Large-scale evolution and extinction in a hierarchically structured environment

Abstract
A class of models for large-scale evolution and mass extinctions is presented. These models incorporate environmental changes on all scales, from influences on a single species to global effects. This is a step towards a unified picture of mass extinctions, which enables one to study coevolutionary effects and external abiotic influences with the same means. The generic features of such models are studied in a simple version, in which all environmental changes are generated at random and without feedback from other parts of the system.

arXiv:adap-org/9803001v1 10 Mar 1998
Large-scale evolution and extinction in a hierarchically s tructured
environment
C. Wilke, S. Altmeyer and T. Martinetz
Ruhr-Universit¨ at Bochum
D-44780 Bochum, Germany
e-mail: Claus.Wilke@neuroinformatik.ruhr-uni-bochum.de
Abstract
A class of models for large-scale evolution and mass
extinctions is presented. These models incorporate en-
vironmental changes on all scales, from inﬂuences on a
single species to global eﬀects. This is a step towards
a uniﬁed picture of mass extinctions, which enables
one to study coevolutionary eﬀects and external abi-
otic inﬂuences with the same means. The generic fea-
tures of such models are studied in a simple version,
in which all environmental changes are generated at
random and without feedback from other parts of the
system.
Introduction
In the history of the Earth, there have been several
catastrophic events which in a short period of time
have wiped out large parts of the existing species. The
amount of species annihilated in such events has been
up to 96% of the biodiversity at that time (Raup 1986).
It has often been argued that these mass extinctions
must have been caused by some disastrous abiotic inci-
dences like extraterrestrial impacts. Evidence in favor
of that has been put forward (Alvarez 1987), but on
the other hand, only 5% of the total loss of biodiversity
in the fossil record can be connected to mass extinc-
tions. The rest are the so-called background extinc-
tions, which happen on much smaller scales. Interest-
ingly, the two types of extinction cannot clearly be dis-
tinguished from another in the frequency distribution
of extinction event sizes. The event sizes’ distribution
forms a smooth curve, very close to a power-law (Sol´ e
& Bascompte 1996).
In order to explain a single smooth distribution,
the idea of coevolutionary avalanches has been de-
veloped (Kauﬀman 1992). The extinction of a single
species might cause another species to die out, which
might drive a third species into extinction and so on,
producing an avalanche that in principle could span
the whole system. Because of the diverging mean
avalanche size, the distribution of extinction events
would then be a power-law, similar to the situation
of thermodynamical systems at the point of a phase-
transition. Nevertheless, this mechanism, called self-
organized criticality, completely neglects external in-
ﬂuences that certainly are present.
On the contrary, as it has recently been shown, a
power-law distribution of extinction events can appear
even in a system in which species are wiped out solely
because of external inﬂuences (Newman 1996). How-
ever, this eﬀect depends crucially on inﬂuences that
are imposed on all species coherently.
From the point of view of a single species it does
not really matter whether it has to struggle with bad
conditions imposed externally, e.g., a global shift in
temperature, or with bad conditions due to heavy com-
petition with other species. All that counts for a single
species is whether it can keep up with its environment
or not.
A species goes extinct when its population decreases
to zero. This can happen for several reasons. One is a
loss of habitat. Climatic or tectonic changes aﬀect the
location and the size of a species’ habitat. If the size
decreases rapidly, the species may not be able to adapt
fast enough to ﬁnd a new niche. Then the population
will drop below a level at which it can sustain itself and
the species will die out. Another reason for species’
extinction is the invasion of new competitors or new
predators. Competitors that invade a territory may be
better adapted to a niche than the species originally
occupying this niche. In this case, the population of
the native species can be decimated so eﬀectively that
it is wiped out. The same thing can happen because
of an invading predator superior to the defense mech-
anisms of the species. Similarly, new parasites can sig-
niﬁcantly reduce the population of a species and drive
it to extinction.
From the species point of view, all the above cases
can be subsumed under the notion of stress. A species
suﬀers stress of various kinds, stress because of climatic
changes, stress because of competition and predation

. . .
. . .
global stress
stress on large regions,
like continents
stress on smaller regions
stress on a single species
Figure 1: Stress is generated in a tree structure.
etc. If the stress exceeds the level a species can sustain,
it will go extinct.
We are going to develop a model in which all causes
for the extinction of a species will be regarded as stress.
Every species i has a threshold xi, or in general a vector
xi, against stress. If a species suﬀers a stress ηi > x i,
or in the general case a vector of stresses η i, where at
least one component exceeds the corresponding com-
ponent of the threshold vector xi, it dies out. So far,
this is a very general approach for a model in which
species are the smallest units considered, i.e. a model
that does not work with individuals or populations.
Clearly, in such a model there will be stress on sev-
eral scales. We have global stress like a global shift in
temperature due to a slight change in the orbit of the
Earth around the sun, or the impact of a very large
meteor. Then we have stress that spans large parts of
the Earth, e.g. a continent or a hemisphere, like the El
Ni˜ no phenomenon that roughly spans the region about
the tropical Paciﬁc Ocean. And ﬁnally, we have stress
that aﬀects smaller regions, or only a single species.
This leads us to a hierarchically ordered system of en-
vironmental stresses. The simplest way to model it is
to generate stress in a tree structure, as it is shown in
Fig. 1.
On all scales, the stress can be abiotic or biotic.
This may sound a bit counter-intuitive, since abiotic
changes are usually taken as large-scale phenomena,
and biotic factors are usually taken as local phenom-
ena. Abiotic changes happen often on a global scale,
like the above mentioned examples of the orbit shift
of the Earth or the meteor impact. But clearly there
are more localized events. A small meteor or a small
vulcano may aﬀect only a limited number of species. If
a species happens to live only in a very small territory,
and this territory gets destroyed by a meteor impact,
the species may be the only one that goes extinct be-
cause of the impact.
On the other hand, biotic phenomena are not nec-
essarily localized. Although direct species competition
will usually be a local phenomenon, there can be also
global biotic phenomena. The composition of the at-
mosphere, for example, depends strongly on biotic fac-
tors, and it can change signiﬁcantly due to biotic ef-
fects.
So far, we have a model which represents the bio-
sphere as a tree, with species situated at the leafs, and
environmental stress generated at the nodes. Now we
have to choose the rules that determine how stress
is generated and what thresholds against stress the
species are given. This is the crucial part where we
decide what mechanisms we want to investigate. If
we were interested mainly in coevolutionary eﬀects, we
would choose rules that link the properties and actions
of the species directly to the generation of the stress.
In such a model, for example, the global stress at time
t could be some sort of a sum over all the adaptive
moves of the species at time t − 1. In this work, how-
ever, we are mainly interested in the generic features
we can expect from the hierarchical structure of the
biosphere. Therefore, we will focus on a version of
the model where the stresses and the species’ thresh-
olds are simply random variables. Species’ interactions
and abiotic eﬀects can be so complicated and so un-
predictable that in a ﬁrst approximation we want to
assume them to be completely random.
The model we study here is probably the simplest
possible. Yet it has some intriguing features which are
very similar to characteristics seen in the fossil record.
To keep our model simple, we choose a homogenous
tree, with l layers and n subtrees per node. In general,
of course, one has to deal with inhomogenous trees.
To each node of the ﬁnal layer we connect exactly one
leaf, where we put m species. An example of such a
tree with l = 4 and n = 2 is displayed in Fig. 2. The
total number of stresses that have to be generated in
one time step is
Nstress =
l−1∑
i=0
ni , (1)
and the total number of species in the model is
Nspecies = mnl−1 . (2)
Every species i has a single threshold xi, chosen at
random from the uniform distribution on the intervall
[0; 1). At every node j, the stress ηj generated in one
time step is a positive, real random variable drawn
from a distribution with probability densitiy function
(pdf) pj(x). It is a reasonable assumption to expect
smaller stresses to happen much more often than larger

