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Sorry if this is too much of a soft question or if it is too broad for this site.

As a person who has much more experience with math than biology, I've always been really interested in the mathematical side of biology. As I study math in biology, however, I am left with the question of "how is this actually useful?"

What kind of unique insight can mathematical models give us about biological systems that we couldn't just find experimentally?

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closed as too broad by fileunderwater, Chris, MattDMo, AliceD, The Last Word Jun 3 '15 at 4:15

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Experiments are challenging. Thus all sciences model experimental results, often mathematically, so that future study can be simplified. $\endgroup$ – canadianer Jun 2 '15 at 19:38
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    $\begingroup$ This is actually broad. There are a lot of applications of mathematical models in biology and there have been many books written on this topic. See this related meta post $\endgroup$ – WYSIWYG Jun 2 '15 at 19:38
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    $\begingroup$ I also think this is too broad and unspecific. Mathematical biology is a huge topic with many different types of applications. However, these two questions might be useful to you: What physics knowledge can be applied to biology of organisms and ecosystems? and When are population dynamics models useful?. $\endgroup$ – fileunderwater Jun 2 '15 at 19:55
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This question is broad.

There are plenty of uses of mathematical modelling in biology.

Wikipedia

You may want to read the wikipedia article: Mathematical and Theoretical Biology.

Books

There are tons of books on theoretical biology. I am personally working in the field of population genetics, a field where much of the theory has been expressed mathematically. Here are book recommendations for this particular field.

On Biology.SE

We have a tag for mathematical-models (you used it in your post). You may want to see what kind of posts are tagged for mathematical-models.

To repeat @fileunderwater's comment, What physics knowledge can be applied to biology of organisms and ecosystems and When are population dynamics models useful? are examples of posts you may want to read.

What prevents predator overpopulation? will give you an example of an easy and classical mathematical model to describe population dynamics of predators and preys.


Personal historical/philosophical note: At some point physicists have started to describe the laws of nature with mathematical expressions and at that time many physicists didn't quite like the idea that physics could be described by math. Biology has gone through the same phase but much later. Early mathematical biologists developed the modern science of statistics (Francis Galton, Karl Pearson, Ronald Fisher). Biologists used existing mathematical tools such as diffusion processes (Kimura). Note that part of the work in the field of diffusion equations have been done by physicist (Einstein solved the 2D equations for diffusion processes). Note also that the dynamic of diffusion processes called brownian motion have been named after a biologist. Today, biology produces massive amount of data that require good computational skills to treat. The time when biologists had only to know how to recognize species is over, now there are few biologists left that aren't able to code in at least one programming language.

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Models are often used to calculate the probability of certain biological outcomes. Such predictions are great help for prioritization of further studies or making other decisions.

Example: EPA uses an array of in vitro assays for fast and cheap analysis of thousands of potentially toxic environmental compounds (ToxCast and Tox21 projects). Next, they use the obtained data to build mathematical models to prioritize chemicals for further more expensive and slow animal studies or to decide on administrative actions to restrict usage of high priority (most dangerous) chemicals. Their prioritization models integrate toxicity data and complex exposure models (including manufacturing and usage logistics, possibility of accidental exposure etc).

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