  We study the degree of success of a single predator hunting a herd of prey on
a two dimensional square lattice landscape. We explicitly consider the self
volume of the prey restraining their dynamics on the lattice. The movement of
both predator and prey is chosen to include an intelligent, decision making
step based on their respective sighting ranges, the radius in which they can
detect the other species (prey cannot recognise each other besides the self
volume interaction): after spotting each other the motion of prey and predator
turns from a nearest neighbour random walk into direct escape or chase,
respectively. We consider a large range of prey densities and sighting ranges
and compute the mean first passage time for a predator to catch a prey as well
as characterise the effective dynamics of the hunted prey. We find that the
prey's sighting range dominates their life expectancy and the predator profits
more from a bad eyesight of the prey than from his own good eye sight. We
characterise the dynamics in terms of the mean distance between the predator
and the nearest prey. It turns out that effectively the dynamics of this
distance coordinate can be captured in terms of a simple Ornstein-Uhlenbeck
picture. Reducing the many-body problem to a simple two-body problem by
imagining predator and nearest prey to be connected by a Hookean bond, all
features of the model such as prey density and sighting ranges merge into the
effective binding constant.