stresses. Therefore, we use pdf’s that fall oﬀ relatively
fast with x → ∞ . An Exponential or Gaussian de-
crease should be a good choice, but the exact form of
the pdf is not really important. We choose the pdf’s
pj(x) at the beginning of the simulation at random
from some family of distribution functions and keep
this choice ﬁxed throughout the course of the simula-
tion.
Finally, we have to ﬁx the way a species is aﬀected
by stress generated on diﬀerent levels of the tree. We
simply take the maximum of all the stress values gen-
erated at nodes that lie above the species in the tree:
if at any of these nodes a stress ηj is generated which
exceeds the species threshold xi, this species goes ex-
tinct. It is then immediately replaced by a new species
with new random threshold.
In addition to the extinction dynamic, we introduce
some sort of adaption. In agreement with our idea of a
ﬁrst, simple model, the adaption is a random walk: in
every time step, a fraction f of the species is selected
at random and given new thresholds.
There are certainly some oversimpliﬁcations in this
model, such as the ﬁxed number of species or the fact
that all species have only one trait. We will return to
this later and explain why we can still expect to cover
the basic features of the extinction dynamic.
Analysis
The behaviour of the above introduced model can be
understood to a large extent from analytical calcula-
tions. But before we begin with our analysis, we note
that the mechanism for species extinction and adap-
tion presented here is similar to the one of the so-
called ’coherent-noise’ models introduced by Newman
and Sneppen (Newman & Sneppen 1996). These mod-
els display a distribution of extinction events that fol-
lows a power-law with exponent ≈ − 2, which is in good
agreement with the fossil record. For this reason, they
have already been used to study macroevolutionary
phenomena (Newman 1996; Wilke & Martinetz 1997).
The diﬀerence to our actual approach lies in the fact
that we use a multitude of stresses in a hierarchically
ordered system, whereas in the coherent-noise models
there is only a single stress, acting on the whole system
at once. Therefore, in the previous works the idea of
stress imposed on the species has been linked to exter-
nal inﬂuences like meteor impacts and was opposed to
coevolutionary eﬀects.
Note that we have eﬀectively a coherent-noise model
at every leaf of the tree if the number m of species
located at one leaf is large.
The eﬀective stress-distribution at a leaf
of the tree
Every leaf of the tree feels a stress-distribution which
depends on the distributions of the nodes above it. Let
there be N nodes above a leaf. Then the N stress
values having inﬂuence on this leaf are N random
variables X1, . . . , X N with pdf’s p1(x), . . . , p N (x). We
have to calculate the pdf pmax(x) of the random vari-
able Xmax = max{X1, . . . , X N }, i.e.,
pmax(x) dx = P (x ≤ max{X1, . . . , X N } < x + dx) .
(3)
With the partition theorem we can write the prob-
ability on the right-hand side as a weighted sum of
conditional probabilities:
P (x ≤ max{X1, . . . , X N } < x + dx)
=
N∑
i=1
P (x ≤ max{X1, . . . , X N } < x + dx
⏐
⏐
⏐x ≤ Xi < x + dx)
× P (x ≤ Xi < x + dx) . (4)
The conditional probabilities read
P (x ≤ max{X1, . . . , X N } < x + dx⏐
⏐
⏐x ≤ Xi < x + dx)
= 1
P (x ≤ Xi < x + dx)
× P (x ≤ max{X1, . . . , X N } < x + dx
∧ x ≤ Xi < x + dx)
=
P (x ≤ Xi < x + dx) ∏ N
j=1,j ̸=i P (x > X j)
P (x ≤ Xi < x + dx)
=
N∏
j=1,j ̸=i
P (x > X j) . (5)
After inserting Eq. (5) into Eq. (4) we ﬁnd
P (x ≤ max{X1, . . . , X N } < x + dx)
=
N∑
i=1
P (x ≤ Xi < x + dx)
N∏
j=1,j ̸=i
P (x > X j) .
(6)
Consequently, for the pdf pmax(x) we have
pmax(x) =
N∑
i=1
pi(x)
N∏
j=1,j ̸=i
P (x > X j)
=
N∑
i=1
pi(x)
N∏
j=1,j ̸=i
x∫
pj(x′) dx′ . (7)

