Evolution as Feedback?

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.

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.

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.

Biologists, please help me out here.

Re "Biologists, please help me out here."

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

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

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.

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.

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).

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.

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.

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.

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.

Anyway, Thanks Matt.

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.

"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."

— Gould

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.

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

exponential population growth

Shenda: Next time, try to read the thread before commenting.

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.

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.

TL wrote:

"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."

Yeah, that's pretty much what I was saying from my perspective as a physicist.

There can be feedback in a feed forward process, but in the case of the evolution of a phylogenic trait, I'm not aware of any kind of feedback mechanism that could change this within the current generation such that it would be passed to the next. Maybe biologists know of some creatures that could do this, but I haven't heard of them. They would have to be pretty strange. There would have to be some modification of the underlying genetic information associated with the trait that gets passed forward. What is more, in order to qualify as a FEEDBACK mechanism, it would have to be the trait (or its genetic precursors) that would be fed back in some way to further enhance the trait. From my layman's perspective, this makes no sense.

I think biologists still reject the notion of inheritance of acquired characteristics. Am I correct?

Maybe there could, for example, be changes in the genetic information due to things like radiation, or chemical contamination, or virus infections, etc.. These wouldn't necessarily change the organism's current phylogenic traits, but could affect its fitness. I don't know how deformities due to diseases are handled. Are there any cases where such deformities get passed on as a result of some genetic modification (maybe in the microscopic world)? I would like to hear what a biologist says about this.

In the cases of population growth, avalanches, chain reactions and other cascading processes, nothing is fed back upstream. It cascades downstream with numbers in the current generation determining numbers in the succeeding generation, and so on. And it is always important to remember the external energy sources that are feeding into these processes.

TL wrote:

"Next time, try to read the thread before commenting."

:-) I did a double take on this also. I think Shenda was doing a parody of an ID/Creationist. Maybe Ken Ham or Duane Gish, who are themselves parodies (I'd guess Gish because his contortions show more anger). Quite funny, actually. I had made a comment in one of my posts to the effect that the ID/Creationist crowd would not hesitate to exploit confusions in the discussions of scientific ideas.

However, from a signal flow graph perspective, any cycle in the flow graph is a feedback, either positive or negative.

In the population dynamics example, there is a population p(t) with a loop by which p(t+1) is determined,

p(t+1) = Ap(t)

for some growth parameter A. This, then, can be fairly called a feedback.

Whether the terminology is useful is another matter. What is always useful, IMO, is drawing a signal (or energy) flow graph. No matter what terms are used for the nodes and arcs.

David B. Benson wrote:

"In the population dynamics example, there is a population p(t) with a loop by which

p(t+1) is determined,

p(t+1) = Ap(t)

for some growth parameter A. This, then, can be fairly called a feedback."

Actually, David, you are correct in noting that it is often referred to as feedback, but this is one of the specific examples I was suggesting has picked up the habitual reference to positive feedback when it may not be that. It certainly mimics the behavior of positive feedback (or vice versa).

Another way to state the same thing is to say that dp/dt = kp. In order to call it positive feedback, you would have to point to a positive feedback loop, containing some part of an amplified p, that feeds upstream into p and in phase with p. If one can do this, then yes, it is positive feedback. Positive feedback can produce exponential increases which are described by the same differential equations as some feed forward processes. There are numerous examples from the physical world in which the same differential equations emerge from entirely different kinds of processes.

I suppose as long as people can keep things straight in their heads, it is ok to call this positive feedback. It has become such a custom that it is hard to correct, and I'm not going to fight it. However, sometimes these subtle changes in perspective can make very large differences in one's understanding of a phenomena and the accurate modeling of it.

Mat young said:

"...but feedforward I know from nothing."

I spent some time this afternoon googling "feed forward processes" It is evident that the concept has evolved and speciated greatly since I first encountered the ideas over 40 years ago. It makes me feel like I'm a dinosaur.

I can easily recognize many of the applications, but some of them seem to me to bear no resemblance to feed forward. Interestingly, the biologists have picked up things that I recognize. I found papers on the optic nerve that Torbjorn Larsson mentioned in comment # 162476. It also appears frequently in neural network stuff.

Shaffer may be able to comment on feed forward in control systems. As I recall, it was not common 40 years ago, but is quite common now. It primarily enhances the response of a system.

Parody aside, I believe that some of the types of feedback being discussed have already been observed. The Irish Elk comes to mind. In this case there appears to have been a very strong intra-species selection for very large antlers. When the environment changed at the end of the last glaciation, the large antlers were a liability and the species went extinct.

Another example *may* be human intelligence which evolved within a fairly short period. I have read some articles that speculate that sexual selection may have accelerated this development in humans.

I would also think that most types of intra-species feedback mechanisms would be severely dampened by external (non species-specific) selection pressures. If taller giraffes have an advantage over shorter giraffes in food acquisition, but a lessor advantage in predator defense, the tradeoff may not be worth it.

BTW, the IDers look like they have really stopped trying.... 2 years ago, a thread like this would have had at least one or two IDers leaving posts similar to my parody, but actually being serious about it. How times have changed!

realistically, there are always tradeoffs, hence the likely lack of interest in the ultimate predictions of this model.

"realistically, there are always tradeoffs..."

True, but not all tradeoffs are equal. I would gladly accept a tradeoff that got rid of my appendix and gave me teeth that didn't wear out so easily!

"...hence the likely lack of interest in the ultimate predictions of this model."

Agreed, but it was definitely worth taking a look at, as are most models that honestly attempt explanations and predictions. Even if they don't pan out, they help keep the mind open and in gear.

turn on your irony meter, TL.

Yeah, that's pretty much what I was saying from my perspective as a physicist.

feed forward in control systems

I think biologists still reject the notion of inheritance of acquired characteristics.

Maybe there could, for example, be changes in the genetic information due to things like radiation, or chemical contamination, or virus infections, etc.. These wouldn't necessarily change the organism's current phylogenic traits, but could affect its fitness.

True, but not all tradeoffs are equal. I would gladly accept a tradeoff that got rid of my appendix and gave me teeth that didn't wear out so easily!

Agreed, but it was definitely worth taking a look at, as are most models that honestly attempt explanations and predictions. Even if they don't pan out, they help keep the mind open and in gear.

Looking at my last response to David B. Benson, I see I didn't give an example. My apologies David, if I seemed to be brushing you off.

Here is an example with money (I can think of a number of others):

Scenario I: Positive Feedback perspective

You put money in a savings account where the rule is that a certain percent of your money will be added to what is already your account at regular intervals (compound interest). At the end of a certain time you will double your money.

Scenario II: Positive Feed forward perspective

You buy a run-down house and fix it up. At the end of a certain time, you sell at a profit by getting more money than you put into the house (cost of house plus cost of fix-up). You then use this money to buy another house and get the same percentage increase in your money. At the end of a certain amount of time (and whatever number of houses), you double your money.

Scenario III: Phenomenological perspective

The Internal Revenue Service tracks your earnings and notes that your income follows the rule dP/dt = kP (they could plot this on a semi-log plot and get a straight line). They use the data to solve for k and learn what your capital gains are for any given time.

The IRS doesn't care about the underlying mechanism for your capital gains; the differential equation is sufficient for them to check your gains.

In all these cases, the external "energy" source is the economy, which in any realistic economic model, should be connected to real energy and resources and ultimately back to the Sun. The path by which it feeds into your capital gains is different, but the phenomenological result is the same.

In this example, money has enough flexibility in the way it can multiply that it can fit any of these scenarioes. In Nature, most physical processes can't do that.

