How do you learn how to use applied math not only as a modeling tool/statistics tool/numerical analysis tool, etc., but as a theoretical tool? How do you know when a proof holds true in applied math/physics/biology? For example:

From what I'm getting at, you basically have a really good mathematical model or law, and you are able to rephrase simple and inherently mathematical relationships in nature in terms of mathematical language. All I've seen from mathematical biology so far are differential equation models that were found long ago in physics, or geometry/topology being applied to molecules. I think I'm misunderstanding something about this topic.