We are interested in the tail of pmax(x). For coherent-
noise models we know that a power-law distribution
of event-sizes will appear if the stress-distribution
pstress(x) satisﬁes
∞∫
η
pstress(x) dx ≈ Cp α
stress(η) for η → ∞ , (8)
where C and α are positive constants which depend on
pstress(x) (Sneppen & Newman 1997). Therefore, we
assume this condition to hold also for the distributions
pj(x) in Eq. (7), with constants Cj and α j, respec-
tively. Then we can approximate the tail of pmax(x)
by
pmax(x) ≈
N∑
i=1
pi(x)
N∏
j=1,j ̸=i
(
1− Cjpα j
j (x)
)
for x → ∞ .
(9)
We proceed further by taking only linear terms in pi(x)
and obtain
pmax(x) ≈
N∑
i=1
pi(x) for x → ∞ . (10)
For large x, this sum will be dominated by the pi(x)
that is falling oﬀ slowest. We say that distribution
pi(x) falls oﬀ slower than distribution pj(x) if there
exists a x0 such that
pi(x) > p j(x) for all x > x 0. (11)
For a set of reasonable stress-distributions it is always
possible to identify one that is falling oﬀ slowest ac-
cording to this deﬁnition.
The fact that the sum in Eq. (11) will asymptotically
be dominated by a single term leads to the situation
depicted in Fig. 2. The tree breaks down into several
independent subsystems. The meaning of the numbers
in the ﬁgure will be explained in detail later. In a
nutshell, they indicate how slow a stress distribution
is falling oﬀ. What interests us here is the breakup of
the tree into several independent parts in the regime
of large stresses. If these parts are not too small, they
will behave like independent coherent-noise systems.
An ensemble of a ﬁnite number of
independent coherent-noise systems
If the stress-distributions close to the root dominate
the behaviour of the system, the tree will break down
into independent coherent-noise systems, as we have
mentioned above. Consequently, we proceed with the
calculation of the distribution of extinction events in
a system consisting of independent coherent-noise sub-
systems. In the calculation, however, we will deviate
3 14
12 5 1 11
9 2 13 8 15 10 4 2
12 12 13 8 15 14 14 14
Figure 2: The tree breaks down into virtually indepen-
dent parts in the limit of large stresses.
slightly from the actual situation in the tree model by
assuming the subsystems to have each an inﬁnite size.
This allows for an easy calculation, and the main re-
sults should also hold for large but ﬁnite sizes.
In the case of an inﬁnite system size, the event distri-
bution of a coherent-noise model possesses a power-law
tail that extends to arbitrary large events. Therefore,
the task of calculating the event distribution of the
compound system equals to the task of calculating the
sum of a ﬁnite number of nonidentically distributed
random variables with power-law tail. The latter can
be treated mathematically exact under relatively weak
assumptions (Wilke unpublished). But since the ex-
act calculations are too extensive to be included in
this work, we will here give only an intuitive argument
about the tail behaviour of the sum.
We begin with the sum of two positive, real random
variables X1 and X2, where the pdf’s p1(x) and p2(x)
have a power-law tail x−τ1 and x−τ2 , respectively. We
assume the pdf’s to be continuous, non-singular, and
reasonably smooth. Under these conditions, we can
write p1(x) and p2(x) in the form
p1(x) = f1(x)
(x + 1)τ1
, (12)
p2(x) = f2(x)
(x + 1)τ2
, (13)
where f1(x) and f2(x) are continuous, non-singular,
and reasonably smooth functions which tend towards
a positive constant for x → ∞ . The pdf psum(x) of
the sum X = X1 + X2 is the convolution of p1(x) and
p2(x):
psum(x) =
p1(x′)p2(x − x′) dx′

=
f1(x′)
(x′ + 1)τ1
f2(x − x′)
(x − x′ + 1)τ2
dx′ .
(14)
After a change of the integration variable to z = x′/x
we obtain
psum(x) =
∫ 1
f1(xz)
(xz + 1)τ1
f2(x(1 − z))
(x(1 − z) + 1)τ2
x dx′
= x1−τ1−τ2
∫ 1
f1(xz)
(z + 1
x )τ1
f2(x(1 − z))
(1 − z + 1
x )τ2
dx′ .
(15)
For large x, there are two main contributions to this
integral, at z ≈ 0 and at z ≈ 1, which stem from the
ﬁrst and from the second term in the denominator.
Since the denominators will become arbitrarily large
for large x, we can assume the other terms to be con-
stant in the regions where the main contributions come
from. Therefore, we ﬁnd
psum(x) ≈ x1−τ1−τ2
[
C1xτ2−1 + C2xτ1−1
]
, (16)
where C1 and C2 are positive constants. Obviously for
large x the term with the largest exponent will domi-
nate. Hence we have
psum(x) ∼ x− min{τ1,τ 2} . (17)
This result can be easily extended to the case of
an arbitrary ﬁnite number of random variables with
power-law tail by iteration. Asymptotically, the tail of
psum(s) will always be dominated by the contribution
from the term with the smallest exponent.
Back to the ensemble of inﬁnitely large coherent-
noise systems, we ﬁnd that it will display power-law
distributed event sizes, as its single constituents do.
If the subsystems’ stress-distributions are functionally
diﬀerent, the exponent of the compound system’s event
distribution will be the smallest of the subsystems’ ex-
ponents.
The above result should also hold in the situation
of ﬁnite coherent noise systems, as long as their total
number is small compared to their typical size.
Trees with random stress distributions
We argue above that in the limit of large stresses the
tree will break down into subsystems, virtually inde-
pendent of each other. The behaviour of our model
depends heavily on the size of the parts we ﬁnd. If
the diﬀerent parts are all very small, the system will
loose its coherent-noise characteristics. Instead of a
power-law distribution the extinction events will then
follow a gaussian distribution because of the central-
limit theorem. Therefore, in this section we will study
the distribution of the subsystems’ sizes that arises if
we randomly assign stress distributions to the tree’s
nodes.
We assume that the propability for a certain stress
distribution to be assigned to a certain node does not
depend on the position of the node in the tree. In other
words, we use the same set of stress distributions on all
levels of the tree. Furthermore, we assume that for any
two stress distributions we use we can identify one of
the two that falls oﬀ faster than the other one. Under
these conditions, we can study the structure of such
trees by simply assigning integers to the nodes of the
tree, where larger integers stand for distributions that
are falling oﬀ slower. If the set of possible stress distri-
butions is inﬁnite, the probability of ﬁnding two nodes
with the same distribution is zero. Consequently, in
a tree with n nodes, we will assign every integer from
1 . . . n to exactly one node. This is displayed in Fig. 2
for a tree with 15 nodes. For every leaf i of the tree
we can then deﬁne a characteristic number ai. This
number is the maximum of the nodes’ numbers en-
countered on the way from the leaf up to the root. All
the leafs with the same characteristic number belong
to the same subsystem. In the example of Fig. 2, we
have ﬁve subsystems in total. Three of them contain
only one leaf, one contains two and one contains three
leafs.
In general, we are interested in the distribution of
subsystems arising in large trees. Therefore, we have
done simulations in which we have several thousand
times assigned random integers to the nodes of a large
tree. For every single realization of the tree, we have
computed a histogram of the frequency of the diﬀer-
ent parts’ sizes. Finally, we have calculated the aver-
age of all the histograms. Fig. 3 shows the result of
such simulations for two diﬀerent trees with 10000 his-
tograms each. We ﬁnd the expected frequency f (k) of
large independent parts in the tree decreasing as a saw-
tooth function that follows approximately a power-law
with exponent − 2, independent of l and n. The sharp
peaks in the distribution arise whenever the size of a
complete subtree is reached. Therefore, we observe in
Fig. 3, e.g., the peaks in the distribution of the tree
with n = 10 appearing at powers of 10.
The power-law can be explained easily if we assume
the main contributions to come from complete sub-
trees. The expected frequency f (k) to ﬁnd an inde-
pendent subtree with b layers, which corresponds to a
subsystem of size k = nb, can be written as the number
of such subtrees in the whole system, N (b), times the
probability that any of these subtrees will be indepen-

