A stochastic game model of searching predators and hiding prey.

When the spatial density of both prey and predators is very low, the problem they face may be modelled as a two-person game (called a 'search game') between one member of each type. Following recent models of search and pursuit, we assume the prey has a fixed number of heterogeneous 'hiding' places (for example, ice holes for a seal to breathe) and that the predator (maybe polar bear) has the time or energy to search a fixed number of these. If he searches the actual hiding location and also successfully pursues the prey there, he wins the game. If he fails to find the prey, he loses. In this paper, we modify the outcome in the case that he finds but does not catch the prey. The prey is now vulnerable to capture while relocating with risk depending on the intervening terrain. This generalizes the original games to a stochastic game framework, a first for search and pursuit games. We outline a general solution and also compute particular solutions. This modified model now has implications for the question of when to stay or leave the lair and by what routes. In particular, we find the counterintuitive result that in some cases adding risk of predation during prey relocation may result in more relocation. We also model the process by which the players can learn about the properties of the different hiding locations and find that having to learn the capture probabilities is favourable to the prey.