ABSTRACT: In 1859 Charles Darwin
published his masterpiece, On the Origin of Species. The critical
innovation in Darwin's work was the idea that natural selection is the
sufficient driving force behind the origin of species. But Darwin may have
wrought far better than he knew. Natural selection is a force that is powerful
enough to explain not only the origin of species but also the regular
extinction of species that is observed in the fossil record. It also explains
other conundrums of the fossil record such as Cope's rule and punctuated
equilibrium.
The main point that Darwin and
his distinguished successors failed to apprehend is that natural selection in
biological systems is a mathematically unstable process (Robertson,
1994). Of course it is no criticism of Darwin to say that he did not fully
understand all of the implications of his discovery. In Darwin's time the state
of biological knowledge as well as the state of many other disciplines,
especially mathematics, was inadequate to allow a complete investigation of all
of the implications of the radical new idea of natural selection. In
particular, Darwin did not have the computational resources that are available
with modern computers. This raw computational power allows us to develop
numerical models that facilitate making detailed explorations of the
mathematical implications of Darwin's momentous idea.
Darwinian natural selection is
based on the concept that individual organisms differ in a quantity called
"fitness," which is basically a measure of reproductive success. The
properties of fitness functions (what Sewall Wright called fitness
"landscapes") are therefore of central importance to any evolutionary
theory. But the study of the properties of these fitness functions has been
hampered by the difficulty of measuring this critical property, particularly in
the fossil record. Fitness is not a parameter like voltage or barometric
pressure that can be easily measured. The difficulty in measuring and observing
fitness values has caused a variety of theoretical problems, some of which can
be illustrated in the first two animations shown here.
Both animations show a schematic
fitness function as a red curve and a population distribution as a blue curve.
The x-axis represents a single parameter or dimension in phenotype space. This
parameter might be simply the size of the organism, perhaps its total weight,
height or length. But it could equally represent any one of a number of other
possibilities for selectively significant parameters including such things as
running speed or drought tolerance. The parameter is scaled to take on values
between 0 and 1.
Animation
number 1 shows a fitness curve that is fixed in time. The population
distribution begins with parameter values that are offset to the left of the
maximum fitness value; as time progresses the process of natural selection
moves the population closer to the extremum of the fitness function. (Notice
that in most web browsers the animation repeats indefinitely.) This is the
conventional picture of the operation of Darwinian natural selection; indeed,
there are many who think that this is the only way that selection operates. As
McMenamin put it: "Many paleontologists seem to assume that the adaptive
landscape is a fixed and static surface, underneath which plant and animal taxa
jockey for access to the highest peaks" (1990, p. 97).
The assumption of constant or
fixed fitness functions is not only common, it is often unnoticed. For example,
Mayr makes the statement: ". . . it is nevertheless evident that there are
definite limits to the effectiveness of selection. Nothing demonstrates this
more convincingly than the fact that 99.99 or more percent of all evolutionary
lines have become extinct. We must ask ourselves, therefore, why is natural
selection so often unable to produce perfection?" (Mayr, 2001, p 140).
Mayr's comment makes sense only in the context of fixed fitness functions. If
populations are "chasing" the extrema of a constantly changing
fitness function (as they commonly are) then the concept of
"perfection" has no meaning. And, as we shall see, the fact that
99.99 or more percent of all evolutionary lines have gone extinct is exactly
what we should expect from the action of Darwinian natural selection for
increased fitness.
Animation
number 2 shows a sequence that exhibits exactly the same change in
population distribution as the first one, but this time the fitness function is
not fixed in time. Its extremum is moving to the right, and the population,
again under pure Darwinian selection, is following the moving extremum of the
fitness functions.
This second animation presents us
with a serious problem because it shows that there are at least two distinct and
different processes that are both perfectly consistent with pure Darwinian
natural selection, and they are both able to produce exactly the same change in
observed population distributions. But the dynamics of the two processes are
totally different. If we observe changes in a population that are caused by
changing fitness functions, as in animation number 2,
and we try to interpret those changes as if they were produced by a constant
fitness function, as in animation number 1, we will be
making a serious error that will lead to a major misunderstanding of the
Darwinian selection process. Further, I suggest that exactly this error is both
common and frequently unsuspected in the literature.
