Can a father get full custody if the mother left the son again?

My son’s mother has left for another man in a different state (1 month this time). This is the second time in three years and second man. She had a baby with the first one and when she came back, I raised the child lovingly as my own along with our first child. Since then she has constantly cheated and left us many times for the first man and now three years later she met another man in Arkansas (her home state) and has left with her child for him. My question is can I try to get full custody of my own child? I don't mind her visiting (if she ever wants too). In the past three years she has left on and off and she never calls or helps with his bills/school etc. There are many other things but for now, this is it. I want to give my son a stable no drama life. The only reason we ever fought was because I kept catching her on Facebook or on phone with other men. This isn't how a mother with two kids and a family is supposed to act right? We were never married. I have lived and taken care of my son since his birth. I am on birth certificate. No domestic violence. I have an apartment, car, job and health Insurance.