Chapter 2 Limitations of Classical Models

Stochastic models in ecology are designed to study ecological systems by simulating the underlying processes and then studying multiple realizations of a simulation model. We focus here on models of life history events. Interestingly, the life table approximation of dividing time into discrete ``quanta’ migrated early into stochastic models in ecology. The Lotka-Volterra model is typically developed in this way [ref]. Modern simulation studies have advanced by considering smaller and smaller time increments, but within this same fixed time step framework [refs].

The shortcoming of this approach is that while one must have finer scale time increments in order to capture more intricate events, more and more time is spent simulating no activity. Alternatively, one can develop models based on the actual time of transitions. The difficulty with this shift in perspective is that events for individuals are no longer synchronized. Further, generations may not be synchronized, making life table summaries problematic.

The development below of a simulation structure for competing risks in a biological system is built upon the concept of potential lifetimes using the cumulative risks Mj(t) as a basic building block. This approach necessitates detailed knowledge of the ecosystem under study, which is precisely what we want. The purpose here is to provide a framework for biologists to incorporate great detail about known and suspected aspects of an ecosystem. Biologists inherently recognize that their knowledge is incomplete and may even be wrong in part. However, it is extremely difficult for them to test their hypotheses about ecosystem-level and population-level properties that may depend on processes that affect individuals. Work in complexity [ref on cellular automata] suggests that higher level structure can emerge from local structure. These models are promising, but to date suffer from the same quantization problem found with life-table derived methods

2.1 Mathematical Biology

18.2

Another thing that was going on at the time this work was done. Everybody was an optimization freak. You had to be optimizer, you had to optimize –. It was a whole world that was supposed to be optimized. There’s no way in hell that you – simple thing like doing a macro optimization. It put – level – wants to behave or go down the tubes and the thing become more extinct because – long term strategy, and so on. The point’s made.

12.22

Mathematical Biology

+The point is, when Bland was doing this work, we were basically doing it in the heyday of the Robert May group.

*Mathematical biology.

+Mathematical biology. The 40s was the first time that somebody tried to apply mathematical techniques to biology. What was very interesting was that they didn’t try to take a look at the biology. They took a look at the mathematics that they understood and then they forced the biology into the mold.

+OK, now we’re into the situation where we’re talking about a 5-pound box and 25 pounds of jello. We’re trying to get the 25 pounds of jello into the 5-pound box. Typical of what mathematicians and physicists do is you chop off the corners until you got the 5-pound box that you wanted. Unfortunately we threw away all of the biology.

*Arms and legs go.

+We threw away the arms and legs, the eyeballs, the ears, all of that was gone. We’re back down to a skinned ameba and we’re saying, ``OK, now we understand the biology.’

*You had the guts left.

+Yeah, we had something like a beating heart not connected to anything and not being triggered by anything. What finally happened, and what we realized pretty really on, was that if we were going to try to do what we wanted to do, we had to take a complete reversal of the ideas.

+We had to take a look at the biology. They applied the appropriate mathematical techniques to it. The problem was that the appropriate mathematical techniques didn’t exist. There wer no techniques for handling small populations. There was no technique for handling the population of more than one. When we got two interacting systems, we were screwed.

*You could do some —.

+Yeah, yeah.

*– time series that were correlated, but that’s about it.

+Yeah, when creditor A at — B, the problem was over. But that’s not a simple problem. Well it is a simple problem. It’s one single event. So we spent a lot of time trying to solve that particular problem. We tried to come up with a mathematical technique where applicable.

Nonlinearity and Modeling

15.6

What I would like to talk about is the problem of linearity and nonlinearity. Models and the reality in models and it’s good stuff. Testing models and model design, things like that, because the problems seem to have not gone away. They have not been – by time. The issues that were standing 20 years ago when we built our first set of models are still with us today. The people that want to throw out their deterministic models and things like that, I want to throw a challenge to them. I think I haven’t pulled a chemistry book but do you remember what, isn’t it femto-seconds, what is that? An incredibly small thing that – manage – in certain optical systems, isn’t that right?

*That sounds right, yeah.

So they developed – basis for prediction of systems that are optically based and experimentally based – into that measure. Why not give the people the abilities kind of –? Let’s give them a full second. Let’s give them a full second to – experimental – into the –. What I want to know to the nearest second is the time that the red spot on Jupiter will disappear. Ok? I want to know to the nearest second. I factored with the kind of resolution and the kind of resuscitation today experimental – just is mind-bogglingly large. I want them, before they give me any – modeling and –, give me to the nearest second when the reddish spot on Jupiter is going to disappear. Is that ok? Is that a reasonable question?

*It’s a well-defined question.

So does that make one point at least. Because one problem, there’s a quasi— that keeps resurfacing. The last time I went to the library and I actually dug through, I found the people busily working on the concepts for competing risk structures. The rate for the alpha beta things and the modeling of the deterministic models.

*The growth models?

Yeah, the growth model stuff.

*The r and K?

15.14

So that’s the other aspect that needs to be considered is not only, that was 20 years ago, but it’s true today in terms of what’s practical to do. Whatever time it seems that if you try to throw in much real biology, or real evolutionary theory, or things where people or animals or plants evolve in a natural system, you end up with some incredibly heavily high structured –.

You sure won’t be able to solve it. At least the techniques for – and the modeling – was in the paper from LBL. I mean the original one in Livermore lab was the original – together with Forest Service and also Dave Wood. At this point it seems to still be a viable approach. It has the potential of creating buffer experiments on the one hand that used those models of that type to do some relations. On the one hand you would be helping to design experiments that would be a better structured, more quantitative and better, by the use of the modeling – and at the other end, you – models as a predictive source of –. Ok, that’s a big sweep of that stuff.

*Well, it seems like a lot of the modeling until very recently was looking at things between these nonlinear events you’re talking about. So you assume that they’re far enough part that you can predictably model in the middle. It seems that in the last-well in the last 5 years, less than 10 years-that people have been developing models-toy models-where they’re interested in the properties of the nonlinearity and in trying to get it to structure like what you’re talking about. Structure the models, structure the nonlinearities, not trying to predict when the next tornado is going to be and where it’s going to hit, but trying to look at the pattern of tornadoes and what happens under different situations.

*So it seems like that kind of modeling is very much in its infancy. People are doing weather modeling, trying to do very sophisticated things with it. I guess what I see what you’re talking about is sort of looking at a much bigger scale. Looking at a lot of different scales at the same time, all the way from what’s happening in a few seconds to what’s happening in an organism’s lifetime to what’s happening on an evolutionary scale.

Yeah.

*And how these are all tied together. There may be some things that an organism does which don’t make sense in terms of its lifetime but might make sense when you look at it on an evolutionary scale. When it has to deal with flooding and natural disasters, it had to respond to that in order for the species to be here today. It had to have the ability to disperse or move quickly under adverse situations.

*The kind of sense I’ve had from reading and from thinking about how people think about models is that to think, Oh well, we don't have the data for that,' orYou know that’s too complicated. Let me look at something where I can do all the pieces.’ When they get to that point, they often are getting to linear models or a very simple type of nonlinearity. That doesn’t seem to constrain your thinking. I mean that’s more, well let’s look at what’s going on. Let’s look at what reality is, and then let’s think about what makes sense in terms of modeling reality. Then we can step in that and see where can we collect data that is going to help us model reality, where we might choose to do that. I mean it doesn’t make sense to try to model all of reality.

Watt tried to model first all the country and then when he failed that, he went out to model the California for his deterministic model for all of California. Remember him doing that?

*Who is this?

Watt. He was one of the early deterministic modelers that was in the same group with Holling and Slobodkin, that’s another name.

*Ok, that one I know.

Slobodkin is another and so what happened is Watt looked at what Holling and Slobodkin were looking at and gee whiz, I will take these – these model and deterministic models and model all of some county which I can’t remember which. And then it was a disaster. So what did he do? He applied for a grant for all of California for his model. You wouldn’t say that he, he didn’t have any trouble of ego.

*I don’t know this person’s work, so I’ll have to look that up.

I think, that’s just totally off the wall, ok?

*That’s probably right.

Whitaker

Totally off the wall, but so there was also a time when everybody was jumping on the bandwagon and some people were doing really quite sophisticated both – experiments like Slobodkin, and other people were – biology like Holling. He was more careful of biology, and there were other people that came along that really were very good biologists like Whitaker that just lived as best he could with normal distribution and so forth. That’s all the guy gave him.

*I have an interesting story about Whitaker. So this student from Vietnam that I mentioned earlier who is working on the Mekong Delta. He is using a package for canonical correlation and one of the justifications in the package is that it allows you to get away from linearity, from linear relationships, and to look at unimodality. Unimodality. So this kind of a shape. And thinking with linearity, you know if you’ve got a certain range, you can think of it as linear. But if you look at his stack of counts, species counts, over the whole range it’s going to be low and then high and then low again.

*I was thinking this is very strange. Where is this coming from? So I asked him to bring in the manual. Well it turns out that there’s a paper of Whitaker’s from the 1950s. He’s basically talking about species densities over an environmental gradient, and how you’re going to tend to have an area where the species can’t really live very well, you get low counts, and then there’s the ideal area and then it drops off again. He used the normal distribution as a convenient approximation for this. These people seem to have taken that idea and totally turned it inside out. If you look at the math of what they’re doing, it’s totally linear, so they’re claiming it’s nonlinear but it’s totally linear and behind it they’re assuming normality.

