Plant growth is influenced by various biotic and abiotic factors, such as light, nutrients, and competitors. In most plants, organic carbon is produced in leaves, whereas nutrients are absorbed by roots. To acquire those elements effectively, plants are considered to optimize the growth of shoot and root.
We theoretically investigate the resource allocation schedule to optimize growth in plants. Resources are produced from the elements and allocated to shoot and root with certation fractions. The success of plants is measured by shoot biomass at the end of season. We analyze the optimal dynamic allocation strategy in solitary plants by using the Pontryagin's maximum principle.
We found that singular control for the resource allocation schedule can occur during a growing season, in which plant allocates resources to shoot and root simultaneously. Before the singular control starts, it was determined whether plants would allocate to shoot or root according to the proportions of their growth efficiency. We examined parameter dependences of the period of the singular control. We will extend this approach in competitive situations and investigate ESS resource allocation strategy. It may provide clues to interpret the tragedy of the commons.