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Critical Biodiversity
J. H. KAUFMAN,* D. BRODBECK,† AND O. R. MELROY*
*IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, CA 95120-6099, U.S.A.,
email [email protected]
†Union Bank of Switzerland, Bahnhofstrausse 45, CH-8021 Zurich, Switzerland
Abstract: Ecosystems are dynamic systems in which organisms survive subject to a complex web of interactions. Are ecosystems intrinsically stable or do they naturally develop into a chaotic state where mass extinction is an unavoidable consequence of the dynamics? To study this problem we developed a computer model
in which the organisms and their interactions “evolve” by a “natural selection” process. The organisms exist
on a multi-dimensional lattice defined both by a diverse physical landscape that does not change and by the
presence of other species that are evolving. This multidimensional lattice defines a dynamic vector of “niches.”
The possible niches include the fixed physical landscape and all of the species themselves. Species may evolve
that specialize or that are adapted to many niches. The particular niches that individual species are adapted
to occupy are not built into the model. These interactions develop as a consequence of the selection process. As
species in the model evolve, a complex food web develops. We found evidence for a “critical” level of biodiversity at which ecosystems are highly susceptible to extinction. Our model suggests the critical biodiversity point
is not a point of attraction in the evolutionary process. Our system naturally reaches an ordered state where
global perturbations are required to cause mass extinction. Reaching the ordered state beyond the critical
point, however, is kinetically limited because the susceptibility to extinction is so high near the critical biodiversity. We quantify this behavior as analagous to a physical phase transition and suggest model independent measures for the susceptibility to extinction, order parameter, and effective temperature. These measures
may also be applied to natural (real) ecosystems to study evolution and extinction on Earth as well as the influence of human activity on ecosystem stability.
Biodiversidad Crítica
Resumen: Los ecosistemas son sistemas dinámicos en los organismos sobreviven a una compleja red de interacciones. ¿Son intrínsecamente estables los ecosistemas o se desarrollan naturalmente hasta un estado
caótico en el que la extinción masiva es una consecuencia inevitable de la dinámica? Para estudiar este problema desarrollamos un modelo de computadora en el que los organismos y sus interacciones “evolucionan”
por un proceso de “selección natural.” Los organismos existen en un enrejado multidimensional definido por
un diverso paisaje físico que no cambia y por la presencia de otras especies que están evolucionando. Este enrejado multidimensional define un vector de “nichos” dinámico. Los nichos posibles incluyen al paisaje físico
fijo y a todas las especies. Las especies pueden evolucionar para especializarse o para adaptarse a muchos
nichos. El modelo no incluye los nichos particulares a los que se adaptan especies individuales. Estas interacciones se desarrollan como consecuencia del proceso de selección. A medida que evolucionan las especies en
el modelo, se desarrolla una compleja red alimenticia. Encontramos evidencia de un nivel “crítico” de biodiversidad en el que los ecosistemas son altamente susceptibles de extinción. Nuestro modelo sugiere que el
punto crítico de biodiversidad no es un punto de atracción durante el proceso evolutivo. Nuestro sistema alcanza un estado ordenado naturalmente en el que se requieren perturbaciones globales para causar estinción masiva. Sin embargo, alcanzar el estado ordenado después del punto crítico esta limitado cinéticamente
porque la susceptibilidad de extinción es muy alta cerca del punto crítico. Cuantificamos este comportamiento como análogo al de fase de transición física y sugerimos parámetros independientes del modelo
para medir susceptibilidad de extinción, orden y temperatura efectiva. Estas medidas también se pueden aplicar a ecosistemas naturales (reales) para estudiar la evolución y extinción sobre la Tierra, así como la influencia de la actividad humana sobre la estabilidad del ecosistema.
Paper submitted April 22, 1996; revised manuscript accepted July 1, 1997.
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Introduction
E. O. Wilson said we are in “the midst of one of the great
extinction spasms of geological history.” Although this
mass extinction is due to human activity, the fossil
record reveals several previous mass extinctions near
the ends of the Silurian, Devonian, Permian, Triassic,
and most recently in the Cretaceous periods (Wilson
1992; Benton 1995). Understanding the origins and dynamics of these extinctions is a topic of considerable interest both for fundamental reasons and because of the
implications for the extent of the extinction events we
are now triggering (Simberloff 1986).
Two major classes of theories have been put forward
to explain these mass extinction events. The first invokes a major external event such as the collision of a
large asteroid or comet with the Earth (Alvarez et al.
1980; Raup 1991), global climate changes (Vrba 1985),
sea level changes (Newell 1952), or volcanism (Moses
1989). The second class of theories postulates that ecosystems, like many other natural dynamic systems, evolve
toward a chaotic or “critical” state (Gleick 1987; Bak &
Paczuski 1995; Sole & Manrubia 1995; Drake et al. 1992).