dent of the rest, P (b). Hence we write
f (nb) = N (b)P (b) . (18)
The number of subtrees of size nb is N (b) = nl−b. For
the probability P (b) we ﬁnd
P (b) =
(
l − b +
b−1∑
i=0
ni
) −1
, (19)
which is simply the probability for the integer assigned
to the node at the root of the subtree to be larger
than all the other integers which are assigned to the
remaining nodes of the subtree and to the nodes above
the subtree. If we increase b by one, we get N (b + 1) =
nl−b−1 = N (b)/n . With slightly more eﬀort, we ﬁnd
also
P (b + 1) =
(
l − b − 1 +
b∑
i=0
ni
) −1
=
(
l − b + n
b−1∑
i=0
ni
) −1
≈ 1
n P (b) . (20)
Therefore, we can write
f (nk) ≈ N (k)
n
P (k)
n = n−2f (k) , (21)
which implies f (k) ∼ k−2.
The peaks in Fig. 3 appear whenever the size of a
complete subtree is reached, as we have noted above.
This means they are connected to the extremely regu-
lar structure of the trees we use in this work. There-
fore, we are currently investigating trees with irreg-
ular structure. For these trees, the spikes disappear
and, in log-log plot, the function f (k) becomes almost
a straight line with slope -2. From the simulations we
have done so far, we can say that this result is very gen-
eral and seems to be independent of the special trees’
properties.
Simulation results
Since we are interested in the typical behaviour of our
model, we have to do many simulation runs with dif-
ferent tree sizes and diﬀerent stress distributions at the
tree’s nodes. But the simulation of large trees is very
slow, and therefore it is hard to get a good sample of
the parameter-space. To overcome this diﬃculty we
have also done simulations based on the arguments of
the previous sections. As we have seen there, in the
limit of large stresses it is possible to map the leafs of
the tree onto a system consisting of several indepen-
dent coherent-noise models, with the sizes k of these
subsystems distributed according to k−2.
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frequency
size
Figure 3: The expected frequency for the occurence of
large independent subsystems decreases as a sawtooth
function that follows approximately a power-law with
exponent − 2. The upper curve stems from a tree with
l = 18 and n = 2. It has been rescaled by a factor
of 100 so as not to overlap with the lower curve. The
lower curve stems from a tree with l = 6 and n = 10.
In Fig. 4 we show a comparison between the full
simulation and the approximation. To come as close
as possible to the full simulation, we use the maxi-
mum of 5 independent, exponentially distributed ran-
dom variables as stresses for the independent coherent-
noise models, since for the tree we have likewise chosen
l = 5 and exponential stress-distributions. Clearly the
behaviour of the approximation is close to the one of
the full simulation, which veriﬁes the analytical reason-
ing of the previous sections. Both simulations display
power-law distributed extinction events. For the full
tree, we ﬁnd an exponent τtree = 2. 35 ± 0. 05, while for
the approximation, we ﬁnd τapprox = 2. 30 ± 0. 05. If we
consider the high level of abstraction from the tree to
an ensemble of coherent-noise systems, this agreement
is excellent.
Note that in comparison to a normal coherent-noise
model with only a single stress variable, the tree model
produces a signiﬁcantly larger exponent τ (If we run a
normal coherent-noise model with the stress distribu-
tion of the approximaton in Fig. 4, we get an exponent
τ ≈ 1. 8). The increased exponent τ has its origin in the
distribution of the subsystems’ sizes. The sizes scale
themselves, thus modifying the scale-invariant behav-
ior of the ensemble, compared to the one of a single
coherent-noise system.
Discussion
We have presented a model of large-scale evolution and
extinction that combines biotic and abiotic causes for
extinction within a single mathematical framework.

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frequency p /( s /)
extinction ev en t size s
Figure 4: The frequency of extinction events in the
tree model and in an ensemble of coherent-noise mod-
els. The lower curve stems from the simulation of a
tree with l = 5, n = 10 and m = 1, which amounts
to a total of 10 5 species. Stress distributions were as-
signed at random to the nodes of the tree. We used
exponentially distributed stress with σ between 0.03
and 0.05. The upper curve corresponds to the simu-
lation of an ensemble of coherent-noise models with a
total of 10 5 species, and with the sizes k of the sub-
systems distributed according to k−2. As stresses we
used the maximum of 5 exponentially distributed ran-
dom variables.
Furthermore, the model takes into account the hi-
erarchical structure of the biosphere. To the best
of our knowledge, the implications of environmental
changes happening on diﬀerent scales have not been
studied previously in macroevolutionary models. De-
spite the choice of completely random environmental
changes, the model has some interesting features. The
distribution of extinction events follows a power-law
with exponent in the region of 2 (note that the ex-
ponent depends on the choice of the stress distribu-
tion, as it is the case with coherent-noise models).
From the fossil record, a power-law distribution with
exponent τ ≈ 2 is reported for the extinction event
sizes of taxonomical families (Sol´ e & Bascompte 1996;
Newman 1996). Moreover, it is interesting to observe
the breakup of the tree into subsystems with sizes k dis-
tributed according to k−2. The power-law distribution
of the subsytem sizes implies that even in very large
trees we will ﬁnd large subsystems, governed mainly by
only a single stress distribution. Intuitively, we would
expect the subsystems to have roughly similar sizes,
and to enter the dynamic of the whole system on an
equal basis. But we observe exactly the opposite. The
subsystems’ sizes are scale-invariant, thus producing
a scale-invariant distribution of contributions to the
overall system’s behavior. In particular, only a small
number of large subsystems produces events on large
scales. This might be an explanation for the fact that
in such large and complex systems like the biosphere
we ﬁnd usually smooth frequency distributions of typ-
ical objects or events.
The model we have studied in this work is certainly
oversimpliﬁed. For that reason, we will close this paper
with some remarks about extensions to the model that
should be examined in a next step closer to biologi-
cal reality. First of all, it is certainly a severe restric-
tion to keep the number of species ﬁxed throughout
the simulation. Nevertheless, this is a restriction used
very often in models of macroevolution (Peliti 1997).
Only recently, work has been done where a change
in biodiversity is considered (Head & Rodgers 1997;
Wilke & Martinetz 1997). The behaviour of the model
we study here is governed by the coherent-noise dy-
namic. For this dynamic, it has been shown that it
can be generalized to include a variable system size
without loss of it’s main features (Wilke & Martinetz
1997). Therefore, we believe a ﬁxed system-size can be
justiﬁed in the present work. It should be possible to
extend our tree model to a model with variable system
size. Another severe restriction is the usage of only one
trait. But here a similar argument holds as in the case
of the ﬁxed number of species. A multi-trait version
of the original coherent-noise model has already been
studied (Newman in press). It behaves very similar to
the single-trait version.
Finally, we want to discuss the way we compute the
stress on a single species out of the multitude of stress
values, generated at the diﬀerent levels of the tree.
Throughout this paper, we have used the maximum of
the stress values. This allows for an easy and very gen-
eral analytical investigation. Another natural choice,
however, would be to sum up all the stresses. We have
also done some simulations in this fashion. The be-
havior of the system remains roughly the same. This
happens because in a ﬁnite sum of non-identically dis-
tributed random variables, we expect large values to be
dominated by a single term of the sum, similar to the
case of the maximum of several random variables. For
the sum of exponentially distributed random variables,
an easy calculation shows that this conjecture is indeed
true. With some more eﬀort, we can prove the same
for the sum of power-law distributed random variables,
as we have already done in this paper. Nevertheless,
in the general case with arbitrary distributions, the
conjecture is hard to demonstrate.

