The stability of atomic nuclei seem to be related to the ratio of _____?

The answer is protons and neutrons. Stable ratios are called "islands of stability" and as the atoms get larger, the islands of stability get farther apart. That is important because some people will try to tell you that electrons and protons can be squeezed together to make something that is only composed of neutrons. That is balderdash. Get the ratio wrong by even one particle and the nucleus flies apart.