What is the field of study that examines patterns of growth, change, and stability in behavior?

It depends in what particular field the study is. Broadly, it would be included in a Psychology or a Behavioural Sciences course but it might be in one of many specialities within those. If it were human behaviour, then it would be part of a Psychology course; if it were animal behaviour, you would be more likely to find it in a Zoology course.