References
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24–33, July 1987.
[Head & Rodgers 1997] Head, D. A., and Rodgers,
G. J. 1997. Speciation and extinction in a simple
model of evolution. Phys. Rev. E 55:3312.
[Kauﬀman 1992] Kauﬀman, S. A. 1992. The Origins
of Order . Oxford: Oxford University Press.
[Newman & Sneppen 1996] Newman, M. E. J., and
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[Newman 1996] Newman, M. E. J. 1996. Self-
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[Newman in press] Newman, M. E. J. (in press). A
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org/9702003.
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[Sol´ e & Bascompte 1996] Sol´ e, R. V., and Bascompte,
J. 1996. Are critical phenomena relevant to large-
scale evolution? Proc. R. Soc. Lond. B 263:161.
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[Wilke unpublished] Wilke, C. unpublished. On the
sum of random variables with power-law tail.


Critical Exponent of Species-Size Distribution in Evolution

Abstract
We analyze the geometry of the species- and genotype-size distribution in evolving and adapting populations of single-stranded self-replicating genomes: here programs in the Avida world. We find that a scale-free distribution (power law) emerges in complex landscapes that achieve a separation of two fundamental time scales: the relaxation time (time for population to return to equilibrium after a perturbation) and the time between mutations that produce fitter genotypes. The latter can be dialed by changing the mutation rate. In the scaling regime, we determine the critical exponent of the distribution of sizes and strengths of avalanches in a system without coevolution, described by first-order phase transitions in single finite niches.

arXiv:adap-org/9804002v1 2 Apr 1998
Critical Exponent of Species-Size Distribution in Evoluti on
Chris Adami 1, Ryoichi Seki 1,2 and Robel Yirdaw 2
1California Institute of Technology, Pasadena, CA 91125
2California State University, Northridge, CA 91330
Abstract
We analyze the geometry of the species– and genotype-
size distribution in evolving and adapting populations of
single-stranded self-replicating genomes: here programs
in the Avida world. We ﬁnd that a scale-free distribution
(power law) emerges in complex landscapes that achieve
a separation of two fundamental time scales: the relax-
ation time (time for population to return to equilibrium
after a perturbation) and the time between mutations
that produce ﬁtter genotypes. The latter can be dialed
by changing the mutation rate. In the scaling regime,
we determine the critical exponent of the distribution
of sizes and strengths of avalanches in a system without
coevolution, described by ﬁrst-order phase transitions in
single ﬁnite niches.
Introduction
Power law distributions in Nature usually signal the
absence of a scale in the region where the scaling is
observed, and sometimes point to critical dynamics.
In Self-Organized-Criticality (SOC) (Bak, Tang, and
Wiesenfeld 1987, 1988), for example, power law distribu-
tions reveal the dynamics of an unstable critical point,
brought about by slow driving and a feed-back mecha-
nism between order parameter and critical parameter.
The critical dynamics is usually described within the
language of second-order phase transitions in condensed
matter systems (Sornette, Johansen and Dornic 1996),
but it can be shown that SOC-type behavior also oc-
curs within a dual description in terms of the Landau-
Ginzburg equation as ﬁrst-order transitions (Gil and Sor-
nette 1996). Indeed, it was shown that a power law
distribution of epoch-lengths, that is, the time a par-
ticular species dominates the dynamics of an adapting
population, is explained by a self-organized critical sce-
nario (Adami 1995) that carries the hallmark of ﬁrst-
order phase transitions. Here, we measure the distribu-
tion of abundances of species and genotypes in an artiﬁ-
cial chemistry, (the Avida Artiﬁcial Life system, Adami
and Brown 1995, Ofria, Brown and Adami 1998) and
show that the distribution is scale-free under a broad
class of circumstances, conﬁrming the results reported in
Adami (1995). In the next section, we discuss the ﬁrst-
order dynamics in more detail and examine “avalanches
of invention” from the point of view of a thermodynam-
ics of information. In Section III, we measure the critical
exponent of the power law of genotype abundances in
the limit of inﬁnitesimal driving, i.e., inﬁnitesimal mu-
tation rate, and discuss the role of the ﬁtness landscape
in shaping the distribution. In Section IV, we repeat the
analysis for a higher taxonomic level (that of species)
and discuss its relation to the geometric distributions
found by Burlando (1990, 1993). Conclusions about the
evolutionary process drawn from the data obtained in
this paper are presented in Section V.
Self-Organization in Evolution
The idea that the evolutionary process occurs in spurts,
jumps, and bursts rather than gradual, slow and contin-
uous changes has been around for over 75 years (Willis
1922), but has gained prominence as “punctuated equi-
librium” through the work of Gould and Eldredge (1977,
1993). The general idea is that evolutionary innovations
are not bestowed upon an existing species as a whole,
gradually, but rather by the emergence of one better
adapted mutant which, by its superiority, serves as the
seed of a new breed that sweeps through an ecological
niche and supplants the species previously occupying it.
The global dynamics thus has a microscopic origin, as
shown experimentally, e.g., in populations of E. Coli by
Elena, Cooper and Lenski (1996).
Such avalanches can be viewed in two apparently con-
tradictory ways. On the one hand we may consider the
wave of extinction touching all species that are connected
by their ecological relations, a process akin to percola-
tion and therefore suitably described by the language
of second-order critical phenomena (Bak and Sneppen
1993). Such a scenario relies on the coevolution of species
(to build their ecological relations) and successfully de-
scribes power-law distributions obtained from the fos-
sil record (Sol´ e and Bascompte 1996, Bak and Paczuski
1996). There is, on the other hand, a description in
terms of informational avalanches that does not require
coevolution and leads to the same statistics, as we show

