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Physics has been immensely successful in finding mathematical laws that
help it describe the Universe. All matter obeys Einstein’s equations of
general relativity. The way elementary particles interact are described
mathematically by the Standard Model. And Schrodinger’s equation describes
atoms, their nuclei, chemical compounds, crystals and many other states
of matter. Even the life and death of stars have well-established formulas.

But when it comes to the most complex system, mathematics has so far
failed. Life on Earth involves a myriad species interacting with each other
in ways that constantly change as they evolve, differentiate and become
extinct. There are no ‘Darwin’s equations’ to describe the evolution of
life on Earth into complex, interacting ecologies.

¿ìè¶ÌÊÓƵs accept that Darwin and his followers have convincingly described
the principles governing the evolution of life. Darwin’s theory of natural
selection identifies in a qualitative way the cause of evolutionary change:
natural selection operating through a struggle among individual organisms
for reproductive success. But the absence of a more quantitative framework
means that Darwin’s principles have not been tested against observation
in the same rigorous manner as physical laws. This also means that there
is no way of predicting the outcome of Darwin’s theory.

BURSTS OF CHANGE

Nevertheless, most scientists believe that life – or evolution – can,
in principle, be described by the laws of physics. So far, the problem has
simply seemed too big and scientists have concentrated on parts of the system,
creating such disciplines as population biology and biogeography. This approach
has been extremely successful in explaining many of the details of evolution.
But we argue that evolution must be viewed as a whole, and that an understanding
of it may not lie in those details. Over the past decade, for the first
time, new mathematical ideas used to describe the behaviour of large, interacting,
dynamical systems have begun to make this holistic approach possible.

The big new idea in this approach was conceived in the late 1980s by
Per Bak, a physicist at Brookhaven National Laboratory in New York state,
and Stuart Kauffman, a theoretical biologist at the Santa Fe Institute
in New Mexico. They argued that life is a dynamical system which, far from
existing in a state of balance or equilibrium, organises spontaneously into
a characteristic and much more precarious ‘critical’ state.

Surprisingly, this model of evolution predicts that life does not evolve
gradually but intermittently, with long periods of inactivity or stasis,
interrupted by bursts of change which are characterised by mass extinctions
and the emergence of many new species. It is just this pattern that many
palaeontologists say exists in nature. In 1972, Niles Eldredge of the American
Museum of Natural History and Stephen J. Gould of Harvard University proposed
from their study of fossil records that the evolution of single species
takes place in steps separated by long periods of stability. They named
this phenomenon ‘punctuated equilibrium’. During the 1980s, David Raup and
John Sepkoski at the University of Chicago found from their studies of records
of thousands of fossil genera that extinctions occur in waves.

Evolutionary biologists have always assumed that rapid changes in the
rate of evolution are caused by external events – which is why, for example,
they have sought an explanation for the demise of the dinosaurs in a meteorite
impact. On the other hand, if life organises into a critical state, catastrophes,
no matter how large, are a natural part of evolution. External causes, such
as a meteorite impact, are not necessary to explain, for example, those
cataclysmic mass extinctions in the late Permian, 245 million years ago,
or the extinction of the dinosaurs 65 million years ago.

This new insight has come from recent advances in understanding how
large, interacting systems work. In 1987, Bak, working with Kurt Wiesenfeld
and Chao Tang, also physicists from Brookhaven, simulated on a computer
some simple examples of complex systems, which are ‘driven’ by feeding energy
into them at a constant rate. These are ‘open’ systems – they are kept far
from equilibrium. Such systems are ubiquitous in nature. For example, the
continuous motion of tectonic plates builds tensions in the Earth’s crust,
and these tensions are intermittently relieved in earthquakes. The Sun shines
continuously on the rotating Earth, thereby creating complex patterns of
currents and winds, climates and weather.

The computer simulation is best imagined in terms of a sand pile, formed
by constantly adding sand to an existing pile (see ‘The self-organising
sand pile’, ¿ìè¶ÌÊÓƵ, 15 June 1991). The grains of sand in the computer
simulation behaved in a ‘cooperative’ manner that differed radically from
the state of equilibrium of flat sand. The behaviour of a single grain affected
that of all the others. As sand was continuously added, the system evolved
into a critical state characterised by large periods of static behaviour,
or stasis, interrupted by intermittent bursts of activity. The heap gradually
became steeper, and there were bigger and bigger avalanches, until the heap
built up a critical slope that produced avalanches of all sizes.

If the computer model is adapted to use wet instead of dry sand, the
pile will grow steeper for a short time – until it reaches a new critical
state with avalanches of all sizes. If one tries to prevent avalanches by
putting up snow screens, the pile will reorganise itself for some time –
until it reaches the critical state again. It is this resilience of the
self-organised critical state which made us think it might be applicable
to a wide range of natural phenomena.

