# Evolution as Feedback?

Last week I went to a colloquium given by Douglas Robertson of the University of Colorado. Professor Robertson began with two observations:

Changes in fitness functions can cause changes in the distributions of phenotypes.

Changes in the distribution of phenotypes can cause changes in fitness functions.

Biologists, according to Professor Robertson, agree with the statements but yawn. Electrical engineers, by contrast, immediately recognize the possibility for positive feedback and announce, “That population is toast.” I am not an electrical engineer, but I am a fellow traveler, and Professor Robertson’s work, um, resonated with me.

For the uninitiated, positive feedback is what you get when the lecturer wanders too close to the loudspeaker, and the microphone picks up sounds from the loudspeaker. As the sounds are amplified and repeatedly fed back into the loudspeaker, you hear a loud shriek. Even in a quiet room, if the gain of the amplifier is high enough, a very small fluctuation in the amplifier voltage can set a positive-feedback loop into action. The electrical engineers are suggesting that something similar may happen to a species, and runaway amplification of one or more features of the phenotype will lead the species to decreased fitness and extinction.

With his colleague, Michael Grant, Professor Robertson has developed a simple mathematical model, which you can see animated here, http://cires.colorado.edu/~doug/extinct/ . The model includes a fitness function (a graph of fitness as a function of some feature such as size) and certain assumptions about the population. One of the more interesting simulations concerns a broad fitness function with a secondary spike on the high side of the peak.

To explain the secondary spike, Professor Robertson notes that the optimum height of a giraffe in isolation might be, say, 4 m. But in the presence of other giraffes, maybe there is an advantage to being 4.5 m tall, so you can get at leaves that other giraffes cannot. If that advantage is enough, then it can overcome the fact the 4.5-m giraffe has otherwise lower fitness than the rest of the herd. The simulation shows that the average height of the giraffes increases monotonically, even as average fitness decreases, and the population heads for extinction.

Another simulation uses a fitness function that consists of two peaks separated by a short distance. The population begins on the shorter peak, stays there for many generations, then comparatively swiftly makes a transition to the second, taller peak: punctuated equilibrium. Such stasis followed by a sudden shift would presumably be hard to account for with a linear model, but it is a natural consequence of the feedback model.

Though it is only one-dimensional and very preliminary, the model seems to account for Cope’s law (the observation that with time most species increase in size), punctuated equilibrium, periodic extinctions, and outlandish sexually selected adaptations like the peacock’s tail and the elk’s antlers. I am mildly surprised that biologists have shown little interest in the model since it was developed in the mid-90’s. I found the simulations intriguing and would be curious to hear informed opinions from others.

Evolution can be seen as a kind of parameterized curve fitting, where the model is the environmental factors and the fitted curve is the consequences of the resulting phenotype-based characteristic of the modified organism. I think this is not news to most people here.

Its interesting, though that most curve fitting algorithms require some amount of damping so as to prevent the system from going into wide swings at each iteration. Given the wrong time constants and evaluation functions, a system could easily become oscilatory and even drive itself to its limits.

On the surface it would seem that evolution would be a very overdamped system, but when one considers competition within the same species, such as the giraffe neck length example, then that assumption might be too simplistic.

Another analogy is a neural network. An evolving species under environmental pressure is like a neural network in training. Anyone who has studied neural networks realize right off that the crock that no new information can be introduced into a simple system from its environment is totally bogus.

Is there a field of study that models evolutionary systems as systems described by differential equations? Has it been successful in predicting the rate of evolution under different conditions?

I am mildly surprised that biologists have shown little interest in the model since it was developed in the mid-90’s.

and furthermore, Dawkins was talking about R.A. Fisher’s ideas.

yup.

having spent a bit of time recently going back over some of Fisher’s papers, I think it worthwile for anybody studying evolution to revisit this work from time to time.

Much of it was the basis for not only what Dawkin’s writes about, but a lot of WD Hamilton’s work as well.

standing on shoulders, indeed.

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As an electronic engineer, I should point out there are TWO kinds of feedback generally considered, positive and negative. By way of illustration, consider a toilet.…

Positive feedback is like the flushing of a toilet - once a tiny action is initiated, a bit of water flows, because more water is now flowing, more water flows - run-away POSITIVE feedback. The system is unstable, because the output was only limited by the tank´s capacity.

Negative feedback is like the re-filling of the tank ! Initially the tank is empty (we just flushed it) The rate of fill depends on the level. The empty tank fills fastest. As the tank fills, the fill valve slows the filling down, as the tank gets more full, the filling slows down even further. This is negative feedback. The system is stable, the tank fills and stops filling.

Without careful analysis systems can exhibit runaway positive feedback, like the “howling speaker” analogy, even when they were designed to be stable, negative feedback systems.

Feedback studies involve a lot of maths and detailed knowledge of loop-delays and loop-amplification.

Steve

I am a physicist, so I am not certain about the subtle details that could go into a rapid evolutionary shift. But resonance from positive feedback is something I understand, and this is something that can occur suddenly when there is a collusion of circumstances that can bring about the rapid enhancement of a particular harmonic or state of a system. Think of something simple, such as driving on a bumpy road and a crack or deformation occurs in the suspension of the car and suddenly you have a resonance that causes a large amplitude oscillation in a fender which then falls off. After the fender falls off, the car responds more quietly to the road. That would correspond to a phenotypic change in the car. ;-)

Are there any examples of changing multiple characteristics of individuals within a population that, within a relative short period of time, collude in some way to give a sudden spurt in phenotype that would appear in the fossil record as a jump? Could such changes accumulate at the cell level without any outward manifestation and then suddenly “kick in”?

Maybe positive feedback or resonance are the wrong words to use here. How are such conditions described more technically in the biology community?

I’ll add something I only fairly recently learned about: the Mathieu equation. This equation describes frequency modulation of a harmonic oscillator. The response is actually amazingly complex. For some pairs of values of the two parameters, the system is unstable.

