Explaining Clumps via Transient Simulations

Following up the previous post on Batterman and mathematical explanation, here is a case where a mathematical explanation has been offered of a physical phenomenon and the explanation depends on not taking asymptotic limits. The phenomenon in question is the "clumping" of species around a given ecological niche. This is widely observed, but conflicts with equilibrium analyses of the relevant ecological models which instead predict a single species to occupy a single niche.

As Nee & Colgrave reported in 2006 (Nature. 441(7092):417-418), Scheffer & van Nes (DOI: 10.1073/pnas.0508024103) overcame this unsatisfactory state of affairs by running simulations that examine the long-term, but still transient, behavior of the same ecological models. This successfully reproduced the clumping observed in ecological systems:

Analytical work looks at the long-term equilibria of models, whereas a simulation study allows the system to be observed as it moves towards these equilibria ... The clumps they observe are transient, and each will ultimately be thinned out to a single species. But 'ultimately' can be a very long time indeed: we now know that transient phenomena can be very long-lasting, and hence, important in ecology, and such phenomena can be studied effectively only by simulation (417).

While the distinction between analysis and simulation seems to me to be a bit exaggerated, the basic point remains: we can sometimes explain natural phenomena using mathematics only by not taking limits. Limits can lead to misrepresentations just as much as any other mathematical technique. More to the point, explanatory power can arise from examining the non-limiting behavior of the system.