Shoot/root balance of plants: Optimal growth of a system with many vegetative organs

The optimal growth schedule of a plant with two vegetative parts is studied to investigate the balance between shoot and root. An intuitive justification of optimization procedures used in Pontryagin's maximum principle is obtained by defining the marginal values of shoot size, root size, and reproductive activity at various times of the season and deriving their differential equations and terminal conditions. The optimal growth pattern which maximizes the total reproductive activity during the season is composed of the convergence of a plant's shape to a balanced growth path, followed by simultaneous growth of shoot and root (balanced growth), ending with the reproductive growth. Along the balanced growth path, a plant has a root/shoot ratio which maximizes the daily net photosynthesis for a given total biomass. The model also shows a simultaneous stop of shoot and root growth when the reproduction begins, the dependence of root/shoot ratio on age, water and light availability, etc., the convergence of a plant's shape to the balanced growth after pruning or an environmental change.