I think I can be a little more explicit about what is bothering me about “positive feedback” in the context of evolution. What is being “fed back”? There seems to be an implicit assumption of acquired characteristics being fed back to produce more of the characteristic.

In another sense, one could say that as organisms travel down the rough road of life, certain characteristics they have are “resonating” to the bumps in the road so these are enhanced if they are in phase with feed back from the road. I have two problems with this. First, it is a questionable use of feedback and resonance as an analogy for what is happening. Secondly, it is the implication that characteristics are enhanced through use or through massaging by the environment.

The giraffe illustration seems forced to me. Competition with members of one’s species is part of the environment in which characteristics are selected. I don’t see how the use of the term “feedback” clarifies anything.

While we’re waiting for biologists to answer, I’ll put in my two cents.

One kind of feedback might occur when there’s some advantage to the members of a species that have more of something (height, limb length, whatever) than their relatives. So the ones with more produce more offspring, so the average goes up. Then that cycle repeats, until some limiting factor balances out the advantage that more of that feature would produce.

Another type is when one species getting more of something (e.g., prey getting faster at running) causes another (e.g., a predator) to also evolve more speed. Which in turns causes the prey to evolve more speed as well. Then that cycle repeats, until some limiting factor undoes advantage of even greater speed. (The Cambrian “explosion” might have been a result of this kind of feedback.)

Henry

sez Larry Moran:

What Robertson’s simulation describes is anagenesis and that is not punctuated equilibria.

From this simulation (I refer to animation 4), it’s not possible to determine whether biological speciation (i.e., reproductive isolation) is occurring. However, given the shift in phenotype, paleontologists would probably call the two populations separate species. (the paleospecies concept works entirely on morphology.)

Don’t forget, it’s a simplistic 2-dimensional (x, and time) simulation. What’s interesting is the implication that phenopytic change can occur quickly. I’d be interested in a simulation where the fitness function varies over space and time.

p.s. on another note, i seem to have set a record for being banned on uncommon descent: first posting! (“Kindly explain the nature and origin of the Designer.”)

To explain the secondary spike, Professor Robertson notes that the optimum height of a giraffe in isolation might be, say, 4 m. But in the presence of other giraffes, maybe there is an advantage to being 4.5 m tall, so you can get at leaves that other giraffes cannot. If that advantage is enough, then it can overcome the fact the 4.5-m giraffe has otherwise lower fitness than the rest of the herd. The simulation shows that the average height of the giraffes increases monotonically, even as average fitness decreases, and the population heads for extinction.

I think the extinction aspect isn’t robust: a model of density dependence should stabilise the population, as would spatial structure in the population (so that local populations can go extinct, but are then re-colonised).

What could cause extinctions in situations like this would be a degradation of the environment, but that’s another matter.

Another simulation uses a fitness function that consists of two peaks separated by a short distance. The population begins on the shorter peak, stays there for many generations, then comparatively swiftly makes a transition to the second, taller peak: punctuated equilibrium.

This is hardly new: it’s a simple case of shifting balance: it goes back to Sewell Wright in the 1930s.

Though it is only one-dimensional and very preliminary, the model seems to account for Cope’s law (the observation that with time most species increase in size), punctuated equilibrium, periodic extinctions, and outlandish sexually selected adaptations like the peacock’s tail and the elk’s antlers. I am mildly surprised that biologists have shown little interest in the model since it was developed in the mid-90’s. I found the simulations intriguing and would be curious to hear informed opinions from others.

For me, there’s nothing new: the feedback component is simple here (have a look at some of the work in adaptive dynamics for more complex behaviour).

Incidentally, one criticism I have is that the simulations don’t exhaust genetic variation, which is obviously unrealistic.

Bob

I think the extinction aspect isn’t robust: a model of density dependence should stabilise the population, as would spatial structure in the population (so that local populations can go extinct, but are then re-colonised).

Yes, I can’t see the difference between instabilities due to the factors Robertson puts in and instabilities in simple population models in ecology. I believe Robertson mentions them in his referenced paper but thinks his model subsumes them - in which case I understand even less why he thinks he can forget earlier models.

I also don’t understand the time scales he uses. If populations so easily hunt after a certain phenotype, we would see many drastic changes. Which btw we do in some cases AFAIK, for example for fishes moving into crowded or uncrowded environments, but I don’t think we see extinctions coupled to these changes.

As an electrical engineer (who, back in college, specialized in feedback control), I have lurked here for a while but only very rarely felt even remotely qualified to comment. Here, though, if I understand the post at all, then the analogies to an electrical situation of positive feedback, while very useful for describing analogies to the evolutionary situation, seem to break down when talking about how these changes eventually lead to extinction - or at least, seem like a slightly oversimplified mathematical model of the evolutionary process.

If I’m understanding the concept correctly, using the giraffe example (oversimplifying as suits my non-biological background), let me see if the following statements follow with what is being said:

1) On a whole, a herd of 4.0-meter-tall giraffes has greater overall fitness than a herd of 4.5-meter-tall giraffes (for reasons of better mobility for evasion of predators, a more robust skeletal structure, whatever)

2) Within a herd of 4.0-meter-tall giraffes, the taller ones have better individual fitness, because of a better ability to win the contest for food on tree limbs

3) Therefore, a herd of 4.0-meter-tall giraffes will gradually grow taller over time, as the taller giraffes are regularly selected in favor of the shorter ones, thus decreasing the overall fitness of the herd.

This all seems well and good, and describes a mathematically unsound system quite accurately; the analogy to the microphone too close to the speaker causing an infinite positive feedback loop (infinite in that it is limited only by the nonlinear characteristics of the speaker and/or microphone, at least), seems apt.

But, in following through with this example, if the height of the giraffe is a problem for the fitness of a herd, it follows, I would think, that it would be a problem for the fitness of an individual, and that there would be a sort of equilibrium reached where the positive fitness relative to the herd and the negative fitness of the individual would reach a balance. This is not a characteristic of a positive feedback loop.