I would also think that most types of intra-species feedback mechanisms would be severely dampened by external (non species-specific) selection pressures. If taller giraffes have an advantage over shorter giraffes in food acquisition, but a lessor advantage in predator defense, the tradeoff may not be worth it.

Here is a more complex model: There are just a population of rabbits, r(t), and foxes, f(t). The rabbit populations changes according to rate parameter a and the fox population,

dr/dt = af(t)

and the fox population changes according to the rabbit population,

df/dt = r(t)

This describes a harmonic oscillator with natural period a. The two populations oscillate in anti-phase about the mean(average) population.

So far it is simple. But now change the rate parameter a to have oscillations, (a + 2qcos 2t) for oscillatory parameter q. We now have the Mathieu equation

dr/dt = (a+2qcos(2t))f(t); df/dt = r(t)

which describes frequency modulation of the harmonic oscillator. If a is much smaller than the period implied by cos(2t), there are small amplitude changes imposed on the oscillations as in an FM radio transmitter.

More interesting is when a is much larger than the period implied by the cos(2t) term. This leads, for sufficiently large 2q, to complex behavior, with oscillations of many different periods shorter than a.

Finally, there are pairs (a,q) such that the Mathieu equation escapes in that the response goes to infinity as time does. This is usually taken to mean the Mathieu equation becomes inapplicable in that something breaks. For rabbits and foxes, the population of one or the other goes to zero.

There is nothing special about the Mathieu equation other than its antiquity. One can easily specify other systems of first order, non-linear differential equations which behave in even more complex fashions. An example is a system of three such equations which approximately explain ice age climate (Saltzman).

But what should be called feedback or just influences appears to depend upon the tradition in any particular subject, I opine.

Interesting that there are technical differences between what I've been thinking of as feedback in evolution, as compare to feedback as understood in electronics. When put in terms of individuals (or generations), "feed forward" may be more applicable. But I wonder, if it's put in terms of feeding back into the gene pool for the species (or population), maybe "feedback" might still apply?

Henry

Parody aside, I believe that some of the types of feedback being discussed have already been observed. The Irish Elk comes to mind. In this case there appears to have been a very strong intra-species selection for very large antlers. When the environment changed at the end of the last glaciation, the large antlers were a liability and the species went extinct.

— Shenda

Henry J --- Feeding back into the gene pool seems appropriate to this non-biologist.

Anton Mates --- While Irish elk is perhaps not the best example, consider Garrett Hardin's Tragedy of the Commons in the context of an isolated population that must maintain a certain minimum size to avoid going extinct. It is, however, to the advantage of each individual to consume more than its share of the resources. So the population evolves towards fewer, bigger individuals and then goes extinct.

It is easy to write equations which will behave this way. But perhaps such are too simple to appeal to biologists.

David B. Benson wrote:

"But what should be called feedback or just influences appears to depend upon the tradition in any particular subject, I opine.

I tried to pick a very simple example of how the same differential equation can represent different phenomena, but maybe it was too simple. I like your example much better.

Take the derivative (with respect to time) of both sides of your first equation and then substitute the second equation into the result (you could also differentiate the second equation and substitute in the first). You now have a linear second-order differential equation that could represent an undamped simple harmonic oscillator. But there are literally dozens of phenomena from mechanical, to electrical, to fluidic, to gravitational, to quantum mechanical, to (you name it), that can generate this same equation from entirely different underlying mechanisms. Simple harmonic "oscillation" shows up in many places.

A differential equation is a nice way to summarize in a compact mathematical form what we have learned about a natural process but, once it is in that form, the knowledge, though easy to use, is now at a phenomenological level (some philosophers of science would argue that if we go deep enough in our probing of Nature, all we have is phenomenology, but that is a whole other topic). You don't need to know the underlying details in order to work with the equation. And, as long as the equation faithfully represents the process in question, you are ok, and nobody cares. Besides, differential equations are only a small subset of the mathematical methods of expressing natural phenomena. Then there is the whole practical area of empirical equations.

In the physics community, for the past 30 years or so, there has been a big push in what is known as Physics Education Research. This has formally studied and documented many of the experiences physics instructors have had with their students' understanding of physics. One of the most studied and well documented areas of this research is the ability of students to work with the equations, appearing to understand a physical phenomenon, but on further probing, demonstrating serious misconceptions about the physics. Part of the reason for this is what I have been trying to illustrate with the phenomenological perspective that equations place on phenomena. Student misconceptions (and preconceptions) can be difficult to dislodge, and then they often return after the effort to do so has ended.

Some corporations have learned this as well. During my time in industry, various engineering departments would need to hire a computer programmer. They learned not to hire people with computer science degrees, but instead, they looked for people with science or engineering degrees who also had extensive experience with computers. The reason was that they had learned that knowing computer theory and how to code was not the same as understanding physical phenomena.

I would bet that there are many engineers and physicists who can tell stories about mathematical whizzes who can do math extremely easily, but can't tell you what it means. It doesn't mean the guy is an airhead, it means that the equations can have multiple meanings depending on context. Mathematicians are attracted to the properties of the equations regardless of contextual meaning (my degrees are in both physics and math, by the way).

Phenomenological approaches to understanding a natural phenomenon are often the best way to get started. They can set the boundaries of the problem and try to capture gross features. But ultimately, one must dig deeper to get at mechanisms. If putting in the postulated mechanisms into a theoretical model produces the phenomenological results, there is hope (but not yet certainty).

Mike Elzinga --- I wrote the undamped harmonic oscillator as a pair of first order, linear differential equations mainly to show off the foxes and hares that must be in every beginning textbook on population biology.

Also, to solve the normally written second order, linear differential equation numerically, one needs rewrite it into the form I used. This, of course, is not needed for the linear harmonic oscillator, for which analytic techniques exist. It is needed for the Mathieu equation which is non-linear.

I don't agree about hiring computer scientists to write the programs. Have the physicists and engineers do the maths. Have the computer scientists, who need then have taken numerical analysis, etc., actually write the programs, paying attention to avoiding bugs and numerical instabilities, producing nice graphics, and all that good stuff.

I wrote the undamped harmonic oscillator as a pair of first order, linear differential equations mainly to show off the foxes and hares that must be in every beginning textbook on population biology.

Anton Mates --- While Irish elk is perhaps not the best example, consider Garrett Hardin's Tragedy of the Commons in the context of an isolated population that must maintain a certain minimum size to avoid going extinct. It is, however, to the advantage of each individual to consume more than its share of the resources. So the population evolves towards fewer, bigger individuals and then goes extinct.

— David B. Benson

It is easy to write equations which will behave this way. But perhaps such are too simple to appeal to biologists.

David B. Benson wrote;

"I don't agree about hiring computer scientists to write the programs. Have the physicists and engineers do the maths. Have the computer scientists, who need then have taken numerical analysis, etc., actually write the programs, paying attention to avoiding bugs and numerical instabilities, producing nice graphics, and all that good stuff."

I hope this wasn't taken as an insult; it wasn't meant that way. The companies I know about that did this, in my estimation, probably misunderstood what computer scientists were really good for. They initially figured they just had to tell the computer scientist they wanted some industrial or physical process programmed and could go away and let the computer scientist deal with it. My point was that they discovered that engineers or scientists with knowledge of the area and computer experience would more likely be able to do that.

It reminds me of a funny story a computer scientist friend of mine told me happened to him. This was back in the days of the IBM 1620 computer and punch cards. His computer facility was housed in one of the science buildings on campus. One day an experimental behavioral psychologist came up to the window of the facility with a huge pile of lab notebooks and papers on rat experiments and asked him to put this into the computer for analysis.

