# The E-look-ahead Rule and its Application to Foraging Theory

Amy Lynn Biesterfeld

Ph.D., 1996

Advisor: Thomas S. Ferguson

The thesis is in two parts. In the first part, regularity conditions are given under which an optimal stopping rule exists. It is shown that if the stopping rule problem satisfies some additional regularity conditions and the condition of monotonicity, then a certain stopping rule, called here the $\epsilon$-look-ahead rule, is optimal. When monotonicity is not satisfied, a suggestion is given for finding a good and computationally feasible, but suboptimal, rule.

The second part of the thesis considers the application of sequential analysis in Foraging Theory. A generalized foraging model is presented which combines two important classes of foraging problems together: finding the optimal selection rule and stopping rule which jointly maximize the forager's expected rate of energy intake. The model also allows all environmental parameters to be random, with very loose restrictions placed on the distributions. This provides a very flexible model, more realistic than previously proposed models, that should be of great use in biological applications. This foraging model encompasses many stopping rule problems and selection rule problems. For this generalized model, the optimal selection rule is given and, under some restrictions for the distributions of the environment parameters, it is shown that the $\epsilon$-look-ahead rule is the optimal stopping rule.

1996