To resolve the difficulty created
by the existence of two different Darwinian mechanisms as illustrated in the
first two animations, we need to begin by recognizing that both of these
processes can occur, and it is quite likely that both of them actually do occur
in the real world. And distinguishing between them is not an easy task because,
as we noted, measuring fitness functions is not easy. To distinguish them in
the real world we need to analyze and understand the behavior of fitness
functions in some detail. The principal difference between the two processes, i.e.,
the parameter that makes it possible to decide which process is operating in a
particular case, is the timescale involved. The timescale for the process in
the animation number 1 will generally be extremely
short: As Gould noted: "millions of years is too generous, too much for
sustained unidirectional change . . . All known and empirically studied cases
of gradualism at ecological scales would be completed in a geological twinkling
of an eye" (1990, p. 7).
On the other hand, there is no
natural timescale for the changes in fitness functions--they can and will occur
on nearly any and all timescales. Thus, ironically, natural selection is seen
to be too powerful a force to actually observe directly in the fossil record,
and nearly all of the observed changes in populations in the fossil record must
therefore result from changes in fitness functions as portrayed in animation number 2. There may be some exceptions, cases in
which natural selection operates on a slower scale, perhaps impeded by the
difficulty of producing the particular genetic combinations needed for a
complex phenotype character. But it is probably true that in most cases natural
selection operates on a timescale too short to observe.
If this is true then
"selective gradients" must be nearly always close to zero, i.e.,
populations are always close to the extrema of fitness functions (where the
gradient is exactly zero). In other words essentially all of the changes that
are observed in the fossil record are actually tracking changes in the
locations of the extrema of fitness functions, instead of the common assumption
that they are tracking the effects of fitness gradients. As Gould put it:
"The assumption of adaptive advantage for traits defining a trend has been
at the same time, both the most pervasive assumption of our literature and the
most frustrating and refractory to adequate demonstration" (Gould, 1990,
p. 20). Under the zero-fitness-gradient idea, Gould's "assumption of
adaptive advantage" is "refractory to adequate demonstration"
because it is false. Fitness values can remain constant while traits
"define a trend" as in animation number 2.
This idea will lead directly to the resolution of a number of conundrums in the
fossil record, including periodic extinction, Cope's rule and punctuated
equilibrium.
To demonstrate how these problems
are resolved we will need a detailed numerical analysis of the behavior of
fitness functions. This analysis will have to begin with a mathematically
precise definition of the term "fitness." Fortunately this is
straightforward: We will define the fitness of an individual as the number of
progeny that an individual produces, or, more precisely, the number of progeny
that survive and reproduce; progeny that do not survive and reproduce make no
contribution to Darwinian fitness. In mathematical terms this is a recursive
definition of fitness.
This definition has several
virtues, not the least of which is that it is commonly in use. Also it is
mathematically precise. And furthermore it provides exactly the numerical
quantity that is needed to develop computer models of the process of natural
selection. For if we are given a population distribution and a set of fitness
values for each member of that population distribution, then the product of
population value times the fitness value tells us exactly what the population
of the next generation will be.
One more thing is needed to complete
our numerical model of the behavior of fitness functions. We need to model the
essential property of heritable variation. To do that, we simply assume that
each individual's offspring (i.e., the product of the population value and the
fitness value) have phenotype values that are distributed in a Gaussian
("bell-shaped") fashion centered on the phenotype value of the parent
organism. Thus by definition the variation is unbiased. (In the following
animations the Gaussian function is shown in green a the lower left. The
function is symmetric, so only the right-half is shown) We also need a feedback
loop that scales the fitness values up if the population gets too small, and
down if the population becomes too large; this feedback loop prevents exponential
growth or decay of the population. The mathematical details of this model can
be found in Robertson and Grant (1996b).
This model contains nothing about
genetics or the details of the process of reproduction with heritable variation.
But neither does it exclude such detailed processes. Indeed, the model is
oversimplified in many ways, and it should be extended and its properties
explored in more biologically realistic contexts. However, there is reason to
think the results shown here are sufficiently robust that such exploration will
generally not contradict these results, and that any genetic model that
exhibits reproduction with heritable variation should be consistent with the
results shown here. Preliminary results with a multiple-allele model using
Hardy-Weinberg equilibrium inheritance statistics generally support this claim
(M.C. Grant, private communication, 2005).
