How to make mathematical models (even at home)

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*:

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!

Footnotes:

*For someone with a background in mathematics some excellent textbooks are:

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.

The Geometry Selfish Herd (Hamilton, 1971)

No apology, therefore, need be made even for the rather ridiculous behavior that tends to arise in the later stages of the model process, in which frogs supposedly fly right around the circular rim… The model gives a hint which I wish to develop: …the selfish avoidance of a predator can lead to aggregation.

— W.D. Hamilton (1971)

Sheep cyclone got me thinking about Hamilton’s selfish herd (pdf). It’s been a while since I’ve commented on model derivation with reference to the published literature, and so it seemed like a good idea to re-read Geometry of the selfish herd (1971) with the goal of discussing Hamilton’s decision-making process regarding his model assumptions.

To quickly summarize, the model from Section 1 of the Selfish Herd involves a water snake that feeds once a day on frogs. If they stay in the pond, the frogs will certainly be eaten and so they exit the pond and arrange themselves around the rim. The snake will then strike at a random location and eat the nearest frog. A frog’s predation risk is described by its ‘domain of danger’, which is half the distance to the nearest frog on either side (see Figure). Lone frogs have the highest risk of predation, which leads to the formation of frog aggregations (like the one in the lower left corner of the Figure). The model from Section 3 of the Selfish Herd presents the same problem, but in two-dimensional space, and so now the domains of danger are polygons. In relation to Section 3, I think Hamilton foreshadows the sheep cyclone because he considers the case where the predator is concealed in the centre of a herd before pouncing, as well as discussing (the probably more common) predation on the margins (when the predator starts on the outside).

As you can see from the quote above, Hamilton makes no apologies for unrealistic qualities because his model gives him some helpful insight. This insight is that aggregations could arise from a selfish desire to diffuse predation risk. In terms of the model derivation, a helpful construct is the domain of danger, whereby minimizing the domain of danger likely corresponds to minimizing predation risk, assuming the predator only takes one prey and starts searching in a random location.

Overall, the one-dimensional scenario seems contrived and I’m not sure if I understand how the insight from one-dimension carries over into two-dimensions.  In two-dimensions, what incentive is there for initially-far-away-from-the-lion-ungulates to let the initially-near-the-lion-ungulates catch-up and form a herd? The model in Section 1, is what I would call a ‘toy model’ – it acts as a proof of concept, but is so simple that its value is more as ‘something to play around with’ rather than something intended as a legitimate instrument. I wonder about the relevancy of edge effects – in Section 1, the model is not just one-dimensional, but the frogs are limited to the rim of the pond which is of finite length. The more realistic two-dimensional example of ungulates in the plain should consider, I think, a near infinite expanse. If the one-dimensional problem was instead ‘frogs on a river bank’, would this all play out the same? Would frogs on the river bank aggregate?

Pre-1970’s computing efficiency probably made it quite difficult for Hamilton to investigate this question to the level that we could today, but none-the-less, I’m going to put this model in the ‘Too Simple’ category. For me, this paper never reaches the minimum level of complexity that I need it to – that would be: (a) simultaneous movement of prey; (b) in response to a pursuing predator; and (c) in an expansive two-dimensional space. Aside from this paper, Hamilton sounds like he was an entertaining fellow whose other work made highly substantive contributions.