That’s what I found in every paper that I actually fought through to the end. People would always claim – linearity, multidimensionality, heterogeneity. Heterogeneity was a big paper that got thrown at me by the group – Washington while we were applying for grants and stuff. You know I dug the paper out and there are, what are they called? Without loss of generality?

*WLOG.

Thank you. Without loss of generality. After you thrash everything out, they would throw this thing where they threw it out all like having – one dimension. Of course it was horrendous math but ok, so none of the business people I ever thrashed it through –. So it was – if you were willing to dig. – so many times over I got very discouraged with that sort of thing. I was burning up too much of my lifetime without meaning much. Especially not getting any useful tools.

*And that’s fine in certain situations and over a certain range of conditions. But it’s when you get to those nonlinear events that it breaks down. You either need to make assumptions to get around that or else actually build that into your model. It seems like there are some of those ideas that are coming out now in some of these Markov chain models. There’s some models of individual behavior where people are actually tracking.

It’s ingrained, it’s basically a Markov chain that I basically throw out as a starting point and then from that – history and evolutionary information going to keep true Markov properly. Am I right on that one?

*You can’t have arbitrary information back into the past.

Right. Yes, with living things, or –.

*Yeah, you can define your past as far back as you want but at some point it ends.

So, this modeling technique in another sense was a fix for Markov-Chains. Not only – dimension.

Robert May et al.

10.2

Critical mostly makes an experiment, the whole idea behind this can make a situation detestable. That was my great fit, with a group of people based on the stuff that Robert May used to do. I’ve yet to see an experiment that could work, or if it did, it was some gigantic artificial chemostat type of experiment where you have a huge population, a rapidly stirred beaker or sitting for a while and it seemed like any time you’d try to hang on, try to hang on to one of those for 10 years. Try to hang onto your chemostat for 10 years. Watch what happens. You know a few people did.

*Right, in England, yeah.

They just want over the map because you have all sorts of weird stuff and awful type of genetic mutation and new strains and everything. It was crazy. So no matter how long you thought you could hang onto something like that, but the longer you hung onto that, there was no variation save. There was supposed to be this – failed totally. I can’t, can you talk of a single exception to the contrary 20-30 years.

*I think people are starting now to look at some models but the stuff that went on in the 60s and 70s and even the early 80s.

I — enough time.

*Yeah, no. It would often be based on things that weren’t measurable. They made nice intuitive sense but then there was no way you could investigate modie. Yeah, I don’t think there’s much out there, even at this point.

OK, so how do you call that science? How do you call what Robert May did science? I thought you were supposed to be able to develop some theory, from the theory a hypothesis, from the hypothesis a series of experiments, from those experiments data, from the data, analyze the data and have a clue on the original theory. Show me where they’ve done it? Anywhere. Find where that group has closed the loop of what you normally would call science. Anybody else would insist on something like that for science. Why shouldn’t they be held to do that for science?

They even taught me that in the introductory course on general science in high school. They pounded that into me a hundred times over. What was it, 7th 8th grade general science course pounded this into your being along with everybody else because it’s absolutely fundamental to doing science. That’s yet to be seen so that’s the other side of some, and I would really like someone to show me the contrary somewhere and to the world if you’re ever going to publish the thing, as a challenge.

*I hear you. It’s a very important point. I know one wildlife ecologist who is doing what he calls individual modeling. He’s trying to model individual behavior. He’s a wildlife ecologist, he goes out, he puts tags on animals and has followed them.

Then I’d put him back in the same loop that we described, of complexity.

*Right.

And then how would he go from there to the next step?

*Well, I want to talk to him and he’s at Madison and I want to show him this stuff. But he’s a wildlife ecologist.

Holling did really nice work and creative biology and he had a biological intuition. He had lots of really nice experiments and the problem, he was always so constrained by the fact that he felt that he had to come back into the same reality things like that, cuz he had no other tools to use as an alternative. And we had some interesting discussions off to the side. He very much wanted me to come back out.

I think I already mentioned the fact that he had new problems of trying to have a monitoring system and it was killing his computer in terms of compute cycles. The thing he found most intriguing was the complete aside on that special method of getting away from calculating trigonometric functions for working out distances, say to trees or how would their animals and plants interact? But also you know you could use his same kind of tables.

He started measuring individual animals daily and other things like that. In every case, if you moved to measurements on individual animals then you came right back to the things that we were describing in this paper, or that Jim was describing in his illustrations using different games. All the images by Jim were just beautiful. I loved it. I really loved it, because Jim found himself right smack in the middle simulation on the one hand and then a bunch of numerological things. He told me he had gotten into a whole new set of them.

10.4

It’s easy with Robert May’s stuff. All I see are gotchas. If you try to take it one step forward like Holling did . . . Important people in my life. The stuff he was doing I thought was really neat.

10.9

I wanted to say something about how it’s hard not only at the deterministic level of Robert May and so forth.

*Yes, May and MacArthur.

May and MacArthur’s approach, but also the fact that I wanted to discuss something about the people that I thought did wonderful work.

20.20

*I wanted to read you something here. This was the Royal Statistical Society, their newsletter, it comes once a month, and there is a picture on the front of Sir Robert May, who is the, he is the science advisor, the key science advisor for the British government. He gave a talk, he first gave some much needed advice on the role of statistics in science and society and then launched into a highly stimulating and enjoyable talk on nonlinear problems in ecology, evolution, and immunology.

Do you realize – that we had with him in his – students and so forth the time we were trying to do some nonlinear stuff?

*No. Why don’t you tell me about that?

Wasn’t he one of the key people that was pushing for linear systems early on? Early on. Young person. Early on.

*Could be. You know, I know that he got into bifurcations which is a nonlinear system, but this is before that.

Yeah, before that. And then when you start looking at systems that bifurcate – there’s a simple system that I could write, the simplest system that involves say a couple of words, it just hasn’t, one has to – squared, and here was this wild behavior. That’s where you – get religion. A different kind. So many of the people thought – systems were against everything and it would provide adequate, there would be some transformation of the kind of modeling technique that would provide them a way of describing natural systems in a reasonable way. That was really – and then later on, pretty soon of, who’s the guy at Cal Berkeley?

*Oster?

Yeah, right. He had intriguing ideas – and he’s the one that passed the thing back to the other finding some wild behavior for the simplest fossil system they could bring up. – from a totally linear system and it, I still have that open question to you. Is there any form, is it possible to form a – that – infinity and yet – infinity of linear distance. Is there any way you are creating the simplest nonlinear system like you just – bifurcation type thing? Ok? Did you ever resolve that one?

*No.

I’ve run it by you a number of times. Over the years. Over the decades actually. – that one worthy of a model theory prism – small simple – problems that are easy to pose, that are simple, not screwed up. So why don’t you hit him with that one?

*It seemed that if it’s going to work, it would have to be a system where you have some system that’s not regular, where you’ve got something that’s getting vanishingly small or vanishingly close to an end point at some rate that leads to a pathological case. I mean you can get degenerate systems but that’s not what you’re talking about. You’re talking about truly nonlinear systems.

Yeah.

*I don’t know.

Why don’t you pass – a friend, model stuff. Didn’t you have a friend who was into model stuff?

*Yes. Keisler.

You had a friend who was into model theory. Why don’t you pass that one on to him?

*Good idea.

Pass it on to him. It’s been – for my entire life. I haven’t been able to find a thing – find a way to prove it otherwise either. So give it to him – keep it clean and neat and see what they say. Anyway May – back and forth each time if you just niche it a little further or went to a little higher value or things like that or – whoa. If – find a way of –. But does he ever get shifted in his points of view and realize – biology reality. But there was a lot of – they really believe that the – at least first approximation. And – we do even today is straight linear theory –.

*Well – approximation if you don’t know anything?

Not only that but there are many things that have – a statistician like – and even – that are not going to become stable and not part of the plant here. It hasn’t been – in place during their time.

*Right, or over a certain range of conditions.

Yeah. However, if you want to throw something into the – asteroid belt is in between Mars and Earth where the asteroid belt is, I’m sure you can find some there – unstable –. So which piece and part of the universe are you talking about? And also what – it’s mind boggling – shot say to Mars are, almost no correction. Or how they missed satellite that didn’t have a, the last part didn’t fire, put it into an orbit that was nonfunctional and I guess eventually that would burn up. What did the people do? They – each time make a further correction in it and finally get it out orbitally by – the correction of the moon – into nothing. – pure linear theory, isn’t it?

*Yeah.

Pure linear theory, all that was done. Built some incredibly clean and – things like that and some things in correction for mistakes and other problems, you’re able to come back – all linear theory –. – you know – linear approximation to occur like a linear theory, approximation type thing.

*Yeah, local linear approximation.

Whatever. And you pretty much ignored the other third body thing out of that one too.

*That’s what I was thinking about.

– that one, you could pretty much ignore it, can’t you? For – time over – space. Or –.

*It’s not solvable.