This model captures in a
simplified way the essential elements of Darwinian evolution, i.e.,
reproduction with heritable variation and natural selection as characterized by
the shape of the fitness (red) curve. Animation number 3
shows how this model operates under the assumption that the extremum of the
fitness curve does not change with time, as in animation
number 1 above. (Notice that in animations 3 through 5 there are two
sequential population distributions shown at one time, the previous generation
in light blue and the next generation in blue. This is done to show explicitly
the one-generation transformation of the computerized model.) In this animation
the population is seen to move toward the extremum of the fitness curve and
stabilize there, just as is commonly expected for Darwinian evolution. Notice that
the population increases as it approaches the extremum of the fitness function,
as we might expect from the definition of fitness. Notice also that the fitness
values decrease as the population increases; they stabilize at a point where
the maximum fitness values are slightly greater than 1, the point where the
population reproduces itself and compensates for a slight "leakage"
in phenotype space caused by the heritable variability.
So far this model has not
produced anything very interesting, anything that would run counter to our
intuitive ideas about how Darwinian natural selection should work. But as we
make the model even slightly more complicated the behavior quickly becomes much
more interesting. Suppose we begin by investigating the case in which the
fitness function has more than one extremum. Animation
number 4 shows this case: the fitness function has a peak at a phenotype
value of 0.2 and a larger one at 0.6, and the initial population distribution is
arbitrarily cut off so that it has values of exactly zero for locations above a
phenotype value of 0.19. In other words, the initial population is entirely to
the left of the first peak in the fitness function. As seen in this animation
the population initially stabilizes under the left fitness peak, and remains
there for about a hundred generations. During this period the variation
programmed into the reproduction process causes the population to spread slowly
across the bottom of the graph. Around generation number 110 the portion of the
population that has developed under the right-hand peak begins to grow, and the
population under the left hand peak decreases sharply. Notice that the
intermediate population between the two peaks, which must exist under any model
that does not include saltation (large jumps in phenotype space), is always
small because its fitness values are small. There is a good chance that the
intermediate population would not be observed in the fossil record because of
its small size. Indeed, it is barely observable in this animation.
If this phenomenon were to be
observed in the fossil record, we would first find a population with a
parameter value centered on 0.2, and then after a time we would find it
replaced by a population centered at 0.6. This is a reasonable description of
what is called "punctuated equilibrium," as described by Eldridge and
Gould (1972). Gould famously argued that the observed fact of punctuated
equilibrium required a non-Darwinian explanation, what he termed
"hierarchial" selection, a selection process that involves
competition among species and higher levels of taxa, rather than competition
among individuals. As Gould put it: "Punctuated equilibrium validates the
hierarchial theory of selection" (Gould, 2002 p. 783). Of course, as this
model demonstrates, punctuated equilibrium does not validate any such thing. As
shown here, a model that contains nothing more than reproduction with heritable
variation plus natural selection is sufficient to produce punctuated
equilibrium. This implies that the observed fact of punctuated equilibrium
tells us essentially nothing about the process of evolution, nothing that we
did not already know from an analysis of the selection process. In particular,
it tells us nothing about hierarchal selection, contrary to Gould's claim.
This model also points up a
common error of logic in classical evolution theory: In evolving from phenotype
A to phenotype B, Darwinian evolution must operate by a sequence of small steps
in phenotype space, as Darwin himself stated. But it does not follow that all
of the intermediate phenotypes must be equally abundant and equally easy to
discover, either alive today or in the fossil record. On the contrary,
Darwinian theory predicts that such intermediate populations must be small,
because their fitness is low. We should therefore expect that the necessary
intermediate forms would not generally be observed. Yet the absence of observed
intermediate forms is one of the most frequently voiced objections to Darwinian
evolution. We should argue instead that the scarcity or absence of observed
intermediate forms is required by the mathematics of pure Darwinian
theory.
It might be noted that a few
hundred "generations" is a very short time, in geological terms.
However, the modeled "generations" do not necessarily correspond to
actual biological generations. There is no mathematical reason why a modeled
"generation" could not represent a large number of actual biological
generations. For that matter, the model parameters could be adjusted so that
the process seen in animation number 4 would take
geologically large numbers of computer cycles. This would be a bit tedious to
watch, however, and the parameters have been adjusted to make the process
reasonably easy to watch.
So far this numerical model has
dealt only with fitness functions that are fixed in time, except for the
scaling that is necessary to prevent runaway exponential population growth or decay.
