Forage
A bird spends its time foraging among patches of berries. When it finds a patch, it stays there until the patch is some-what depleted, and then it goes off in search of a new patch. What it wants to know is how when it should “pick the patch over.” If it stays too long in a patch it will be spending valuable foraging time looking for the hidden berries that remain, but, on the other hand, when it leaves the patch it has to spend some time finding another. When should it leave the patch and go off to find another?
This is an optimal allocation problem. What the bird needs to do is maximize the number of calories it acquires each day, that is, the overall rate at which it gains calories during a foraging cycle (foraging + search).
We need to know how the amount of energy E it gets from a patch depends on the time t spent in the patch and the students are given the following graph. What is the graphical solution.
Another piece we might consider in the model: if the search for a new patch requires flying around for a while, the bird will bur up some of those berries that it ate in the patch and we should account for that.