Patch leaving rules: a stochastic version of a well-known deterministic motivational model.

A famous motivational model proposed for insect parasitoids by Waage (1979) provides a candidate mechanism for patch-leaving decision rules in foragers. In this model, the animal is supposed to enter a patch of resources with an initial tendency to stay in it, which then regularly decreases. Every encounter with a resource item increases or decreases this tendency, and the forager is supposed to leave the patch when this tendency or motivation falls below a given threshold. Evidence of such increases and drops in this tendency to stay were often obtained by analyzing experimental data with a Cox (1972) proportional hazards model. The Waage (1979) model is purely deterministic and predicts a fixed departure time for a fixed set of encounters with foraging items. On the other hand, empirical data show a large variability of departure times under fixed conditions. We present a fully stochastic version which overcomes this problem and gives a quasi close form expression for the distribution of patch residence times as well as a statistical procedure to estimate its parameters. Two examples of the model fitting on experimental data sets are provided. This novel model, although more complicated than Waage (1979) model, improves its realism and provides a stochastic interpretation of motivation as a proximal mechanism leading foragers to optimality.