I think of it as being sort of like comparing a chimp to a gorilla. Same genetic family, evolved in different directions. So, a Dire Tiger may be similar to a Tiger but the two won't hang out together, I'd think. They really just don't have that much in common.
If you accept they're different species, you wouldn't see an "alpha male" effect since they wouldn't be able to crossbreed with non-Dire versions. One might view itself as boss and protector (thinking of the non-Dires as adopted "children"), but it wouldn't have some of the other aspects (the "getting all the chicks" part).
This could vary with the exact type, though; a Dire Squirrel might prefer hanging out with regular squirrels, just to avoid an inferiority complex. But it wouldn't be the sort of instinctive family bond you'd get with members of the same species, IMHO.