That’s what I mean.

*You can approximate it to whatever precision you want but you can’t get an exact – three point problem.

So really I’m throwing a bunch of things like that.

*Interacting system.

Because they are constantly interacting –.

Jerzy Neyman and Flour Beetles

10.6

You have to be talking about events being the total orientation and time as a dependent variable, and the next moment you’re telling about time with events coming in as a dependent variable. Does that make sense?

*Yes, and then it’s really complicated if you try to model it in terms of time because then you’ve got all these interactions. You have a whole bunch of stochastic processes that are going on at the same time and they’re all interacting and you got to model all that jointly.

And if you don’t do it jointly, you lose all the biological reality.

*Exactly. And then you’re back to mathematical biology.

Modelling / Neyman

You know it was interesting to go to, what was the incredibly neat statistician who pioneered a lot of use into biology. He was up there at Berkeley too, in statistics, but.

*Neyman.

Neyman, Jerry Neyman. I went to one of his lectures one time where he was showing how to model biological systems, stochastic models of a biological system. So I was very intrigued because I was also looking into probabilistic models and here he was putting on a whole lecture just treating the whole issue of problems that came out. And he was also doing a lot of pioneering work on calculating life data and things like that, statistics on life tables and stuff like that, kinds of thing from a sophisticated probabilistic point of view. So I go there and right off he says well, the egg and the pupa are both inactive, they don’t move, so they’re stationary. Any the larva and the flower beetles.

*Yeah, it was flower beetles.

Flower beetles, wonderful. You must know the example I’m talking about.

*I know the paper.

Well, incredibly small world. So I went to this lecture.

*I took the course and we went through the paper. I took the course from him so I know the work. He had me come to the board and work stuff out.

And you couldn’t be in his class without going to the board. He nailed everybody. He was very careful about nailing everybody with some amount of time, up in the front.

*Right, Socratic method with a vengeance.

So anyway, here he is describing first the statistics he was gathering from the beetle. It was an active form and the larva was an active form, OK. They both moved around. You had to deal with spatial movement and other things like, and pupa and eggs didn’t move out from under you so you’re going to have to deal with them moving around in the system, like wherever they got laid or whatever, they stayed there, they didn’t budge.

So the thing is, here he is working on flower beetle eggs and pupae that don’t move out from under you, they just lay and so on. So then he goes on to fully develop the stochastic model, this single model will lump them together and since these two move (and its and larva) go on from there, to simplify the calculations and statistics in the two cases. So that’s what he did for the, anyway the lecture that I went to, OK? That was the lecture. This is a wonderful statistics, just terribly sophisticated and neat, he’s just garbaged all the ethology in the system. Ethology just ate it because what he had done.

The eggs, may be stationary but they have a death rate, and all sorts of different predation can happen on them and that predation that will happen to them is totally different than what happens with the pupae. There’s a group of hyper parasites that knock pupae out that are totally dependent on them. They can’t live on eggs, the ones that take out eggs can’t live with pupae. Exactly the same situation for larvae and adults.

And also the other thing was that it garbaged time in the model. They’re very careful not to schedule any events. It records the development time, which is very dependent on temperature and things like heat and humidity and things like that. They hatch in circumstances that we can measure-that is all scheduling larvae. Larvae then goes through a series of instars, growing, requiring certain periods of time to the next instar.

And they also have a behavior with parasites that are totally different from the egg. Then you have to schedule time, the ones that survive. You’ve got to schedule the formation of pupae, but the pupae though is exactly the thing I’ve just been describing, and so on. The details of the adult can be laid out with the same ethological approach.

*Right.

So beetle statistics, but as far as I was concerned, it was garbaged over biology, especially the ethology of individuals. Does this make sense to you?

*Yes it does, and it was beautiful statistics and beautiful mathematics and one of the things that he was able to do by ignoring some of the details you just went through, was he was able to use moment generating functions.

Linearity assumptions.

*Yeah, and collapse things down and come up with a nice way to look at multiple individuals using generating functions. When you have independent events you can multiply probabilities which means in the moment generating functions you’re adding things, so it makes really nice mathematics. That was one problem I had with that material was that once you stepped away from those nice assumptions that he worked with, you couldn’t use that mathematics at all. It broke down. You’re stuck.

Am I right that this, in terms of stochastic modeling, is another gotcha that hopefully we’re working our way across?

*That’s right. That’s exactly right.

The stochastic literature is riddled with what I’ve just described. Am I right, does it appear at least at this point to you, you’ll have to cogitate it more I know, but it at least appears to solve it in a more reasonable and biological way and yet we haven’t eaten it mathematically, have we?

*No. The mathematics in this, in some ways is really very simple in your approach, from event to event. The complication is all in the relationship in the scheduling of the events and in figuring out how to simulate it, but the structures is elegant. I’m biased of course.

That was another path I wanted to treat because I’ve been there a number of times in my life already.

10.9

Modelling / Neyman

It was neat stuff, but it was nice that you had the same experience, we even walked through the thing, the paper on flower beetles, so I guess it puts a new dimension on the test for it, huh?

*Yes.

It’s coming to that again. There are so many types — and everything comes all to pieces, busts it up. But at least at the time you don’t see, do you find that also a problem to you that the particular mathematical convenience that Neyman used makes for great mathematics. It makes for neat statistics, and it makes for some interesting applied problems where it’s better than nothing. It had problems along the line I just described.

*Yeah, and it makes progress on a problem that wasn’t solvable before, but it’s kind of sad because you can’t, you can’t generalize it, you have to go to a totally different approach to generalize it. I thought about that a lot and I couldn’t figure out how to generalize it in the time domain. It’s just too complicated and you can’t use something like the approach he did. It was kind of frustrating because he was in his late 70s probably when you talked to him. Let’s see he died in 81 and he was about 87 I think.

I also talked with him, you know, when was it? Well the talk about the stuff we were doing 20 years ago.

*Yes, so that was probably 5 years earlier, so he was probably in his early 80s at that point, late 70s or early 80s, probably his early 80s. Yeah, I worked for him a couple years. I was his RA and his TA, yeah I was a TA in that course where he had the Socratic method.

Yeah, I thought he was a wonderful teacher.

*Oh yes. One of the best teachers. But it was frustrating because I couldn’t see how I could use his methods on the kind of problems that we encountered together or that I encountered working with you, or the kind of problems that I encountered with ecologists that I worked with. Looking at predator/prey systems or plant/insect interaction is actually more what I was working on. I couldn’t see how to use that stuff and I still can’t see how to use it.

Because this is a way of describing why you can’t.

*Yeah, yeah, right.

18.8

This is the part where the order – things to happen, this is the part when – it was the only thing around with numerical – at that time, the.

*Flour beetle.

Yeah, flour beetle it was called, and then – put together active adults with active larvae – because – and the trouble is when you do that, you’ve just blown the – everything has to take place and to evolve right. In garbage everything in time. So that’s the other point. You can’t, you’ve got to, – you’ve got to – open-ended type of environment so real biology, you can actually happen. And you’ve got to be able to tie to some kind of statistics that you can be formal about in some reasonable way also. So that’s the idea of –.

Bellman (?) / Theory vs. Computation

16.1

They always caught it when the people are very famous and very talented. Was it Bellman who was a numerical analyst at Bell Labs that was fantastically good at writing all sorts of very good programs and other things? I think his name was Bellman. Numerical analysis person at Bell Labs.

*I don’t know.

I’m really off the wall but he had just come out with a new book and it had some really neat techniques and some new proofs for some new mathematical. He was a very innovative sort of person, some day I’ll get the right name and the right person. I’ve garbaged his name. But so what I was doing is I was sent to use one of the techniques that he had and it looked proofs, the proof looked great and I ran some numbers through it and it gave me garbage.

Then I forget the name. He was the primary numerical analysis person there at Cal-Berkeley at the time we were doing this work, but I can’t think of his name. So I went to his office and brought him the book and said do you see any problems with this particular proof? I have a problem with it. Can you find any problems with it? So he sat down and rederived the stuff and said no, Bellman’s stuff here is right on, perfect, no problem.

The problem is that Bellman and I got in myself, all made the same slip on a kind of a mathematical kind of error that is easy to make and was wrong, and you only get garbage when you retry, put the real numbers into something. So I said OK, you’ve just proved it by, why don’t you put a few numbers in it? And he said, `I’ll be damned, I’ll be damned. Well, you know, Bellman does this often.’ Bellman – because he’d just been sucked in by the same one, you know, publicly there. Real story.

So if you think the – you had better be real careful of that being really filled with – and it also has the other point. It’s very much, it’s really crazy, the other aspect was that they’re always throwing around the stuff about mathematics is so exact and so precise and so all this good stuff, except they have, it takes real people to do real math. Real people with limits.

*Make real mistakes, yeah.

So it’s a bit like you demanded a programmer go out there and write lines and lines of programming and never test it. Never run it on a real machine. How well would you – formal, equally well defined, equally precise, equally. Would you believe that that would work in – or do we really prefer the guy actually – debugging software and test programs?

*Yeah, it would be nice.

So that’s another whole piece of, another comment about because of the, and there’s always this, they’re always just discuss the – for things that are – approaching heter– and yet when you find out that you –. Sometimes everything is there but it’s an error, so be careful. You should always – be careful.

