There are global settings to prevent animals from getting older or die, but this also logically includes that they cannot have offspring.
If you want to have animal families, you could at first stop aging, set up your zoo with the species you want, then activate aging and breeding until you have offspring for all of them, and then stop aging again to keep them forever at that stage.
If you do this for individual species at different times, you might have older ones dying and offspring maturing for the other species.