Levin (1980)* is a concise and insightful discussion of where mathematical modelling can go wrong. It is quite relevant to my investigation of The Art of Mathematical Modelling and does a nice job of addressing my ‘why make models?’ question.
This paper answered one of the questions that I had long been wondering about: who is considered to be the father of mathematical biology? Levin’s answer is Vito Volterra** – at least for mathematical biologists who come from a mathematical background. Levin then says that modern day mathematical biologists, as the descendents of Vito Volterra, lack his imagination; too often investigating special cases or making only small extensions of existing theory. It’s a fair point, but thinking takes time and time is often in short supply. My take on Levin’s comment is ‘aspire to be imaginative, but to remember to be productive too’. Furthermore, Levin identifies one of the ingredients that make great models great: imagination – I’m adding that to my notes.
A second piece of advice is that mathematical models that make qualitative predictions are more valuable than those that make quantitative predictions. Levin’s reasoning is that ‘mathematical ecology as a predictive science is weaker than as an explanatory science because ecological principles are usually empirical generalizations that sit uneasily as axioms.’ That is quite eloquent – but is it really quite that simple? For example, if you make a quantitative prediction with a stated level of confidence (i.e., error bars) is that really that much worse than making a qualitative prediction? The sentiment of the quote appears to be to not overstate the exactness of the conclusions, but to me this seems equally applicable to quantitative or qualitative models.
Levin coins the phrase ‘mathematics dressed up as biology’. I have my own version of that, as I like to say ‘that’s just math and a story’, in both cases, for use whenever there are weak links between any empirical observations and the model structure.
To conclude, this paper discusses why the different approaches of biologists and mathematicians to problem solving can result in mathematicians that are keen to analyze awkwardly derived models and in biologists who lack an appreciation for the mathematician’s take on a cleanly formulated problem. Rather than discussing what makes great models great, Levin’s paper reads like advice on how not to make bad models, and because it’s so hard to distill the essence of good models, looking at the art of mathematical modelling from that angle is a constructive line of inquiry.
Levin (1980), Mathematics, ecology and ornithology. Auk 74: 422-425
*Suggested by lowendtheory, see Crowdsourcing from Oikos blog.
**Do you agree? For me, if this is true then the timing is interesting: Vito Volterra (1926), Ronald Ross (1908), Michaelis-Menten (1913), P.F. Verhulst (1838), JBS Haldane (1924) and the Law of Mass Action dates to 1864.
Levin also hits on several items from my ‘why make models’ list and so I have updated that post.