In a chaotic system, the smallest of changes or local perturbations may trigger macroscopic events on global
length scales (Gleick 1987). If chaotic processes were operating in an ecological system, then extinction events of
all sizes should be evident over an appropriate spatiotemporal scale. Furthermore, predicting the dynamics would
be problematic because of the extreme sensitivity of the
system to small perturbations. It is this sensitivity that
makes weather prediction so difficult (Lorenz 1963).
The mathematical properties of chaotic systems have
been shown by Mandelbrot (1983) to be fractal. The spatial and dynamic properties of all fractals scale (i.e., they
obey power laws). Similarly, the properties of physical
systems that undergo second-order phase transitions exhibit scaling behavior at the “critical temperature” where
the phase transition occurs (Ma 1976). This similarity led
to the term “self-organized criticality” to describe systems that naturally evolve to and stay at a critical or chaotic state (Bak & Paczuski 1995; Patterson & Fowler
1996; Perry 1995; Sole & Manrubia 1995).
Several authors have constructed models for evolution
in an “ecosystem” (Plotnick & Gardner 1993; Drake
1990; Drake et al. 1992; Durrett & Levin 1994). Some of
these models lead to chaotic behavior. For example, Plotnick & McKinney (1993) applied percolation theory to
study the extinction process. In the Plotnick–McKinney
model, depending on the relative rates of species creation and species death, the system may be tuned to a
point where extinction events of all sizes may occur.
When tuned to this critical state, the death of a single
species may trigger a mass extinction.
Flyvbjerg et al. (1993) developed a model based on an
evolutionary fitness landscape, the “shifting balance the-
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ory” described by Wright (Wright 1982; Jongeling 1996).
Over time, they observed extinction events (equal to the
number of transformations) of all sizes (Gould & Eldredge
1993). Quiescent periods are characterized by most species having similar “fitness,” and avalanches of extinctions occur when there are large disparities in fitness.
This result is in contrast to the conclusions of Kauffman
& Johnsen (1991) whose “KNC-models” suggest that the
ecology as a whole is “most fit” at the critical point. In
his book The Origins of Order, Kauffman (1993) postulates that natural selection drives biological “adaption to
the edge of chaos.”
These models suggest that natural ecosystems could
be in a critical state. The question is, how can one tell if
chaos is an inevitable consequence of the dynamics? If
the behavior of ecosystems is analogous to the behavior
of other dynamic systems that evolve to or through a
critical state, then a mathematical framework already exists to describe their dynamics. To apply this framework,
one must find measures of ecosystems that are analogous to the variables used to describe the state of other
dynamic systems: an effective temperature, susceptibility, and order parameter (Ma 1976). It is not at all obvious, a priori, what these measures would be.
We set out to develop a model in which both organisms and their interactions evolve into a complex web of
interdependency. In allowing the interactions to develop as a consequence of the model, we sought to
avoid building into the web of interdependency features
that would predetermine the dynamic behavior of the
model. Our goal is neither to reproduce the richness of
natural biological interactions nor to prove that the extinction dynamics exhibited by our model are the same
as natural extinction dynamics. Rather, we hope to use
this model to develop measures and to show how they
might be applied to understand extinction dynamics in
real biological systems.
The Model
Our model world is a square lattice (100 3 100). The lattice, which does not use periodic boundary conditions,
is analogous to an isolated island in a lifeless sea. Each
lattice site is randomly assigned one of six local environment types; therefore, no one environment type occupies an unbroken path spanning the lattice (Plotnick &
Gardner 1993). At every location or site (x,y), the environments on the lattice are labeled by a number, e(x,y)
which denotes the fixed physical environment type.
That variable is meant to represent local physical conditions. At every site (x,y) and instant in time, t, several
“species” can coexist in some dynamic balance.
Every organism on the lattice is labeled by the group
or species it belongs to. For example, at some instant in
time at a particular site (i,j) there may exist at the bot-
Kaufman et al.
tom of the chain of ecological dependence an organism
of a particular species, call it s(17). The existence of species s(17) on site (i,j) may provide a niche for another
species on the same site. This second species also has a
label which might be s(3). The species s(3) also had to
compete for a niche, but the niche that makes it possible for s(3) to exist on site (i,j) is not the environment
e(i,j); it is s(17). Higher on the chain another organism
may occupy the niche created by the presence of s(3).
Instantaneously at a particular site the species are organized in this hierarchical fashion. Over a longer period
of time, and over the world as a whole, organisms compete in a more complex food web. The same species
s(3) living on s(17) at site (i,j) and time t may be living
on a different species (occupying a different niche) at
some other site (m,n) at time t. The model is dynamic so
the chain of interdependency at any site and time t may
be different at time t9. The organisms exist on this multidimensional lattice with local conditions defined both
by the physical landscape (e(x,y)), which does not
change, and by the presence of other species. This multidimensional lattice defines a dynamic vector of available “niches” (Plotnick & Mckinney 1993). The possible
niches include all of the local environment types and all
of the species themselves. The particular niches individual species are adapted to live on are not built into the
model. Adaptation occurs through a natural selection
and ultimate speciation process. The only restriction we
built into the model regarding the food web is that no
species may live on itself. This is our “conservation of
energy” rule. This is not a predator-prey model. Organisms in the model are never consumed by other organisms. In dynamic balance the presence of prey makes it
possible for a predator to survive in the same sense that
the presence of a tree makes it possible for an epiphyte
to survive, but the mathematical model only tracks the
existence of the relationship or dependency. A species
becomes extinct only if there is no niche available for it
to live on or if it cannot successfully compete for an
available niche somewhere on the lattice, at which time
every individual organism of that species disappears.
