Mixing model for the analysis of stable isotopes: spatiotemporal niche identification
Stable isotopic compositions of potential water source pools and plant water provide important information on water sources and water uptake patterns (Dawson, Mambelli, Plamboeck, Templer, & Tu, 2002). Plants may have access to more than one water source pool (e.g. recent rain, soil water, and groundwater) in different proportions. Rooting depth and distribution define the depth and soil volume from where plants potentially extract these water sources. Therefore, water isotopic compositions of plant xylem can be viewed as a mixture of isotopic compositions from different water sources. Vegetation with roots distributed throughout the soil profile, for example, garner water from different soil depths resulting in twig water exhibiting mixed isotopic signature (Cramer, Thorburn, & Fraser, 1999).
A typical formulation using two isotopic signatures (δ1 and δ2) to partition the contributions (ƒ) of three sources (a, b, c) to a mixture (m) is:
\begin{equation} \delta_{m}^{1}=f_{a}\delta_{a}^{1}+f_{b}\delta_{b}^{1}+f_{c}\delta_{c}^{1}\nonumber \\ \end{equation}\begin{equation} \delta_{m}^{2}=f_{a}\delta_{a}^{2}+f_{b}\delta_{b}^{2}+f_{c}\delta_{c}^{2}\nonumber \\ \end{equation}\begin{equation} 1=f_{a}+f_{b}+f_{c}\nonumber \\ \end{equation}
The number of sources that can be partitioned is limited by the number of isotopic signatures employed. For the dual isotope example above, the mixing model is a system of three equations with three unknowns (fa, fb, fc), for which there is a unique solution (Phillips & Gregg, 2003). The relative contributions of different sources to xylem water were estimated by Bayesian mixing modelling using the Stable Isotope Analysis (SIAR) package. Stable isotope mixing models are used extensively for studying food webs but can also be applied to the determination of plant water sources (Beyer, Hamutoko, Wanke, Gaj, & Koeniger, 2018; Evaristo, Jasechko, & McDonnell, 2015; Evaristo & McDonnell, 2017; Voltas, Lucabaugh, Chambel, & Ferrio, 2015). We considered four different sources of water: two soil depths (0–10 cm and 10–25 cm), soil pockets and weathered rock (regolith) and groundwater reservoirs remaining in fractures and fissures within the rhyolitic-rock (see Figure S2) protected from evaporation.