This Java applet
(If you don't see anything above, please install Java
on a 1 dimensional ecology of 100 interacting species S. Every species
assigned a fitness value F between 0 and 1. At each time instant the
species is killed together with its #Neighbors (=2 nearest in standard
version of model). They are replaced by new species with new
random fitnesses. The ecology co-evolves to a highly correlated state,
fractal activity gives burst of originations and extinction, as seen in
fossil record of species on earth.
the mutation rate can be adjusted between 0.001 and 1. A finite mutation
rate will illustrate the different time-scales associated
to evolving species with high or low F values.
evolution of species during the last 540 million years
present some intriguing evidence for cooperative behavior: Often during the
history of life there has been major "revolutions" where many species has been replaced simultaneously.
Spectacular examples are the Cambrian explosion 540 million years ago
where a huge variety of life arose in a short time interval, and the
Cretaceous-Tertiary boundary where mammals took over large parts of the ecosystem.
Inbetween these major revolutions there have been periods of quiescence, where
often all species seemed to live in "the best of all worlds", and only few species suffered extinctions. However, the pattern of life is more subtle than completely on-of transitions of macro-evolutionary revolutions.
When inspecting the extinction record one observes that often there was extinction of smaller size in the more quiet periods.
In fact if one plots the size distribution of ecological events, one observes all sizes of extinctions.
That is, the large events is becoming gradually less frequent
than smaller events. There is no "bump" or enhanced frequency for the large scale extinctions. However,
the distribution is broader than a Gaussian, in fact close to scale
distribution 1/E^1.5. This overall gradual
decline of event size distributions indicates:
1) That large and
small events may be associated to similar type of underlying
dynamics. If extinctions were external due to for example asteroid
impacts (Alvarez et al (1987)), one would expect a
peak at large events.
2) That the probability
distribution for extinction events are non-Gaussian implies that the
for obtaining large events are relatively large. This shows that the
in the ecosystem do not suffer extinction independently of each other.
the overall macroevolutionary pattern supports cooperativity, even on
scale of the global ecosystem.
In the figure we show
the family tree of ammonites from their origination 350 million years
until their extinction 66 million years ago. One observes a tree like
with branchings and killings,
that occasionally undergoes a near total extinction. However, as long
a single species survive, one see that it can diversify and subsequent
a large family. Finally 66 million years ago, not a single ammonite
and we now only know them from their very common fossils. What we
see on the family tree are the interactions between the species, nor
we see which other species these ammonite interacted with. Thus the
dynamics behind their apparent
The toy model:
To model the observed
macro evolutionary pattern we will start with objects on the size of
main players on this scale, lets call them species.
A species of course consists of many individual organisms,
and dynamics of species represent the coarse grained
view of the dynamics of these entire populations. Thus, whereas
population dynamics may be governed by some sort fitness, we propose
that species dynamics
is governed by stabilities.
Given the basic evolutionary
entity, called a species, we characterize this with one number
Fi. This number is the
stability of the species on An ecosystem of species consists of
selecting N numbers Fi
each representing a species. Each of these species are connected to a
few other species.
For simplicity, we put the numbers Fi, i=1,2,...N on a 1-d line,
to a 1-dimension ecosystem (the real ecosystem
will have a very high dimension, but model on such a geometry behave
At each time-step we change the least stable species. In addition
is defined in relative terms, which implies that the fitness of a given
is a function of the species it interacts with. The co-evolutionary
then reads (Bak \&
At each step, the
smallest of the Fi are
located. For this as well as its nearest neighbors one replaces their
F's by new random
numbers in the interval [0,1].
It is the simplest
model which exhibits a phenomenon called self organized criticality. As
system evolves, the smallest of the Fi's is eliminated. After a
period, for a finite system, a statistically stationary distribution of
is obtained. For very large N this distribution is a step function
the selected minimal F
is always below or at a threshold value Fc and therefore the
of F constant above Fc. For the dimension d=1 shown in the applet
one typically we
update two nearest neighbors,
and then obtain a self-organized threshold Fc=0.6670. The upper
panel in the applet panel illustrate how the minimal F sites moves in
space" as the system evolve.
One observes a highly correlated activity, signaling that the system
developed some sort of cooperativity.
The selection of least
stable species to mutate next implicitly assumes a separation of time
in the dynamics. Thus the selection of the least fit of all species to
next is the natural outcome
of the updating model:
At each step: Select each of the Fi with probability proportional
exp(-Fi/r). This selection
defines a list of active sites. For members in this
list, replace them as well
as their nearest neighbors by new
random numbers in the interval [0,1].
Here r represent an
attempt rate for microscopic evolutionary changes, and is proportional
the mutation rate per generation. For low enough r this model
degenerates into the one where
always the minimal Fi is selected. The value of Fmin represents a time
(in fact it is log( time)). Thus if Fmin is large,
then exponentially long times pass without
any activity at all.
If Fmin is small it
is selected practically instantly.
Mutation rates r are
typically chosen to be between 0.001 and 0.1.
and extinction events far beyond species boundaries is unknown. All we
say is that the behavior is far from what we would expect from a random
extinction/origination tree. some rather long evolutionary time
scale (a time scale much longer that the organism reproduction time
S.J. Gould (1972). Punctuated equilibriua:
An alternative to Phyletic Gradualism. in
T.J.M Schopf, J.M. Thomas (eds), Models in Paleobiology,
S. Francisco: Freemanand Cooper.
Ibid. (1993). Nature 366, 223-227.
Character change, speciation, and the higher taxa.
Evolution 36, 427-443.
(1991). Extinction, Bad genes or Bad luck?
Oxford University Press.
\& J.J. Sepkoski Jr. (1992) Mass
extinctions and the marine fossil record. Science
Van Valen (1973). A new evolutionary law. Evolutionary
Theory 1, 1.