To fit or not to fit: Evaluating stable isotope mixing models using simulated mixing polygons

Stable isotope analysis is often used to identify the relative contributions of various food resources to a consumer's diet. Some Bayesian isotopic mixing models now incorporate uncertainty in the isotopic signatures of consumers, sources and trophic enrichment factors (e.g. SIAR, MixSIR). This had made model outputs more comprehensive, but at the expense of simple model evaluation, and there is no quantitative method for determining whether a proposed mixing model is likely to explain the isotopic signatures of all consumers, before the model is run. Earlier linear mixing models (e.g. IsoSource) are easier to evaluate, such that if a consumer's isotopic signature is outside the mixing polygon bounding the proposed dietary sources, then mass balance cannot be established and there is no logical solution. This can be used to identify consumers for exclusion or to reject a model outright. This point-in-polygon assumption is not inherent in the Bayesian mixing models, because the source data are distributions not average values, and these models will quantify source contributions even when the solution is very unlikely. We use a Monte Carlo simulation of mixing polygons to apply the point-in-polygon assumption to these models. Convex hulls ('mixing polygons') are iterated using the distributions of the proposed dietary sources and trophic enrichment factors, and the proportion of polygons that have a solution (i.e. that satisfy point-in-polygon) is calculated. This proportion can be interpreted as the frequentist probability that the proposed mixing model can calculate source contributions to explain a consumer's isotopic signature. The mixing polygon simulation is visualised with a mixing region, which is calculated by testing a grid of values for point-in-polygon. The simulation method enables users to quantitatively explore assumptions of stable isotope analysis in mixing models incorporating uncertainty, for both two- and three-isotope systems. It provides a quantitative basis for model rejection, for consumer exclusion (those outside the 95% mixing region) and for the correction of trophic enrichment factors. The simulation is demonstrated using a two-isotope study (¹⁵N, ¹³C) of an Australian freshwater food web.