Fig. 3

An optimum is usually a function that is minimized or maximized given constraints. In the case of crossbills, the optimal bill depth is the bill depth that minimizes the time necessary to meet daily energy demands. Individuals that can meet their energy requirements more rapidly will have a competitive advantage when food is scarce, minimizing the risk of starvation and allowing them to do other things like find and court a mate. First, we need to consider the relationship between bill depth and the time to remove seeds from the cones. An example is shown for 27 Red Crossbills timed foraging on Douglas-fir cones (Benkman 1993), where the bill depth minimizing the time to extract a seed is approximately 9.4 millimeters (Fig. 3a). In almost all cases, we have found a curvilinear relationship between time per seed and bill depth with an intermediate bill depth requiring the least amount of time to extract a seed. However, the optimal bill depth is not simply the one that requires the least amount of time to extract a seed. Because body size increases allometrically with increases in bill depth in crossbills as in most seed-eating finches and sparrows (Benkman 1993; Benkman et al. 2001), daily energy requirements increase with increases in bill depth. Consequently, the optimal bill depth, which minimizes the product between time per seed and estimated daily energy requirements, is smaller than the time minimizing bill depth. The optimum is 9.3 millimeters in the example of crossbills foraging on Douglas-fir (Fig. 3b). The extent to which the optimum is shifted to a smaller size depends on the extent to which the decrease in energy requirements compensates for the increase in time per seed. In the example here, the optimal bill is only slightly (0.1 mm) smaller than the time minimizing bill depth