Non-parametric upscaling of stochastic simulation models using transition matrices.

Abstract

The problem of scaling up from tractable, small-scale observations and experiments to prediction of large-scale patterns is at the core of ecological theory and application, and one of the central problems in ecology. We present and test a general nonparametric framework to upscale spatially explicit and stochastic simulation models. The idea is to design a state space, defined by the important state variables of the small-scale model, and to divide it into a finite number of discrete states. Transition probabilities are then tallied by monitoring extensive simulation runs of the small-scale model, covering the entire range of initial conditions, states and external drivers that may occur for the desired application. We exemplify our approach by upscaling an individual-based model that simulates the spatiotemporal dynamics of Festuca pallescens steppes under sheep grazing in Western Patagonia, Argentina, with a spatial resolution of 0·3 m × 0·3 m and a 0·15-ha extent. The upscaled model simulates a 2500-ha paddock with 0·15-ha resolution and is enriched with additional rules that describe heterogeneity in the local stocking rate at the paddock scale. We obtained 24 transition matrices that governed the upscaled model for different combinations of stocking rates and annual precipitation. The upscaled model produced excellent predictions for the long-term dynamics, but as expected, it did not fully capture the interannual dynamics of the original model. Rules for heterogeneity in the local stocking rate allowed for emergence of realistic vegetation patterns as commonly observed for water points in arid rangelands. Our general nonparametric upscaling approach can be applied to a wide range of stochastic simulation models in which the dynamics can be approximated by a set of states, transitions and external drivers. Because estimation of the transition probabilities can be done parallel, our approach can be applied to a wide range of models of intermediate complexity. Our approach closes a gap in our ability to scale up from small scales, where the biological knowledge is available, to larger scales that are relevant for management.