> But I haven't thought about it very deeply, and I > haven't formally investigated how the properties of, say, clusters of > self-organizing agents behave in a 2D environment vs. a 3D environment. > > So the other day someone asked me: what are the issues, what are the > differences? Fundamental or superficial? > > Offhand, I sort of assumed the relationship between different simulation > spaces would be pretty much the same in both 2D and 3D. Sure, I assumed > there would be a scaling issue, but not much else. (e.g., a 2D pred-prey > model would show different numbers than a 3D pred-prey model, but the > dynamics would be essentially the same.) > > Is this true?
It's certainly not *obviously* true, and there are sound reasons (I think) behind my off-the-cuff intuition that it's false. (However, if someone has actual *data*, that might trump intuition. Or not.) The "drunkard's walk" in 2-dimensional (flat, infinite) space is recurrent; in 3- or greater- dimensional (flat, infinite) space, it isn't. (Your self-restriction to a torus kills part of the reason that works, but not the whole thing.) More generally, the behavior of limit sets for systems of ODEs in the plane is very, very nice, because the topology of the plane is fairly nice; whereas the behavior of limit sets for sysstems of ODEs in 3- (and much more so, greater)D can be nasty beyond belief. (Well, in the presence of people who are so much into chaos, I guess I shouldn't say *nasty*. How about, complicated?) And here your finiteness assumption wouldn't help at all. I suspect that Ed Lorenz's original numerical calculations that eventually led to a solid mathematical understanding of the Lorenz attractor in 3D could, without much heavy lifting, be embedded into a 3D NetLogo model of something-or-other. For 3D pred-prey models, look for stuff by combinations of Zeeman (E.), Zeeman (M.), and Hirsch. Sorry I can't be more specific without a search I don't have the time for right now. Bye. ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