He said his initial thought was to smile, take the notebooks and papers and put them into the paper shredder, and then with a look of concern, turn back to the psychologist a tell him the computer just malfunctioned and they would need to call for service, sorry.

shows what happens in this common situation, that the presence of a population produces a fitness advantage to being slightly above the average size (although again the same argument applies to other parameters besides size, for example running speed in either a predator or a prey species, or drought tolerance in plant species)."

And following that extinction a remnant population near the original peak of the fitness function begins to grow (this would look like another instance of punctuated equilibrium if it were observed in the fossil record) and the process repeats. In the fossil record this repeating behavior would be a phenomenon sometimes termed "iterated parallelism" (Dobzhansky et al., 1977, p. 327).

This situation should be very common, and it easily explains Mayr's observation that 99.99% of species have gone extinct and that in fact most species do not last more than about 5-10 million years.. Most species should be driven to extinction by Darwinian natural selection as observed in animation number 5.

And the species that do not go extinct will be those that are not strong components of their own selective environment, including brachiopods such as Lingula that are sessile and do not compete strongly with others in their environment.

Henry J wrote:

"Interesting that there are technical differences between what I've been thinking of as feedback in evolution, as compare to feedback as understood in electronics. When put in terms of individuals (or generations), "feed forward" may be more applicable. But I wonder, if it's put in terms of feeding back into the gene pool for the species (or population), maybe "feedback" might still apply?"

That's an interesting way of looking at it. "Gene pool" may be an abstraction that can be thought of a extending over the entire history of the species and changing through time as a result of feedback from itself into itself. We then have to see how the mechanisms that operate on the gene pool work to do that. What do you bet the biologists have already thought of that?

If putting in the postulated mechanisms into a theoretical model produces the phenomenological results, there is hope (but not yet certainty).

Posted by Mike Elzinga on February 26, 2007 12:21 AM [...] What do you bet the biologists have already thought of that?

I'd like to reply to Anton Mates' comments. I'm brand-new at this; this is my first attempt at a response (I'm something of a web dinosaur here). Please forgive me if I am not handling the formatting and other protocols in an optimum or normal fashion.

First, Mates suggests that "if all giraffes have long enough necks to reach almost all the available foliage, it's no longer advantageous to have an even longer one, given the physiological problems that would cause." But giraffes tend to be about 18 feet tall, and trees commonly reach heights of 100 feet (and uncommonly, 350 feet). Thus Mates' scenario does not seem to be a very reasonable possibility for the real world. Perhaps we should worry about it when giraffe populations reach average heights somewhere in the 100-150 foot range, but for physical reasons this is not going to happen.

Similarly, he says "if all gazelles are fast enough to outrace their predators most of the time, the advantage of being even faster is comparatively small." Again, this is a situation that generally does not occur in the real world. If it did, the predators would go extinct. What _does_ happen is that the system sets up a two-component feedback loop: faster predator speeds produce a fitness advantage for faster prey speeds, and vice-versa. Both species are driven by feedback loops to the limits of physical possibility, and these physical limits are not generally in the phenotypic range that would produce optimum fitness in the absence of these feedback loops.

Next Mates comments:

For another, even if a given biotic pressure is always pushing the fitness optimum in one direction, there may be lots of other biotic pressures pushing it in other directions. Sure, it's always better to be faster than average; but it's also always better to be stronger than average, and smarter than average, and more famine-resistant than average. You can't satisfy all of those at once, and in general their relative importance will change depending on the current state of the population.

The above is not an argument against Robertson's model ever applying, of course; I just don't see any reason to expect it to apply to most species, whereas he apparently does.

This argument seems to me to be a simple non-sequitur. The fact that all of the selective pressures cannot be satisfied at the same time (granted) does not mean that the pressures do not exist all the time. And it is those feedback-generated selective pressures that the model focuses on. Those pressures _will_ apply most of the time to most species that are significant components of their own adaptive environment. And species _do_ respond to selective pressures to the extent that biological and physical constraints allow; otherwise Darwinian theory would not function.

The discussion that follows Mates' statement: "Robertson's model predicts, as he says, frequent extinction. His explanation for why everything doesn't go extinct is as follows" does not accurately state my position. I did not give _any_ explanation "for why everything doesn't go extinct." In fact, my point is exactly the opposite: with rare exceptions everything _does_ go extinct, but not all at the same time. There is no reason to expect that feedback loops will be synchronized across millions of species. And many of the species that are driven to extinction by feedback effects _will_ leave remnant populations subject to further evolution. Those that do not leave such remnant populations simply become irrelevant to the future functioning of Darwinian selection.

These remnant populations are _not_ immune to feedback effects, but the feedback effects will be much smaller for smaller population sizes. In fact, the feedback effects generally become significant at exactly the point that the population approaches the carrying capacity of the environment and then continues to try to grow exponentially (as Malthus noted). This is the point where selection effects will dominate and the fittest will have significantly higher probabilities of survival and reproduction, as Darwin recognized. At that point the feedback loops will attain their full power.

Mates goes on to say: "If observed in the fossil record, this would not at all look like punctuated equilibrium." I agree completely. This is _not_ the process that I claimed looks like punctuated equilibrium. Punctuated equilibrium is seen in the model in the case where the fitness function has two peaks, and neither peak is affected by the population except to the extent that there is a limit on total population size.

Mates then says:

However, the model would additionally imply that most species should rapidly go extinct even when none of the traditional culprits (environmental change, the appearance of a competing species or a more effective predator, etc.) apply. We should see inexplicable gaps in the fossil record where a species simply disappears, leaving a perfectly good but unused niche; its prey species are still abundant and no other predator has yet evolved to live on them.

There are many cases in the fossil record where species do go extinct without clear evidence of any of the "traditional culprits." For example, the evolution of horned dinosaurs produced one suite of species in the Judith River age, another similar but different set in the Horseshoe Canyon age a few million years later, and a third set in the Lancian age yet a few more million years later. A similar pattern is seen much earlier, in the successive waves of mammal-like reptiles of the Permian and early Triassic (see Bakker, The Dinosaur Heresies, 1986, pp. 246-247 and 406-424). But the fossil record is not nearly complete enough to try to determine every example of this behavior.

Mates states "What reason is there to think that Lingula doesn't compete strongly with conspecifics? Just because it's sessile (as an adult) doesn't mean it doesn't compete. For one thing, it reproduces by spermcasting, so every male Lingula is in competition with every other male in the ocean!" It seems obvious that spermcasting is a less strong mode of competition than using oversized antlers to drive off competing males. Further, spermcasting is not directly related to any of the phenotypic features that are observed in the fossil record, such as shell size and shape, whereas antler size _is_ one of the parameters observed in fossil deer and elk.

Next, Mates statement: "Basically, Robertson's model predicts that an isolated population of organisms in a very static environment is most likely to go extinct. But we find many long-lived, morphologically primitive taxa in such environments-deep-water seamounts, for instance." completely misstates the point of my arguments. The populations that are most likely to go extinct are those that most strongly affect their own adaptive environment, not necessarily those in isolated populations or in "static" environments. In fact, I object to the very notion of a "static" environment. This idea generally results from focusing attention only on the physical component of the environment (temperature, rainfall), a common mistake, and ignores the very component that is most significant here, the biotic component (i.e., the presence of evolving competitors).

And if Mates is going to cite crocodiles as an exception, he should discuss the many similar and related species, from phytosaurs to champsosaurs, that did go extinct. It is not clear that the present-day crocodilian species are exactly the species that lived 100 million years ago or are merely the latest in a succession of similar species.

