Optimal Level of Chemical Defense Decreasing with Leaf Age

A recent study on a biennial plant demonstrated that pyrrolizidine alkaloids decrease with the age of leaves due to reallocation from old leaves to new leaves. Here we study the optimal age-specific pattern of defense chemical concentration in leaves that achieves the maximum growth rate of the plant. We consider a plant growing exponentially in a constant environment. Assumptions are the following: (1) The loss of leaves due to herbivory decreases with defense chemical concentration (2) The daily net photosynthesis of a leaf decreases with its age. (3) Using photosynthetic products, the plant produces new leaves that may contain defense chemicals. (4) Although there is a cost to producing new defense chemicals, such chemicals can be reallocated without cost. In the optimal schedule calculated using Pontryagin's maximum principle, the chemical defense level decreases with leaf age. The optimal level of chemical defense increases with the cost of leaf production and herbivory intensity but decreases with the cost of defense chemical production, effectiveness of the defense chemical, net productivity, and growth rate of the plant. If both generalist and specialist herbivores attack the same plant, the optimal defense level is dependent only on the generalists' abundance and sensitivity, but is independent of the specialists that are unaffected by the defense chemical.