*Yeah, it’s hard to teach people that though.

You have to get burned a number of times to get it.

2.2 Biology vs. Physics

Model Theory

13.3

+I look back on that, I look back at the time that I spent with Bland up at Berkeley, the times at Giannini, as probably being the most golden, the most productive that I’ve ever had. Not so much because of the papers, but because of the ideas. They were all new. Everything was new, and in a sense we were sort of doomed to immediate failure because they were so new that they were for the most part reasonably unaccepted. If we’d have been Robert May or if we’d have been you know somebody working — Voltaire, we would have probably ended up.

*It would have been published.

+Yeah, we would have published a thousand papers and, but we would have also not done the problem.

*A gotcha. We were talking about gotchas the other day. You get so far with a particular approach and realize that there’s some key assumptions which are untenable.

+Oh yeah.

*You can’t make the measurements, you got some assumption with independents where there isn’t any independents and.

+Well that was another thing that we agonized over a whole lot, again in this paper we talked about. What you needed to get a Poisson distribution, you had to make assumptions that were biologically not what we really wanted to make. But the alternative was no solution to —, so we made assumptions about that.

Bill Hawking(?).

+Yeah.

Sometimes they’d — sometimes by mathematicians, sometimes by biologists, but mathematician and then — way of tightly — the two by doing in a time space and an event space where appropriate. We could do formal math to the next event, the entire event came to a screeching halt. Most of the systems untouched because most of the systems, we don’t have to do anything with it, it’s already been defined and prescheduled, right. But the — the event came — to the reproduction. No 1, 2, 3, or 5. Just like a pregnant woman —.

*7.

7, sometimes, if you take fertility drugs or something —.

21.1

The point I wanted to make was that this general discussion of broad things like this and some of the things – want to do interaction – obviously – some of the things – because some of the stuff is probably less clear why certain things were done –. Now there’s some fill and still – mathematical which is unfortunate – at least a little bit more into some of the first implementing – too. Because that could have been an interaction between a natural – and the mathematical world and the world of – assimilate on the computer in some reasonable way.

Five Parameters

10.1

There was that other piece of the work, where we were looking at a different way of parameterizing the space. I think we had 5 parameters in that parameter space. And they related to things like position and rates and all sorts of stuff like that.

Like gee, 5 parameters to characterize a multidimensional system. Boy, are you even cutting it loose in hopes of making any sense, 5 loose parameters arbitrarily shoved into the way that they’re dealt with in the one non-stationery poison process. So it would yield so you would think that, woah, how can that many loose things in there, just make any sense? You can draw the universe with that amount of degrees of freedom. But the whole idea was to pick parameters that were closely tied to biological meaning. That was the idea. Biological meaning in a behavioral sense of what was happening in the rail.

You’ve got to talk about goals playing on those, and you’ve got certain numbers and events and other things. This tree has got to be born sometime, and it’s going to die some time and has some expected lifetime. But you also have bark beetles and other organisms. They live and die and so on. These parameters have to then be a bridge to the natural biology a person measures anyway and failed in terms of natural things. At least as I just reach back, that was the idea behind them. Does that make sense?

*Yes it does.

And that has the big potential problem that I need to bring up. Is it really there because all those degrees of freedom you have to build in the structure of the problem and the behavior of the animal. Partly structure, partly location, partly temporal stuff, partly hy-age, partly getting born, partly dying. Who, people, animals, they have to get born, they have to live some life they have to develop, they have to reproduce and reproducing they don’t stay around very long, and at some point they die and none lives forever either. And so that’s the idea of having parameters that are tied to things like this. Does that make sense?

*Yeah.

So the free parameters are not free at all. They’re tied to the biology a person measures in their natural experiment hopefully.

10.4

Birth and Death

You see, I remember the process and the things that let you set birth and death, life span, scales of time, scales of space, and things like that. Now usually a reproduction of that then would schedule some kind of sub-event because for most higher animal’s reproduction is just a little bit more complicated than simple cell division so you have to, you need a compiling procedure at that point. For example, it’s a scheduling of the reproduction based on the state of the system and the distributions that you measure and of course that’s a trial of Monte Carlo type simulation and then it, I lost my thread, damn.

*We were talking about, you were talking about birth process and the scheduling of birth process.

Oh, thank you. Birth process then for most animals then you got to figure litter size, sex distribution, maybe even, maybe they’re all one, sometimes you get just one sex in a particular animal at any one time, all female or all males, or whatever. And so that kind of information is crucial to factor in too. But guess what, that biology is directly measurable. This is appropriate to be factoring from ethology or animal behavior anyway.

The whole idea of keeping, treating what happened to plants as an exact analogy to the way that you treat the animal, you see. They have a birth, a growth process, a reproduction process and eventual death. No plants live forever it turns out, and so blue-green algae has a different lifespan than bristlecone pine. And so what has to be developed are the equivalent statistics but again, measurable.

You can measure them, like you get any other statistical data, and also then to tie you in some rigid way to what you were sampling from and you see that was the other thing that was driving them up the tree. We had this big plot of Ponderosa Pine. Here is this beetle interacting, now we’re going to stress it with air pollution, oxidant air pollution, with ozone damage, how do we model this damn plant? We might measure how many trees and what kind of spacing, what they ate. What, you can do aging by boring the tree and so.

And then you follow the actual death. The tree is alive you load the smog levels, all statistical and then you couple them by the actual beetle biology. You find that Dendroctonus doesn’t just get attracted to any tree and it does this only in a certain direction. If it gets stuck in the phloem, it doesn’t even develop because the larva starve for lack of food. At some point the oxidation of pollution probably makes some marks susceptible to beetle damage but at some later stage that it stalls it by starving the larvae for lack of phloem food data, the food, for the parents of larva. The parents are there and then the larvae don’t develop apparently, I remember Dave saying.

Some of the sense that we’re describing here, quite measurable from biology, looking at it as an animal, like applying animal ethology to plants. People who love plants will like that surely, the very thought that plants should behave more like people. People even hug a few occasionally. So, am I totally off the wall there?

*One thing you’re talking about is that you are looking at things at different scales, depending on what aspect you’re looking at. If you’re looking at the bark beetles, you look at them on a scale of days and weeks and years. If you’re looking at the pine tree, you’re going to be looking at a totally different scale, both in terms of space and time. The resolution is totally different, and the span is totally different.

I think when things actually start in real time, — all parameters are there.

*Yeah, I think so.

Anyway, that’s what I had in mind anyway, 20 years ago.

Models Based on Measurements

12.16

+The other thing that we learned and this is something that Bland picked up on earlier is that you can build the best model in the world, OK, it doesn’t make any difference. You can build the absolute best model in the world. If you’ve collected the data wrong to drive the model, you’re dead meat. You’re not going to go anywhere with that, and so from the get-go we were trying to design the database in such a way that it would derive the models that are in this book, in these articles.

+If we were to do it today I think we’d do it in a whole different way, because the tools would be so much different. You have to remember back then, we were archiving data in these little cylinders, that they had this arm that would come out and go chunk, — the cylinder dah dah dah, kachunk. You’d read the data in and then it would go kachunk dah dah dah, kachunk. It was really beautiful. Stupid but beautiful, you know, and archiving data.

*Slow by today’s standards.

+Yeah, archiving data was horrible, you know, where if we had, at that time that particular piece of equipment cost something like $50 million, just to archive data. We’re not doing anything with it, we’re just going kachunk dah dah dah kachunk. That’s all we were doing. Now we could buy a CD Rom Writer for $5,000, hook it up to a PC and we could archive the known universe, literally. We never had that capability back then. We were premature with That kind of modeling. The way that we would do it now is probably very amenable to a network based system with appropriate databases in different locations.

You’re right.

Hilbert Spaces

+I should preface that. We’ve never explored it and I think we’re probably 100 years ahead of time. In that first paper where basically we’re taking a look at projecting out solutions to a biological system from some sort of a Hilbert space, an incomplete Hilbert space. But the idea that we have made a measurement on a system is going to tell us what the solution is going to be because we made the measurement, and if we’d made a different measurement, we may have gotten a different answer. In a sense it’s kind of like, and again Bland and I spent a lot of time eating these French doughnuts over at —, talking about it’s kind of like a Heisenberg uncertainty principle in some ways as applied to biological systems.

+Well that brings in essentially the measurement thing that we have never explored and we really need to go back. We, collective we, really need to go back and take a look at where that’s going to take us at some time in the future.

*Depending on what you choose to measure and how you choose to measure. Examine the way the process works is going to determine what you actually see.

+That’s the difference between us and population biology and the —- equation. They don’t measure anything. They’ve got a population here, they got a population here, they got a coupling constant, they solve a differential equation and guess what, differential equation blows up on them. Why does it blow up on them? Take a look at some of the stuff that Bland was looking at back then. We were taking a look at catastrophe theory and at fractals, and at solutions to NP-complete problems. What we were finding is that the continuous variable techniques that were developed for population biology were totally inappropriate. It’s not surprising that the solution didn’t converge.

*You can’t work with interacting processes.