Finally, Mates last point that species that are competitive components of their own adaptive environment are often spectacularly successful is exactly the point made in several of my papers, and is exactly a prediction of feedback models. In addition to driving species to extinction, runaway feedback loops will drive species to explore vast regions of phenotype space that would be explored only slowly or not at all without runaway feedback loops. We should therefore expect to see them occupying vast swaths of phenotype space. This phenomenon is probably responsible for the rapid development of species in the Cambrian explosion (when metazoans first became significant components of their own adaptive environment) and to the rapid filling of many vacant ecological niches by mammals following the demise of dinosaurs. Far from contradicting feedback theories, these examples constitute major evidence in favor of the theory, and in favor of the universality of feedback effects from Cambrian times through to the Cenozoic.

I'd like to clarify some of the ideas on feedback that are concerns to
Mike Elzinga and S.M. Taylor.

First I want to say that nothing in my work entails any assumption of
(Lamarckian) inheritance of acquired characteristics. I assume that
phenotype characteristics are inherited with random variations, and
those random variations suffer differential selection (death) by
Darwinian mechanisms. The whole point of my work is that the time-
varying phenotype characteristics possessed by populations have an
effect on the fitness of individuals in the populations. But this is a very
pure Darwinian formulation that assumes only inheritance with random
variation plus natural selection.

For another clarification, these feedback theories would work perfectly
well with Lamarckian inheritance, i.e., inheritance of acquired
characteristics. But such inheritance is not required by the theory and
I don't believe it happens very often in biological systems. The only
cases where it would be relevant would be those involving lateral gene
transfer, generally by virus vectors. Lateral gene transfer is important in
bacterial inheritance, and it is uncommon but not unknown in eukaryote
inheritance.

To address Mike's question about "what is being fed back," let's examine
the analogy between an evolving biological system and an electronic
amplifier with feedback. For the amplifier, the signal is the voltage at
the input and output. For the biological system, the "signal" is the
distribution of phenotype characteristics in each generation. The analog
of "gain" in the amplifier is biological reproduction. Now the analogy
breaks down a little here. In electronic feedback, a fraction of the
output voltage is used to modify the input voltage. In the biological
system, the output (the next generation phenotype characteristics) alters
the gain of the amplifier (gain==fitness). So the analogy is not exact,
but it is still reasonable to think of a system where the output signal
modifies the amplifier gain as a system with "feedback." It will have all
of the characteristics of a more conventional feedback system, including
the distinction between positive and negative feedback, as well as the
tendency of positive feedback to produce oscillation and runaway
destruction.

Notice that the word "back" in feedback refers to the physical location
in the system where some fraction the output signal is put "back" into
the amplifier input. It does not refer to "back" in time, and if it did it
would make no physical sense. Thus the discussion of "feed forward"
does not seem to me to be very useful.

P.S. Mike, I notice that you are from Kalamazoo. I was born in Three
Rivers; small world.

Doug:

Thanks for your clarification. Using the analogy of voltage feedback modifying the gain of the amplifier is a nice example. That kind of thing happens all the time, even when it is not suppose to.

Having done a lot of successful theoretical modeling myself, I have become quite conscious of the fact that different mechanisms can give the same phenomenological results. Here is where serious problems can arise. If you put in mechanisms that produce the observed phenomenological results, as I said in one of my posts, there is hope. But the danger is in believing the mechanisms are the real answer to the questions being addressed. They may work for a while, but eventually they lead to the breakdown of the model. Worse, one expends a lot of time working with misconceptions and has, in effect, gone down a wrong path. But I guess we can learn from that as well. I certainly have.

Matt Young mentioned you gave a seminar on this at U. Colorado, Boulder. I know Carl Wieman there at Boulder and his wife Sara Gilbert, who works at NIST, from my days at the University of Michigan. I have been out to Boulder only once to visit NCAR.

Mike

I agree completely with your statement: "I have become quite conscious of the fact that different mechanisms can give the same phenomenological results." That's why I try to start my modeling with ideas that are simple enough that they have no significant probability of being wrong. I started here with the simple idea that organisms are significant components of their own adaptive environment. The corollary for modeling purposes is that there is often a fitness advantage to being bigger than the other guy, but at the same time a disadvantage to being very large. For "large" you can substitute any selectively significant phenotype parameter.

This idea leads immediately to a model that is mathematically unstable, unstable in ways that mirror observed phenomena in the fossil record. I do not think that these instabilities are well understood in the biological community. If they were, then evolution texts would begin with the idea that organisms are significant components of their own adaptive environment. They would also focus on the qualitative difference between the physical component of the environment, which does not generally produce mathematical instabilities, and the biotic component, which does produce significant instabilities.

In working this way I am trying to consciously emulate Euclid, who explored the necessary implications of the simplest axioms he could come up with. I am also cognizant of the fact that he believed that his axioms were absolutely true, and he was wrong about the parallel axiom. It is therefore possible to make a serious error even starting from the simplest axioms. I should note that Euclid was apparently aware that his parallel axiom was different from the others: it is the last one that he used. It is hard to avoid the conclusion that he did not devise it until it was absolutely necessary to continue his sequence of proofs, and in particular it was necessary to the sequence that led to his proof of the Pythagorean theorem.

Nevertheless, Euclid demonstrated that it is possible to do useful work from very simple axioms even if you are wrong about one of more of them. By stripping down to the minimum amounts of information necessary you make it easier for others to find and correct your mistakes. In Euclid's case, it took 2000 years to find the mistake. I hope it doesn't take that long to find my mistakes.

I know of Carl Weiman, of course, but I don't know him personally. I'm in the Geological Sciences Department at CU.

--Doug

Matt Young mentioned you gave a seminar on this at U. Colorado, Boulder.

I started here with the simple idea that organisms are significant components of their own adaptive environment.

"One of the differences between us and the University of Colorado is that our football team has enjoyed some success in recent years (28-18 over the past 4 seasons, undefeated in 2004)."

If I am remembering correctly, some of the U. of Colorado football recruits were enjoying other "successes". ;-)

"Carl Wieman has left Colorado and is now at the University of British Columbia, though he supposedly returns periodically. Unless I am mistaken, Sarah Gilbert has resigned or retired from NIST to accept an assignment at UBC."

Ah. I missed a couple of meetings where I usually run into him, so I didn't know.

To Sir_Toejam

I said that organisms _are_ a significant component of their own adaptive environment, so that there is an adaptive advantage to being a little larger, faster, more drought-tolerant, etc. than average. The question of whether and how they impact their environment is a completely different matter that I did not intend to address.

I said that organisms _are_ a significant component of their own adaptive environment

I'd like to reply to Anton Mates' comments. I'm brand-new at this; this is my first attempt at a response (I'm something of a web dinosaur here). Please forgive me if I am not handling the formatting and other protocols in an optimum or normal fashion.

— Doug Robertson

First, Mates suggests that "if all giraffes have long enough necks to reach almost all the available foliage, it's no longer advantageous to have an even longer one, given the physiological problems that would cause." But giraffes tend to be about 18 feet tall, and trees commonly reach heights of 100 feet (and uncommonly, 350 feet). Thus Mates' scenario does not seem to be a very reasonable possibility for the real world.

Similarly, he says "if all gazelles are fast enough to outrace their predators most of the time, the advantage of being even faster is comparatively small." Again, this is a situation that generally does not occur in the real world. If it did, the predators would go extinct.

What _does_ happen is that the system sets up a two-component feedback loop: faster predator speeds produce a fitness advantage for faster prey speeds, and vice-versa. Both species are driven by feedback loops to the limits of physical possibility, and these physical limits are not generally in the phenotypic range that would produce optimum fitness in the absence of these feedback loops.

