This is mostly just a note to myself, but in case anyone else is interested: Steve Walker has a nice post walking you through the derivation of the continuous time Price equation. The discrete time form is much more commonly seen (e.g., in my own work), but the continuous time form crops up in ecology, for instance in some very nice work by Steve Ellner.
Steve also points out that the continuous time Price equation is basically an instance of a mathematical rule called the chain rule. Which raises the deep and interesting question of the boundary between abstract mathematical rules and concrete mathematical models of specific real-world systems and phenomena. Where does math-as-abstraction or math-as-rules-of-logic stop and math-as-description-of-real-things begin? The line can be rather fuzzy, which is something I’ll probably be posting about more in the near future.
Well, nobody really agree on what mathematics “is”, thus we don’t even know if we should be surprised or not when it shows deep connections between unrelated objects. Is it something we “invent” or something we “discover” (I vote for the former)?
It’s great to see more people embrace the Price equation, even though it’s still a somewhat controversial equation. It’s quite random but I’ll take this occasion to post a link to Rice’s wonderful work on stochastic fitness and migration (http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0007130). It’s a great paper but has been somewhat ignored for some reason (perhaps because the level of mathematics?).
Looking forward for more math-posts!
I should not the ref to Rice’s work is not *completely* random, he uses the Price equation…
I’m well aware of Sean Rice’s work, it’s wonderful stuff. I’ve only used the standard, “retrospective” Price equation myself, the “prospective” version, which takes account of the fact that the future isn’t known and so has to be described with probability distributions rather than single numbers, is really Sean’s baby.
Re: the Price equation being controversial, only with van Veelen & his friends, as far as I can tell. For the life of me, I have no idea what he’s on about. And I’m far from alone in taking that view. I would never invoke proof by authority, but I will say that it’s reassuring to be in agreement with Steven Frank, Alan Grafen, Samir Okasha, Steve Ellner, Andy Gardner…on the utility of the Price equation.