Perhaps a more apt analogy would be a badly-tuned negative feedback loop, or perhaps one with a steady-state error? Negative feedback loops are a common standard in electrical control, where a calculated error is processed - often through a PID calculation that utilizes integral and derivitave control as well as proportional - and then subtracted from a reference signl to provide the input to a control system. Such systems can also be unstable if not tuned correctly - if the proportions in which you take the proportional, integral, and derivitave factors (in jargon, Kp, Ki, and Kd) are out of whack. The characteristics of an unstable feedback control loop are probably different from that of a species headed to extinction (often the unstable loop will wind up being an oscillation to positive and negative extremes of a reference with increasing amplitude over time, something I don’t think would be characteristic of an evolutionary system). Still, that analogy seems a little better than the very oversimplified system of a positive feedback loop, at least to me.

As SM Taylor states above in the comments, and everyone else seems to have missed, feedback is either positive, or (more frequently) negative. It is only positive feedback that is augmentative and can go out of control. Negative feedback is more usual, and in the general case leads to an equilibrium condition: the number of members of a population entering a population equals the number leaving. This is one reason humans persist in having two arms, two legs, yet a single head.

I suspect the evolutionary and population dynamics biologists lurking here are going to get a smile out of watching a physicist and an electrical engineer struggling with biology concepts.

When those of us in physics and engineering think of positive or negative feedback, we are thinking primarily of a system that is composed of an amplification system with an input and an output. The amplifier takes energy from an external source and adds it to a signal on the input thereby increasing the amplitude of the signal on the output (the signal is often the electrical analog of some measurement of a physical quantity such as sound pressure, mechanical motion, etc.). Usually there is some kind of time delay as the signal passes through the amplification system but, to simplify, we don’t need to consider this in order to understand the effects of positive and negative feedback. This delay can be effectively incorporated into the return path from output to input. The amplifier must have a gain greater than 1 and usually it is much higher.

With positive feedback, part of the output (now larger than the input signal) is returned to the input where it passes through the amplifier again, picking up additional gain in the process. In order for this to work, the feedback must be such that, after passing through whatever delays there are in the feedback loop, it returns to the input “in phase” with the input signal. The cycle repeats rapidly, and the output increases exponentially until the amplifier is no longer able to supply the additional energy to the signal, i.e., the amplifier “saturates”.

With the sound system mentioned as an example, there is usually a whole spectrum of frequencies passing through the amplifier, but the howling produced by positive feedback often occurs for a small interval in this spectrum because it is sound waves in this interval that get reflected back to the input with the proper phase to be amplified again and again. Other frequencies get passed through the amplifier out of phase enough to not come out amplified in phase with the output, hence they are suppressed.

Negative feedback returns a portion of the output such that by the time it arrives at the input of the amplifier, it is out of phase with the input signal. This has the effect of making the output of the amplifier very stable at a gain that is determined by the proportion of the output signal that is fed back to the input.

There are other types of feedback in which the rate of change of the output or the cumulative changes in the output can be fed back to the input. These are called differentiators and integrators, and they are the analog of differentiating or integrating a function. These, along with fixed gain negative feedback amplifiers, can be ganged together to represent a differential equation. The whole setup becomes an analog computer. These can be used to simulate such things as population dynamics in predator-prey relationships.

So up to this point, I have no problem with using these ideas to solve the differential equations that calculate the numbers of individuals in a population interacting with their total environment, including with members of their own species as well as with other species.

Where my problem starts is in how this applies to phenotypic changes (or to whatever underlying genetic precursors to these). In a previous post, I asked what is being fed back in a positive feedback loop. Here it gets murky for me. The way I read the positive feedback idea was that a phenotypic characteristic was being fed back into something (individuals? populations?). In order for this to enhance the characteristic, there must be some kind of amplifying mechanism that inputs the characteristic and spits it out with whatever enhancements this mechanism produces and the cycle repeats for positive feedback. Maybe I misinterpreted the context or meaning, but it looked to me like phenotypic changes were being acquired and enhanced through use in the environment.

The example of predator-prey speed enhancement was given as a possible example of positive feedback. But why positive feedback? Isn’t this just a case of faster predators getting fed better and faster prey getting away better, and both doing this in parallel with the fastest getting to produce more offspring? I’m not sure how the idea of positive feedback provides more enlightenment here.

In any population, exponential growth in NUMBERS can be explained by stating that the rate of increase in NUMBERS is proportional the NUMBER of reproducing individuals already in the population. Sometimes this is referred to as positive feedback, but I think that term should be used with caution here. I would think even more caution needs to be exercised when discussing phenotype.

Different underlying physical phenomena can lead to the same kind of differential equations. We can often take a phenomenological approach to understanding the broad outline of things we see in Nature. But just working with the differential equations without understanding the underlying physical mechanisms can get us floundering in loops where we think we understand something that is eluding us.

Maybe this is just a layman’s confusion about biology. This is why I was appealing to the biologists to explain how this concept of positive feedback was being used. I suspect there may be a source of confusion for other layman in the way these terms are used. And any sources of confusion will inevitably be exploited by the ID/Creationism crowd.

Due to fecundity, a population grows exponentially. This might be thought of as ‘positive feedback’.

The population soon eats up all its food supply, as in Garrett Hardin’s Tragedy of the Commons. This might be thought of as negative feedback. In any case the population crashes. If it crashes to zero, the population is extinct.

More typically, a population without significant predation or diseases, at first, that is, in a new habitat, undergoes growth and decline cycles, rather like an underdamped harmonic oscillator.

Now the biologists can come along and straighten me out if I have erred…

In a previous post, I asked what is being fed back in a positive feedback loop.

I am not sure if this is what you are asking, but: They define a fitness function, which is a graph of fitness as a function of some phenotypic characteristic such as height. It is at least roughly Gaussian.

Then they define a population function, or a histogram of population as a function of the same phenotypic characteristic. Also roughly Gaussian at the outset, the histogram is much narrower than the fitness function and perhaps offset from the center of the fitness function.