+We didn’t have to. Now there’s a whole field out there that does nothing but chaos type of stuff. We’re sitting there looking at chaos and we didn’t have any idea what the hell we were looking at. But we knew that the techniques that were being applied by May-the predator play parasite models which you know really are pretty. They make really nice paper, didn’t solve one biological problem that we were aware of.

*Well the stuff that’s going on at Santa Fe doesn’t necessarily. It gets close.

+It gets closer but the point is.

That coin-flipping used for.

*From Feller.

Feller did was — and that, the data, and then the statistical analysis of that, even the expectation is unstable, infinite — much less var— so I had already been looking at chaos theory by way of Feller’s work and I had to factoring that in all the way along, interestingly enough. That was one of the things that so turned me off, the fact that these people are — precisely predict the biological statistics, that they were applying grants for, writing papers on, dah dah dah.

Event-Driven Paradigm

10.4

*One of the things that really caught my attention was switching from thinking about time to thinking about events and scheduling events. The whole concept involves scheduled events. Between right now and the next scheduled event you have a Poisson process. At that next event there may no longer be a Poisson process and it may be totally different. Things may go on at that event.

But then you don’t go back.

*Right. Then you reschedule things and take care of the bookkeeping and then you’ve got a Poisson process again. That’s brilliant.

Of course that’s the core of the whole thing that makes it work.

*That’s right.

And if you don’t do it jointly, you lose all the biological reality.

12.18

+I think one of the things that Bland did which is really important from a philosophical standpoint, from a philosophical and mathematical standpoint, is the simple fact that when we started to do this work it was really clear that we needed a major conceptual change. How do I relate that major conceptual change? Probably the best way to relate it would be that Albert Einstein in 1905 made the conjecture that time and space were related and that is the basis of special relativity.

+Bland — the completion that the measurement and the state of the given small population of animals to relate it and you couldn’t diverge the two of them, that you had to work within that arena to solve a problem. If you didn’t, you may be solving a problem but it had nothing to do with the real world.

*Yeah, so you move from a time-driven paradigm to an event-driven paradigm.

+We moved to an event-driven paradigm for a number of reasons. One of them being that if we had done the time-driven solution-I mean it was like to the three-hundred thousands, and if you multiply that times the amount of compute power that we had, we’d still be doing the first time step. Just for ten individuals. So we moved to an event-driven domain for the very simple reason that it got rid of the time variable.

*It also gets rid of the interacting process structure which kills you from an analytical point of view in the time domain.

+But if you think about what it does, it projects out the solution. You literally project out the solution that you’re going to be working on.

*That’s beautiful.

+One of the things that Bland did was the chess and backgammon games. Basically we used those 3 particular analogies to define how a biological system worked. Even if the system was completely deterministic like chess, something worse than an NP-complete problem, then you’re dead meat.

*So the — does pretty well but, yes, having to project so many pathways.

13.3

There had to be real biology in some way — for measurement in a very direct —. You exercise all the complex — and then all the link — they were tied to —. And guess what? You’re right back to a purely defined currency — stationery — to the next event and parallel with 10s or 100s of thousands, even, you could even carry 10 a thousand — incredibly enough. So that was the driving force behind everything else.

+That was our key. That was our key.

*That’s a brilliant, that’s one of the brilliant points.

+That was our key. That was the key to the whole thing that we did, — Poisson distribution and the idea of independent events. We could treat each individual as essentially an individual until he interacted. And that interaction was basically defined in the event space. In fact, I remember going through that. We looked for the minimum of the minimum of the minimum of all events and the next sucker that got it, got it. If he died, he was erased from the rest of them and the whole event structure changed at that particular point.

+But we weren’t tracking individual times for each one of the persons out there. When I can remember going through discussions. ``Bland, is he doing anything yet? No. Let’s go to the next time step. Is he doing anything yet? No, let’s go to the next time step.’ And in the normal solution for time-based systems, that’s what you would be doing. Most of the time you would be just iterating until the next event occurred.

+Think of the distinction of basically what was being done by Robert May and groups with basically the population biology sort of system. Everything was a continuous event, so things were changing based upon a set of continuous variables that may or may not be semi-continuous. There were spikes that we were allowing and things like that but the point is is that’s not how the biological system works.

+The best example that Bland ever gave me was the rabbit and the fox example. In a physical system, if the fox moves toward the rabbit, the rabbit takes one step back. If the fox takes one more step to the rabbit, the rabbit takes another step back. That’s what it would be in the physical system. Some continuous function would describe the relationship between this fox moving and this guy moving, and it would be a continuous function.

*And linear.

+Yeah, linear. What normally happens is the fox goes tick, tick, tick. The rabbit is seeing him but then there’s this zone where the rabbit says I’m out of here. Boom. He doesn’t run two steps. He runs to Fresno. That example tells you that you’re in an event-driven system. You know the fox is not going doink, doink, doink, has anything happened? No the fox is moving, but in terms of the interaction biology part of it there is no interaction at this particular point.

*We’re not modeling the interactions happening, cuz there’s perceptions going on, but we’re not modeling the perceptions, we’re modeling the movement.

+No, but at that particular point where the fox gets to the point where the rabbit says OK, I’m out of here, we’ve had an event. We have not had a continuous function. We’ve had an event and from that particular point we have a new problem, so what we were looking at.

Span and Resolution

12.16

+The other thing that was really key, and I’ve used this the rest of my life because Bland and I have had some amazing discussions on this, was the concept of span and resolution. Before you do the problem, you ought to know what problem you’re doing. If you plan on doing the problem from the ameba to the gorilla, you got a problem. You have to limit the time and space that you’re looking at. In this third paper there when we basically started to take a look at the Blodgett data and the distribution of fees and draw — functions.

+That was the first attempt to really defining a problem that was amenable. You had stochastic events coming in at one end, because your resolution was not short enough, and you had constant problems coming in at the other end which changed the dynamics very, very slowly. Like weather conditions, or maybe climate conditions, something like that. Something that had a 20-30 year time frame, or 300-year time frame. If you take a look at most of the work that is currently being done in almost any field, the first thing they forget to define is this time, the resolution and span of the problem. It’s like the two words never got invented and we were looking at that 20 years ago.

*25 years ago.

18.6

Then the other thing is it’s about that time you start looking at this thing a little more closely and I’ll be damned all those things out there – you’re now finding – systems – on bunch of truncated –. When you notch it down one thing and – revolution. So – here pops out another kind of structure that’s really important to deal with and if you just randomly assign things across the entire – you have a mind-blowingly big – there.

Also I just very, very, very faintly remember. I used to go – model theory too from time to time, to go find out what made sense in terms of this or that or other things in mathematics, all kinds of things – doing in the world. There was this one, what do you call this thing? – problems – pathways or all possible pathways, a set of all possible pathways over all possible events, or all possible ways to think –. It’s – problems –. What do you call them? They’re – fact of life. You know you can do them for 2 or 3 or 4 and then you – go to a hundred and the other part – and the number is so large there’s no way of ever presenting them.

*Common torque problems? NP hard problems? [Not solvable in polynomial time-they take exponential time in the number of terms.]

Yeah, sort of like an NP problem, things like that. That class of problem. So, that’s the other thing – because – immediately this would be – into a problem that is so uncomputable it just blows your mind.

Non-homogeneous Poisson Process

And so part of this whole trip was to – let’s take a look – non-homogeneous Poisson processes. Have I got the right thing?

*Yeah.

I can’t believe it. I really can’t – non-homogeneous Poisson process – the woman that was head of the whole program under – and said was invalid. In spite of the fact.

*Christine Shoemaker?

Yeah. In spite of the fact that we had all the detailed proof in the papers that we gave her. So life goes. But at any rate, the idea there being that you could have done – intervening by taking a minimum – find that the minimum – in computationally pretty and then some of the heterogeneous algorithms — graduate students to work out just on mind-bogglingly – using priority cues. So you have a very small group – biology could make more sense to process it as – information, go at it from the direction – you process without getting – say in the thousand range or the two thousand range, you –. Because it grew exponentially or something absurd like that one, or something close to that. Exponential plus some small factor.

So that was the idea but then you would simply – linear math, with improved linear system – because it was a non-homogeneous process it meant that you could feed into it – data from other statistics, the other – completely distribution free. So all the time usually had enough of tying everything together that was – sensitive, you could actually run on the system in terms of efficiency. It was – cost of all the intermediate events and even if you had a huge problem – rather than using –, use priority queues –. So then you – guess what you have to do? – of millimeters – by using just –.

18.7

But it’s an – the consultation and – but at the same time you’ve got to – what things you got to, you’re going to end up eventually being able to have to put it into the form of a linear system, what do you know, sooner or later – generally. So they would come back to linear systems. And then you would have to – is no longer a linear system. It’s been – reality. So – going to have so many young, it could have zero young, or one young, two young, probably won’t have a hundred young. – it does that, – a hundred young. How many beetles will have a hundred young? So you see how crucial it is to start factoring things into real biology, real reality at this point, to make things work right.

This is the part where the order – things to happen, this is the part when – it was the only thing around with numerical –.