None of the species-species interactions are built into
the model. The food web evolves by “mutation” or,
more precisely, speciation. What is required in the
model is a “data structure” that keeps track of the relationships between all existing species and all environment types. Mutation will be accomplished by small random changes to this data structure. The data structure is
not a “genetic” code. It is an accounting scheme that
tracks how well every species is adapted to every existing type of niche. For all species we store two vectors:
the “environment” vector and the “species” vector.
They measure how well each species is adapted to the
existing environment types and to live on the other existing species, respectively. The use of two separate vectors is for convenience because (1) the environments do
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not change in our model and (2) the species vector dynamically changes length as new species come into existence (through mutation and speciation) and as old
species die.
Data Structure
The numbers or adaptive weights stored in these two
vectors reflect the relative “fitness” of species for different possible niches and are used to determine the outcome of competition between individuals. Consider
again species s(3). Call it an orchid. The available niches
in the world include the local environment types and all
of the existing species. The orchid is not adapted to
most of these niches. It may be adapted to only one type
of local environment. Suppose the label for this environment type is e(x,y) 5 4 on all sites (x,y) where it occurs.
The environment vector for species s(3) is then zero except for the fourth element. The magnitude of that element reflects how strongly the orchid is adapted to the
environment. These adaptive weights are relative and
the orchid may be adapted to environment type four
with a weight of two. If there are six different local environment types, the environment vector for the orchid is
then (0,0,0,2,0,0). As before, the orchid is also adapted
to live on species s(17), which we will call a fig tree. For
simplicity in this example, suppose s(17) is the only
other species on which the orchid is adapted to live.
Then the species vector for the orchid is zero except for
element 17 which measures how strongly the orchid is
adapted to the fig tree. That element might have a value
(weight) of nine. The orchid may speciate into a new organism with new adaptations or different adaptive weights.
Competition Process
Given the vectors of relative competitive weights for every species, the competition process between individual
organisms is straightforward. At every time cycle of the
program, each individual on the grid competes to exist
again on the sites where it existed in the previous time
cycle and competes for sites that are nearest neighbors
to the sites where it existed. The nearest neighbors to
any site on a square lattice are, by definition, the four
sites offset by 61 unit in x or y. The competition takes
place simultaneously at all sites (x,y) on the map. Consider a specific example of organisms competing to live
on a particular site (i,j) at time t. The individuals of various species at site (i,j) and at the four nearest neighbor
sites to (i,j) are put on a list. Even if a species is represented at more than one of the nearest neighbor sites, it
is only added to the list once. A competition process follows, first for the spot at the bottom of the chain of eco-
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logical dependence (the environment e(i,j)). Organisms
that exist higher on the chain of ecological dependence
on a neighboring site may also be able to compete for
the bottom niche on site (i,j). Species not adapted to
e(i,j) have zero adaptive weight for that element of their
environment vectors and do not enter this round of
competition. The organism that wins the competition
for the first niche at time t, is chosen probabilistically
based on the relative adaptive weights. The single winner is not necessarily representative of the most well
adapted species at a particular site and time, but that is a
more likely event than selection of a relatively less fit
competitor. Having selected the organism that will fill
the bottom niche, the organism is removed from the list
of competitors for that site and added to a map matrix
m(i,j,s), where s is the species label and the elements of
m are zero when species s is absent from i,j and one
when it is present. Species that lost the competition for
the first niche then seek to occupy the niche created by
the presence of the chosen species that now resides at
(i,j). The selection process is the same except, of course,
that the adaptive weights are taken from their respective
species vectors. Each species selected in this manner becomes, itself, the next available niche. The species not
yet selected compete to fill the next niche. The process
continues until either all represented species have been
chosen or until none of the organisms awaiting a niche
are adapted to the last available niche. Any species remaining have then lost the competition for (i,j) at time t.
Creation Event
At the beginning of the simulation, the world (grid) is
seeded with a single individual of a single species, s(0),
which is adapted to live at a single type of environment
with an adaptive weight of one. A specific site, (i,j), for
this creation event is chosen at random on the edge of
the grid. Because no environment e(i,j) provides a continuous path across the map (i.e., it does not percolate),
the seed species s(0) will spread to at most a few neighboring sites. To spread further, the seed organism must
speciate.