The population multiplies (all organisms at the same time), and more-fit organisms produce more viable offspring than than less-fit. They assume that each organism produces a smallish range of phenotypes and then calculate a new population histogram. To calculate the number of organisms in the next generation, they multiply the new population histogram by the fitness function.

Thus, the population histogram changes from generation to generation, and the population on average acquires new phenotypic characteristics, such as greater height. In some calculations, they also allow the population to influence the fitness function. I think that is properly described as feedback.

You can see the simulations in the link I gave in the original article. The reference [Robertson and Grant, 1996a] outlines their procedure.

I too would like some biologists to weigh in, partly to prevent us physicists and EE’s from venturing where angels fear to tread.

More typically, a population without significant predation or diseases, at first, that is, in a new habitat, undergoes growth and decline cycles, rather like an underdamped harmonic oscillator.

No, you’re more likely to get damped cycles. There are populations which cycle, but the explanations tends to involve competition between species (e.g. predator-prey interactions) or delayed effects (OK, I don’t know these latter explanations in detail). You can get cycles in simple population models, but I haven’t seen any convincing cases of these in the real world: the more realistic choice of parameters leads to an equilibrium. In reality, there is environmental noise which will make the population size jump around a bit.

I think a problem for biologists in explaining this to engineers is that it’s not clear that we would describe this as feedback, so I think a discussion might, well, decline into mud slinging (hm, and that would be due to positive feedback!). Just to give one illustration: David B. Benson suggests that exponential growth is due to positive feedback, but for population biologists, it’s the baseline case without feedback. I’m not saying David is wrong, just that there are different conceptions of feedback.

Bob

Yeeeehaah! Here I am, a physicist, making a fool of myself in front of a bunch of biologists and loving every minute of it because I am learning something.

After I read Robertson’s paper a few more times I finally realized where I was hung up. It was on a single word and what that word means to a physicist. The word is feedBACK.

In the case of the fitness “function” and the population distribution in phenotype, I would have preferred a somewhat more accurate but less frequently used term, feed FORWARD. The fitness function carries a phenotypic trait into the FUTURE generations. Feed forward has quite a different connotation, because there is an accumulation (or diminution) of a trait or property that continues to be carried down the line or into the future as a result of multiplications taking place in the present. That is partly why I was expressing caution about using the term positive feedback for the example I used in which the rate of increase of a population was proportional to the number of reproducing members of the current population.

I realize that there is a kind of convention in using positive or negative feedback in cases where there is really information or energy, etc. being transferred down the line or into the future (I have slipped into this form of expression myself). But try feed forward in Robertson’s paper and see if it doesn’t make things clearer. All the while these things are taking place in the environment in which the organisms are acting, the energy source they draw on is coming from the sun by way of the environment. Once the rate at which these energy recourses becomes insufficient to sustain the individuals in the population (for whatever reason), the population collapses.

I suspect the evolutionary and population dynamics biologists lurking here are going to get a smile out of watching a physicist and an electrical engineer struggling with biology concepts.

True, but then again, that’s got to be a better feeling for all involved than being told, by physicists, mathematicians, and engineers, that they’re totally wrong about this whole evolution thing…

It was a little late last night to summarize the source of the misconceptions I, as a physicist, and some of the engineers posting here were dealing with. I think we were all agreeing that the use of terms positive or negative feedback was leading to some inconsistencies, as well they should given what those terms mean to us.

In my last post I suggested that the evolution of a phylogenic trait would be better described as a “feed forward” process. This term is not use as often as it should be, but I think most engineers and physicists (especially if they are my age and older) know what it involves. Mathematically it behaves quite differently from feedback, although there are effects that mimic positive and negative feedback. In fact, there is a way to transform some feed forward problems into feedback problems with a little effort. It involves running the time axis toward the negative direction while watching the phenomena of interest morphing in front of you (somewhat like a coordinate transformation into a moving frame).

The main difference between feedback and feed forward is where the feed information is injected. In feedback, it is injected upstream at a later time. In feed forward it is injected downstream at a later time (the time delays can be insignificant but, in most cases, we don’t think of them as being injected into the past). There are also variations on these depending on the source and phase of the feedback information.

I already gave examples of feedback in my previous posts. Examples of feed forward include things like the synthesis of the heavier elements in the shock wave of a supernova, an avalanche, exponential population growth, some kinds of industrial processes where changes are fed in down line. Autocatalytic reactions are another.

I suspect the reason that explicit use of the term “feed forward” is seldom used is that most people are referring to specific instances of it and use the terms that seem more appropriate to the particular situation (e.g., avalanche). Problems arise when the terms positive or negative feedback are used for phenomena that mimic the effects of feedback but, in reality, are not due to feedback. The example I used about exponential population growth is one. So is avalanche. In fact, exponential population growth is much like a runaway avalanche. But there can also be steady-state examples of these depending on what kinds of inhibitors are fed in.

Lurking in the background of all of these cases, whether feed forward or feedback, is the energy source that drives them. This is often easy to overlook.

When I finally made the mental flip from feedback to feed forward, I began to see Robertson’s paper as a description of a relentlessly driven process that occasionally avalanches briefly at some points, but then gets snuffed out as the energy sources become insufficient to sustain it or some inhibiting factor creeps in. From this perspective, evolution is very much like the synthesis of heavy elements, or the population of less probable states in a system that is driven hard by external energy sources. Maybe evolution is not as improbable as it seems when viewed from the perspective of a system with positive or negative feedback mechanisms. This is where a perspective can make a big difference in how one understands a problem.

Nick: Thanks very much for the post. I think I learned more from this one than I have from all the others.

Bob O’H — Thanks for the clarifications. Now it is my turn. :-)

When you flip the DC power switch on an underdamped harmonic oscillator you get damped cycles which fade away. If the harmonic oscillator is critically damped or overdamped, there is just one pulse which fades away.

So I think we are in agreement. It seems to me that a population just entering a new habitat, without predation or diseases, will eventually acquire both. Either the population completely collapses or else adapts to the circumstances. In the latter case, I would expect eventually a state of quasi-equilibrium for population density.

