Optimal Foraging in Nonpatchy Habitats. 2: Unbounded One-Dimensional Habitat

Abstract

We want to determine the trajectory that an animal must follow in order to maximize its food intake. In this paper, the habitat is supposed to be one-dimensional and infinite. The food distribution on this habitat can be arbitrary (continuous or not). The animal has a limited time T available to exploit the food resource and to return to its starting point. We find explicitly the optimal strategy, i.e., the stopping point $\bar x$ and the velocity at each point of the traversed segment $[ 0,\bar x ]$. This segment is divided into two subsets according to the food density. In the richer subset the animal equalizes the density distribution; in the poorer one, it travels as fast as possible.Mathematically, we approximate the food distribution by a piecewise constant distribution, and we solve explicitly the approximate problem by using the techniques of the calculus of variations based on convexity hypotheses. Using then a density argument we recover the solution of the general problem.