How Do You Use Applied Math as a Theoretical Tool in Biology?

[vicsmithvic]

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:

http://www.nature.com/nature/journal/v410/n6825/full/410268a0.html

This thread began with Peskin's model of the sinoatrial node, the heart's natural pacemaker, as a collection of N identical integrate-and-fire oscillators38. For the simple case where each oscillator is connected to all the others, Peskin conjectured that they would all end up firing in unison, no matter how they started. He gave a proof for N = 2 oscillators; it was later demonstrated39 that the conjecture holds for all N. Peskin also conjectured that synchronization would occur even if the oscillators were not quite identical, but that problem remains unproven.

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.

 

[R.Mesri]

In order to learn how to use applied math as a theoretical tool, you should start by studying the basics of mathematics, such as calculus, linear algebra, and probability theory. This will give you the necessary background knowledge to understand more advanced mathematical concepts. When it comes to understanding when a proof holds true in applied math/physics/biology, you should familiarize yourself with the subject and its related theories. You should also look for relevant research papers and books to gain an understanding of the current state of the field. Additionally, you can look for tutorials, online courses, and other learning resources to supplement your understanding.