In response to the OP, and it's been said: The game is designed so one player may only have claim on one mob at any one time. It's designed to prevent hoarding.

And also as was said, and completely more to the point: SE need to either add more mobs to Dynamis or lower the repop times on mobs in Dynamis, especially with the upcoming update. Making it similar to Abyssea repops would be a good start.