here. Rather than contradicting the aforementioned pic-
ture (Newman et al. 1997), we believe it to be comple-
mentary.
In the following, we set up a scenario in which informa-
tion is viewed as the agent of self-organization in evolving
and adapting populations. Information is, in the strict
sense of Shannon theory, a measure of correlation be-
tween two ensembles: here a population of genomes and
the environment it is adapting to. As described else-
where (Adami 1998), this correlation grows as the popu-
lation stores more and more information about the envi-
ronment via random measurements, implementing a very
eﬀective natural Maxwell demon . Any time a stochas-
tic event increases the information stored in the pop-
ulation, a wave of extinction removes the less adapted
genomes and establishes a new era. Yet, information
cannot leave the population as a whole, which there-
fore may be thought of as protected by a semi-permeable
membrane for information, the hallmark of the Maxwell
demon. Let us consider this dynamics in more detail.
The simple living systems we consider here are popu-
lations of self-replicating strings of instructions, coded in
an alphabet of dimension D with variable string length
ℓ. The total number of possible strings is exponentially
large. Here, we consider the subset of all strings cur-
rently in existence in a ﬁnite population of size N , har-
boring Ng diﬀerent types, where Ng ≪ D ℓ. Each geno-
type (particular sequence of instructions) is character-
ized by its replication rate ǫi, which depends on the se-
quence only, while its survival rate is given by ǫi/⟨ǫ⟩, in
a “stirred-reactor” environment that allows a mean-ﬁeld
picture. This average replication rate ⟨ǫ⟩ characterizes
the ﬁtness of the population as a whole, and is given by
⟨ǫ⟩ =
Ng∑
i
ni
N ǫi , (1)
where ni is the occupation number, or frequency, of geno-
type i in the population. As Ng is not ﬁxed in time, the
average depends on time also, and is to be taken over
all genotypes currently living. The total abundance, or
size, of a genotype is then
si =
∫ ∞
ni(t) dt =
∫ Te
Tc
ni(t) dt , (2)
where Tc is the time of creation of this particular geno-
type, and Te the moment of extinction. Before we obtain
this distribution in Avida, let us delve further into the
statistical description of the extinction events.
At any point in time, the fate of every string in the
population is determined by the craftiness of the best
adapted member of the population, described by ǫbest.
In this simple, ﬁnite, world, which does not permit
strings to aﬀect other members of the population ex-
cept by replacing them, not being the best reduces a
Figure 1: “Energies” (inferiorities) of strings in a ﬁrst-
order phase transition with latent heat ∆ ǫ.
old= 0
E new= 0
E
E
∆ε
string to an ephemeral existence. Thus, every string is
characterized by a relative ﬁtness, or inferiority
Ei = ǫbest − ǫi (3)
which plays the role of an energy variable for strings of
information (Adami 1998). Naturally, ⟨E⟩ = 0 charac-
terizes the ground state , or vacuum, of the population,
and strings with Ei > 0 can be viewed as occupying
excited states, soon to “decay” to the ground state (by
being replaced by a string with vanishing inferiority).
Through such processes, the dynamics of the system tend
to minimize the average inferiority of the population, and
the ﬁtness landscape of replication rates thus provides a
Lyapunov function. Consequently, we are allowed to pro-
ceed with our statistical analysis. Imagine a population
in equilibrium, at minimal average inferiority as allowed
by the “temperature”: the rate (or more precisely, the
probability) of mutation. Imagine further that a muta-
tion event produces a new genotype, ﬁtter than the oth-
ers, exploiting the environment in novel ways, replicating
faster than all the others. It is thus endowed with a new
best replication rate, ǫnew
best, larger than the old “best” by
an amount ∆ ǫ, and redeﬁning what it means to be infe-
rior. Indeed, all inferiorities must now be renormalized:
what passed as a ground state ( E = 0) string before
now suddenly ﬁnds itself in an excited state. The seed
of a new generation has been sown, a phase transition
must occur. In the picture just described, this is a ﬁrst-
order phase transition with latent heat ∆ ǫ (see Fig. 1),
starting at the “nucleation” point, and leading to an ex-
panding bubble of “new phase”. This bubble expands
with a speed given by the Fisher velocity
v ∼
√
D∆ ǫ , (4)
where D is the diﬀusion coeﬃcient (of information) in
this medium, until the entire population has been con-
verted (Chu and Adami 1997). This marks the end of

the phase transition, as the population returns to equi-
librium via mutations acting on the new species, creating
new diversity and restoring the entropy of the population
to its previous value. This prepares the stage for a new
avalanche, as only an equilibrated population is vulner-
able to even the smallest perturbation. The system has
returned to a critical point, driven by mutations, self-
organized by information.
Thus we see how a ﬁrst-order scenario, without coevo-
lution, can lead to self-organized and critical dynamics.
It takes place within a single, ﬁnite, ecological niche, and
thus does not contradict the dynamics taking place for
populations that span many niches. Rather, we must
conclude that the descriptions complement each other,
from the single-niche level to the ecological web. Let us
now take a closer look at the statistics of avalanches in
this model, i.e., at the distribution of genotype sizes.
Exponents and Power Laws
The size of an avalanche in this particular system can be
approximated by the size s of the genotype that gave rise
to it, Eq. (2). We shall measure the distribution of these
sizes P (s) in the Artiﬁcial Life system Avida, which im-
plements a population of self-replicating computer pro-
grams written in a simple machine language-like instruc-
tion set of D = 24 instructions, with programs of varying
sequence length. In the course of self-replication, these
programs produce mutant oﬀ-spring because the copy
instruction they use is ﬂawed at a rate R errors per in-
struction copied, and adapt to an environment in which
the performance of logical computations on externally
provided numbers is akin to the catalysis of chemical
reactions (Ofria, Brown and Adami 1998). In this ar-
tiﬁcial chemistry therefore, successful computations ac-
celerate the metabolism (i.e., the CPU) of those strings
that carry the gene (code) necessary to perform the trick,
and any program discovering a new trick is the seed of
another avalanche.
Avida is not a stirred-reactor environment (although
one can be simulated). Rather, the programs live on a
two-dimensional grid, each program occupying one site.
The size of the grid is ﬁnite, and chosen in these experi-
ments to be small enough that avalanches are generally
over before a new one starts. As is well-known, this is
the condition sine qua non for the observation of SOC
behavior, a separation of time scales which implies that
the system is driven at inﬁnitesimal rates.
Let τ denote the average duration of an avalanche.
Then, a separation of time scales occurs if the average
time between the production of new seeds of avalanches
is much larger than τ . New seeds, in turn, are produced
with a frequency ⟨ǫ⟩P , where ⟨ǫ⟩ is again the average
replication rate, and P is the mutation probability (per
replication period) for an average sequence of length ℓ,
P = 1 − (1 − R)ℓ . (5)
Figure 2: Fitness of the dominant genotype in the pop-
ulation, ǫbest as a function of time (in updates).
For small enough R and not too large ℓ (so that the
product Rℓ is smaller than unity) we can approximate
P ≈ Rℓ, and inﬁnitesimal driving occurs in the limit
⟨ǫ⟩Rℓ ≪ 1
τ . (6)
Furthermore
τ ∼ L
v (7)
with L the diameter of the system and v a typical Fisher
velocity. The fastest waves are those for which the latent
heat is of the order of the new ﬁtness, i.e., ∆ ǫ ∼ ǫ, in
which case v ≈ ǫ (because D ∼ ǫ in Eq. (4), see Chu and
Adami 1995) and a separation of time scales is assured
whenever
Rℓ ≫ L , (8)
that is, in the limit of vanishing mutation rate or small
population sizes. For the L = 60 system used here, this
condition is obeyed (for the fastest waves) only for the
smallest mutation rate tested and sequence lengths of
the order of the ancestor.
In the following, we keep the population size constant
(a 60 × 60 grid) and vary the mutation rate. From the
previous arguments, we expect true scale-free dynamics
only to appear in the limit of small mutation rates. As in
this limit avalanches occur less and less frequently, this
is also the limit where data are increasingly diﬃcult to
obtain, and other ﬁnite size eﬀects can come into play.
We shall try to isolate the scale-free regime by ﬁtting the
distribution to a power law
P (s) ∼ s− D(R) (9)

