Who Has The Right Over The Child

Who would get the rights over a child if the mother moves from state to state, works part time and goes to school but lives with her parents but neither one of them work. she has moved to three states in the last two years and keeps the child away from the father. the father has a full time job, has been liven at the same address for the past year, supports both of his two kids but never signed the birth certificate and has seen the other child in the past year but that's because the mother keeps moving and the mother of the other child has medical problems.