Hope I have said the biological part correctly…

Oops! I meant Matt. Thanks. Nick for your response also.

Sheesh! Sorry Nick if you are lurking here. I’m starting to mix names from different threads. I think I’ll to take a nap.

Anyway, Thanks Matt.

I suspect the evolutionary and population dynamics biologists lurking here are going to get a smile out of watching a physicist and an electrical engineer struggling with biology concepts.

True, but then again, that’s got to be a better feeling for all involved than being told, by physicists, mathematicians, and engineers, that they’re totally wrong about this whole evolution thing…

And in either case better than lawyers.

Anyway, Thanks Matt.

You are very welcome. This discussion has been enlightening.

Thank you all for providing yet ANOTHER PROOF that Intellegent Design is correct!!!

By showing that any Positive feedback will inevitably destroy a system unless it is countered by a Negative feedback, and because if there is a SINGLE POSITIVE feedback WITHOUT a countering NEGATIVE feedback, life would not be POSSIBLE because it would spirall out of control and destroy itself!!! Because there are SO MANY different systems that has positive feedback, it is UTTERY IMPOSSIBLE that they ALL have MATCHING FEEDBACKS due to chance alone!!!

The ONLY POSSIBLE explanation for all of these being in balance is that they HAD TO BE DESIGNED!!!

Mendaciously yours,

Shenda

Biologists, according to Professor Robertson, agree with the statements but yawn.

Perhaps because there are so few examples of what Robertson models. Giraffes don’t keep getting taller AFAIK. Population ecology and speciation theory certainly involve math but it’s more complicated than Just say feedback. Runaway selection on one character leading to extinction of a population is rare at best. Robertson’s model may be a better model of a selective sweep of a mutation that confers resistance to a pathogen (the other allele may become extinct). Even here the change may not occur in all populations of the species. Setting aside the matter of multiple populations, where are the examples of the model?

It was said that it shouldn’t be called Punctuated Equilibrium (PE) if all of the species undergoes whatever the change is. This is a detail of semantics; however, a species usually has multiple populations and not all are expected to be subject to whatever sparks speciation or a large morphological change in one population. Perhaps this is another hint that there is more to it than Robertson’s model.

Whenever PE is brought up some clarification is in order: PE explained.

A comment on the oft misunderstood PE from Gould’s essay in Natural History (12/97- 1/98):

Gould Wrote:

“To illustrate how poorly we grasp this central point of time’s immensity, the reporter for _Science_ magazine called me when my ‘Cerion’ article, coauthored with Glen Goodfriend, appeared. He wanted to write an accompanying news story about how I had found an exception to my own theory of punctuated equilibrium - an insensibly gradual change over 10,000 to 20,000 years. I told him that, although exceptions abound, this case does not lie among them but actually represents a strong confirmation of punctuated equilibrium. We had all 20,000 years’ worth of snails on a single mudflat - that is, on what would become a single bedding plane in the geological record. Our entire transition occured in a geological moment and represented a punctuation, not a gradual sequence, of fossils. We were able to ‘dissect’ the punctuation in this unusual case - hence the value of our publication - because we could determine ages for the individual shells. The reporter, to his credit, completely revised his originally intended theme and published an excellent account.”

The abstract:

Science. 1996 Dec 13;274(5294):1894-7. Paleontology and Chronology of Two Evolutionary Transitions by Hybridization in the Bahamian Land Snail Cerion

Goodfriend GA, Gould SJ.

The late Quaternary fossil record of the Bahamian land snail Cerion on Great Inagua documents two transitions apparently resulting from hybridization. In the first, a localized modern population represents the hybrid descendants of a 13,000-year-old fossil form from the same area, introgressed with the modern form now characteristic of the adjacent regions. In the second case, a chronocline spanning 15,000 to 20,000 years and expressing the transition of an extinct fossil form to the modern form found on the south coast was documented by morphometry of fossils dated by amino acid racemization and radiocarbon. Hybrid intermediates persisted for many thousands of years.

As the sounds are amplified and repeatedly fed back into the loudspeaker, you hear a loud shriek.

Sure, I’ve seen this. But I’ve yet to see a loudspeaker or amp be “toast” as a result. I am not an electrical engineer, but I would presume in those cases that some process other than “toastness” must cause the system to approach a regime of null feedback. Likewise I don’t think decreased fitness and/or extinction is necessarily an inevitable result of positive feedback. But it is an interesting possibility.

Mike Wrote:

But try feed forward in Robertson’s paper and see if it doesn’t make things clearer.

Well, since a change in phenotype (at birth) in one generation affects the fitness function (at death) for the same generation, I think you are right. I missed that too.

Feedforward seems to be common in biology. IIRC some parts of the brain works this way. While a signal, say from the optic nerve bundle, is processed by layers of neurons some parts of the signal is fed forward to prime the next part of the process chain.

Mike Wrote:

exponential population growth

Since all of the “signal” is fed forward in making the next generation, this must be formally correct. So no feedback means automatically feedforward in a model of population growth, I take it.

turn on your irony meter, TL.

I probably won’t have much time in the next couple of days to lurk here. I have a few chores to do before leaving for Hawaii for a week or so. So this is just a note of thanks.

Doug:

Thank you for taking the time to respond to our questions about your work. I learned a new perspective on this type of modeling, and that is always fun. I may try a little more of this now that I am retired and have some discretionary time.

Anton:

Thanks for raising a lot of knotty issues about evolution. I don’t know as much about this topic as I would like, but I’m learning. I am optimistic that this kind of modeling can help us fill in an understanding of the many entangled mechanisms. The important part is to have good observers who can notice and report them accurately. Then the models have something realistic to shoot for.

And Matt:

Thanks again for putting up this topic.

Mike

Have a good time in Hawaii. I’ll be spending the week watching migrating Sandhill Cranes in southern Colorado, so I won’t be on line again until the end of the week.

–Doug

It seems the discussion is taking a break, but it has been interesting. I will put in some comments for the next round.