But the formalism developed in Robertson and Grant (1996b)
lets us analyze the effects of much more interesting changes in fitness
functions. Of course there are many causes of changes in fitness, but there is
a fundamental dichotomy between two very different classes of causes of fitness
changes. The dichotomy is between changes in the physical environment and
changes in the biotic environment. Changes in the physical environment entail
such things as temperature and rainfall fluctuations, sea-level variations and
even asteroid impacts. These changes tend to be quasi-random; they also are not
very interesting mathematically, although they may be interesting biologically.
The changes in the biotic
environment are mathematically interesting because they engender mathematical
instabilities whose effects can be observed in the fossil record. In addition,
the biotic component of fitness often dominates over the physical. As Van Valen
noted: "There is evidence that most environmental pressure is biotic
rather than physical" (Van Valen, 1985).
The effect of the biotic
component of fitness functions can be expressed in a simple and self-evident
statement (Robertson, 1991, p. 470): The populations that are being altered by
evolution to adapt to their environment are themselves a significant component
of that environment. This indisputable fact has serious consequences: It
introduces the possibility of feedback loops in the process of Darwinian
selection, and the feedback loops introduce the mathematical instability from
which many of the observed properties of the fossil record can be derived. The
phenomenon can be restated in two deliberately symmetric statements (Robertson,
1994, p. 265):
1. Changes in fitness functions
can cause changes in the distributions of phenotypes.
2. Changes in the distributions
of phenotypes can cause changes in fitness functions.
The first statement is merely an
assertion of natural selection. The second is a direct consequence of the
effect of organisms on their own selective environment. Neither of these two
statements alone causes any problem; it is only when they are taken together
that they imply a significant instability in Darwinian processes.
To understand this instability we
need to model the effect that organisms have on their own adaptive environment,
i.e., on their own fitness values.
But how to we express this effect quantitatively? An important clue comes from
an observation made by George Gaylord Simpson:
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, 1949, p. 86).
The next animation 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).
This animation starts with a
population at an extremum in its fitness function, the extremum that would
exist if the population were absent, as shown the Figure 1.
The existence of this population creates an additional fitness component which
produces a new fitness extremum that is to the right of the maximum population
size, as shown in Figure 2, because there is a significant
fitness advantage to being slightly larger than average. The per capita fitness
advantage is shown in the magenta curve at the bottom left in both figures.
The further time development of
this model is shown in animation number 5. The
population, responding to Darwinian natural selection, moves toward the local
fitness maximum, but that maximum always stays to the right of the population
extremum because its location is controlled by that population. This is a classic
runaway feedback loop similar to those that cause poorly designed electrical
circuits to "smoke." It is also similar to a horse chasing a carrot
that is tied to a stick that its rider is holding in front of it. The
population moves steadily far to the right, following the local extremum of the
fitness function, and away from the fitness extremum that would exist in the
absence of that population. As Simpson noted, this model produces a monotonic
increase in phenotype size, a phenomenon which is observed sufficiently often
in the fossil record that it has been given a name: It is called Cope's rule of
phyletic size increase.
As the population moves in animation number 5, the fitness decreases because there is
some point where large size has significant disadvantages. (If this were not
true, then populations would evolve toward infinite size; this is clearly not a
physically realistic possibility.) At that point the population crashes. In
other words, the population has been driven directly to extinction through the
action of Darwinian natural selection. This is the most counter-intuitive
result from this model. 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 is perhaps best
illustrated by a thought-experiment: Suppose the optimum height for a giraffe
is twelve feet tall. Then by Darwinian selection, we would expect to find a
population of twelve-foot-tall giraffes. But as soon as that population exists,
there would suddenly be a fitness advantage to being thirteen feet tall because
the taller giraffes would have access to forage that is unavailable to their
shorter relatives. So we should expect that a population of thirteen-foot-tall
giraffes would develop, and the process would then repeat with still taller
giraffes until the overall population has been driven to low fitness values and
eventually directly to extinction.
It has often been noted that
conventional density-dependent or frequency-dependent selection models also
exhibit a decrease in fitness under pure Darwinian selection. Both of these
models fit the critical criteria described here, that biotic components of the
fitness function dominate over physical, and the populations that are adapting
under natural selection are themselves critical components of their own
adaptive environment. This condition is also met under Darwin's concept of
sexual selection, as well as Vermeij's arms-race models, and Van Valen's Red
Queen hypothesis. The mistake that is often made is to consider all of these
phenomena as special-case exceptions to the general behavior of Darwinian
selection, rather than as explicit examples of the general behavior of
selection under biotic selection pressure. This simple model involving feedback
effects under biotic selection pressure puts all of these phenomena into a
unified theoretical framework. Further, given that biotic selective pressure
generally dominates over physical, this is the way that Darwinian selection
should be expected to behave in general.
