Yes, "留, 住 and 待 (呆)*. They all mean to stay or live."
留 has the meaning of "to keep", and "to stay", and "待" has the meaning of "wait", "keep in" and "stay in". they do not equal "住", which means physically "to live in" or "to stay over" in a defined ...
They are not interchangeable
留在 = stay at
住在 = live in
呆在 = stay at
Although both 留在 and 呆在 mean 'stay at', 呆(stay) as a verb implies unproductively
We say 留在北京工作, but not 呆在北京工作 (unless you hate the job, and consider it unproductive for yourself)
we say 留在家中照顧生病的孩子, but not 呆在家中照顧生病的孩子 (unless you don't want to take care of the sick kid, and ...