Physics

13.3

+You see that the problem is that Feynman had it easy. Feynman really had it very, very easy. When he wrote down the Feynman path integral, as you know, he basically integrated over all possible paths that a particular particle could travel. He essentially used the classical action and basically through interference processes everything sort of dropped out. He got the classical paths in the extreme.

+We don’t get that. We got 10 individuals out there, all of whom are interacting, none linearly, in time-delayed systems. We’re screwed, so we can’t use the standard techniques that even Feynman used. We need to have essentially the event-driven system that we came up with. The problem, and the cornerstone of the second paper, is the fact that we basically merged biological information with mathematical technique, showed the limits that the mathematical techniques could take us to, and said from this particular point it has to be calculational technique.

+If you take a look at the solution of the four-color problem, it’s the exact same thing. They took a look at the four-color problem which is a classic problem in mathematics, they found a mathematical solution which doesn’t work but ultimately the problem was solved on a computer and what Bland was proposing was that biological problems are not amenable to analytic solutions.

12.22

+Again, to go back to the only thing I really know and that’s physics. If you take a look at what’s happened in physics, you see the same sort of thing about the time that Einstein was doing his stuff. You saw these various ideas of explanations of how to handle systems that essentially went relativistic on you, with the speed of light. What you found was that everybody had a solution for it. But in each case that they found a solution, they also found something that wasn’t a solution.

+Then along comes essentially Lorentz. He says ``OK, I don’t know what the hell is going on here but I’m going to invent a thing called the Lorentz transformation,’ which Bland is very, very familiar with. And guess what? All of a sudden everything that had to do with Maxwell worked and everything that had to do with classical mechanics worked, but he had no idea why. All he knew is that it worked. It took somebody like Einstein to come along and say this is what you did. We are going to relate time and space together and everything falls out.

Population Ethology

+Bland has said you cannot divorce the measurement technique from the answer that you get. If you do, you get population biology which is by definition the wrong answer except in those wonderful cases where it just kind of happened to work out. Or you get something where you can’t do anything. Basically what you’re faced with then is doing the Charles Darwin routine of collecting a lot of data and hoping this thing works.

+Bland gave an intermediate point. You can take a look at small populations. This is what a biologist looks at, not 10 to the 23rd animals, but you can look at 20 animals, 100 animals. You can put collars on things. You now have a technique developed where you can basically start some sophisticated analysis, not just doing means and standard deviations.

-What Bland puts in this paper is also individual, how the individual in the group relates to all of it and without that individual’s own — characteristics.

+We define the individual —.

— applying this to like — what is community and why people exist in a community by looking at this paper so that’s pretty far out but I found it fascinating.

*Interacting individuals and each individual has its own uniqueness.

-Because I just studied this same.

+You remember? You remember when we drew up the list of what an individual was?

-Yeah —.

It’s right there at the bottom.

+We spent something like 3 months over that stupid list of what an individual was.

-But that is a really important list. I wish I would have had that list just about 2 weeks ago, I’d put it in my paper. Because what I was studying was the difference of law in China, China’s law base and — law base. One’s based on the individual, one’s based on the group, it’s two different basic concepts and it’s right here in a whole different way.

+We’re so cool.

-I could have plugged all that stuff in.

I’m totally cool.

-Yeah, you’re cool Bland. That’s right. It’s fascinating.

We are a group, and yet we’re four interacting individuals with a totally different mix of skills and personalities.

-That’s for sure.

How well does that match your, the idea that — the entire population ethology —. Ethology is the study of individual behavior at the individual level — the uniqueness of animals interacting with people on a one-by-one basis — each individual is unique in his own way.

-And there’s the dichotomy.

*Yeah.

— that first — how many — out there — deterministic — so I — the entire world but it’s also very true — with all living organisms, all with living systems.

13.1

+A lot of what we have gone through has in some sense just been taking the fundamental ideas that we’d developed here and then sitting there doing the Dutch boy and the dyke routine. We plug this hole here and then we plug this hole here because we didn’t have the techniques available to do it. The philosophical and theoretical stuff I think for the most part is reasonably complete here.

It’s from a biologist’s perspective, Bland Ewing’s perspective. It is not from the perspective of the mathematician, though Bland is probably a better mathematician even now than most of the people that work in the field of mathematics. I just think that we’re getting to the point, probably within the next 5 to 10 years, that this kind of a technique is going to become one of the most valuable techniques out there.

*See the explosion that’s happening with Markov chain Monte Carlo that’s moving around, which is a lot of the same aspects of what we were doing.

+Of course all of what I said has to meet with Bland’s approval.

No.

Modeling on a Budget

11.8

Yeah, sounds great to me. It was interesting to have talked with Jim about some of the issues. he’s also – from a conversational point of view, a computer point of view. That’s the other thing that eventually depends on — at a very low level, probably — assembly language or — machine where somebody’s going to be using —.

Because you just can’t take much overhead, you can get debugged — but you’ve already seen the problems, at very high level languages and trying to get any efficiency out of it. And the thing is that it’s not your job — to do that — and not really Jim’s job either or anybody, or mine not even, it’s, we’re going to have to find places and pieces and things —. Actually the — thing was laid up and all the pieces are there and.

I could even use this machine right here — have such a problem with trying to type or getting, like buying — overtime. There’s a lot of work that I can’t do —.

*Yeah, well you’ve got the ideas and that’s the key at this point. It sounds like Jim wants to take a crack at it, the computing. I assume he’s got some support staff that he can use to help out. I mentioned earlier that I have an ecologist friend who has a student who wants to work with me. I’ve given him a copy of that document to look at. I gave it to him about 2 weeks ago. Actually I gave it to him just before I left on this trip and so I’m curious what his reaction will be about this stuff.

Building a Model

14.3

– how that could be implemented. But the thing is that that, with Don Dahlsten’s data, and also some of Bob Luck’s data, you see they were both also involved with generating numbers of counts, where counts were really the basic data that you had to build everything else from. It was a quantitative anchor into the world.

*Yeah. Well Bob Luck and Jim Barbieri have been talking and making some plans to do some modeling, particularly on the scale on the oranges. Orange scale.

It would certainly seem to me that would be another thing they would have probably would be, if you can do it for the bark beetle, then that should just be a small thing to –. The bark beetle wouldn’t work right for his system. OK, so the other question I have is that OK, let me, trying to pull a number of things. Things, what I want to pull out is – but there’s also.

*Gamma?

Gamma, thank you. Gamma function, and there were also some other general kinds of – that people go to – over as I remember. Everybody has their own favorite linear distribution for numbers. There used to be a blood issue a long time ago.

*Yeah, I think it still is for some people. I think the computing power has.

Made it totally silly.

*Yeah.

Running Model

16.4

I actually got it up and running later on, showed that to Jim. It was also a time that the evolvement – trained to do the first things, again plots of this stuff over things like both their own data and also things like the, what was that? The moose wolf predation thing that we were using as an example in the first place. But that part, yeah, part of it got out what needed to be done that never got done. Where the parts I did get done were some of the summary – shaped curve and manipulating error and so that – generating hand numbers for people like – really wanted to work with. People like my major professor Justice wanted to work with too.

Dave Baasch

17.4

Basically that’s why I got started using the CDCs on campus, and that’s really where I got started doing the modeling with Bland. We started working on new models that weren’t just regression analysis models, and really working on stochastic modeling. I liked it because it made sense.

With this we can build a model that says this is how we think the system works. Now let’s see if the model works. If the model doesn’t work, that means we don’t understand what’s going on. So we need to change the model to reflect our learning and our new understanding and see how it correlates to the real world. The whole idea behind this, to me, was that the models were a test or a verification of our understanding of the system as opposed to well, I can measure 2200 things and come up with a prediction based on who knows what. This is the correlation.

There was some interesting stuff going on with taxonomy at the time. Numerical taxonomy was the big thing where you don’t weight characteristics. You just measure as many characteristics as you can. Give them all an equal weight and then compare them to get a measure of differences. Well, in one sense, if you can measure enough things it probably matters. But people were measuring 20 things, such as has feathers or doesn’t have feathers, and that’s a significant characteristic. It’s probably more significant than has red feather shafts or has yellow feather shafts. People would just choose the things they could measure.

I had a problem with this kind of correlative modeling, I guess. The stuff Bland was doing and the stuff he was looking at with stochastic modeling and stochastic processes made more sense. Everything I know about stochastic statistics and modeling was driven by Bland and the modeling he was doing. I liked it because it made intellectual sense in terms of trying to describe the system.

He was only using it for small populations and he wasn’t trying to use it to model the Pacific Ocean. He was using it in a relevant manner that intellectually appealed to me. The idea behind the models and the reason for using them, and the reason for doing them that way, felt intellectually clean to me. It made sense and it was really different than what everyone else was doing.

Now he went off on a lot of tangents. He did in terms of data representation. I mean he was always amazing me with what he was trying to do. I had never seen stereo photography before. He used stereo glasses to look at aerial imaging and played with that.

Small World Networks

20.5

So all these things going in parallel(?). But also at the same time, this thing of fractal stuff, not only that but – as though the theory of nearest neighbor and other things like that. – run a lot of things but there’s the type of theory that – like to do model theory and things like that. Some of the far out mathematicians – proof for lunch.

