Optimal phenology of annual plants under grazing pressure.

Plants show phenological responses to herbivory. Some enclosure experiments have demonstrated that the onset of the peak flowering season is dependent on grazing pressure. We constructed a mathematical model using Pontryargin's maximum principle to investigate changes in flowering time by examining shifts in resource allocation from vegetative to reproductive plant components. We represented a primary production of a plant individual by two types of function of vegetative part size, a linear function and a convex non-linear function. The results of a linear production model indicate that optimal phenology follows a schedule that switches from the production of vegetative parts to that of reproductive parts at a given time ('bang-bang' control). However, in a non-linear model, a singular control, wherein the plant invests in both productive and reproductive parts, may be included between obligate production and reproduction periods. We assumed that the peak of the flowering season occurs immediately following the exclusive investment in reproduction. In a linear production model, differential herbivory rates on the vegetative and reproductive parts of a plant resulted in shifts in the peak flowering time. A higher herbivory rate on the vegetative components advanced the peak, whereas it was delayed when grazing pressure focused on reproductive components of the plant. In the non-linear production model, increased grazing pressure tended to postpone the flowering peak. These results corresponded well with results of enclosure experiments, thus suggesting adaptive control of flowering time in plants.