I don't understand exactly what you mean by deep mathematics but you should watch Numb3rs. And before I get yelled at for recommending a TV series when it comes to mathematics, here http://numb3rs.wolfram.com/
To answer you question, an example of shallow math is using an ODE to model population biology. This is mainly because the math doesn't actually tell you something that you qualitatively didn't already know. An example of deep mathematics is using representation theory to analyze the quantum charge, spin etc of fundamental particles.
Yes, it enumerates real-world applications of mathematics in fields other than Physics/Chemistry, of which you said you were skeptical.
This is mainly because the math doesn't actually tell you something that you qualitatively didn't already know.
So I take it that the depth of mathematics is determined by its ability to predict?
You say physics and chemistry are the only deep uses of math, and illustrate that with a physics example to define "deep", and a biology example to define "shallow". That's not meaningful at all: it's circular logic (i.e. bad mathematics).
Without math, we would have a much poorer understanding of population biology. Your population biology example was from 1926 and the state-of-the-art has moved on - so you have badly represented the subject. I suggest that your personal understanding of biology is where the shallowness really lies.
The parent commenter has made a genuine effort to understand you and offer you some advice, despite your bullshit. You should thank them.
oooookay.