Next Mates comments: For another, even if a given biotic pressure is always pushing the fitness optimum in one direction, there may be lots of other biotic pressures pushing it in other directions. Sure, it's always better to be faster than average; but it's also always better to be stronger than average, and smarter than average, and more famine-resistant than average. You can't satisfy all of those at once, and in general their relative importance will change depending on the current state of the population. The above is not an argument against Robertson's model ever applying, of course; I just don't see any reason to expect it to apply to most species, whereas he apparently does. This argument seems to me to be a simple non-sequitur. The fact that all of the selective pressures cannot be satisfied at the same time (granted) does not mean that the pressures do not exist all the time.

The discussion that follows Mates' statement: "Robertson's model predicts, as he says, frequent extinction. His explanation for why everything doesn't go extinct is as follows" does not accurately state my position. I did not give _any_ explanation "for why everything doesn't go extinct." In fact, my point is exactly the opposite: with rare exceptions everything _does_ go extinct, but not all at the same time. There is no reason to expect that feedback loops will be synchronized across millions of species.

And many of the species that are driven to extinction by feedback effects _will_ leave remnant populations subject to further evolution. Those that do not leave such remnant populations simply become irrelevant to the future functioning of Darwinian selection.

These remnant populations are _not_ immune to feedback effects, but the feedback effects will be much smaller for smaller population sizes. In fact, the feedback effects generally become significant at exactly the point that the population approaches the carrying capacity of the environment and then continues to try to grow exponentially (as Malthus noted).

Mates goes on to say: "If observed in the fossil record, this would not at all look like punctuated equilibrium." I agree completely. This is _not_ the process that I claimed looks like punctuated equilibrium. Punctuated equilibrium is seen in the model in the case where the fitness function has two peaks, and neither peak is affected by the population except to the extent that there is a limit on total population size.

Mates then says: However, the model would additionally imply that most species should rapidly go extinct even when none of the traditional culprits (environmental change, the appearance of a competing species or a more effective predator, etc.) apply. We should see inexplicable gaps in the fossil record where a species simply disappears, leaving a perfectly good but unused niche; its prey species are still abundant and no other predator has yet evolved to live on them. There are many cases in the fossil record where species do go extinct without clear evidence of any of the "traditional culprits."

Mates states "What reason is there to think that Lingula doesn't compete strongly with conspecifics? Just because it's sessile (as an adult) doesn't mean it doesn't compete. For one thing, it reproduces by spermcasting, so every male Lingula is in competition with every other male in the ocean!" It seems obvious that spermcasting is a less strong mode of competition than using oversized antlers to drive off competing males.

Further, spermcasting is not directly related to any of the phenotypic features that are observed in the fossil record, such as shell size and shape, whereas antler size _is_ one of the parameters observed in fossil deer and elk.

Next, Mates statement: "Basically, Robertson's model predicts that an isolated population of organisms in a very static environment is most likely to go extinct. But we find many long-lived, morphologically primitive taxa in such environments-deep-water seamounts, for instance." completely misstates the point of my arguments. The populations that are most likely to go extinct are those that most strongly affect their own adaptive environment, not necessarily those in isolated populations or in "static" environments. In fact, I object to the very notion of a "static" environment. This idea generally results from focusing attention only on the physical component of the environment (temperature, rainfall), a common mistake, and ignores the very component that is most significant here, the biotic component (i.e., the presence of evolving competitors).

And if Mates is going to cite crocodiles as an exception, he should discuss the many similar and related species, from phytosaurs to champsosaurs, that did go extinct. It is not clear that the present-day crocodilian species are exactly the species that lived 100 million years ago or are merely the latest in a succession of similar species.

Finally, Mates last point that species that are competitive components of their own adaptive environment are often spectacularly successful is exactly the point made in several of my papers, and is exactly a prediction of feedback models. In addition to driving species to extinction, runaway feedback loops will drive species to explore vast regions of phenotype space that would be explored only slowly or not at all without runaway feedback loops.

We should therefore expect to see them occupying vast swaths of phenotype space.

Anton

Given that you wrote a 2500 word reply, I don't think there is any need for you to apologize for the length of time that it took, particularly given the level of careful thought that went into your reply. It's probably going to take me a couple of days to respond fully. But I think I can make that response. I think we are actually not as far apart in our opinions as it may appear at first.

In the meantime if you would like to read my four papers on the subject, they can all be downloaded by mouse-clicks on the references section of the website noted earlier:

http://cires.colorado.edu/~doug/extinct/

I do not claim that these papers are perfect or have all the answers you are looking for. I'm still learning, myself. And your critiques are forcing me to think things through a little farther yet.

Doug and Anton:

I am wondering if ANY model can account for all the details Anton raises. Doug mentions that his initial approach is to start with a simple enough model that it has a low probability of being wrong. To me this means a phenomenological model that captures SOME of the gross features that more detailed models with understood mechanisms should replicate.

Biological systems, especially evolutionary systems, have much more contingency in them than do any of the models of physical systems that engineers and physicists deal with. This suggests to me that any model one tries to build to represent an evolutionary system must necessarily restrict itself to an idealization in which contingency is excluded.

Then the question becomes, does the model EXPLAIN anything in the absence of contingency? This places a very heavy burden on understanding the underlying mechanisms that went into the model. It also makes comparing results with real world examples, as Anton is doing, much more difficult because we don't have a really good way to account for those possible contingencies. In order to do this successfully, one would have to find populations that have existed in environments relatively free of contingency (how would we know what these are?), but then we would probably be dealing with a population that was not very interesting and for which a model doesn't add much to our understanding.

This not to say, of course, that modeling is useless. My take on it is that it gives us a chance to see if proposed mechanisms produce what we see. But that means getting beyond phenomenological models. In biological systems, this is a kind of dilemma.

In the biological system, the output (the next generation phenotype characteristics) alters the gain of the amplifier (gain==fitness).

It does not refer to "back" in time, and if it did it would make no physical sense. Thus the discussion of "feed forward" does not seem to me to be very useful.

as Darwin noted; even highly specialized creatures like elephants could populate a world very swiftly if space allowed.

Anton:

Let's see if we can cut through some of the clutter and try to find common ground, a set of ideas that we can agree on, and we can then argue from there.

It might also be useful to proceed in small steps, to focus on agreement on one or two points at a time, rather than scatter our time and arguments all over the block.

It seems to me that the points that we most need to agree (or disagree) on are first, whether organisms are indeed critical components of their own adaptive environment (I don't see that there could be much room for argument here) and second, whether that implies a steady, ubiquitous selection pressure in certain particular directions in phenotype space, toward larger size, faster running speed, more drought tolerance, and so forth, _irrespective of the shape of the underlying fitness surface_.

I think that George Gaylord Simpson makes this point rather well:

"Even though individual animals may be perfectly adapted at a particular size level, in the population as a whole there is a constant tendency to favor a size slightly above the mean. The slightly larger animals have a very small but in the long run, in large populations, decisive advantage in competition for food and for reproductive opportunities and in escaping enemies. . . . This is, I believe, the causal background of the empirical paleontological principle that most phyla have a steady trend toward larger size." [Simpson, G.G., (1949). Tempo and Mode in Evolution, Columbia U. Press, New York, NY, p 86.]

It seems to me that the giraffe height example and the gazelle/cheetah feedback running speed feedback loops should probably be treated as gedanken experiments ("thought experiments," in English), in the sense that Einstein used the word. The point of a gedanken experiment is to strip away the overwhelming complexity of the real world and thereby try to focus only on the factors that are of fundamental importance.