Speciation
In our model every mutation event creates a new species from the parent organism. We have included two
types of mutations. The first is an increase in fitness, by
one, of one element of the environment or species vector. The element selected for this net increase in fitness
is random. It is equally likely to occur in any possible
niche (environment or species vector element). The second type of mutation involves a shift of adaptive weight.
In this shift, the fitness (adaptive weight) for one niche
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is decreased by one, and the competitiveness for a different niche is increased by one. These niches are chosen
at random with one restriction. The niche in which the
species becomes less competitive must initially have
nonzero weight. The second mutation process allows
new species to “explore” the space of possible interactions without increasing overall fitness. Of course the actual fitness of individuals is a relative quantity depending
both on the characteristics of the species and on the local conditions in the multidimensional lattice. These
conditions vary from site to site and as a function of time
within the model.
So far we have defined no tuning parameters in this
model. Because we want the system to select the “favorable” interactions while increasing fitness quasi-statically, we set 60% of mutations to result in a shift in competitiveness (“neutral” speciation) and 40% to an increase
in competitiveness (“advantageous” speciation). In both
types of mutation processes, the parent species is a
“possible” niche for the child species. Recall that no species can “live on itself.”
Finally, one must decide what absolute rate of mutation to allow. The speciation process is intended to introduce small local changes. One can then measure any
responses or “avalanches” of mass extinction that occur
in response to a new species. For this reason we adopt a
quasi-static approximation. On average, we assume there
is enough time between speciation events so that new
species that can spread across the grid will before another speciation event occurs. If speciation occurs at an
overall higher average rate, then this system will never
be in steady state and an arbitrary length scale would be
built into model (namely how far a new species can
spread before it speciates again). To avoid this we dynamically reduce the mutation rate as the population of
the grid increases. Speciation occurs just before each
competition, or spreading, event. A “child” species then
competes for the site on which it was created as well as
the four nearest neighbors. The parent species is represented in this competition as well. The mutation probability per competition event per species per site is defined as
0.75
1
R =  ---------- ×  -------------------------- ,
 L   population
where L, the width of the grid, equals 100. In reality, genetic mutation occurs with constant probability for every reproductive event (independent of population).
The quasi-static approximation implies the actual rate of
mutation is always small compared to the rate species
can spread by competition. In this regime it is possible
to study the response of a system in steady state to single, local perturbations.
Our goal in choosing mutation rules is to incorporate
in the model a means for the “ecosystem” to explore different interaction spaces through a series of small per-
Kaufman et al.
turbations. By running the program many times one can
sample the most likely possible food web organizations.
The absolute mutation rate was made sufficiently small
so that it is an “irrelevant tuning parameter.” In the
quasi-static approximation the absolute frequency of
mutation is low enough to allow new species to spread
to their steady state range. Lowering the mutation rate
further would only increase the time in the simulation
when nothing happens.
Effect of New Species on Old
The appearance of a new species creates a new niche.
Can this new niche be a viable niche for the old organisms on the grid? Because the particular speciation process we chose will slowly increase overall fitness, new
species are likely to replace parent species over time.
Whether or not new keystone species support the hierarchical chain of ecological dependence determines the
stability in this model. The question of whether an existing species provides a niche for a new mutant defines
the most important parameter in our model. To resolve
the question we turn to Darwin. We assume speciation
arises from a mutation process in a population of organisms of the same species present on a given site. As we
discussed, one can consider that a species that provides
a niche for another species is being preyed upon and
that the populations are in dynamic balance. With regard to the “prey,” survival of the fittest implies that
those individuals that evolve to resist predation are more
likely to survive and reproduce. If there are no predators
present, however, there is no evolutionary advantage to
those individuals that resist predation. To incorporate
this phenomenon in the model, we established the following rules.
(1)
(2)
(3)
If an existing species could “prey” upon the parent
of a new mutant and if the existing species is not
present on the site when and where the mutation
occurred, then it can survive on the niche created
by the child species with probability P1.
If an existing species could prey upon the parent
of a new species and if the existing species is
present on the site when and where the speciation
occurred, then it can survive on the niche created
by the child species with probability P2 , P1. It is
less likely to survive on the niche created by the
child because the new species may have evolved
to resist predation.
If an existing species could not prey upon the parent, then it cannot prey upon the child. In this last
rule we treat the mutation as a small perturbation
from the parent species (which was not a viable
niche for the predator).
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525
These three conditions cover all possibilities. If P1 5
P2, species would never evolve to resist predation. In
this case the model never produced specialization and
all species evolved to live everywhere. Because we don’t
know a priori how to define P1 and P2, we studied both
P1 5 1.0, P2 5 0.0 and P1 5 0.75, P2 5 0.25. These
conditions produce the same dynamics. On first inspection this rule seems quite destabilizing, making it extremely likely that a keystone species would be replaced
by a variant that would not support existing predators.