And for people like – I guess the whole thing where you – nearest neighbor thing and it turns out that the thing about the fact, that all you needed is one strange linkage into a system and it will totally shortcut distances, interconnecting ports. So – all these strange linkage, compared to the rest of the world out there. Or it’s the old hairdresser type thing where the hairdresser seems to be the center of gossip and – cross-connects stuff.

*Small world networks.

Yeah, small networks. Beautiful. And it turns out that if you have a totally random system, it doesn’t do zip if you have a totally, even – not do that, but if you have a system that had all the – and bits and pieces, and – connection, it makes it a very small world. So it’s going to be interesting to see how some of the – kick this stuff around in the future too.

*Well, that’s being used for comparison of protein sequences, DNA.

Oh, I didn’t realize that. So you have a practical application, I’ll be damned –.

*Yeah. Also with fractals, I’ve heard a couple thoughts of using fractals to model internet traffic.

Yeah, and I bet it’s going to do an incredibly better job.

*Yeah.

Especially if you’re go – any type of selecting process –one in human brain, where people have tried to emulate that and it’s been a total disaster in terms of any real quality of the work, you know, even Jim Barbieri – some stuff. I looked at the number of mathematicians. Boy they like to throw a lot of curves at –.

On the other hand, he showed me a book by the guy down there at Caltech.

*Carver Mead.

Carver Mead, thank you, Carver Mead. Sometimes I can grab his name, sometimes I can’t.

*Yeah, it’s funny. That’s one name you have a lot of trouble with.

Yeah, Carver Mead. Anyway. So the theory – his work and some of the things that that was – and so they, I guess they found some interrelation of stuff, but the main thing – what the system really needed was a completely nonlinear – stochastic for some problem, and certainly deterministic – nonlinear systems to chaos. Chaos, build a system on – selection, then based on, then my guess was that they – systems that actually worked – so that was when I pretty much lost Jim on that kind of stuff. – I’ll be damned – chaos again –. Chaos and practice –.

*Yeah, well sometimes, yeah, chaos and complexity. I’d like to go down to the Santa Fe Institute some time. That would be, you’ve heard about the Santa Fe Institute?

No, I haven’t.

*Uri Gelman set that up and he got a bunch of people together, including Metropolis who was one of the original Monte Carlo people, and they studied, Holland, a guy named Holland. Anyway a bunch of people who, he brought them together to study chaos and complexity and they got together I think it was about ten years ago or less than that. I don’t, maybe about ten years ago, and they have been meeting ever since and they have conferences and it continues to be an interesting hot bed of ideas in this area.

This is interesting also because Monte Carlo was one of the key pieces of making his modeling technique work.

*Right.

There were about a dozen – and he keeps – Jim Barbieri. There was a stack of at least a dozen fundamental pieces, any one of them was reasonably simple and standard within its own field and – something a little off the wall, where the second – type thing. Like – using the simple Monte Carlo, I mean simplest thing we are using the generalize – processing center –.

*Right.

– rather than just a simple – generalized integral type form of it. Is that correct?

*Yeah. Sounds right.

– that wasn’t just in the first form but still it was something very straightforward. It wasn’t very off the wall either. But then it took a – all put together at the same time to make a modeling system work. Of course, you were one of the first people that took some of the – math and then of course Jim Barbieri was the first person to try to program some – and also – writing disability. He – if I had that – writing disability – and no matter which technique or any, I had a writing dyslexia that just, I’ve always had to find somebody else to help me do the writing because I never could. At any rate, so it based these on Jim. He takes after – going after that meeting, where was it?

Follow Up

20.9

*Yeah, I think that concept of modeling to the precision of your system is really, I don’t think that’s well understood. I don’t think, you know, if you have, if you can’t measure things very well or if it doesn’t even matter, I mean if there’s enough.

One or the other or both.

*Right, if there’s slop in the system, you know, so that, I mean the environmental system or animals or plants can adapt to it a 10% or a 50% slop, then maybe that’s the way it should be modeled to.

Otherwise you end up with a – that has no relationship to the system you should be modeling.

*Yeah, right.

To say nothing, it may be a little harder problem than you really need to have. The problem many of the existence – logarithmic – of an order of magnitude –. And a high precision system might, you might have some that can say you really could measure, it’s a heavy duty, whether – system but 10% or something like that. But I just couldn’t make it go much beyond that.

20.10

I’m rather curious though about one thing I would like to – the question. Have you seen any glitches in the system? Any problems with the system – into and so forth? Or does it all seem to hang together still after all these years and decades?

*It all seems to hang together and I think there are people doing something approaching aspects but I’m not aware that anyone has really picked up the depth and the subtleties of what you developed 25 years ago. I haven’t seen that and I don’t.

2.3 Predator-Prey Models

We now examine why classical differential equation models do not work well for biological systems. The global properties and dynamics of biological systems are typically described by a system of coupled, first order nonlinear differential equations (Nicholson and Bailey 1935; Holling 1961; Watt 1962; Leslie et al. 1970). However, they impose unacceptable restrictions such as piecewise continuity that are problematic with small populations. Further, they cannot readily incorporate individual behavior.

Consider a predator-prey relationship (Nicholson and Bailey 1935) between two interacting species, which can be written as a pair of coupled ordinary differential equations, \[ \frac{dH(t)}{dt}=H(t)[a+\alpha P(t)] \mbox{ and } \frac{dP(t)}{dt}=P(t)[-b+\beta H(t)] ~, \] with $H(t)$ and $P(t)$ the prey and predator populations, respectively, at time $t$. The prey birth ($a$) and predator death ($b$) rates and coupling parameters $(\alpha,\beta)$ may be constants or complicated functions of time and/or environmental factors. Recasting time trajectories into phase space (Strang 1986), \[ \frac{dP}{dH}=\frac{dP/dt}{dH/dt} =\frac{\beta HP-bP}{aH-\alpha HP} \mbox{ or } dP\left(\frac{a}{P}-\alpha\right)=dH\left(\beta-\frac{b}{H}\right) ~, \] shows how these predator-prey equations describe elliptical orbits in the $H-P$ phase plane. Avoiding degenerate cases, $H = b/\beta$ or $P = a/\alpha$, and integrating gives $a\log(P)-\alpha P = bH-\beta \log(H)$, a family of ellipses (Figure 1) that are cyclic, stable, closed curves, depending strongly on initial conditions. These equations lack spatial heterogeneity, and temporal changes must be explicitly modeled. The rate parameters can seldom be directly measured but instead are indirectly based on summaries from disparate field data, perhaps from several studies. The continuity requirement restricts use to large population sizes.

<<Figure 1 about here>>

Quadratic terms in predator-prey equations model internal competition, or density dependence, \[ \frac{dH(t)}{dt}=H(t)[a(1-H(t))+\alpha P(t)] \mbox{ and } \frac{dP(t)}{dt}=P(t)[-b(1-P(t))+\beta H(t)] ~, \] which can lead to chaotic behavior or stable limit cycles, depending on the rate parameters (Strang 1986). That is, a modest change in parameters could have profound effects, which would be appropriate only in a biological system pushed to its extreme. Other trajectories are possible with more complicated equations or parameters, including strange attractors with multiple attractor points and inherently complex phase space trajectories (Ruelle 1987).

No deterministic mathematical model can hope to describe the population dynamics in terms of individuals exhibiting the properties described in the previous section. It is intractable to describe all important variables and interactions that determine dynamics in a biological system. It is only in the limit of populations of extremely high numbers that the deterministic approximation is suitable. May (1971, 1973ab) investigated the mathematical structure of such models, in particular the stability conditions for a set of coupled, nonlinear differential equations that simulate the properties of interacting populations. In general, any model that tries to handle many interacting species is unlikely to be stable (May 1973a). Henson et al. (2001) extended this argument to discrete population counts of hosts and predators, showing chaotic behavior that blends continuous and discrete state models.

2.4 Classical Differential Equation Models

Consider a community comprised of a number of interacting species. The global properties and dynamics of this community can be described by a system of coupled, first order nonlinear differential equations. The dynamics of this multi-species community is given by $m$ equations (May, 1973a) \[ \frac{dN_i(t)}{dt} = f_i(N_1(t),N_2(t),\cdots,N_M(t)) \] where $F_i$ is the growth rate of the $i$th species at time $t$. $F_i$ is usually nonlinear and represents all of the relevant interactions that affect the population $N_i(t)$. The assertion that this adequately describes the dynamics of a given community forces certain restriction onto the biological system. For instance, this describes a system that changes piecewise continuously, which is problematic at low population numbers.

Other trajectories are possible with more complicate equations or parameters. Ruelle (1987) details four general types: (1) stable equilibrium solutions, (2) stable limit cycles, (3) strange attractors and (4) chaotic or unstable solutions. Strange attractor solutions have multiple attractor points, and the phase space trajectories are inherently complex. Classic strange attractors include the Henon and Lorenz attractors. Simply allowing a quadratic term can be potentially chaotic. Consider for a moment the quadratic system given by Strang (1986), \[ u_{n+1}=\gamma u_n(1+u_n)~. \] That is, the solution for $u_{n+1}$ is dependent on previous values of both $u$ and $u^2$ and the coefficient $\gamma$. When $\gamma$ = 3.45, the 2-cycle becomes unstable, but it is otherwise stable between 0 and 5. In the context of describing interacting populations, this is a desperate situation since any seemingly modest change in the parameters would force the solution into chaos.

