Butt problem

Hey guys
I bought 4 lovebirds 1 month ago and showed them to a vet they all were healthy 1 month ago.I don't know what is their gender I seperated them into 2 cages(cages are a little bit small 32w 45l and 60h centimeters). one ofthe lovebirds is realy stronger and bigger than other, he/she is at least between 10 to 20 percent bigger than his/her cage mate and eats normaly. one time I decide to put all lovebirds into their cage to play each other but the big one imediately attacked so badly and bit the new cage mate. and I understand I shouldn't put them all in one cage she/he just like her/his cage mate. also I can grab all other lovebirds but this one is realy wild she/he bites my fingers 10 times a second. recently ( 3 or 4 ) days I noticed that his/her butt doesn't have feathers and it's not normal but he/she acts totaly normal and healty and as I said he/she stronger than all others and is much better in acrobatic actions in comparing with my other lovebirds. in photos you can see him/her(I put my finger on top of the cage she/he scare from my finger and did this action in photos to see his/her problem) if someone guess what's the problem please help me.

Your Love Birds are lovely ... Is she a female could she possibly be preparing to lay an egg...Im not an expert on love birds by any means but I think perhaps it might be a good idea to have your AV check her as it could also be any number of other things and its always best to be safe than sorry.

Thank you very much
Yes you are right I think I must bring her to a vet also it's good to know exprienced people idea about this problem ( that's my first time having birds).