How does population stability evolve?
I guess you meant the population size stability. 

It is considered that the biosystems will increase their capacity of adaptation when evolving in very fluctuating environments. I believe the population stability is embedded in the adaptability of individuals. 

There is a measurement about it, evolvability, when the environment changes, the faster the population adapted to the new environment, the higher evolvability it has. This means that if the population has higher evolvability then the population may be more stable to environmental variation.

Even so, there might be plenty of underlying mechanisms to enable a certain population stability. I recently read a paper about two mechanism: phenotypic switching and sensing machinery, which are common in bacteria and fungi. Please refer to Edo Kussell and Stanislas Leibler's science paper (2005).

As the organism increase its complexity, biosystems invented many more tools to make the individual survival from disasters. Such as human's intelligence. Well it would be an endless issue.