As a WordPress blogger, I get a handy list of search terms that have led people to my blog. A particularly memorable search term that showed up on my feed was ‘how to make mathematical models at home’. What I liked about this query was that it suggests mathematical modelling as a recreational hobby: at home, in one’s spare time; just for fun. This speaks to an under-appreciated quality of mathematical modelling – that it’s really quite accessible once the core principles have been mastered.
To get started, I would suggest any of the following textbooks*:
- A Biologist’s Guide to Mathematical Modeling by Sally Otto and Troy Day
- Modeling Biological Systems: Principles and Applications by James Haefner
- A Course in Mathematical Biology by Gerda de Vries, Thomas Hillen, Mark Lewis, Birgitt Schonfisch and Johannes Muller
- Dynamic Models in Biology by Stephen P. Ellner and John Guckenheimer
Now, I know, you want to make your own mathematical model, not just read about other people’s mathematical models in a textbook. To start down this road, I think you should pay attention to two things:
- How to make a diagram that represents your understanding of how the quantities you want to model change and interact, and;
- Developing a basic knowledge of the classic models in the ecology, evolution and epidemiology including developing an understanding of what these models assume.
This would correspond to reading Chapters 2 and 3 of A Biologist’s Guide to Mathematical Modeling.
A good way to start towards developing your own model would be to identify the ‘classic model’ which is closest to the particular problem you want to look at. If you’re interested in predator-prey interactions, this would be the Lotka-Volterra model, or if you’re asking a question about disease spread, then you need to read about Kermack and McKendrick and the SIR model. Whatever your question, it should fall within one of the basic types of biological interactions, and the corresponding classic model is then the starting point for developing your mathematical model. From there, the next step is to think about how the classic model you’ve chosen should be made more complicated (but not too complicated!) so that your extended model best captures the nuances of your particular question.
Remember that the classic model usually represents the most simple model that will be appropriate, and only in rare circumstances, might you be able to justify using a more simple model. For example, if the level of predation or disease spread for your population of interest is very low, then you might be able to use a model for single species population growth (exponential/logistic/Ricker) instead of the Lotka-Volterra or SIR models, however, if predation and disease spread are negligible, then it arguably wasn’t appropriate to call your problem ‘predator-prey’ or ‘disease spread’ in the first place. Almost by definition, it’s usually not possible to go much simpler than the dynamics represented by the appropriate classic model.
That should get you started. You can do this at the university library. You can do this for a project for a class. And, yes, you can even do this at home!
*For someone with a background in mathematics some excellent textbooks are:
- Mathematical Models in Biology by Leah Edelstein-Keshet
- Mathematical Models in Population Biology and Epidemiology by Fred Brauer and Carlos Castillo-Chavez
- Mathematical Biology by J. D. Murray Part I and Part II
but while the above textbooks will give you a better understanding of how to perform model analysis, the ‘For Biologist’s’ textbooks listed in this post are still the recommended reading to learn about model derivation and interpretation.