> I can't begin to think of the resource constraints a population would have if none of them have to die.

population growth is caused by procreation which is just other side of the coin to mortality. If (big "if" actually) species of immortals (who thus wouldn't need any significant procreation) would have higher evolution speed than mortal&procreation species then immortals would take over. We'll see this split (and resulting competition between subspecies) in human species in the coming decades.