Another way of saying this is what would my pet cat/s do for eternity, what mental processes would they have or physical needs. Would they have appetites and what would they eat, what would motivate them?.
I might be able to envision (just) my pet cat catching a grasshopper and eating it in heaven but not really a bird. I can't see heavenly cats killing heavenly birds.
I have 2 cats, the younger one (6yrs) loves going out on a warm night and catching grasshoppers, and small insects, etc, birds and mice too if she can. The older one (12yrs) has been there & done that & doesn't bother with insects, though will still catch the other stuff if he can (he's great with rats).
If the younger one ends up losing interest in insects as i suspect she will then don't know what they'd do in heaven. The things that amuse her, used to amuse him but don't now.
I can imagine them running around & exploring the fields & pastures of heaven, ie new Earth, interacting with all sorts of other animals etc, they'd be in no danger, the could go on months\years long adventures and explorations, I'd always now where they were & be able to go to them quickly, if not with the click of my fingers.
They would sleep for hours (but would they??) and explore but yeah, after a couple of thousand years, would they still be okay mentally and have that curiosity etc.
Its hard for us to imagine what we would do after a few hundred years of living in heaven, even harder to think what pets would do.