The major instability in
Darwinian selection as shown here results whenever there is a disadvantage to
being very large, but at the same time there is an advantage to being slightly
larger than the average of the extant population. Generalizations on this idea
for other types of selective parameter are easily made. 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.
This feedback model predicts both general extinction for most species and
exceptions for a small number of other species under pure Darwinian natural
selection, and is in good agreement with the fossil record. This is the
principal reason for the title of this website. Of course other causes of
extinction exist, including physical changes in the environment such as are
caused by asteroid impacts. But this analysis shows that Darwinian selection
should produce regular extinctions even in the absence of changes in the
physical environment.
This model illustrates why Mayr's
concept (in the quote cited above) that natural selection always approaches
phenotypic perfection is nonsense. If perfection is defined as the extremum of
the fitness function in the absence of biotic selective pressure, e.g., the 12-foot giraffes of the
thought-model above, then natural selection drives populations away from this
"perfection," and toward nothing other than the limits of physical
possibility.
Notice also that this model
provides a mechanism by which the natural selection process can
"explore" phenotype space rapidly and efficiently. In every direction
in phenotype space in which reproductive variation and biotic selective
pressure exist, these feedback loops will drive population components very
quickly to the limits of what is physically possible. We do not have to rely on
purely random variation to explore all of phenotype space. Purely random
explorations would be expected to move at a rate that scales roughly with the
square root of elapsed time (expressed in number of generations), while these
feedback processes could scale as linear rates in time, a significant speed-up
whose advantage grows with the number of generations.
Further development of these
ideas can be found in the references by Robertson and Robertson and Grant,
below. Additional numerical exploration of models of this sort is clearly
needed to further clarify our understanding of the operation and of the
underlying mathematical structures inherent in Darwinian natural selection.
Different functional forms for the shape of the fitness function and the
feedback functions could easily be explored, for example. And more biologically
realistic reproduction models might prove useful and informative; the present
model can be thought of as asexual reproduction, or nearest-neighbor mating in
sexual reproduction models. Finally the assumption that the entire fitness
function can be scaled by a single multiplicative parameter that adjusts to the
population size needs to be examined carefully. In all of these cases the
difficulty is that the number of possible alternatives is literally infinite,
and the challenge will be to find modeling options that are biologically
reasonable and computationally tractable.
References:
Dobzhansky, T., F.J. Ayala, G.L. Stebbins, and J.W.
Valentine (1977). Evolution, W.H. Freeman,
San Francisco.
Eldredge, N. and S.J. Gould, (1972). Punctuated
Equilibria: an Alternative to Phyletic Gradualism. In Schopf, T.M. (ed.) 1972: Models in Palaeobiology. San Francisco:
Freeman Cooper, pp. 82-115.
Gould, S.J., (1990). Speciation and Sorting as the Source
of Evolutionary Trends, or `Things Are Seldom What They Seem,' in Evolutionary
Trends, K.J. McNamara, ed., U. of Arizona Press, Tucson.
Gould, S.J., (2002). The Structure of Evolutionary Theory, Belknap Press, Harvard U.
Press,Cambridge, MA.
Mayr, E., (2001).What Evolution Is, Basic Books,
New York, NY.
McMenamin, M., (1990). The Cambrian Explosion, Palaios,
5, # 2, 97.
Robertson, D.S., (1991). Feedback
Theory and Darwinian Evolution, Journal of Theoretical Biology, 152/4,
469-484.
Robertson, D.S., (1994). Is
Darwinian Evolution a Mathematically Stable Process?, Evolutionary Theory,
10 (5), 261-272.
Robertson, D.S. and M.C. Grant,
(1996a). Feedback and Chaos in Darwinian Evolution: I. Theoretical
Considerations, Complexity, 2, #1, 10-14.
Robertson, D.S. and M.C. Grant,
(1996b). Feedback and Chaos in Darwinian Evolution: II. Numerical Modeling, Complexity,
2, #2, 18-30.
Simpson, G.G., (1949). Tempo and Mode in Evolution,
Columbia University Press, New York, NY.
Van Valen, L.M., (1985). A Theory of Origination and
Extinction, Evolutionary Theory, 7,
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