The Existence and Uniqueness of Solutions in the Leaf Photosynthesis-Transpiration-Stomatal Conductance Model for C3...

The leaf photosynthesis-transpiration-stomatal conductance model, which consistently describes leaf photosynthesis, transpiration, and stomatal conductance, has been widely used as a standard for quantifying these processes in terrestrial plants. However, since its proposal more than 30 years ago, the model has faced a fundamental mathematical problem: Does a solution always exist? And even if a solution is obtained, can it be guaranteed to be the correct one among potentially multiple mathematical solutions-that is, the one actually realized in nature? Here, we resolve this problem by mathematically proving that the model always yields a unique solution satisfying biologically and physically meaningful criteria. This result establishes a rigorous mathematical theorem on the existence and uniqueness of solutions in the model, thereby ruling out concerns about the non-existence of solutions and ensuring that past and future estimates satisfying the criteria are correct. These findings provide a robust theoretical foundation for the model and have far-reaching implications for a broad range of fields, spanning plant and ecosystem research to climate and Earth system studies.