In the famous example he used to illustrate relativity, Einstein envisioned a set of experiments taking place inside a closed railroad car moving in a straight line at a constant speed. Now critics could have argued that real railroad cars never travel at an exactly constant speed, that there are vibrations, that you have to allow for the curvature of the Earth as well as its rotation and motion around the sun, not to mention the rotation of the Milky Way galaxy, that there is wind coming through cracks in the boxcar doors which cannot be perfectly sealed, and so forth. All of these corrections would indeed make the gedanken experiment more realistic, but all of them would tend to obscure the fundamental points that Einstein was trying to make.

Similarly, we could argue that for real giraffes there are two sexes with different average heights, and that a sexual competition feedback loop may be involved, and some giraffes are juveniles of shorter height, and the giraffes also need to find water (a serious problem for giraffes), and there are lions lurking at the water holes, and so forth. There is literally no limit to the factors that could be considered here, an endless regress of "improvements," all of which tend to obscure rather than illuminate the main point.

Similarly, for the running-speed gedanken experiment, it isn't actually necessary for the gazelle to run faster than the cheetah when you add in the factor of endurance. The gazelle can be slower, so long as he is not caught before the cheetah has to stop to catch his breath. And this is a common trade-off: The gazelles are slower but have greater endurance. This raises the question of the distance between the animals at the start of the chase, which determines how long the chase must continue for a given speed differential. And this raises the question of how good the gazelle's sensory apparatus is (can the cheetah get close enough before he is seen or smelled?), and how much cover is available to the cheetah, and so forth, again an endless regression of complications that obscure the fundamental fact that there is almost always an advantage to being a faster-than-average cheetah or gazelle.

In the real world these feedback loops tend to be quite finely tuned, so that gazelles are just fast enough that only the slowest are generally caught, and the acacia trees (as you note) are just a little bit taller than the average giraffe. And this is exactly the point where the selective pressures generated by feedback loops will be greatest, because the slightly faster gazelle or taller giraffe will have access to more foodstuff than the average, and the slightly higher leaf or faster gazelle will have a significantly improved probability of survival.

The feedback loops are finely tuned because there is a disadvantage to being too far away from the mean; if the tree is too tall, as you note, there is a cost to being tall with no comparable benefit in avoiding predation. I think this is related to your initial point, but rather than preventing the feedback loop, as you suggest, this factor will only ensure that the feedback loop is generally tuned closely so that it operates with maximum effectiveness.

If we can agree on just these points, then I think we might have the substance of a very interesting publication here. The factor of population size that is raised in my previous submission is one that has not been covered in any of my previous publications, and it raises the interesting question of whether Darwinian selection might operate in radically different modes and directions, depending only on the size of the population relative to the carrying capacity of the environment. I think this shift (from no-feedback to feedback) probably happens, and is not understood or covered in the literature. In other words, selection works in the naive fashion that most biologists expect when populations are small, and then goes of in different and perhaps opposite directions when populations are large. This is an idea that came to me as I considered your question as to why the remnant populations are not subject to feedback effects.

Doug said:

"There is literally no limit to the factors that could be considered here, an endless regress of "improvements," all of which tend to obscure rather than illuminate the main point."

I think his captures many of the issues that are being discussed here. In the physics examples you gave, those other effects were small perturbations which can be quantified and shown to be insignificant compared to the effects being studied.

In biological systems, this is much more difficult to discern. Even in restricting our study to a single parameter (e.g., a single phenotypic trait), we don't necessarily know if other traits are "orthogonal" (as we say in math and physics) to the one we are studying. In fact, as Doug points out in some of his examples, they clearly are not. I think we are agreeing here. So in these instances, other traits could be major contingencies (in addition to those in the environment) that we are ignoring to our peril.

Now, I agreed that Doug's equating gain to fitness makes the feedback analogy more apt in his model. But this is still a phenomenological abstraction, and I am still thinking that causality in inherited systems like biological organisms still argues for the feed forward ideas (perhaps there may be exceptions with lateral transfer in microscopic systems but, not being a biologist, I don't know).

Torbjorg Larsson brings up state machines. I have had only very limited exposure to these, but I believe the way they are constructed focuses on causality throughout (at least in the construction stages of these machines). Feedback in these systems arises from the phenomenological perspective one can take on them AFTER they are complete.

Perhaps the clue in effective modeling of evolutionary systems in biology is to pick systems for which there is enough qualitative information to allow decisions about what effects are perturbations, what effects are sufficiently orthogonal, and what contingencies can be ignored. Since I have not been immersed in the biology community, I am not qualified to suggest such systems. And I don't know how the modeling theorists and the field biologists (analogous to the experimentalists in physics) coordinate their efforts to do this. Are there any biologists lurking out there who can weigh in on this discussion?

(By the way, as Torbjorg pointed out, I did note in an earlier post that we usually don't think of feedback as going backward in time. I did, however, have cases in mind in which we do this with general relativistic models that explore the properties of spacetime under extreme conditions. But I don't know of any analogies in biological systems for which this would be appropriate. :-) )

I have had only very limited exposure to these, but I believe the way they are constructed focuses on causality throughout

In the meantime if you would like to read my four papers on the subject, they can all be downloaded by mouse-clicks on the references section of the website noted earlier:

— Doug Robertson

I am wondering if ANY model can account for all the details Anton raises.

— Mike Elzinga

Doug mentions that his initial approach is to start with a simple enough model that it has a low probability of being wrong.

It seems to me that Anton Makes is making a serious conceptual error when he says: "The trouble with that, to my mind, is that a sufficiently simple model almost certainly is wrong" and in the arguments surrounding this statement.

If we transpose the argument into physics, a simple Newtonian model says that an object in motion tends to stay in motion with the same velocity unless acted on by an external force. In contrast, Aristotelian physics says that an object in motion tends to come to rest. Every experiment we could try is actually in better agreement with the Aristotelian model than the Newtonian one (Aristotle was wrong, but he wasn't stupid). But by Makes' argument we should discard the Newtonian model because it is not in perfect agreement with any experiment. The point here is that in physics we need to correctly separate two effects, Newtonian inertia and friction. Aristotle lumps them together into a theory that is qualitatively correct but sterile. Separating the two effects in the Newtonian fashion leads to a much more fruitful quantitative understanding of classical dynamics.

Thus to say that Malthusian models predict "completely wrong long-term behavior" is not the right way to look at things. As Makes notes, the Malthusian model is correct in the case that the population is small compared to the carrying capacity of its environment. As the population approaches the carrying capacity additional effects begin to kick in, such as differential starvation or lack of whatever the limiting resource happens to be, and the Malthusian exponential growth is modified to avoid an unsustainable unlimited population growth.

I am arguing that, in a Newtonian fashion, we need to attempt to separate out all of the competing forces in Darwinian theory much as Newton did for classical mechanics. And admittedly this is a more difficult problem than the one Newton solved in physics. I agree that we will then start out with simple models that may be in poor accord with experimental data, but as we learn to combine the various effects correctly we should approach a model that agrees well with observations and at the same time gives us a better understanding of the underlying dynamics.

Separating out the various evolutionary forces is a difficult problem but it is not completely intractable if we set up our simple "gedanken" experiments correctly. By "correctly" I mean that we should start out with ideas that are simple enough that there is no significant probability of their being wrong. Obviously Darwin took the first critical step in this direction by identifying natural selection as the principal driving force in evolution. But even Darwin recognized that there are other effects that need to be added to the theory, most obviously in his discussion of sexual selection. Later biologists added similar corrections with names such as frequency-dependent selection, density-dependent selection, Vermeij's "arms race" arguments, Van Valen's "Red Queen," and other names.

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.