One might expect this instability to be the source of extinctions on all scales, and perhaps the cause of self organized critical behavior. The actual behavior of the
model was a surprise. Even for P1 5 1.0 and P2 5 0.0
(the parameters used for the data reported here) we
found evolution to a dynamically stable state.
Results and Discussion
We ran the simulation 14 times (using different initial
random number seeds) for 200,000 cycles. A cycle, or
time step, is defined by the competition and spreading
process. In one time step an organism may spread from
a site it occupies to a nearest neighbor site. The minimum time required for an organism to spread across the
lattice is t 5 L cycles where L (5100) is the linear size of
the grid. The data were analyzed after 100,000 and
200,000 cycles (typical program, Figs. 1 & 2).
It is useful to discuss, qualitatively, the dynamics observed in a typical run, or evolution time series, on the
grid. In a typical simulation, there is a period of time
where evolution takes place in a localized region (upper
left-hand corner of Fig. 1 at t 5 2090). Eventually, one or
two species acquire, through mutation, the ability to
spread throughout the grid (first frame in Fig. 1, initial
spreading event). Often we found that two species
spread together. In this example (Fig. 1), 15 species exist and two of them are in the process of spreading
across the grid together. In frame 3, over 10,000 time
steps later, some of the primitive children of the initial
seed are still evident in the upper left corner. This frame
also captures the almost coincident birth of two new
species capable of spreading throughout the grid. This is
a rare event given the quasi-static mutation rate (one mutation per 133 time steps on average). The last two
frames of Figure 1 are representative of a persistent state
of the ecosystem. We call it the “primitive” state. There
is a fair degree of diversity on much of the grid area, but
there is no overall spatial structure or organization. This
state may persist for long periods of time. It is highly unstable and susceptible to sudden periods of extinction
activity where one or more keystone species are replaced, making large numbers of species unfit. In this
state two types of global changes can occur: mass extinction where large numbers of species die, and species
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Critical Biodiversity
Kaufman et al.
Figure 1. The early stages of evolution in one ecosystem at various times (t) and diversity (S). Each species
is assigned a different color at random. The frame at
t 5 2090 shows the first two species to spread across
the island. In the “primitive state,” there is no longrange or persistent spatial structure. The primitive
state is subject to mass extinctions and to frequent
replacements of species by more fit species.
Figure 2. Above a critical level of diversity mass extinctions no longer occur. There is well defined spatial
structure which persists over long periods of time. This
structure is not determined by the distribution of environment types. It reflects specialization and formation
of separate “communities” of species. Mutation and
extinction continue as before but the spatial organization persists.
replacement where dominant species on the grid are replaced. The primitive state is characterized by long periods of quiescence when the total number of species
does not change and by sudden mass extinctions where
large numbers of species die. This is exactly the intermittent behavior discussed by Flyvbjerg et al. (1993).
Relative fitness of many species on the grid may be affected by a single mutation effect. With few local exceptions, no species is isolated or protected. In this
primitive state, many of the species percolate throughout the grid.
In the “ordered” state species on the grid have organized themselves into separate subecosystems. Specialization has occurred creating well defined communities.
This is evident both in Figure 2 and in detailed examination of the species and environment vectors. Not all evolutionary time series reach this state in 200,000 cycles
(more than half do). In this state the species do not be-
come generalized over time but instead specialize for
competition within well defined communities. The collection of these separate communities forms well defined spatial patterns on the map. These patterns are uncorrelated with the local environments. The species
themselves define these large-scale patterns or environments which persist over long periods of time (Fig. 2).
Each community undergoes changes, and species are replaced by more fit species at the same rate as before.
These changes, however, are localized to the clusters
that contain the communities and do not lead to massive species loss. Once a system evolves to this ordered
state, we never observe a collapse back to the primitive
state.
For each run of the simulation it is possible to measure
the number of species versus time (Fig. 3). From these
time series it is possible to make some general observations about the dynamics. Extinction events of different
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Critical Biodiversity
sizes occur. Time series C (Fig. 3) corresponds to the
run depicted in Figs. 1 and 2. It exhibits three mass extinctions before 100,000 cycles where almost half the
species on the island die. Large extinctions are evident
in each of these time series, especially early in the emergence of each ecosystem. Over longer periods of time
the fluctuations grow more slowly than the overall diversity. The fraction of species extinguished, however, decreases as the diversity of life increases. The model produces ecosystems that seem to pass through a state
characterized by fluctuations on all scales. Is this state a
critical point? It seems not to be a point of attraction for
the ecosystems. Eventually, diversity and specialization
produce dynamic stability.
