Scribbles on the blackboard in a physics department … look like art.
Whatever all those markings might mean, it just looks damn cool. Like a doorway into some other world. Messages scratched in the sand.
Lotka-Volterra? What the heck is that?? I have NO IDEA. But there it is! In pink and green chalk!
Schools still use chalk boards?! Holy crap! I thought everyone used white boards now.
A whiteboard just doesn’t have the same effect, does it?
I had never heard of the Lotka-Volterra equation, so I googled it…and all the hits refer to the predator-prey model, which I *have* heard of. But the equations don’t look anything like the one on the blackboard, so I guess we have a mystery….
..in case anyone’s interested, the predator-prey model attempts to mathematically capture the relationship between two species, one of which preys on the other. The predators gobble up the prey until the numbers decline so much that the predators start starving…then, as the number of predators declines, the prey species begins rebuilding its numbers…and it all starts over again.
Lotka-Volterra is interesting, but it’s not physics, and not that complicated.
I don’t know what kind of physics they’re teaching at UW Madison, but it is apparently the kind of physics that involves biology.
In general, Lotka-Volterra always seemed to describe to me the relationship between deer and deer hunters. If there are a lot of deer hunters, the deer numbers are depleted, so the hunters stop hunting, and the deer come back. Then the hunters start hunting again, and deer numbers go down, etc, etc, etc.
There’s another equation to describe population behavior in the absence of a predatory relationship, where numbers grow until the ecosystem cannot support the population density. Then the population declines due to disease, lack of resources, etc. Population goes down, achieves balance with ecosystem, and starts growing again… etc etc etc… e.g. deer in areas with no hunting, or humans on earth.
1. Lotka and Volterra (it was two guys, and I think Lotka actually wound up as Volterra’s son-in-law, IIRC) also derived an equation to explain interspecific competition – the effect that two species, each going after the same resource (say, deer and squirrels with acorns) have on each other. It’s not a *perfect* model (neither is the L-V predator-prey model) but it’s an interesting starting point to talk about what’s important in these relationships.
1a. I think what CW is referring to are the Logistic Growth Equations, which predict what will happen to a population if intraspecific competition (competition between members of the species) is the main factor playing a role. There are actually tons of ecological models out there to explain stuff about populations. Most of them don’t work that hot in the real world because the real world is stochastic and most of the older models are deterministic (i.e., there are weird random things that happen in the real world that a cut-and-dried model doesn’t account for – like brucellosis outbreaks or ice storms or meteorites hitting the earth).
2. I am allergic to the solvent in whiteboard pens. When my department was going to move to a newly-renovated building, I (and one other faculty member who was also allergic) raised a big stink about the whiteboards, so we got chalkboards instead. (Besides, whiteboards smack too much of the corporate world to me. Blackboards belong in academia).
That said, the equation purported to be the Lotka-Volterra is not a formulation of it that’s familiar to me. I’ll have to go back and look at it again and see what the guy is trying to say with it. (And generally, in physics-for-the-life-sciences, a lot more time is spent on stuff like the dynamics of blood flow rather than on ecology).
I think Ricki is correct about the Logistics Growth Equation to describe population trends. I couldn’t remember the name – I wanted to call it the Loggins-Messina equation or something like that.
Also the scribbles proported to be Lotka-Volterra in the picture didn’t look like it to me either, but I didn’t say anything because I’m not that smart on this stuff and didn’t want to look stupid…:o
CW:
Well, God, you ALL make me look stupid!! What do I have to say about the equation? “It looks like art.”
I’m an idiot.
Carry on – this is a fascinating conversation, even though I have no idea what you all are talking about.
See why I dropped out of engineering school? I’d rather feel like an idiot in history than science.
“The Loggins-Messina Equation”
OK. Now I want to call it that. Or wait, how about the “Hall-Oates Equation” for something.
Oh, man, how we could mess with the minds of our students – give crazy, 80s-band-based names to equations that don’t have an actual “name” name.
Of course, that would probably be called academic malpractice in some circles.
I’m glad someone else concurs with me on what’s being called Lotka-Volterra not looking like it – I didn’t want to look stupid either. I’m really a field-person, not a chalk-and-blackboard-and-numbers person. I can tell you what a plant IS, but I’m often hardpressed to tell you the dynamics of how its population grows or responds to other things in the area. (I’d rather be an old-time Natural Historian than anything else, but there are precious few positions for true Natural Historians any more).
The Lauper-Madonna Equation would be good, too. Those two alone were responsible for my high school music choices.
