  We consider adaptive change of diet of a predator population that switches
its feeding between two prey populations. We develop a novel 1 fast--3 slow
dynamical system to describe the dynamics of the three populations amidst
continuous but rapid evolution of the predator's diet choice. The two extremes
at which the predator's diet is composed solely of one prey correspond to two
branches of the three-branch critical manifold of the fast--slow system. By
calculating the points at which there is a fast transition between these two
feeding choices (i.e., branches of the critical manifold), we prove that the
system has a two-parameter family of periodic orbits for sufficiently large
separation of the time scales between the evolutionary and ecological dynamics.
Using numerical simulations, we show that these periodic orbits exist, and that
their phase difference and oscillation patterns persist, when ecological and
evolutionary interactions occur on comparable time scales. Our model also
exhibits periodic orbits that agree qualitatively with oscillation patterns
observed in experimental studies of the coupling between rapid evolution and
ecological interactions.