It is desirable to eliminate the time variable in analysis
of this problem. What are meaningful time-independent
measures? For these we look to standard problems in
condensed matter physics. We are interested in determining if there is a critical point associated with this dynamic system and if that critical point is the point of attraction. Critical points are associated with second-order
phase transitions (Ma 1976). Some textbook examples
are ferromagnetism and percolation (Ma 1976; Stauffer
1985). Formally, these concepts apply to systems in
equilibrium. Critical behavior, however, can often be observed in dynamic systems in “steady state” as they are
“tuned” through or near a critical point. In a secondorder phase transition, the order parameter varies smoothly
from zero to a finite value with a power law:
β
M = T – Tc ,
where T is an effective temperature and Tc is the critical
point. In percolation the effective temperature is the
527
density of points on the lattice and the critical temperature is the percolation threshold (that density where the
largest cluster first spans the system). At any critical
point many properties exhibit power law behavior or
“scaling.” Often the observation of scaling is used as evidence that a dynamic system is at a critical point. Many
properties, however, exhibit scaling over some length
scale even when T is not tuned to Tc. This length scale
over which the power law behavior is observed is the
correlation length, which diverges at Tc and decreases
with a Boltzman Law as the system is tuned away from
the critical point.
To determine if extinction dynamics in ecosystems is a
true critical phenomenon, it is important not only to
look for scaling, but to find an order parameter, susceptibility, and effective temperature for the system. Measures of these quantities need not be unique (several
quantities can often be used to measure the degree of order in a system, for example). The measures should be
general and apply to real ecosystems as well as different
models. Because diversity determines the overall organization (or lack thereof) in an ecosystem, we use as an effective temperature the total diversity, S (number of species living). This is a quantity we can measure in
analyzing time series (Fig. 3) to study how susceptibility
to extinction varies with biodiversity. To measure susceptibility to extinction, x(S), we measure average extinction size as a function of diversity S. This is an integral, the sum of total number of extinctions (species
deaths) E (S9) for S9 , S for each S. This sum for each
particular diversity, S, cannot begin until the system has
first evolved to diversity S:
∞
1
χ ( S ) ≡ --S
∫
E ( t, S′ ( t ) )θ ( S′, S ) dt
{
θ = 0 : S′ ≥ S
θ = 1 : S′ < S
.
t0
s' ( t 0 ) > s
Figure 3. Three typical time series showing the biodiversity as a function of time. The curve C is the ecosystem evolution depicted in Figs. 1 and 2. Note the absence of mass extinction above a diversity of about 20
species. The fluctuation or extinction size does not
grow as fast as the overall diversity.
This quantity is not necessarily finite. In fact one expects susceptibility to be singular at a critical point. For
a finite simulation it typically exhibits a cusp. Because
the integral cannot practically be carried out to t 5 ` we
invoke the law of large numbers and plot x(S) averaged
over 14 runs of the simulation computed at t 5 100,000
and t 5 200,000 (Fig. 4). A cusp is observed near S 5 18
where the susceptibility to extinction peaks. The peak
does not move as the simulation time is doubled.
To define the order parameter we need first to understand what we mean by the ordered state. We are interested in the intrinsic relationship between diversity and
extinction. The ordered state is the state with the smallest (relative) extinction probability per species per perturbation event. The perturbation events are the speciation events. We define E(S) as the number of extinctions
that occur at diversity S integrated over all time. C(S) is
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528
Critical Biodiversity
Kaufman et al.
Figure 4. The susceptibility to extinction as a function
of diversity has a cusp or maximum at a critical biodiversity near 18 species. This critical point does not
shift as the simulation time is doubled.
the number of creation or mutation events as a function
of diversity. The probability of an extinction event at diversity S per creation event per species is
ρ 1 ( S ) = E ( S ) ⁄ γSC ( S ) ,
where the normalization constant is
γ =
E(S)
-.
∑ -------------SC ( S )
S
Similarly, the probability of an extinction event at diversity S per creation event is
ρ 2 ( S ) = E ( S ) ⁄ αC ( S ) ,
where the normalization constant is
α =
E(S)
-.
∑ ----------C(S)
S
The probability of surviving C(S) perturbation events is
then
P ( C ( S ), S ) = { 1 – ρ 1, 2 ( S ) }
C(S)
,
where r1,2 5 r1 or r2. This then defines two different
measures for an order parameter, M: one measures the
average survival probability at diversity S and one measures the survival probability per species at diversity S.
C(S)

E(S) 
M 1 =  1 – ------------------ 
γSC
(S) 

and
,
C(S)

E(S) 
M 2 =  1 – ---------------- 
αC ( S ) 

.
Neither order parameter is a total survival probability for
individual species. If a species survives at diversity S,
there is still the probability it will become extinct at di-
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Volume 12, No. 3, June 1998
Figure 5. Three measures of the order parameter,
M: M1 measures the average survival probability at diversity S, M2 measures the survival probability per species at diversity S, and M3 the integrated extinctions
that occur before the system first evolves to diversity S.
versity S 1 1. All species eventually become extinct. The
sensitivity to fluctuations that is lost in the averaging
process that defines the order parameter is contained in
the definition of susceptibility.
