A discrete-time model is introduced to discuss the existence and stability of periodic orbits. In particular, square-shaped and parallelogram-shaped cycles on square and more generically rectangular domains are analyzed. In addition, a figure-eight-shaped cycle on a rectangular domain is also studied. A discrete-time model is formally derived in the limit where the domain size tends to infinity. Numerical simulation suggests that the discrete-time model approximates the corresponding particle model when the domain size is sufficiently large. An observable gap between the orbits for these two models in cases where the domain is of medium scale is also discussed.