Optimal patch use in a stochastic environment

This paper considers an animal foraging on prey which are distributed in well-defined patches. It is assumed that the environment may be stochastic and that the animal can gain information on patch type as it forages. The foraging policy which maximises mean reward rate for the environment is characterised in terms of a function of state called the potential function. This policy is shown to be given by the rule: continue foraging on the present patch while the potential is positive, when the potential falls to zero move on to the next patch. Let r denote the current reward rate on a patch and let γ denote the maximum mean reward rate for the environment. It is shown that r ? γ if it is optimal to leave. Conditions which ensure r < γ are also given. For a large class of environments the optimal policy is stated in terms of a revised reward rate r?, and is given by the rule: continue on the present patch while $\text{r}\text{?}\text{> γ}$, when r? falls to γ move on to the next patch. Finally, it is shown that the stay time on a patch is a decreasing function of γ.