The Existence and Uniqueness of Solutions in the Leaf Photosynthesis-Transpiration-Stomatal Conductance Model

The An-E-gs model, which consistently describes leaf photosynthesis (An), transpiration (E), and stomatal conductance (gs), is widely recognized and utilized as a "standard model" for quantifying these processes in terrestrial plants. However, since its proposal over 30 years ago, the model has faced a longstanding challenge: the "solution selection" problem, arising from the existence of multiple solutions with no guarantee that the obtained solution is correct. In this study, we mathematically proved that the An-E-gs model always has a unique solution satisfying the criteria gs > 0 and Ci > 0, where Ci represents the CO2 concentration inside the leaf. This result establishes a rigorous mathematical theorem on the existence and uniqueness of solutions in the model. The theorem resolves the longstanding "solution selection" problem by enabling the unambiguous identification of the correct solution through selecting the unique solution that satisfies these criteria. Furthermore, the theorem ensures the validity of past estimations that satisfy these criteria and guarantees that future studies applying these criteria will yield correct estimations. These findings provide a robust mathematical foundation for the An-E-gs model, reinforcing its role as the standard model for estimating leaf photosynthesis, transpiration, and stomatal conductance across diverse disciplines, from plant biology to climate science and beyond.