The order parameters defined above are finite. We
plotted M1(S) and M2(S) versus S, for 14 trials run to t 5
200,000 (Fig. 5). As a further check that we were close
to the steady state value for M1,2, we also plotted a third
measure of the order parameter, M3(S) (Fig. 5). The M3 is
defined as the integrated extinctions that occur before
the system first evolves to diversity S. This measure is always in equilibrium for values of S where it is defined.
Unlike M1,2, the absolute scale of M3 is undetermined.
The entire curve should normalize to total extinctions so
as S→`, M2→1. Normalization at finite diversity trivially
sets M3(Smax) 5 1.
The three measures of the order parameter are consistent. The M1,2 peaks near S 5 0 because the initial seeding is a singular process. All of these possible order parameters suggest a critical point near S 5 18. The tail in
the order parameter, M3, below S 5 18 is a finite size effect intrinsic to a finite system. To understand this effect
consider the example of percolation. As points are added
to a lattice, the first “infinite” cluster of adjacent points
forms at the critical concentration pc. For a finite lattice,
there is a nonzero probability of forming a finite cluster
that spans the lattice at a concentration below pc.
The data suggest that there is a critical biodiversity Sc,
where the susceptibility to extinction diverges and below which the species survival probability falls exponentially (Figs. 4 & 5). The particular value of Sc (near 18
species) is not universal and scales with system area.
What is special about this point? Does the extinction
rate increase when the diversity is near Sc? The average
number of deaths per time cycle is constant (about 133
deaths/cycle) and just below the constant mutation rate.
Kaufman et al.
There is no change in death rate or mutation rate near
the critical diversity. What, then, causes the susceptibility to extinction to change with diversity? We plotted
the average time the system spends in a state with diversity S and the average number of species deaths as a
function of diversity (Fig. 6). This data also seem to be
singular near Sc. Near the critical biodiversity evolution
slows down. Mutation and extinction rates do not slow
down; the system simply “gets stuck” in the primitive
state depicted in Figure 1. Evolution to a diversity higher
than that of the primitive state is kinetically limited. Evolution slows down because the ecosystem is subject to
mass extinctions and collapses back to lower diversity.
Hence the measured E(S) and C(S) also peak near S 5 18
and define the critical point as evidenced in the order
parameter and susceptibility.
Critical Slowing Down
This kinetic barrier may be a manifestation of “critical
slowing down” (Ma 1976). As a system evolves to a critical point, it takes longer and longer to come to equilibrium. Near Tc the system is very sensitive to fluctuations.
A system in such a state could be mistaken for an example of a self-organized critical phenomenon as it could
spend exponentially long times in a state that exhibits
scaling and “avalanches” of all sizes. If stuck near Sc an
ecosystem would truly be perched on the edge of chaos.
However, that is not necessarily the point of attraction
or ultimate ordered state of the system. In our model
evolution eventually produces diversity greater than Sc
leading to specialization and true dynamic stability.
There are still extinction events where S decreases but
these do not grow with S. Note that the ordered state is
not characterized by an explosion in population with
more and more species living per site. The diversity per
site increases more slowly than S. As the number of spe-
Figure 6. The average rate at which species die is independent of time. The critical point occurs because the
system spends more time near the critical biodiversity.
Critical Biodiversity
529
cies on the island increases, specialization leads to the
formation of smaller and smaller clusters or communities with separate local food webs. Local mutation
events or species loss are limited to these smaller clusters. Massive extinction would require global changes affecting the entire grid.
Comparison with Zoogeographical Evidence
The conclusion that life evolves through a critical state
to a dynamically stable state may seem natural to biologists and naturalists. The generality of this conclusion is
the key question. A computer model can never reproduce all of the dynamic behavior observed in nature.
Nevertheless, physical systems driven by mathematically
similar forces and interactions often fall into the same
“universality class.” That is, the power laws that govern
the scaling behavior of the system variables near the critical point have common “critical exponents.” Sometimes even very simple models will produce the same
exponents as the more complex natural systems. To
compare our model to real ecosystems evolving through
natural selection, we can compare the behavior of the
model with experimental observation of real ecosystems. These data are abundant due to the hard work of
conservation biologists (Darlington 1957; Williamson
1989).
Darlington (1957) compiled data from several authors
who measured the increase in species diversity with island area in the Antilles Islands. MacArthur and Wilson
(1963) first recognized the scaling behavior of species
number with island area. This species-area relationship,
S 5 CAz, has been found by several investigators to hold
for numerous types of species over numerous island
groups (Darlington 1957; Boecklen & Simberloff 1986).
Experimentally, z ranges from 0.1 to 0.5, often with the
power law valid over many decades in area (Wilson
1992; MacArthur & Wilson 1963; Diamond 1984). The
fact that the scaling law holds but the exponent varies
from island group to island group has been attributed to
how far various island groups are from continental
masses. This separation distance influences the introduction rate of new species to the different island groups
and perhaps determines how far the ecosystems of
those groups are from steady state or dynamic equilibrium. More isolated island groups seem to exhibit large
exponents in the species-area relationship. Regions farther from equilibrium exhibit smaller exponents (Wilson 1992; Diamond 1984).
