A foraging problem: Sit-and-wait versus active predation

The literature on foraging shows that some predators use a combination of ambush and active search to locate a prey. Let us suppose that a prey must go every day to some determined places to feed, and to another place, 0, to drink. A predator can stay at zone 0 waiting for the prey (sit-and-wait strategy) or it can move between the different places where the prey will go to eat (search strategy). If predator and prey meet each other in the same place, prey will be caught with a probability depending on the place. We study this problem in different situations, modelling them as two-person zero-sum games. We solve them in closed form, giving optimal strategies for prey and for predator and the value of the games.