In addition, there is also an implicit assertion that these rates have at least quasi-global properties, that is, they represent processes defined over the entire population. From the perspective of a field biologist however, individuals drive the dynamics of a population and predation, for example, is not realistically represented as a global property. This does not mean that global properties are not applicable or necessary. Global meteorological conditions might affect a population’s ability to function, and can be imbedded throughout the model in any number of different layers.

2.4.1 Early Work: The Holling Predation Studies

Holling (1966) analyzed the functional response of the praying mantis (Hierodula crossa, Giglio-Tos.) to a population of houseflies, Musca domestica, finding that hunting behavior is intrinsically probabilistic in nature. Holling decomposed the mantid predation process into three basic components:

Rate of successful search, depending on (a) reactive distance of the predator for prey, movement speed of the (b) predator and (c) prey, and (d) capture success (proportion of prey successfully attacked).
Time prey are exposed to predator, depending on activities (a) not related and (b) related to prey feeding.
Time spent in handling each prey, including (a) pursuing and subduing, (b) eating, and (c) digesting afterward.

From Holling’s analysis, it is apparent that the predator-prey interaction is a sequence of events involving individuals, not densities. Secondly, for the interactions to occur, certain conditions must be satisfied which are dependent on the morphological and ethological characteristics of both the predator and the prey. The rate functions described by Holling should properly be interpreted as probabilities; e.g. the probability of a successful search is dependent on a given prey’s availability, the predator’s state, etc. From the standpoint of an individual, the predator-prey interaction is an event whose occurrence is dependent on a large number of conditional probabilities that are in turn dependent upon the states of both a given predator and a given prey.

Holling’s work implies that interaction is an event that is inherently probabilistic and uniquely dependent upon the individuals involved. How can one develop a method of transforming inherently discontinuous, heterogeneous, probabilistic data into piecewise continuous rate functions for the predator-prey equations? The biological information available appears to be incompatible with this brand of mathematical formalism for analyzing structure in population biology.

How can we find a more suitable mathematical formalism in which to express the structure of a biological community? Ewing (Ewing et al. 1974; Ewing et al. 2001) defined a system that is inherently probabilistic in which the structure of the biological system determines a suitable mathematical framework for research. In the development of this approach certain properties of the biological system became apparent when viewed at the level of resolution of the individual. We illustrate this in the next section.

2.4.2 Lotka-Volterra, Leslie matrices

2.4.3 Time and phase space

2.4.4 Stable cycles and chaos

2.4.5 Quadratic and Linear Equations in Optimization Research

When using a MR/C (multiple regression/multiple regression) approach to optimization research, we prefer linear equations rather than quadratic equations. This does not exclude use of quadratic equations in the MR/C approach; however, we view a linear equation that provides a satisfactory solution (i.e., very high predictive value and compliance with norms for several residual diagnostics tests) to be more useful and reliable than a comparable quadratic equation. The primary rationale for our preference for linear solutions follows:

2.4.6 Limitations

2.5 Life tables and competing risks

2.6 Synchronized generations

2.7 Time Driven Simulation Models

Faster and faster computers have driven efforts to model populations of interacting species using global population characteristics such as birth, death and migration rates. Stochastic dynamical systems have incorporated individual-based behavior (Mangel and Clark 1988; Wolff 1994; Ruxton 1996; Broom and Ruxton 1998; Ruxton and Saravia 1998; Wilson 1998; Wiegand et al. 1998; Gronenwold and Sonnenschein 1998; Hutchinson and McNamara 2000).

These simulations are built with equilibrium in mind. There is, however, reason to believe that biological systems resolved to individuals are far from equilbrium (Prignogine 19xx), capable of self-organizing into unexpected new states. Kauffman (1993) suggests that a basic condition for an organism to be considered living is its existence as a continual non-equilibrium process. Only non-equilibrium, self-organizing systems have a real possibility of extinction (Paczuski and Bak 1999; Paczuski et al. 1995).

Unfortunately, these simulations typically divide time and space into discrete `quanta’ of equal size. Even recent successful simulations have been severely limited in population size and in temporal and spatial resolution relative to the span to compensate for the vast number of computations. Discrete time introduces probabilistic artifacts such as periodicity and synchronicity that do not disappear as time steps get smaller. Similarly, discrete spatial patterns artificially constrict the type of interactions that can be modeled.

Suppose it were possible to define a finite biological system at the individual level. Even for a small population, the number of possible solutions becomes operationally intractable rather rapidly. Standard analytical techniques that do not exploit system structure have many states to investigate, with most of them empty. The system is sparse, and na?ve simulations based on space and time grids are quite inefficient. The magnitude of this problem is illustrated below. This discussion ignores unequal time increments between successive events for individuals, which introduce asynchrony into the system. It further ignores environmental factors that might affect some or all individuals in a community.

Suppose a reduced and aggregated state description of an organism has n variables and that each variable has m states for mn possible individual states. Since this state-specific description has limited resolution, the organism can alter slightly without changing state. Thus at each time step it can move to mn distinct states, including no change. Any particular organism might associate with any other organisms in the population. Thus the complete state-space representation for a population of r organisms is an array of dimension N=nxr with M=mN=mnr possible states. Thus for the population there are Mk=mnrk possible k-step trajectories, but only r trajectories actually selected, one per organism. Organisms with complete memory could select the kth state transition, based on their previous (k-1)-step trajectories, requiring a full Mk matrix of probabilities. At the other extreme, organisms with no memory would require only M2 probabilities at each step. In practice, organisms have partial memory, falling between these extremes. Time-driven models typically use memoryless calculations, thus have kM2=km2nr computations in k steps. While some economies are possible using sparse matrix techniques, calculations are still done for all r individuals at every step.

Rather than stepping through time, checking every individual at each point, event-driven quantitative population ethology steps through events. We only follow a few events per individual, depending on their memory capabilities. Calculations are reduced dramatically. Identifying the individual with the next future event is a log(r) calculation using priority queues (Knuth 19xx). That individual has an n-dimensional state description with up to mn-1 possible future events, although typically only a few are relevant given the current state. Thus the number of calculations need for k events per individual in a memoryless system is kmn(mn-1)rlog(r). Note in particular that our computational complexity is almost linear in the numbers of individuals and events. Partial memory can be added selectively to individuals or to types of events without dramatically increasing this number.

As a modest example of the high dimensionality, consider a simulation in which r=1,000 individuals are followed, say 200 per generation over 5 generations. Suppose each individual has an average of n=10 attributes. The dimensionality of the system is N=n?r=10?(5?200)=10,000. If on average each variable has m=10 states, then a time-driven simulation must handle a state space of M=mN=1010,000. If the model is run over k=1,000 time steps, the number of computations is kM2=1020,003. Taking advantage of sparse possible trajectories, mn=1010 might be reduced to m=10, yielding 102,003. For time-driven models, which is obviously far beyond the capacity of any computer system, now or in the foreseeable future. For event-driven models, the number of computations 1026log(10), or 108log(10) taking advantage of sparse trajectories, which is computationally feasible. Even so, this is unfairly weighted toward time-step models, since not every time step is likely to yield an event for any individual. Time-driven computations can be simplified further, of course, by clever updating schemes. However, time-driven computations are still exponential in the number of individuals.

2.8 Population size: continuous or discrete

2.8.1 Time almost always one unit

2.8.2 Stochastic dynamic systems

2.8.3 Discrete time stochastic Petri nets

We were not previously aware of Petri nets and appreciate learning about them. We have carefully read extensive literature on Petri nets, finding only two articles with biological applications: Gronewold and Sonnenschein (1998) and Genrich, K|ffner and Voss (2000). Both use discrete-time Petri nets, analogous to stochastic dynamic programming. We believe our approach has close connections to continuous-time stochastic Petri nets (see references in paper), but our emphasis is different. Three distinctions are important: (a) the goal in many Petri nets applications to date is to model a system that requires some synchronization (e.g. network packets to be reconstructed as a coherent message), while biological systems are inherently asynchronous, albeit driven in part by diurnal and seasonal cycles; (b) path choices are viewed as problematic conflicts to be resolved rather than fundamental competing risks central to the modelling effort; (c) Petri net analysis tools aimed at global properties do not appear capable of uncovering simulation properties when mean value functions M are nonlinear. Distinction (a) is crucial, leading for instance to unhelpful complexity in the Petri net implementation of Gronewold and Sonnenschein (1998) for movement of larva among trees. Our approach would lead to a more natural migration process, with probabilities based on distances between trees.

2.9 Event-driven Petri nets

Our approach has close connections to continuous time stochastic Petri nets (Ajmone Marsan et al. 1995; Lindemann 1998) that are event driven. However, we believe our perspective offers a much simpler way to develop ecological models, focusing on the next scheduled event and the local structure of competing risks for event transitions. It is helpful to draw connections to Petri nets (cf. Ajmone Marsan et al. 1995; Lindemann 1998), which put equal emphasis on states (places) and events (transitions) connected by arcs.