Hi,
If there is blood in the cage, and the bird is still bleeding, then it probably has a broken blood feather. This will not stop bleeding until the rest of the feather has been removed, so you would need to take it to an emergency clinic. If there is no more fresh blood, then the bird is probably OK unless you see a wound.
It’s important to understand that you can’t chase your birds around trying to catch them. Parakeets are prey animals, which means predator animals catch and eat them. When you chase a bird, he is literally flying to save his life. He thinks you want to harm him. It is terrifying for the bird, and the bird can get stressed to the point that he dies. If the birds will not go back in the cage for a treat, then you can’t let them out. You may want to get a larger cage that gives them room to fly. I recommend that you rad our pages on behavior and training so you can better understand how to work with your birds. You could try training them to get on a stick and then go in the cage, but you would have to get their winds trimmed in order to train them. The feathers will grown back in the next time the bird molts.
Pet Bird and Parrot Behavior
Teaching Your Bird
Thank you for asking Lafeber,
Brenda