Anton Wrote:

But it’s also simple enough that the premise is unlikely to be universally true; and therefore, you cannot say that it correctly describes the long-term evolution of life on Earth. Rather, it usefully approximates the evolution of certain populations…and you’ll have to give some criteria for which ones qualify.

This is exactly the type of developing critique I hoped for. It would of course also help if someone could evaluate the use of the model for such populations in a case study.

Doug Wrote:

I am arguing that all of these ideas can be subsumed under a simple and unified theory based on natural selection plus the unassailable observation that organisms are often significant components of their own adaptive environment.

I’m not happy with the term unassailable. In fact, I think I now remember this discussion from an earlier thread.

First on principle, since all assumptions must be possible to critique. Second on contingency, since it isn’t certain that the assumption will be important in all cases while the term seems to imply it is. And here it is contentious.

Doug Wrote:

It is true that biological generations often overlap (although there are some species, such as annual plants, whose generations do not overlap). This overlapping would tend to make things more complex, not less.

I am not knowledgeable enough to tell whether overlap will tend to inject structures into a potentially chaotic system or if it will can smooth things out.

The later is of course an heuristic from differential equations, which Anton warns us for. But it seems a simple enough problem that one of us can study how some simple concurrent and non-equiperiodic discrete iterative maps will behave.

Doug Wrote:

But when a large population develops in the vicinity of one of these underlying fitness extrema the individuals there will begin to compete strongly for limiting resources and will thereby become significant components of their own adaptive environments.

I see three potential problems with this.

The first problem is likely wrong, since I quoting something I have understood vaguely. But I think I have read that there is a problem with gene fixation close to an extrema. I.e. it isn’t expected that a population comes close to an extrema in all characteristics since that would stop evolution.

The second problem is that it seems that multidimensional fitness spaces flattens out - there is always a direction for a population to move. (Which avoids the first problem, incidentally.) This result has been mentioned a lot in connection with analyzing Dembski’s NFL histories, so it should be easy to find.

The third problem is neutral drift that will work to remove local extrema by allowing drift away from them. (Again avoiding the first problem, as I understand it.)

Doug Wrote:

I wish I knew how to put quotes into little boxes.

There are many variants of HTML and XML formatting, and some blogs implements some small subsets, often fewer in the comments (no pictures, for example) than in the posts.

The exact implementation will unfortunately also vary due to ambiguousness in the standards and due to the implementation script. Many blogs show applicable format codes close to the comment box. Here it is displayed under the “KwickXML Formatting” link on the front page.

In fact, I can use HTML formatting to show how it is done without the comment box script grabbing the code. KwickXML lets us embellish the basic blockquote with an author if useful:

Code:

NN Wrote:

Blah blah blah

Test:

NN Wrote:

Blah blah blah

Doug Robertson Wrote:

You write: “But what you’re arguing, I think, is that the pressure points in a sufficiently constant direction that it consistently favors one or a few of those properties at the expense of others.”

You persist in putting words into my mouth, which I must object to as being unsanitary. I don’t believe that I ever wrote “at the expense of others.”

Respectfully, had I intended to put words in your mouth, I would have presented them as a quote rather than as my interpretation of your argument. Evidently that interpretation was off. But it seems to me that enhancing one ability would have to be at the expense of others, if the net effect is to move the population off an absolute fitness peak. If our hypothetical gazelle simply gained the ability to run 10 mph faster without reduced endurance or sensory abilities or impact resistance; if a hypothetical peacock had a double-sized tail but was nevertheless physically on par with the existing population, its absolute fitness wouldn’t go down, would it?

But I think that all of this discussion of the multi-dimensionality of both phenotype space and selective pressures, as well as the resulting constraints and trade-offs, is beside the point entirely. Let me try to make my argument in a rigorous fashion in arbitrary numbers of dimensions: We first assume that there is a fitness “surface” (a fitness function or set of (positive) fitness values, one for each point in phenotype space), and that even in the absence of feedback the function will have a number of local extremal values or “peaks”. Under conventional evolution theory without feedback, such as might be expected to occur at low population levels, the various populations will move toward these fitness peaks by pure Darwinian selection. I don’t think we disagree at this point.

Well, I really don’t get a kick out of being contrary, but I think the initial assumption itself implies a certain limitation on the model.

Real-world phenotype space, due to its high dimensionality, does not generally have peaks (there have been a number of Panda’s Thumb posts on this which I can dig up if necessary.) Low-dimensional slices of phenotype space have peaks. For instance, if you’re talking about gazelles and your only axes are “body mass” and “leg length,” then sure, there’s probably an ideal combo of both which gives them the best running speed, endurance, etc. But if you throw in more axes–details of leg morphology, lung morphology, bone structure, metabolism, etc.–your peak will almost certainly become a ridge which does permit a further increase in fitness along some direction.

Now working with a low-dimensional slice of phenotype space is often useful if, for instance, you’re talking about a relatively short timespan, so mutation can be ignored and the only axes of variation are those for which polymorphisms already exist. But I have my doubts whether it’s a workable approximation over the typical lifetime of a species.

But when a large population develops in the vicinity of one of these underlying fitness extrema the individuals there will begin to compete strongly for limiting resources and will thereby become significant components of their own adaptive environments. Feedback loops will thereby be set up that will generate selective pressures that will tend to move the population away from the underlying fitness extremum in whatever direction(s) is (are) possible, again by pure Darwinian selection. And because the fitness has been at a local extremum, any motion at all in phenotype space will move the population toward lower fitness values. These lower fitness values imply an increased probability of extinction. (We are always dealing only with probabilities here, a point that I often fail to make explicitly, and I apologize for that omission.)

Now that I fully agree with. Even in high-dimensional space, where there exists a curve which is nondecreasing with respect to absolute fitness for the population to follow, feedback loops will cause its evolutionary trajectory to diverge from that curve to some degree, lowering its fitness from what it would be otherwise.

By the way, you may have already discussed this elsewhere, but sex ratio theory seems like another area in which your model could be very fruitful; not only because sex ratios have such a dramatic impact on absolute fitness but also because they’re often quite sensitive to population density IIRC.