This simple model reverses the expectations of a Darwinian model that does not include this particular observation. The feedback loops that are implied by this idea will tend to drive species away from the underlying peaks in fitness landscapes and directly toward extinction, exactly as is commonly observed in the fossil record.

This is the model that I am arguing provides the simplest basis for further unraveling the forces involved in real biological systems. And yes, it needs further corrections, most obviously realistic models for genetics and sexual reproduction, not to mention contingencies such as geographic separation of populations. But if we try to graft these corrections onto a model that lacks either natural selection or feedback effects, we will be in serious trouble, headed down Aristotelian paths that lump together effects that are better separated at the conceptual level.

If you read my papers on the subject, you will find that there is a third mathematical concept that must be added to the basic theory of Darwinian selection. This new idea is the concept of an iterated nonlinear mapping.

It is based on the unassailable observation that biological reproduction takes place in discreet generations, and the phenotypic parameters of the next generation are some nonlinear function of the current generation and environment.

This leads directly to the iterated nonlinear mapping theories developed by Robert May, Mitchell Feigenbaum, Benoit Mandelbrot, and others, and whose roots trace back to work by Henri Poincare and his student, Gaston Julia. This also takes us into the heart of modern chaos theory. Iterated nonlinear mappings are some of the strangest things ever encountered in mathematics. The famous Mandelbrot set is one of the simplest examples.

The immediate implication here is that evolution should be expected to behave in ways that are both complicated and counter-intuitive. No mathematician before Poincare imagined that an iteration of something as simple as (z^2 + c) on the complex plane would produce Julia sets and the Mandelbrot set. And biological evolution involves iterating a function that is not at all simple.

But one of the fundamental insights of chaos theory is that chaos is produced by iterating almost any nonlinear function. It doesn't matter what nonlinear function you choose to iterate. It is the process of iterating that is fundamental, that causes chaos in all sorts of mappings. And iteration is the concept that carries over directly into evolutionary biology.

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.

Thus it seems to me that the theoretical foundation for Darwinian evolution must rest on at least three fundamental conceptual ideas:

1. Reproduction with heritable variation plus natural selection (classical Darwinism).

2. Feedback loops that are produced when organisms are significant components of their own adaptive environment.

3. Iterated nonlinear mappings, because evolution takes place by discreet generations and each generation is some (complicated) nonlinear function of the previous generation and environment.

There may be other fundamental ideas required--I make no claims to omniscience. But it seems to me that these three concepts provide the minimum theoretical background against which the further details of evolution, including, as I said, genetics and sexual reproduction, geographic contingencies, physical limitations, and so forth can be studied.

Mike

Equating "gain" to "fitness" is more than just a phenomenological abstraction. In the computerized models that I have described in my publications, "fitness" is defined as the number of offspring produced, or more precisely as the number of offspring that survive and reproduce. This is a definition that is in common use, and it provides exactly the numerical value that I need to make the numerical models work. Thus "fitness," the number of offspring, is exactly the "gain" of the evolving system.

The other concepts that you discuss, such as orthogonality, perturbations and so forth are ideas that are extremely useful in the context of linear or linearized systems, but they carry over badly or not at all into the wildly nonlinear systems that are characteristic of evolutionary biology. I would say as a first approximation that orthogonality doesn't exist in biological systems (except for those systems that do not interact with each other at all, such as ecosystems on distant islands), and if there are rare cases where orthogonality does exist, it would be extremely difficult to demonstrate that fact.

So far as I am aware, none of the models I use entail any violations of the fundamental physical principle of causality. If they did, they would be in need of serious revision.

biological reproduction takes place in discreet generations

Dang--that passed my spell checker. I meant "discrete" of course. Most biological systems are not very discreet when it comes to reproduction. For that matter, neither are human populations any more, if TV and movies are a representative sample.

I am reminded of H.G. Wells' comment about editing: "No passion on Earth, not love nor hate, equals the desire to change someone else's draft."

Doug Robertson --- Perhaps more simply stated, evolution can be studied as a dynamical system.

I suggest you consider the Mathieu equation which I posted about earlier in this thread. It has escaping properties that appear to be of interest to you.

Also you may care to read The Simple Genetic Algorithm which offers a dynamical system perspective.

David Benson--I'm a geologist, and I have only limited experience with differential equations. (In the computer age they seem so nineteenth-century :) ).

I'm not sure what you mean by "consider the Mathieu equation." Are you asking me to find an analytic solution? Or write a FORTRAN package to integrate it numerically? The last time I looked at a differential equation, in analyzing the behavior of a gravity meter, the equation had a polynomial solution.

Doug --- The Wikipedia page on the Mathieu equation offers good guidance and references some of the literature. The point that might be of use to you is that for various pairs of parameters (a,q), solutions are unstable. Right at the critical boundary, the instability grows vary slowly. But once the solutions diverge enough, this is can be taken to represent extinction.

The Wikipedia page will be enough to convince you that writing a program is better than attempting analytical solutions in terms of the Matthieu functions, otherwise know as elliptic sines and elliptic sines.

As I mentioned in previous posts, this offers a variation on the foxes and hares so beloved of the population biology textbooks, offering extinction possibilities and also some rather bizarre frequency halving and doubling behavior, all without deterministic chaos (the problem is in only two dimensions).

Doug:

I thought I was saying that orthogonality is a difficult concept to apply in biological systems. In bringing this up, I was really directing much of my reply at Anton who was raising all of these issues. My apologies for not being clear.

The iterated nonlinear mappings you mention are what I was thinking were more appropriate for the kind of things we see in evolution. I didn't think to mention them in my reply to Anton.

In some of the modeling I am more familiar with, the rules start out simple, but then "emergent" rules evolve, and the iterated system develops with the emergent rules as they become more dominant in the succeeding generations. The emergent rules don't necessarily apply to the lower level entities that form in the iterative process, but to the emerging structures. In my mind, these are what I think of as involving feed forward ideas. Nothing gets fed into the current generation, and emergent phenomena do not appear until the accumulation of enough structure in subsequent generations. I would not have though of feedback as being possible in such a system. Your example caught me by surprise.

More interestingly, one can build iterative structures like this to see if there are emerging phenomena that lend themselves to simpler phenomenological models. In effect, we build a training model for other analytical techniques. An example would be a material in which an array of superconducting paths percolates with time throughout a material. The objective would be to see how an analytical technique of detecting the phenomenological magnetic susceptibility would appear in an actual measurement. I've also done this in the context of noisy environments in which one tries to image objects out there in a changing noisy environment. What phenomenological effects can be used to track objects that fade in and out and blend in with the noise?

It seems to me that the points that we most need to agree (or disagree) on are first, whether organisms are indeed critical components of their own adaptive environment (I don't see that there could be much room for argument here) and second, whether that implies a steady, ubiquitous selection pressure in certain particular directions in phenotype space, toward larger size, faster running speed, more drought tolerance, and so forth, _irrespective of the shape of the underlying fitness surface_.

— Doug Robertson

I think that George Gaylord Simpson makes this point rather well: "Even though individual animals may be perfectly adapted at a particular size level, in the population as a whole there is a constant tendency to favor a size slightly above the mean. The slightly larger animals have a very small but in the long run, in large populations, decisive advantage in competition for food and for reproductive opportunities and in escaping enemies.... This is, I believe, the causal background of the empirical paleontological principle that most phyla have a steady trend toward larger size." [Simpson, G.G., (1949). Tempo and Mode in Evolution, Columbia U. Press, New York, NY, p 86.]

