and analyzed their relative importance in shaping system (in‐)stability. Bistability, the presence of a grass dominated and a shrub dominated state, emerges for intermediate levels of farmer support due to positive feedback that arises from competition between shrubs and grasses and from herbivory. We furthermore demonstrate that disturbances, such as drought events, trigger abrupt transitions from the grass dominated to the shrub dominated state and that the system becomes more susceptible to disturbances with increasing farmer support. Our results thus highlight the potential of interaction networks in combinations with feedback loop analysis for improving our understanding of critical transitions in general, and bush encroachment in particular. [ABSTRACT FROM AUTHOR]

1. Sustainable land use under climate change requires detailed knowledge of the system dynamics. This applies particularly for the management of domestic livestock in semi-arid and arid grazing systems, where the risk of degradation is high and likely climate change may have a strong impact. A suitable way to assess potential future trends of these complex systems is through the application of simulation models. 2. We reviewed 41 models published between 1995 and 2005 simulating semi-arid and arid livestock grazing systems. The models were categorized according to the model aim and type, their temporal and spatial scale, and several indicators of model complexity. Additionally, we developed a list of model requirements for adequately simulating the effects of climate change. Based on these requirements, we evaluated the potential of current models to simulate impacts of climate change and determine important shortcomings. 3. Three general model types could be distinguished, namely state and transition, difference and differential equation, and rule-based models. Over time, we found that the number of models aiming to improve management strategies increased, while there were fewer models that aimed to understand system dynamics. This was accompanied by a trend to simplify model descriptions of hydrological relationships. 4. Important shortcomings of current models included the impact of increased CO2 levels on plant productivity and the ability to resolve changes in intra-annual precipitation patterns. The consideration of both external drivers is crucial under climate change, hence sustainable long-term decision making is currently lacking important information. 5. Synthesis and applications. Sustainable livestock management in semi-arid and arid systems requires knowledge about effects of future climate change to adjust livestock density adequately. Producers’ experiences from current weather conditions are not necessarily transferable to future conditions, thus models could help to support management. However, an analysis of current models has shown that few existing models are able to assess the impacts of the predicted climate change. Therefore, we call for the development of new dynamic grazing models that provide land managers with the necessary tools to face the threat of future climate changes. [ABSTRACT FROM AUTHOR]