Optimal stomatal behaviour under stochastic rainfall

Vegetation CO₂ uptake is always accompanied by water loss. The balance in this gas exchange is controlled by the stomata, through which CO₂ and water vapour diffuse between the leaf and the atmosphere. The optimal stomatal behaviour theory proposes that vegetation should optimise its stomatal behaviour such that, for given water availability, photosynthesis is maximised. In this paper, we optimise stomatal conductance as a function of soil water content for the maximum expected value of photosynthesis rate. This optimisation process is considered under stochastic rainfall. The optimal solution is largely shaped by two constraints: the risks of soil water exhaustion and surface runoff, which results in an inverse S-shaped curve of stomatal conductance along the soil water gradient. We derive the optimal functional relationship between stomatal conductance and soil water content under varying rainfall frequency, mean annual precipitation and atmospheric CO₂ concentration. Comparisons with large-scale observational data show that the model is able to broadly capture responses of photosynthesis, transpiration, and water use efficiency along rainfall gradients, although notable discrepancies suggest additional factors are important in shaping these responses. Our work provides a theoretical framework for analysing the vegetation gas exchange under environmental change.