Alternative approaches for solving underdetermined isotope mixing problems

Statistical mixing models have been developed to help ecologists deal with isotope tracer data and to estimate source contributions in complex systems such as food webs and sediments. However, there are often too few tracer measurements and too many sources, so that unique solutions are not possi- ble in underdetermined mixing models. This review highlights 3 approaches for solving otherwise under- determined mixing models. The approaches include frequency-based statistics, calculations based on sec- tors measured in mixing polygons, and linear mixing between central and sidewall points in the mixing polygons. All approaches have some assumptions that allow extrapolation of mean solutions from measured data, with the simplest assumption being that any uncertainty in source contributions is divided in an even-handed manner among sources. A new graphical approach is proposed that allows scientists to critically recognize and separate data- supported aspects of solutions from any assumed aspects of solutions. The data-supported aspects of solutions can be tracked conservatively as the sum of the minimum source contributions, ΣMIN, and for the many cases where ΣMIN is low, additional ways to approach mixing problems are summarized from the published literature. Many underdetermined mixing problems do not have robust mean solutions with tracers employed thus far, so that there is a longer- term need for additional tracers and methodologies to really solve these complex ecological problems. This review concludes with several practical steps one can take to interpret isotope tracer information from underdetermined systems.