  We propose a minimal model of predator-swarm interactions which captures many
of the essential dynamics observed in nature. Different outcomes are observed
depending on the predator strength. For a "weak" predator, the swarm is able to
escape the predator completely. As the strength is increased, the predator is
able to catch up with the swarm as a whole, but the individual prey are able to
escape by "confusing" the predator: the prey forms a ring with the predator at
the center. For higher predator strength, complex chasing dynamics are observed
which can become chaotic. For even higher strength, the predator is able to
successfully capture the prey. Our model is simple enough to be amenable to a
full mathematical analysis which is used to predict the shape of the swarm as
well as the resulting predator-prey dynamics as a function of model parameters.
We show that as the predator strength is increased, there is a transition (due
to a Hopf bifurcation) from confusion state to chasing dynamics, and we compute
the threshold analytically. Our analysis indicates that the swarming behaviour
is not helpful in avoiding the predator, suggesting that there are other
reasons why the species may swarm. The complex shape of the swarm in our model
during the chasing dynamics is similar to the shape of a flock of sheep
avoiding a shepherd.
