Omega, the final multiplier

The application of optimisation theory to vegetation processes has rarely extended beyond the context of diurnal to intra-annual gas exchange of individual leaves and crowns. One reason is that the Lagrange multipliers in the leaf-scale solutions, which are marginal products for allocatable photosynthetic resource inputs (water and nitrogen), are mysterious in origin, and their numerical values are difficult to measure -- let alone to predict or interpret in concrete physiological or ecological terms. These difficulties disappear, however, when the optimisation paradigm itself is extended to encompass carbon allocation and growth at the lifespan scale. The trajectories of leaf (and canopy) level marginal products are then implicit in the trajectory of plant and stand structure predicted by optimal carbon allocation. Furthermore, because the input and product are the same resource -- carbon -- in the whole plant optimisation, the product in one time step defines the input constraint, and hence implicitly the marginal product for carbon, in the next time step. This effectively converts the problem from a constrained optimisation of a definite integral, in which the multipliers are undetermined, to an unconstrained maximisation of a state, in which the multipliers are all implicit. This talk will explore how the marginal products for photosynthetic inputs as well as the marginal product for carbon -- i.e., the 'final multiplier,' omega -- are predicted to vary over time and in relation to environmental change during tree growth.