Optimal allocation of two resources in annual plants

The fitness of an annual plant can be thought of as how much fruit is produced by the end of its growing season. Working under the assumption that annual plants grow to maximize fitness, we use optimal control theory to understand this process. We introduce a model for resource allocation in annual plants that extends classical work by Iwasa and Roughgarden to a case where both carbohydrates and mineral nutrients are allocated to shoots, roots, and fruits. We use optimal control theory to determine the optimal resource allocation strategy for the plant throughout its growing season as well as develop a numerical scheme to implement the model. We find that fitness is maximized when the plant undergoes a period of mixed vegetative and reproductive growth prior to switching to reproductive-only growth at the end of the growing season. Our results further suggest that what is optimal for an individual plant is highly dependent on initial conditions, and optimal growth has the effect of driving a wide range of initial conditions toward common configurations of biomass by the end of a growing season.