any and all of them will work. some just hit harder and shoot longer than others. i have a 22-250 that i really love! but i think if i bought a predator rifle right now, it would be a .204 since i really dont need a ton of range and it wont tear fox and bobcats up so bad.
the .223 is probally a universal round, becuase it kills easily, hardly any fur damage and cheap ammo. and shoots accurate
the .22-250 will hit harder and shoot striaghter and futher but that means more fur damage. it will eat up a fox or bobcat. little higher ammo
the .204 will not give quite the distance, but it will kill w/ no fur damage. it shoots striaght and is pretty good against the wind. very accurate. higher ammo but not sure how much higher
other rounds include: 220 swift, 17 remington and fireball, .243, and others.
all of them will kill predators easily and effectivly, but some just do different things