quotes

Daniel H. Burnham, Chicago architect and city planner (1846-1912):

Make no little plans; they have no magic to stir men's blood and probably themselves will not be realized. Make big plans; aim high in hope and work, remembering that a noble, logical diagram once recorded will not die, but long after we are gone be a living thing, asserting itself with ever-growing insistence. Remember that our sons and our grandsons are going to do things that would stagger us. Let your watchword be order and your beacon beauty.

Nero Wolfe, in The Rubber Band, by Rex Stout:

What is it that he has been trying so desperately to preserve, with all his ruthless cunning? His position in society, his high repute among his fellow men, his nimbus as a master biped. Well, he will lose all that, which should be enough for any law.

This, in a book that used "biped" once earlier. It's got me thinking of lost hominin nimbuses.

Nimbi?

R. A. Fisher and Sewall Wright introduced diffusion approximation methods into genetics; Fisher (1937) was the first to consider spatial disperal using a reaction-diffusion model. I found this quote a useful expression of his acknowledgment of the limits of the model:

The use of the analogy of physical diffusion will only be satisfactory when the distances of dispersion in a single generation are small compared with the length of the wave. In reality diffusion is a complex process, compounded often of the diffusion of gametes, and that of larvae, in addition to adult forms; a more exact treatment than that supplied by a simple coefficient would involve the interaction of these components, and the stages at which the selective advantage was enjoyed. So far as it is applicable, the analogy of physical diffusion, therefore, greatly simplifies the problem (355-356).

The paper has no references.

I'll probably have some more material on quantitative analysis of dispersal in the few days. Here's a quote from Peter Turchin (1998:17-18):

Of course, we do not know that animals truly move at random, like flipping coins to decide whether to turn right or left. Each individual could be a perfect automaton, rigidly reacting to environmental cues and its internatl states in accordance with some set of behavioral rules. However, even if this were true, we might still choose to model behavior of such animals stochastically, because we would not have the perfect knowledge of all the deterministic rules driving these animals. Even if we did, we might not want to include them all in our dispersal model, since such a model would have an enormous number of parameters and would require a very accurate representation of all environmental "micro-cues." The point is that randomness is a modeling convention. Because it is impractical, and not even helpful, to attempt to model individual movement deterministically, we use a more parsimonious probabilistic model.

I'm pausing the quote to point out my boldface. It has become computationally feasible in the last few years to model enormously complicated scenarios with individuals acting pseudo-deterministically. The most popular use of such modeling is to try to constrain dispersal models by some geographic conditions, such as local habitat richness, rainfall, or altitude (see also, "One model, hold the extra parameters"). Of course, animals really do disperse in ways that depend on such geographic parameters. The question is whether any datasets are sufficient to test models involving so many parameters.

This approach is aptly termed behavioral minimalism (Lima and Zollner 1996). In essence, we adopt a thermodynamic approach: the behavior of individuals is erratic, or irregular, but the redistibution process at the population level has many regular features. There is a direct analogy with with thermodynamic theory. The motion of each gas molecule is chaotic and essentially unpredictable, and can only be described probabilistically. When dealing with large numbers of molecules, however, the laws at the aggregate level are for all intents and purposes deterministic. Similarly, the problem of biological dispersal can be treated by starting with a probabilistic description of individual movements (in other words, formulating the problem as a random walk), and then approximating the redistribution process of the ensemble of individuals with a deterministic equation, diffusion.

The effective scale of stochastic versus deterministic processes is important. I'm chiefly interested in the dispersal of adaptive genes in human populations, for which the deterministic approximation may be considered to have become more and more relevant over time, as the population sizes of regional populations grew. Still, the present pattern in many cases may reflect the stochasticity of populations from earlier time periods, when they were smaller. And formerly important deterministic processes, such as the adoption of agriculture, may no longer be directly observable. So how do we model variance?

The thermodynamic approach to dispersal does not have to assume that the movement of each "particle" is completely random. The important feature of this approach is that we can control the degree of realism in the model. Environmental factors that have strong effects on movement can be included explicitly in the model, while other factors that have weak effects (or about which we have no information) are included in the stochastic component.

This would incorporate the geographic modeling approaches mentioned above -- deterministic processes related to spatial variance of habitat or dispersal potential. But then the important step must be to find a minimal deterministic model to account for the data, and then test it with other observations -- such as more extensive genetic sampling, archaeological information, or historical documentation.

References:

Turchin P. 1998. Quantitative Analysis of Movement. Sinauer, Sunderland MA.