Density-Dependent Harvest Rates by Optimal Foragers

Abstract

We develop and test a theory of an optimal functional response, a function that maps resource density into the number of resources harvested over a period of time which includes foraging, dormancy, and alternative (non-foraging) activities. We characterize fitness as a function of a vector of inputs which may be complementary. The inputs depend upon the allocation of time among activities. We use the technique of Lagrange multipliers to characterize the allocation of time at a constrained optimum. The time constraint causes a foraging animal to incur a missed opportunity cost, which, counter to intuition, implies that the marginal cost, as well as the marginal benefit, of foraging time may increase with resource density. Consequently, the relationship between the proportion of resource consumed and resource density is indeterminate unless further assumptions are made. We consider two assumptions which ensure that, over a range of resource densities, the optimal functional response will be "stabilizing" in the sense that the per capital mortlaity rate of the resource increases with resource density. These assumptions are: (1) there is a variable energetic cost of foraging, and (2) resource depletion occurs within the foraging period. We experimentally tested predicitons of the theory by offering to free-living desert rodents (Dipodomys merriama) millet seeds which had been mixed with dirt and placed in aluminum trays. The results support the theory''s predictions. The proportion of seeds eaten increased with seed density at relatively low seed densities. The rodents'' optimal functional response indicated a threshold density of seeds below which foraging time was unprofitable. Above this threshold, the slope of the optimal functional response decreased from an initial value of one, indicating that missed opportunity costs increased with an increase in initial seed density.