Ants, Crickets and Frogs in Cyclic Pursuit

We consider a deterministic continuous pursuit, in which n ants chase each other in cyclic order and with preassigned, varying speeds. We also consider two discrete analogs, in which crickets or frogs are engaged in cyclic pursuit with constant and equal speeds. We examine the possible evolutions of these pursuits as time goes to innnity: collision, limit points, equilibrium states and periodic motion.