Meta-model for Patagonian grass steppe dynamics - library of transition matrices for steppe dynamics under sheep grazing

One of the central problems in ecology is how to scale from small-scale observations and experiments to large-scale patterns and processes. One approach to such upscaling is to use dynamic simulation models, but their application to large scales relevant for management is limited by computational costs, and their outputs are difficult to analyse without a systematic strategy. Our general objective is to propose such a strategy. The idea is to approximate the dynamics of detailed simulation models through a set of states, external drivers, and transition matrices, and then use Markov chain and network analysis of the resulting transition matrices to gain insights into the dynamics of the underlying detailed model. We used the individual-based model COIRON, which simulates the dynamics of semiarid grass steppes in Patagonia (Argentina) under alternative grazing management, as example. Our specific objectives are to identify pathways of degradation and rehabilitation, as well as critical grazing thresholds and early-warning vegetation states to guide sustainable grazing management in these steppes. Our results indicate nonlinear effects of stocking rate and grazing season on steppe dynamics. Markov chain analysis suggests benefits of seasonal over continuous grazing at intermediate stocking rates, and network analysis of recovery and degradation trajectories shows that intermediate stocking rates maximize differences between grazing seasons. Finally, our analysis identified specific vegetation states as early warning signals that indicate a high risk of irreversible vegetation changes. Patagonian grass steppes should ideally be managed with multi-paddock grazing at moderate stocking rates around 0.5 sheep·ha-1. The transition matrices summarize the relevant key features of the detailed model for larger scales, and applying Markov and network theory provides a systematic strategy to analyse its dynamics to respond to biological questions, both are often difficult to obtain by direct analysis of the detailed model.

Methods
The meta-model was built according to the upscaling procedure for agent-based models presented in the publication. In this way, we produced a library of state and transition models to represent the community dynamics of western grass steppes in Patagonia across different grazing seasons and stocking rates.