Tuesday, February 19, 2008

the mathematical theory of the struggle for life

"If sharks were people,” the small daughter of his landlady asked Mr. K., would they then be nicer to the small fish?” – Brecht, Wenn die Haifische Menschen wären

Continuing LI’s notes on the predator/prey relationship – we discovered, through one of Machery’s essays, that a famous essay by Volterra had caught Raymond Queneau’s eye, and was mentioned in his 1943 essay, The place of mathematics in the classification of the sciences, which begins like this: “In its relations with mathemtics, every science passes through the following four phases (four as of now, perhaps five tomorrow)” – which elegantly combines the academic and the Groucho Marxian. Queneau briefly surveys the sciences, claims that physics has gone through three of his stages, already, and then writes: “This is the ideal stage for the scientists of the late nineteenth century. The other sciences reamin far behind in this regard. Only very limited subjects are treated by this method: in biology, the theory of the fight for life; in sociology, econometrics. These two examples furthermore show that there is no incompatibility between the analytic method and the life sciences. The delay is in part explained by the fact that such an application apparently offers no problem to be resolved from the mathematical point of view, and so no potential for discovery; mathematicians thus soon lost interest in these theoretical fields, which offered no grist for their mill. if the theory of the fight for life was developed by Volterra, it’s because it ultimately led to integro-differential equations worthy of interest.”

The explanation from mathematical banality might not quite have been the whole story, or at least LI can’t see it. But it is a nice story, nonetheless. The essay arose out of a project Queneau was working on in July 1942 (terrible months to encounter the struggle for life): Brouillon projet d’une atteinte a une science absolue de l’histoire, (sketch for a project for an attempt at an absolute science of history) of which Voltarra’s Lecons sur la theorie methematique de la lutte pour la vie was going to be one of the main sources, and Vico, Bruck, William Flinders Petrie, Spengler, “authors who believed they could discern rhythms or cycles in history” would be the other.” In the Model History in which Queneau jotted down these reminiscences, he also wrote: “if there had never been any wars or revolutions, there would never have been any history; there wouldn’t be any matter for history; history would be without an object… happy people have no history. History is the science of the misery of mankind.”

Most people have heard of the Volterra-Lotka equations, which show how predator-prey relations should oscillate around an ideal equilibrium over time, all other conditions being equal, and how that oscillation takes on four states. Queneau’s idea that, perhaps, these predator-prey states are strung out over human history is startling.


Scissors MacGillicutty said...

Some nitpicks:
—the Lotka-Volterra are periodic; maybe everyone uses oscillation as a synonym for periodicity these days, but I have to draw the line somewhere.
—Lotka-Volterra depends on four parameters. I have no idea what you might mean when you say the "oscillation (sic) takes on four states."
—Depending on the values of the parameters—which are the increase in the prey population sans predators, the decrease in prey population due to predators, the decrease in the predator population due to natural causes, and the increase in predator population due to consuming prey—the prey and predator populations can
(1) increase and shrink periodically, or
(2) go extinct, or
(3) remain constant.
(3) is a special case that only occurs when the prey population is equal to the predator population death rate divided by the predator population's increase from eating prey, and the predator population is equal to the growth rate of prey divided by the rate at which the predators eat 'em.

I don't know why I'm being so cranky. None of this changes your larger point....although since Lotka-Volterra has to do with two difference species, it's application to intraspecies killing seems dubious.

Sorry. I either need coffee or a stiff drink. Maybe a little of both? I really hated both diff eqs classes I took, and the thought that someone somewhere is enjoying thinking about them is driving me mad.

Anonymous said...

Man is an animal which, if it lives among others of its kind, requires a master...But then the master is himself an animal, and needs a master.

(Kant channeling Kafka, from "history from a cosmopolitan point of view", sixth thesis.)


roger said...

Wow, S.M. Good explanation. I was translating what I understood about the different relations of predator-prey to reflect the fact that they “vary periodically with time” In, Mathematical modeling for the Life Sciences, the author talks about steady states of the system given by the equation, which can change with the number of predators. This is really all I meant. Sorry I wasn’t more specific. I don’t think the model gives us any information about the species type – whether the predator is of a different species than the prey. It is true that the species difference would prevent any mixed states – for instance, Europeans and Amerindians could reproduce, so that messes up applying the model to humans without major modification.

But maybe this is why Queneau never finished his work!