I may need to modify this argument to deal with the transient fitness extrema generated by the feedback effects, but I can do that and will, unless you’d like to work it out yourself.

Ha! I’m afraid I lack both time and talent. I’ve always had an interest in evolutionary modeling, but it’ll be years before I can actually get into it, if ever.

I agree with Kingsolver and Pfennig that there may be uncommon cases where there is a fitness advantage to smaller size (or lower running speed, or whatever). I’m a little surprised and skeptical that it would occur in as many as 20% of species, but the exact percentage is not particularly important. In fact, the sign of the effect (toward larger or smaller) is important to the biology but not to the underlying mathematics. You could just as easily have feedback loops that drive species toward lower fitness values (higher probability of extinction) in the direction of small size as well as large size. I believe that the drive toward large size (faster speed, etc.) will predominate for biological reasons, but the mathematics of feedback loops does not care about the direction.

Quite true, but the extinction likelihood is affected by the biology as well; large organisms have vulnerabilities to sudden changes in their environment which small organisms don’t. Which, come to think of it, suggests that large organisms’ fitness will drop faster than small organisms’ fitness–they’ll be less able to cope with even the environmental changes for which their own evolution is responsible.

I wish I knew how to put quotes into little boxes.

Hopefully Torbj�rn has made it clear; [left angle bracket]quote[right angle bracket] Quotation here [left angle bracket]/quote[right angle bracket]

[left angle bracket]quote author =”Author Name”[right angle bracket] Quotation here [left angle bracket]/quote[right angle bracket]

Doug Robertson Wrote:

And I am not sure that arthropod sizes peaked in the paleozoic. There are few arthropods anywhere, anytime that are as large as a modern Alaskan King Crab, perhaps a few Silurian Eurypterids, but they are an exceptional case (as are the King Crabs). Of course we do not have a complete census of modern arthropod sizes, especially in the ocean, and still less a complete census of extinct arthropod sizes.

I believe the Japanese spider crab is the largest known modern arthropod. But it’s dwarfed not only by the larger eurypterids but also the terrestrial myriapods of the late Palaeozoic; the millipede-like Arthropleurids could grow to over two meters long and half a meter wide! Then, of course, you have much larger arthropods in virtually every niche than their modern correspondents–protodonates with 70-cm wingspans, megasecopterans with meter-wide wingspans, the 20-inch cockroach Apthoroblattina, the meter-long scorpion Brontoscorpio. Arthopod gigantism in the Silurian through Permian is very well-recognized; Robert Dudley’s lab at U.T. Austin specializes in flying insect gigantism, if you’d like to read more.

The largest modern arthropods, moreover, provide evidence for general constraints on arthropod size due to respiration and molting times. They’re marine, permitting much faster molting due to absorption of salts from the water (the spider and king crabs complete their molt in a matter of days, whereas the coconut crab, largest of the terrestrial arthropods, takes over a year). And they typically live in cold, highly oxygenated water. The deeper king crab species, for instance, which live in lower-oxygen environments, are much smaller. Chapelle and Peck published a Nature paper on this topic in 1999, “Polar gigantism dictated by oxygen availability.”

All of which is just to say–we have good reason to think arthropods aren’t getting any bigger, and good reason to think this is because they’ve already hit their evolutionarily optimum size given post-Palaeozoic oxygen levels and vertebrate competition.

Now I think we may be getting somewhere. The arguments of Larsson and Mates center on the properties of fitness functions in higher dimensioned spaces:

The second problem is that it seems that multidimensional fitness spaces flattens out - there is always a direction for a population to move. (Which avoids the first problem, incidentally.) This result has been mentioned a lot in connection with analyzing Dembski’s NFL histories, so it should be easy to find.

The third problem is neutral drift that will work to remove local extrema by allowing drift away from them. (Again avoiding the first problem, as I understand it.)

Anton Mates Wrote:

Real-world phenotype space, due to its high dimensionality, does not generally have peaks (there have been a number of Panda’s Thumb posts on this which I can dig up if necessary.) Low-dimensional slices of phenotype space have peaks. For instance, if you’re talking about gazelles and your only axes are “body mass” and “leg length,” then sure, there’s probably an ideal combo of both which gives them the best running speed, endurance, etc. But if you throw in more axes-details of leg morphology, lung morphology, bone structure, metabolism, etc.-your peak will almost certainly become a ridge which does permit a further increase in fitness along some direction.

Now working with a low-dimensional slice of phenotype space is often useful if, for instance, you’re talking about a relatively short timespan, so mutation can be ignored and the only axes of variation are those for which polymorphisms already exist. But I have my doubts whether it’s a workable approximation over the typical lifetime of a species.

I think I need to clarify exactly what I am assuming about the properties of fitness functions in large dimensional spaces in the absence of feedback effects.

First, I assume that the fitness values are always non-negative, having an absolute minimum at zero.

Second, I am only interested in phenotype argument values between 0 and infinity, i.e., the right half of the real line (including zero), the “first quadrant” in two dimensions, and similar assumptions in higher dimensions.

I don’t think there is any argument so far.

Third, I assume that fitness values are zero for phenotype argument values of zero, i.e., there is no fitness at zero body size. In higher dimensions, this means that the fitness function is zero along both axes in a two-dimensional fitness functions, along the planes defined by zero argument values in three dimensions, and so forth.

Fourth, I assume that fitness values take on significant non-zero values only “near” the origin, and that they always approach zero as the distance from the origin approaches infinity in any direction. (This may be a point of contention.)

Now in one dimension these assumptions are sufficient to establish the existence of at least one local extremum between zero and infinity by Rolle’s theorem. (I think this is what Mates’ means by a “low dimensional slice”) We have to further assume that “fitness” has the necessary properties of “continuity” to make Rolle’s theorem apply, but I see no good reason to think that fitness is discontinuous at any point in phenotype space.