We monitored the avalanches and counted how many were of a particular
size. Despite the complexity of the system, this distribution follows a
simple law: the number of avalanches in which s grains topple is just a
power of s,^s-r, where r is approximately equal to 1. This law
shows that if in a certain time period there is one event of size 10 000,
there will be approximately 10 events of size 1000, approximately 100 events
of size 100, approximately 1000 events of size 10, and so on. The ‘power’
law means that the physics of small avalanches is the same as that of large
ones. There is no characteristic timescale to separate large and small-scale
behaviour.

Earthquakes follow precisely this type of pattern. It is known as the
Gutenberg-Richter law and it suggests that the crust of Earth on which,
for instance, Californians are living, is in the critical state. In 1991,
an Italian team led by Paulo Diodati of the University of Perugia in Italy
measured the acoustic activity around the volcano Stromboli. They found
that the volcanic activity also followed the laws of self-organised criticality.
X-ray bursts emitted from solar flares, sudden variations in the frequencies
of pulsars, and light emitted from quasars follow a similar pattern, with
bursts of all sizes. In these cases, the variable s can be attached to the
amount of energy released.

Less surprisingly, real sand piles also follow the sand pile model.
In 1992, at the University of Michigan, Michael Bretz, Franco Nori and colleagues
used a video camera to study sliding sand where sand on a tray was ‘driven’
to produce avalanches by being slowly tilted. Information from the camera
was transformed into a digital signal and fed into a computer, which measured
the size of the avalanches. These were of all sizes, as the sand pile model
predicts.

CRITICAL LEAP

The main purpose of studying self-organised criticality, however, is
not to understand the physics of sand but to apply the idea to other systems:
to an ecology of interacting species, for example, or perhaps to an economy
of interacting elements (see Complexity, ¿ìè¶ÌÊÓƵ supplements, 6/13
February 1993). As long ago as 1966, Benoit Mandelbrot of IBM’s Thomas
J. Watson Research Center in New York suggested that fluctuations in economics
follow the same type of law as earthquakes. Do economies operate in a critical
state, far from the traditional view that they are in equilibrium? The observations
of Eldredge, Gould and Raup in the 1970s and 1980s of punctuated equilibrium
behaviour in biology are perhaps more interesting still. Is it possible
to leap from sand piles to biology? In fact, constructing a model to describe
punctuated equilibrium in biology turned out to be much more complicated
than merely changing the language of the sand pile model.

A good place to start seemed to be the Game of Life, a computer model
invented in the 1970s by John Conway that consists of a chessboard-like
arrangement in which squares can be either ‘alive’ or ‘dead’. Starting from
a random setup, the pattern of squares evolves according to simple rules.
Live squares die if they have too few, or too many, live neighbours. New
live squares are born when the number of live neighbours is just right,
allowing for a balance between creation and extinction.

In 1989, Michael Creutz and Kan Chen, physicists from Brookhaven National
Laboratory, and Bak discovered that they could produce criticality in the
model by adding new live squares at random to the Game of Life at some low
rate,: punctuated equilibrium occurs in the form of avalanches of activity
of death and birth of all sizes. But the Game of Life does not have much
to do with real biology. And unfortunately, changing the rules of the game
only slightly made the criticality disappear: it was not a robust property
of the system.

In 1990, Kauffman and Sonke Johnsen, a research associate from Norway,
were also hunting for self-organised criticality, this time in computer
models of ecologies of interacting species. The environment of a real species
depends on its interaction with other species in the system: for instance,
the genes of a zebra determine how it is built, and hence how fast it can
run. But its fitness to its environment depends just as much on how fast
the local lions can run. In Kauffman and Johnsen’s model, each species is
characterised by a ‘genetic code’, and a species’ fitness depends on both
its own ‘genes’ and on a number of genes – let’s call this number C – in
other species.

If C is small, a species will experience its environment as essentially
unchanging. It will stop evolving once it has adapted to it. In this way
it becomes an unchanging part of the environment of those other species
that depend on it. So they are even more prone to stop evolving, and so
on. This ecology evolves to a state of arrested evolution, in which species
are unable to improve their fitness. It remains low. On the other hand,
if C is large, each species depends heavily on the state of many other species
and is continually trying to maximise its fitness to an ever-changing environment.
The ecology never stops evolving: it is in a state of continued chaotic
evolution. The fitness species acquire in this state is low, too.

So, assuming that the goal of evolution is to optimise the mutual adaption
of species, does it self-organise to a critical point somewhere between
these two extremes? When Bak and Henrik Flyvbjerg working with Benny Lautrup
of the Niels Bohr Institute at the University of Copenhagen, applied mathematics
to this model, it appeared that here, too, the criticality relied on a judicious
choice of model – in this case the value of C. Increasingly, it seemed impossible
to find a model of evolution that mirrored the real world without ‘cheating’.

