A mathematical model is a ~~set of mathematical equations~~ system of mathematical constructs that represent the nature of biological, physical, and/or chemical process, or even processes that occur in other science or non-science disciplines. But, how and why do we make mathematical models?

Below, I have provided my view on how we make mathematical models (Figure 1). The steps involved are model derivation, model analysis, and interpretation. The middle ‘analysis’ step is the one that mathematicians and physicists excel at. This step may draw on skills that are developed in probability, statistics, dynamical systems and other mathematics classes. In fact, considering just the two bottom boxes alone (from Model to Model results) would constitute a problem in pure mathematics. Applied mathematics involves two extra steps, which are the transformation of the biology into the mathematics, and the inverse transformation back into the biology after the mathematical results are obtained.

Figure 1. Mathematical modelling involves the steps of model derivation, analysis and interpretation.

The skill of model derivation is taught, to some degree, in mathematical modelling classes: we learn that stochastic models are appropriate for questions involving small population sizes and that models should be *just simple enough*. The skill of model derivation is also what I refer to as the art of mathematical modelling. Compared to the abundance of classes, books, and theorems that are available to help us with model analysis, there are relatively few directives on how to go about model derivation, and the directives that exist are subjective and vague.

Having said that, I am not criticizing the *just simple enough *advice: actually, I think this is fantastic advice; quite possibly, the-very-best-holy-grail of all advice for aspiring modellers. And yet, this advice is distinctly hard to pin down. This brings me to my main point. Just as painters and sculptures take courses in art appreciation, perhaps us modellers could take some lessons from a (blog) journey on model appreciation. See you next time? I hope so, because I know that I, for one, would like to get to the bottom of this!

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t should be d in the figure

Thanks for that.

I LOVE the figure! The froggies are lovely and it depicts the need to really understand the question and the interpretation, rather than just being technicians who can run a model.

This is a great start. However, have you considered a broader definition of mathematical modelling? Mathematical models that are a “set of equations” limit us to quantitative analysis. There are many other constructs of mathematics that are used in modeling such as, simulation, real-time bio-feedback and cybernetics. Maybe use “systems of mathematical constructs” to open up other reasons that we model. With the explosion of computer game technology we now have many more tools to study the world around us.

That’s a great point. Thanks! This is one of the reasons that I started the blog, I was hoping that comments like this would help us to arrive at some good definitions of some basic concepts. For this reason, too, I really liked Florian’s post at Theoretical Ecology about ‘what is error?’ It’s a basic concept that’s so taken for granted.