Similarly, for the running-speed gedanken experiment, it isn't actually necessary for the gazelle to run faster than the cheetah when you add in the factor of endurance. The gazelle can be slower, so long as he is not caught before the cheetah has to stop to catch his breath. And this is a common trade-off: The gazelles are slower but have greater endurance. This raises the question of the distance between the animals at the start of the chase, which determines how long the chase must continue for a given speed differential. And this raises the question of how good the gazelle's sensory apparatus is (can the cheetah get close enough before he is seen or smelled?), and how much cover is available to the cheetah, and so forth, again an endless regression of complications that obscure the fundamental fact that there is almost always an advantage to being a faster-than-average cheetah or gazelle.

The factor of population size that is raised in my previous submission is one that has not been covered in any of my previous publications, and it raises the interesting question of whether Darwinian selection might operate in radically different modes and directions, depending only on the size of the population relative to the carrying capacity of the environment. I think this shift (from no-feedback to feedback) probably happens, and is not understood or covered in the literature. In other words, selection works in the naive fashion that most biologists expect when populations are small, and then goes of in different and perhaps opposite directions when populations are large.

This is an idea that came to me as I considered your question as to why the remnant populations are not subject to feedback effects.

David--I'm inclined to agree with your earlier comment:

"There is nothing special about the Mathieu equation other than its antiquity. One can easily specify other systems of first order, non-linear differential equations which behave in even more complex fashions. An example is a system of three such equations which approximately explain ice age climate (Saltzman)."

I'm definitely not a mathematical genius, and I have trouble seeing the relevance of this particular equation to evolution theory. I can see that it's helpful if you need to analyze an elliptical bongo drum.

Also I have serious problems with using differential equations to model biological processes in general. Almost no biological process has the kind of analyticity ("smoothness") that would make a differential equation approach rigorous. That does not mean that they cannot make useful approximations. But it seems to me that there is a better way to approach biological problems, one that begins with a better approximation and further is much easier to deal with conceptually and computationally. I'm referring to iterated nonlinear mappings modeled in computer software. They are inherently discrete and thus do not need any continuity assumptions.

Further, they are much easier to implement and much easier to interpret. You can start with an array that specifies population size as a function of body size (or any other selectively significant phenotype parameter), another array that specifies fitness or number of offspring as a function of body size, a parameter that specifies the width of the distribution of those offspring in phenotype parameter space, and so forth. Having done that, and implemented a few other computational and bookkeeping details, the iterations of the model are pretty well specified, and you can run the model and experiment with the parameter values that determine its behavior. All this is very much simpler and closer to the underlying biology than any differential equation approach that I am aware of.

I really do not mean to denigrate differential equations, which are inherently powerful and have vast applications, but I do mean to suggest that for problems in biology there are better modeling techniques, partly borrowed from chaos theory, that seem to me to be both easier to implement and easier to interpret.

I also mean to suggest that one reason that these techniques are not more widely used is that they were not feasible in the pre-computer age, and much of the presently understood mathematics of evolution theory was derived by individuals such as Wright and Fisher in the pre-computer era. Things are so much easier today than they were back then.

If we transpose the argument into physics, a simple Newtonian model says that an object in motion tends to stay in motion with the same velocity unless acted on by an external force. In contrast, Aristotelian physics says that an object in motion tends to come to rest. Every experiment we could try is actually in better agreement with the Aristotelian model than the Newtonian one (Aristotle was wrong, but he wasn't stupid).

— Doug Robertson

But by Makes' argument we should discard the Newtonian model because it is not in perfect agreement with any experiment. The point here is that in physics we need to correctly separate two effects, Newtonian inertia and friction. Aristotle lumps them together into a theory that is qualitatively correct but sterile. Separating the two effects in the Newtonian fashion leads to a much more fruitful quantitative understanding of classical dynamics.

Thus to say that Malthusian models predict "completely wrong long-term behavior" is not the right way to look at things. As Makes notes, the Malthusian model is correct in the case that the population is small compared to the carrying capacity of its environment. As the population approaches the carrying capacity additional effects begin to kick in, such as differential starvation or lack of whatever the limiting resource happens to be, and the Malthusian exponential growth is modified to avoid an unsustainable unlimited population growth.

Separating out the various evolutionary forces is a difficult problem but it is not completely intractable if we set up our simple "gedanken" experiments correctly. By "correctly" I mean that we should start out with ideas that are simple enough that there is no significant probability of their being wrong.

Obviously Darwin took the first critical step in this direction by identifying natural selection as the principal driving force in evolution. But even Darwin recognized that there are other effects that need to be added to the theory, most obviously in his discussion of sexual selection. Later biologists added similar corrections with names such as frequency-dependent selection, density-dependent selection, Vermeij's "arms race" arguments, Van Valen's "Red Queen," and other names. 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.

This simple model reverses the expectations of a Darwinian model that does not include this particular observation. The feedback loops that are implied by this idea will tend to drive species away from the underlying peaks in fitness landscapes and directly toward extinction, exactly as is commonly observed in the fossil record.

This is the model that I am arguing provides the simplest basis for further unraveling the forces involved in real biological systems. And yes, it needs further corrections, most obviously realistic models for genetics and sexual reproduction, not to mention contingencies such as geographic separation of populations. But if we try to graft these corrections onto a model that lacks either natural selection or feedback effects, we will be in serious trouble, headed down Aristotelian paths that lump together effects that are better separated at the conceptual level.

Also I have serious problems with using differential equations to model biological processes in general. Almost no biological process has the kind of analyticity ("smoothness") that would make a differential equation approach rigorous. That does not mean that they cannot make useful approximations.

— Doug Robertson

Anton

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." It is true that I tend to focus on one trait at a time, but that is largely because of the limits on my computer simulations. It seems to me quite likely that most organisms can be and are subject to selective pressures in many directions at once, larger size, faster running speed, more drought-tolerant, and so forth. The question of whether all of these pressures can be satisfied simultaneously is another matter altogether. There are always trade-offs, of course, and the trade-offs are not always obvious. For example, it might appear that there is a trade-off of larger body size for lower running speed, but a moose manages to be both larger and faster than I am, and has about the same running speed as the much smaller coyote. I think it is clear that species do manage to respond to selective pressures, never perfectly but often with surprising effectiveness and even with what might be called ingenuity (if you will ignore the anthropomorphizing implication of that last word, which is not intended).

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. 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.)

I think that this argument is rigorous, or at least not easily assailed. Questions about which direction the population moves, and how far, and how many competing pressures exist, and what biological/physical constraints exist, are all important to understanding the detailed biology but irrelevant to the basic argument here.

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. The argument is easier to see after watching the computer models operate.

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.

I think this addresses your discussion of running speed as well.

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.

As for your last question: "Why are the remnant populations not subject to feedback effects even though the primary population, which necessarily attains low size/density on its way to extinction, is still subject to them? It seems to me there would have to be some factor forcing the primarily population crash to happen relatively quickly, so that there's no time for the population to adjust to the no-feedback condition before going extinct." I think the answer is that when the fitness (reproduction rate) becomes low enough, the population decays exponentially and that exponential behavior is what makes the population crash quickly. That is certainly what happens in my mathematical models, and probably in the real world as well. (But remember we are only talking about increased probability of extinction in the real world.) The fact that both growth and decay occur exponentially is probably the main reason that behavior resembling punctuated equilibrium is so commonly observed in my models. And such exponential behavior is also characteristic of real biological growth and decay.

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

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

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.

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.

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.

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 wish I knew how to put quotes into little boxes.

Blah blah blah

Blah blah blah

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."

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.

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.)

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.

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.

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

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.

— Doug Robertson

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.)

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.

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.

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.

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

By the usual definition of analytic function

r exp(-r)

does indeed have a power series expansion and is analytic.

By the usual definition of analytic function r exp(-r) does indeed have a power series expansion and is 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.