and monitor the behavior of D from low to high mutation
rates.
In Fig. 2, we display a typical history of ǫbest, i.e.,
the ﬁtness of the dominant genotype 1. Note the “stair-
case” structure of the curve reﬂecting the “punctuated”
dynamics, where each step reﬂects a new avalanche and
concurrently an extinction event. Staircases very much
like these are also observed in adapting populations of
E. Coli (Lenski and Travisano 1994).
As touched upon earlier, the Avida world represents
an environment replete with information, which we en-
code by providing bonuses for performing logical compu-
tations on externally provided (random) numbers. The
computations rewarded usually involve two inputs A and
B, are ﬁnite in number and listed in Table 1. At the end
of a typical run (such as Fig. 2) the population of pro-
grams is usually proﬁcient in almost all tasks for which
bonuses are given out, and the genome length has grown
to several multiples of the initial size to accommodate
the acquired information.
Table 1: Logical calculations on random inputs A and B
rewarded, bonuses, and diﬃculty (in minimum number
of nand instructions required). Bonuses bi increase the
speed of a CPU by a factor νi = 1 + 2 bi− 3.
Name Result Bonus bi Diﬃculty
Echo I/O 1 –
Not ¬A 2 1
Nand ¬(A ∧ B) 2 1
Not Or ¬A ∨ B 3 2
And A ∧ B 3 2
Or A ∨ B 4 3
And Not A ∧ ¬B 4 3
Nor ¬(A ∨ B) 5 4
Xor A xor B 6 4
Equals ¬(A xor B) 6 4
Because the amount of information stored in the
landscape is ﬁnite, adaptation, and the associated
avalanches, must stop when the population has ex-
hausted the landscape. However, we shall see that even a
‘ﬂat’ landscape (on which evolution is essentially neutral
after the sequence has optimized its replicative strategy )
gives rise to a power law of genotype sizes, as long as the
programs do not harbor an excessive amount of “junk”
instructions2. A typical abundance distribution (for the
1As the replication rate ǫ is exponential in the bonus ob-
tained for a successful computation, ǫbest increases exponen-
tially with time.
2 “Junk” instructions do not code for any information,
and do not aﬀect the ﬁtness of their bearer. Consequently,
programs with excessive amounts of junk code will give rise
to many “degenerate” genotypes with no competitive advan-
tage. In this regime, the genotype abundance distribution
is exponential rather than of the power-law type, due to a
run depicted in Fig. 2) is shown in Fig. 3. As mentioned
Figure 3: Distribution of genotypes sizes P (s) ﬁtted to
a power law (solid line) at mutation rate R = 0.004.
earlier, we can also turn oﬀ all bonuses listed in Tab. 1, in
which case ﬁtness is related to replicative abilities only.
Still, avalanches occur (within the ﬁrst 50,000 updates
monitored) due to minute improvements in ﬁtness, but
the length of the genomes typically stays in the range of
the ancestor, a program of length 31 instructions. We
expect a change of dynamics once the “true” maximum
of the local ﬁtness landscape is reached, however, we did
not reach this regime in the experiments presented here.
The distribution of genotype sizes for the ﬂat landscape
is depicted in Fig. 4. Clearly then, even such landscapes
(ﬂat with respect to all other activities except replica-
tion) are not neutral. Indeed, it is known that neutral
violation of condition (6).
Figure 4: Distribution of genotypes sizes P (s) for a land-
scape devoid of the bonuses listed in Tab. 1, at mutation
rate R = 0.003.

Figure 5: Fitted exponent of power law for 34 runs at
mutation rates between R = 0 .0005 and R = 0 .01 copy
errors per instruction copied. The error bars reﬂect the
standard deviation across the sample of runs taken at
each mutation rate. The solid line is to guide the eye
only.
evolution, where the chance for a genotype to increase or
decrease in number is even, leads to a power law in the
abundance distribution with exponent D = 1.5 (Adami,
Brown, and Haggerty, 1995).
In order to test the dependence of the ﬁtted expo-
nent D(R) [Eq. (9)] on the mutation rate, we conduct a
set of experiments at varying copy-mutation rates from
0.5×10− 3 to 10 ×10− 3 and take data for 50,000 updates.
Again, a “best” genotype is not reached after this time,
and we must assume that avalanches were still occurring
at the end of these runs. Furthermore, in some runs we
ﬁnd that a genotype comes to dominate the population
(usually after most ‘genes’ have been discovered) which
carries an unusual amount of junk instructions. As men-
tioned earlier, such species produce a distribution that
is exponentially suppressed at large genotype sizes (data
not shown). To avoid contamination from such species,
we stop recording genotypes after a plateau of ﬁtness
was reached, i.e., if the population had discovered most
of the bonuses. Furthermore, in order to minimize ﬁnite
size eﬀects on the determination of the critical exponent,
we excluded from this ﬁt all genotype abundances larger
than 15, i.e., we only ﬁtted the smallest abundances. In-
deed, at larger mutation rates the higher abundances are
contaminated by a pile-up eﬀect due to the toroidal ge-
ometry, while at lower mutation rates a scale appears to
enter which prevents scale-free behavior. We have not,
as yet, been able to determine the origin of this scale.
In the results reported here, we show the dependence
of the ﬁtted exponent D as a function of the mutation
rate R used in the run, which, however, is a good measure
of the mutation probability P only at small R and if the
sequence length is not excessive. As a consequence, data
points at large R, as well as runs where an excessive
sequence length developed, carry a systematic error.
For the 34 runs that we obtained, the power D was
measured for each run (for the low abundances), and an
average was calculated for all the runs at a particular
mutation rate. This data is plotted in Fig. 5 and shows
a plateau in the ﬁtted exponent only at intermediate
mutation rates, with D = 2.0 ± 0.05. A ﬁt of the middle
abundances (10-100) produces a critical coeﬃcient more
or less independent of mutation rate, around D = 2 .0,
but with less accuracy (data not shown). At high R, we
witness a deviation from scale-free behavior (reﬂected in
the rising D for small abundances) which is most likely
due to pile-up, i.e., a ﬁnite toroidal lattice. This eﬀect
may be avoided by using absorbing rather than periodic
boundary conditions. We also see a violation of scaling
at small R, which is due to the emergence of some other
scale. While it is most likely a ﬁnite-size eﬀect, the exact
origin of this scale is as yet unclear. We comment on the
signiﬁcance of these results in Section V.
Still, more control over the spread in exponents for
ﬁxed mutation rate would be desirable. This can obvi-
ously be achieved by plotting D versus P , rather than R,
for example, and by better keeping track of the coding
percentage within a genotype, a variable that we know
signiﬁcantly aﬀects the shape of the distribution. Such
experiments are planned for the near future.
Distribution of Species Sizes
In Avida, it is possible to monitor groups of programs
that display the same “phenotype”, while diﬀering in
genotype. Even though programs in this world are hap-
loid (single-stranded) and do not reproduce sexually, it
is convenient to label such groups taxonomically, i.e., we
refer to them as “species”. Strictly speaking, a species
consists out of all those genotypes that, when executed,
give rise to the same “chemistry”, i.e., such programs
diﬀer only in instructions that are either unexecuted,
or else are neutral. Algorithmically, the determination
whether two genotypes belong to the same species is
complicated by the fact that sequence length is not con-
stant in these experiments. Thus, we need to be able to
compare strings with diﬀering lengths, which is achieved
by lining them up in such a manner that they are identi-
cal in the maximum number of corresponding sites. Sub-
sequently, a cross-over point is chosen randomly and the
genomes above and below this point are swapped. In
other words, we construct a hybrid program from the two
candidates and test it for functionality, but without in-
troducing it in the population (see Adami 1998.) In the
experiments reported here, we actually test two cross-
over points in order to rule out accidental matches. In
retrospect, we ﬁnd that almost all those strings classiﬁed
as belonging to the same species by this method diﬀer
only in “silent”, or at least inconsequential, instructions.