The equation on the blackboard looks like a generalization of Lotka-Volterra to N variables instead of the normal 2 variable predator/prey relations. Though it appears to be a single equation, because of the subscripts I believe it represents a system of N non-linear differential equations and should generate some interesting behavior. I might generalize it even further by replacing the term “1″ in the parenthesized expression by some coupling coefficient, say b(i).
jd – Notice that the “i” subscript in the purple equation is blue, the same color as the equation above involving the hyperbolic tangent function. If he meant to relate those two equations, I don’t know where the tanh function came from, it certainly doesn’t look like the Runge-Kutta solutions to the Lotke-Volterra equations that I remember.
If this is an N order system of DEs, then it looks like he has one prey species and a lot of predators eating the same thing. It might also be an attempt to numerically model N interactions (predation events) that occur over time interval i, could it not? That might be a modification for a model of the Net because one trackback or view does not remove the original page as a predation event would.
By the way Sheila, I just looked in my sophmore Differential Equations textbook and I found that it was copyrighted the year that next year’s incoming college freshemen were born. Thanks for making me feel old ;-). Since I studied this stuff during the psychotic year or two when I actually thought that the best beer was free beer, you can call my “contribution” to this debate the “Pabst-Old German-Red White and Blue solution” to the Loggins-Messina equation.
BTW CW, Lotka was an American biophysicist. A lot of physicists would be familiar with this, having been asked to do math on the behalf of their math-challenged biology colleagues. Because biologists obviously went into biology to do science while simultaneously avoiding math ;-).
There’s an old joke about a guy standing in line with 20 items at the 12 items or less line in a labcoat. The punchline is that he’s either a physicist who can’t read or a biologist who can’t count.
A-HA! I was right. The form of the solution to the Lotka-Volterra DEs for “j technologies operating in the same market niche” can be found here:
http://www.aom.pace.edu/tim/Pistorius_Utterback.pdf
The summation on the chalkboard looks like a variation of equation #4 in that paper.
Sheila – don’t you just love us science tpyes? You see art and we get out the screwdrivers so we can see how it works.
Personally, I don’t know what I would do without people like you. Everyone on this thread. I wish you had been my math and science teachers. Collectively. (Wouldn’t that have been a hoot? You and CW and ricki and David Foster all teaching a class together? With only one student? Me??)
:)
John:
Nice find! Notice that equation (4) in the pdf also includes the suggestion I made for an additional parameter. I also agree with you that the blackboard equation above this one, involving the tanh function, can not be related, since a non-linear DE, which requires Runge-Kutta for a numeric solution, can not have a closed functional solution. I think they were just brainstorming, and the equations are not related. Note also that the equation to the left of these is Shannon’s definition of information.
Just FYI, I believe they were talking about the equation in the context of chaos and complexity theory, specifically with regards to the blogosphere. It was a conference, not in a course. Thus Ann Althouse’s presence in the physics department to take those pictures. I can see how the workings of the blogosphere would be considered a predator/prey relationship.
ricki, my wife has the opposite problem: she’s allergic to chalk dust. She had to buy her own dustless chalk when she was a TA, and I think she would have killed for a whiteboard. I hated coming out of class smelling like chalk dust, myself.
Now that EVERYONE is using PowerPoint, there is much less technology difference between disciplines, but it used to be a general rule that physicists used hand-drawn overheads, chemists used computer-printed transparencies, and biolgists used color in their overhead printout – the further from pure mathematics you got, the less peer pressure to make you overheads look like you sketched them on the back of a cocktail napkin. I was one of the first in my (Chemistry) department to use color overheads.
Sheila, you have no idea what I would have given for a class of motivated students like you, no matter what their backgrounds. What I mostly got were pre-med students kvetching about getting an 89 instead of a 91, and poorly trained in-state students who belonged in a Community College, one of whom literally told my wife to give her an answer because “she was tired of thinking”. No wonder we TAs fought to teach Honors College classes, despite the extra workload.
I’m tired of thinking???
hahahaha
Someone actually said that to a teacher? Damn. What on earth does one say to that.
What my wife said was “if you want to stay in my class you will START thinking right now”. The kid later dropped her class. You’d laugh if you knew my wife: she’s the epitome of every no-nonsense Chinese science TA you ever heard about, except she has no accent in English.
God, that is so AUDACIOUS. I have certainly THOUGHT before, while in class: “Dammit, can’t someone just do this for me? I’m tired of thinking.” But to say it? To the teacher?
bwahahaha
Yeah, well my wife never had a student proposition her on the final course evaluation, leaving a phone number to contact. You know, the evaluations that the TA coordinator and thesis advisors see before the TA does? Embarrasment city, that was.
Once, on a lab exercise, I made the mistake of setting up a situation, describing the outcome, and then asking what do you think? (as in, how do you interpret the set-up, does it test the hypothesis posed).
I had a student come up to me who had a problem with the question. She asked me: “What do you want us to think?”
(Fortunately, I was still young and idealistic to say “I want you TO THINK” instead of just banging my head against the chalkboard like I’d do now).