I’ve never seen Rolle’s theorem extended to multidimensional spaces, but it seems likely to me that it does extend reasonably, so that the assumptions above are sufficient to establish the existence of at least one local extremum in the fitness function somewhere in the “first quadrant” of phenotype space. If this were not true, that no extremum exists then there must always a positive (or zero) fitness gradient (an “uphill direction”) everywhere in phenotype space, and fitness would approach infinity as phenotype values approach infinity, or at least remain non-zero toward infinity, if the gradient is exactly zero, and I think this is not a reasonable property of any realistic fitness function. Any such realistic fitness functions must fit my fourth assumption above.

If we accept these properties of fitness functions in the absence of feedback effects, then I think my arguments about modifying the fitness function by feedback effects follow. Further, Larsson’s comment that “there is always a direction for a population to move” is not correct (my italics). And Mates comment: “Real-world phenotype space, due to its high dimensionality, does not generally have peaks” must have at least one exception, which by itself is sufficient for my arguments. And Mates further comment “your peak will almost certainly become a ridge which does permit a further increase in fitness along some direction” is relevant only if the ridge continues all the way to phenotype values of infinity (or zero) with non-zero fitness values. This does not seem reasonable to me, but I am willing to entertain discussion of the matter.

I am deliberately ignoring the details of biology that produce a fitness extremum for two reasons: First, I do not need the details for my arguments. Second, the exploring detailed reasons for fitness extrema in multiple dimensions appears to me to be a very difficult problem, and I like to avoid difficult problems because I have only a limited amount of brainpower to bring to bear on them. (Feel free to make jokes about exactly how limited my brainpower is, if you like. :) )

Avoiding discussion about the details of the shape of the fitness function and its extrema also avoids long and pointless arguments about whether or not “enhancing one ability would have to be at the expense of others,” One single local extremum of the fitness function in phenotype space is sufficient for my argument. I happen to believe, by the way, that the fitness function will often have a large multiplicity of local extrema, but that is another argument altogether, and not directly relevant to the discussion here.

BTW, I am also making a tacit assumption that in the absence of feedback fitness functions are constant in time. This is obviously not perfectly true but is a reasonable first-order approximation. The effects of time-varying physical changes in the environment (temperature, rainfall, etc.) can be grafted onto the theory without much difficulty and without affecting the properties of feedback loops in general. Extreme and rapid physical changes, such as might be caused by asteroid impacts or nuclear wars, will dominate evolution in the limited time frames where they occur. However it seems reasonable to assume that such extreme effects are uncommon, and we can reasonably model the way that evolution will behave in the time intervals between such uncommon effects.

Finally, Larsson’s point about neutral drift removing local extrema is, I think, confused. My understanding of neutral drift is that it refers to mutations that have no phenotypic expression, and thus no effect on evolution. I think Larsson is trying to argue that ordinary heritable random mutations will move population phenotype values away from fitness extrema, and this is perfectly true but it has no effect on the existence or properties of those fitness extrema. Further, heritable random mutation is only half of the Darwinian process. The other half, natural selection, will tend to remove the effects of such random mutations if they have indeed moved the population away from the fitness peak. This last statement is true only in the absence of feedback effects, which is why feedback produces such an important modification to standard Darwinian theory and why I think it should be placed in a position of central importance in evolution textbooks.

Consider the non-negative quadrant of an n-dimension Euclidean space, that is, only non-negative values in every coordinate. (I only do this because it seems to be what Doug wants.)

Let r denote the distance from the origin. Then

r exp(-r)

is zero at the origin, goes to zero at infinity, and has a ridge of maximal values, none of which is a local maxima.

David B. Benson Wrote:

Let r denote the distance from the origin. Then

r exp(-r)

is zero at the origin, goes to zero at infinity, and has a ridge of maximal values, none of which is a local maxima.

Hmmm … This type of unphysical mathematical curiosity is probably the reason that Rolle’s theorem is not generally extended to high dimensional spaces.

You actually could do better with

r^2 exp(-(r^2))

a spherically symmetric Maxwellian, which is analytic at the origin (as the other example is not), and is not restricted to any quadrant of an n-dimensional space, but has a similar “ridge” of maximal values.Both functions have their “ridge” at r = 1 (thanks, Mathematica).

However, we can rescue the situation if we simply define a set of points on such a “ridge,” that are connected by a perfectly level slope (zero gradient in fitness), as a single extremum for fitness purposes, then I think my arguments still hold.

I think we can agree that extended, perfectly level regions of fitness functions are not likely to occur in the real world, and we can deal with them in this fashion if they do. Such extended regions of fitness functions would still be attractors under Darwinian selection, but selection would not favor any point along such ridges over other points. And feedback loops could drive species off such ridges as easily as off from more conventional extrema.

So I don’t think this type of closed ridge poses any more difficulty for my arguments than an extended flat “table” in fitness space would.

Doug Robertson:

(Feel free to make jokes about exactly how limited my brainpower is, if you like. :) )

Sorry. You’ll have to wait in line. Us pinheads were here first.

By the usual definition of analytic function

r exp(-r)

does indeed have a power series expansion and is analytic.

Sorry, we’re getting into some mathematical arcana of dubious relevance to evolution. I probably should not have raised the subject of analyticity. However, your statement:

David B. Benson Wrote:

By the usual definition of analytic function

r exp(-r)

does indeed have a power series expansion and is analytic.

is perfectly correct if r is an ordinary variable which takes continuous values from -infinity to infinity. But the trouble comes in with your definition of r as the distance from the origin. This is what removes the analyticity at the origin (and only at the origin–the function is perfectly analytic everywhere else).

In effect, this definition of r converts your expression to:

Abs(x) exp(-Abs(x))

(in one dimension) where Abs() denotes absolute value, and x takes on the ordinary set of continuous values from -infinity to infinity. This function has a cusp at the origin, and does not have a derivative there, and so is not analytic.

Doug — Correct as a function defined everywhere on the real line. But when defined only on the non-negative reals there is no cusp and the obvious power series expansion is fine.

This works in any dimension, of course, by using cylindrical, spherical, … coordinates.

I agree this is getting far from biology.