## Summary

In this study, we have formulated a simple mathematical model that gives us very profound results. We found that crows are naturally minimizing the energy that they expend in order to feed. At least for this study, we have verified that optimal foraging theory is applicable. We have also learned the basic steps involved in developing a mathematical model. The benefit of the model we used today was that it was very simple to analyze. More complicated problems give rise to models with systems of multiple equations, and different types of equations for which solutions are not so easy to find. However, the basic steps to formulating most models are very similar to what we have learned today.

There are, in fact, similar examples of optimal foraging that could be studied in this way. For instance, as time goes on, it becomes more and more difficult for a bee to obtain pollen from a flower as the amount of pollen in the flower decreases. At some point, the bee decides to fly to a new flower that has not been depleted of its pollen. However, flying from one flower to the next takes a certain amount of energy. A model similar to the one we created today can help show that optimal foraging theory applies to this situation as well, and can be used to determine approximately when the bee should move on to the next flower.

To close, let's make one more observation about the average energy expended and its dependence on how high the crow chooses to fly. In Figure 9 , we saw that after a height of about 3 metres, the curve becomes almost flat until about 8 metres. This indicates that even though the ideal height to which the crow must fly to use the least amount of energy is about 5.6 metres, it does not need to be very precise. If it flies a bit too high or low, it does not matter much. It will expend a little more energy, but the change will not be very substantial.