Figure 6: Distribution of genotypes within species at
R = 0.004, ﬁtted to a power law with D = 2.44 ± 0.05.
The abundance distribution of genotypes within
species more closely corresponds to the kind of geomet-
ric distributions investigated by Willis (1922) as well as
Burlando (1990, 1993). Indeed, Burlando found, in an
analysis of distributions of subtaxa within taxa obtained
from the fossil record as well as recorded ﬂora and fauna,
that these distributions appear to be scale-free across
taxonomic hierarchies, with critical coeﬃcients between
2.0 < D < 2.5. This distribution can also be viewed as
a distribution of avalanches sizes, if avalanches are re-
deﬁned as events that spawn diﬀerent genotypes of the
same species. Indeed, in this manner it is possible to
investigate hierarchies of avalanches, each higher level
presumably sporting a higher critical coeﬃcient.
In the experiments reported here, we found species
coeﬃcients closer to D ≈ 2.5, but we also found viola-
tions of power-law behavior which are most likely due
to the contribution of species of diﬀerent lengths to the
abundance distribution. Indeed, the amount of “junk”
instructions in a species most likely governs the steep-
ness of the distribution, and several diﬀerent such species
may give rise to a multifractal distribution rather than a
pure power law. In the future, we expect to disentangle
such distributions by appealing to a an even higher level
in taxonomy, reuniting all species of the same sequence
length within a genus. The latter taxonomic level could,
for example, be entirely phenotypic, by keeping track of
which tasks a genus executes (irrespective of its geno-
type).
Still, even though changing sequence lengths aﬀect
the distribution of genotypes within species, those ex-
periments in which the sequence length does not change
signiﬁcantly can give rise to power laws with single ex-
ponents, as shown below in Fig. 6. The data for this
experiment were obtained from the same run as gave
rise to Figs. 2 and 3.
Conclusions
The distribution of avalanche sizes in evolving systems,
which is quite clearly related to the distribution of ex-
tinction events, can reveal a fair amount of information
about the dynamics of the adapting agents. For exam-
ple, purely random systems in which there are no ﬁt-
ness advantages, and where selection does not occur,
can still show power law behavior, as extinction events
are governed by the return-to-zero probability of random
walks (Adami, Brown and Haggerty 1995). In Avida,
we observe a scaling exponent D = 2 .0 in an interme-
diate regime of mutation rates. While it is still unclear
whether the mixing of scales that we have observed at
small and large mutation rates is due to the ﬁnite size
of the lattice or the emergence of another scale, we can
conclude with conﬁdence that scale-free dynamics does
occur. Scaling violations should be investigated by a
thorough ﬁnite lattice-size analysis, and this is planned
for the future along with more reﬁned methods for deal-
ing with explicit neutrality (i.e., “junk” code.)
An interesting hint at what the distribution might be
like in Nature comes from Raup’s analysis of a data set
prepared by Sepkoski (Raup 1991): genera of marine in-
vertebrates from the fossil record. Raup’s “kill-curve”
can be transformed into a distribution of sizes of extinc-
tion evens (as shown by Newman 1996) governed by a
critical exponent close to D = 2 .0. This is tantalizingly
close to the coeﬃcient we found in our genotype abun-
dance distribution, but we must be careful in comparing
these distributions.
The avalanche-size distribution of genotypes gives us
a good indication of the strength of an evolutionary
shock, but also about the length of time the particu-
lar species dominates the dynamics, and therefore, of
the time between evolutionary transitions. Also, each
evolutionary transition brings with it a wave of extinc-
tion, as all previously extant genotypes and species of
lower ﬁtness must disappear on the heels of the new
“discovery”. The size of extinction events proper, how-
ever, is not measured by the “epoch-length” distribution
reﬂected in the avalanche sizes, but rather by the abun-
dance of genotypes within species (or any higher taxo-
nomic abundance distribution) because each species ap-
pearing in this distribution must eventually go extinct,
and thus this distribution must equal the distribution
of extinction sizes. The latter distribution (measured in
Section IV), appears to have a critical exponent around
D ≈ 2.5, higher than the corresponding one from the
fossil record. Furthermore, we must keep in mind the
simplicity of the model treated here when comparing to
actual fossil data. As mentioned in the introduction, co-
evolution does not play a role in the dynamics control-
ling the size of avalanches in this model, while we must
assume that extinctions in Earth history have some co-
evolutionary component. On the other hand, the abun-

dance distribution of genotypes within species is consis-
tent with those obtained by Burlando (1990, 1993), who
argued that they represented evidence for a “fractal ge-
ometry of Nature”.
From the present analysis, it is clear that there is
as yet no reason to jump to conclusions from the evi-
dence extracted either from the fossil record, theoreti-
cal models of extinctions (Newman 1997), or else direct
implementation of the dynamics of adaptive avalanches
as we have done here. We do, however, see clear evi-
dence that avalanches not reigned in by any scale can
and do develop in evolving and adapting systems with-
out co-evolutionary pressures, via ﬁrst-order transitions
in populations occupying single ecological niches. Not
only do we ﬁnd scale-free dynamics for the time between
transitions (as evidenced by the genotype abundance dis-
tribution) but also for the strength of these transitions,
measured by the distribution of species-sizes. It is left
for future experiments to determine how such dynamics,
taking place in interacting ecological niches, gives rise to
power laws for co-evolutionary systems, and how the de-
scription in terms of ﬁrst-order transitions is ipso facto
transmutated into a second-order scenario.
This work was supported by NSF grant No. PHY-
9723972.
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