But then we discovered a key insight from another field. Kim Sneppen,
and Mogens Hogh Jensen at the Niels Bohr Institute had been working on computer
models of the growth of surfaces of crystals. They had developed a model
of self-organised criticality for such a system. Their model differed from
previous such models in a minor, yet important way. It allowed the least
stable site, rather than sites chosen at random, to grow and change. When
Sneppen came to visit Brookhaven, we realised that, translated to the language
of evolution, this modification says that the least fit species is the most
susceptible to change by mutation. We had left this – the most fundamental
component of Darwin’s theory – out of our early studies.

This additional insight, combined with our previous attempts, enabled
us to construct a mathematical model we hope has captured the essence of
biological evolution – not in terms of its details, such as the birth and
death of individual organisms, but in terms of the evolution of species.
Our model is quite simple. We start with a large number, say 1000, random
numbers placed on, say, a circle. The model ‘evolves’ over several time
steps. First, the computer replaces the smallest number on the circle with
a new random number. It also changes the two nearest neighbours of this
random number on the circle to two new random numbers. In the next time
step, the number which happens to be lowest is replaced by a new random
number. This could be one of the three numbers already changed in the previous
step, or it could be another number somewhere else on the circle. Its two
neighbours are also assigned random numbers.

The computer repeats the process again and again. After very many, say
1000, such steps, the numbers reach a more or less stationary distribution.
They stop increasing – they converge on an average value that is not particularly
high. For a time, all the changing numbers cluster together at one part
of the circle. Then suddenly, the position of the next number selected is
somewhere entirely different – a new cluster begins to form. The clusters
may be of any size, large or small: the model has self-organised to the
critical state.

RANDOM NUMBERS

The random numbers in our model represent the fitness of the species
situated on the circle. The selection of the smallest number represents
selection of the species that is least fit for mutation. The change in the
value of the random number on the neighbouring sites represent, at a very
general level, biological connections such as food chains, predator-prey
and parasite-host relationships. In our model, the random changes to the
selected number affect its neighbours – just as in the sand pile model,
where the toppling of a single grain can affect its neighbours. And since
the evolution takes place in terms of avalanches – in mathematical terms,
the formation of clusters of random numbers – our model also shows punctuated
equilibrium behaviour, just as real biological evolution does.

We have studied many different versions of this model, and in all cases
we found self-organised criticality. The behaviour is robust – as it must
be to represent real evolution since our models will certainly differ from
the real thing when it comes to details. In the models, within the large
avalanches of hectic activity, the average fitness is low. The same species
are mutating again and again in search of better fitness. The fitness of
the various species is also low during mass extinctions, and high during
the periods of stasis with low evolutionary activity.

So how far can all this be applied to real biology? According to our
model, large avalanches take place without any external force. So large
events in the history of evolution – such as the Cambrian explosion 570
to 510 million years ago and the extinction of the dinosaurs 65 million
years ago – may have taken place without being triggered by large cataclysmic
events; they may be intrinsic consequences of the dynamics of biology, that
is, they are self-organised.

Another interesting observation is that species with many connections
– that is, those with a high degree of complexity – are more sensitive to
the environment. They are more likely to participate in the next co-evolutionary
avalanche and become extinct. So complex species should exist for a relatively
short time, compared with the time that simpler species can exist before
becoming extinct – according to our model, cockroaches will outlast humans.

But the most interesting feature of the model is its extreme simplicity
and robustness; it has no subtle structure. We believe that only a few conditions
need to be satisfied for the model to work. The picture which seems to emerge
from our model is that while evolution does take ecology to the critical
point by its self-organising dynamics, the fitness of that point is not
particularly high. The critical point is not, as Kauffman once described
it, ‘a nice place to be’. So ‘survival of the fittest’ does not imply evolution
to a state where everybody is well off. On the contrary, individual species
are barely able to hang on – like the grains of sand in the critical sand
pile.

We also noticed that in the critical state all species interact, as
illustrated by the existence of large avalanches. Since all species affected
by any given avalanche share their fate, they might be regarded as a single
‘organism’. As the ecology evolves from its original state towards its critical
state, this kind of organism grows in size until the entire system is effectively
one organism.

We have not found the equivalent of Darwin’s missing equations – we
are not even close to doing that. But we do have a simple model for a vastly
simplified Darwinian evolution. This model is a mathematical model, formulated
as a set of equations that can be solved on a computer. Despite its simplicity,
the model has already taught us that punctuated equilibrium, stasis, and
intermittency can be direct consequences of simple principles of evolution,
as expressed by Darwin. No extra explanations beyond those principles are
required.

Per Bak is a physicist at Brookhaven National Laboratory, New York.
Henrik Flyvbjerg and Kim Sneppen are physicists at the Niels Bohr Institute
at the University of Copenhagen.