The species-area relation is also obeyed in reserves.
Lovejoy et al. (1984, 1986) selected reserves ranging in
size from 1 to 1000 ha for preservation from development north of Manaus. Forest areas around the reserves
were clearcut or burned for establishment of cattle
ranches. Lovejoy et al. measured species diversity before
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Critical Biodiversity
and after isolation of these reserves. We find from these
data that the species-area relation seems to hold for
these reserves. For example, the number of butterfly
species scales with an exponent of 0.18.
It is trivial to numerically simulate the Lovejoy experiment with our computer model. The same grids or “islands” that were allowed to develop to 200,000 time
steps were used as input. Smaller circular areas (ranging
in radius from 1 to 50) were isolated, and the remaining
area outside this “reserve” was numerically “poisoned”
by setting the local environment type to a new value
that no species could live on. The mutation rate was set
to zero and the system allowed to relax. We found that
the species-area relation is obeyed by our simulation
with 0.14 , z , 0.28. The average value is 0.19 (Fig. 7).
Where the species diversity of the “pre-isolate” island
was near Sc, the power law held for over three decades
in area (up to the size of the simulation). After isolation
the collapse to the steady state number of species occurred very fast (in one to three time steps). This is not
surprising given the quasi-static mutation rate of the
original simulation, which guarantees the “pre-isolate”
islands to be near steady state. Given the range of island
area exponents observed in nature it is not possible to
conclude based on an average z 5 0.19 that our model is
in the same universality class as real ecosystems (though
it is consistent). We can, however, take the numerical
experiment one step further. If the spread in natural values for z is not random but actually a reflection of how
far the island ecosystems are from dynamic equilibrium,
it may be useful to study the collapse in species diversity
in numerical models deliberately driven out of equilibrium. In nature this question is important because species will of necessity migrate to reserves from clearcut
areas, but those reserves may not be able to support all
Kaufman et al.
those species looking for refuge. Simberloff (1972),
Boeklin and Simberloff (1986), Diamond (1984), Terborgh (1974), Wilcox (1978), and Soulé (1979) have all
studied relaxation models and faunal collapse models to
lend some predictability to the problem of how biodiversity will decay in reserves of restricted area. Depending on initial assumptions, collapse dynamics were found
to be exponential or algebraic functions in time.
Again we used as input the data from three “islands”
evolved over 200,000 cycles. The “evolution” process
was then continued with seven mutation rates up to 50
times the quasi-static rate for up to 500 time steps. Each
island was thereby driven far from equilibrium. At this
high rate of speciation, the diversity increases linearly in
time. The mutation rate was then “turned off” and either
the entire island or some isolated area within the island
was allowed to relax back to a steady state. The collapse
dynamics always had the functional form
S ( t ) = ( S o – S eq )e
–t ⁄ τ
+ S eq
in agreement with Diamond (1984). The time constant
of collapse, t, scaled linearly with isolate area and was
independent of initial speciation rate. Using the fitted
values for So (the peak diversity) and Seq, it was possible
to determine how each scaled with isolate area. The
data suggest power law behavior only down to a lower
length scale, set by the high mutation rate. The exponents were still in the range of natural species-area data:
So z A0.3 and Seq z A0.5. The peak diversity (system furthest from steady state) scales with area with a larger exponent than the steady state diversity. This is consistent
with the experimental observation of smaller exponents
for continental isolates that experience higher species
migration rates.
Conclusions
Figure 7. The Wilson-MacArthur island area relationship is manifest by this model. As a function of island
size the number of species scales as S 5 CAz with
0.14 , z , 0.28.
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Volume 12, No. 3, June 1998
We set out to discriminate between two classes of theories explaining mass extinction: one predicting that ecosystems evolve to a dynamically stable state with mass
extinction triggered only by global catastrophe, another
predicting that mass extinction is an intrinsic property
of natural dynamic systems. Our model yields yet a third
possibility. Although natural systems may in fact evolve
to an ordered state with a high degree of stability, attaining this level of diversity may require evolution through
a critical point susceptible to fluctuations of all sizes. Because of critical slowing down, a real ecosystem may get
stuck near a point of “critical biodiversity” for long periods of time before evolving to an ordered state.
In the absence of global disruption, ecosystems may
naturally evolve toward stability and order. Some of the
zoogeographical data, however, present a sobering possibility. Whereas the observation of scaling is not proof
Kaufman et al.
that a system is at a critical point, one expects a system
that has evolved to the ordered state to exhibit an upper
length-scale where scaling breaks down. If no cutoff or
maximum correlation length is observed, the islands in
question may be very close to criticality. There are four
obvious possibilities. First, our work may lack a key
property of natural selection and perhaps these islands
have evolved to the critical state. Second, the absence of
a cutoff length may arise if the islands are “too young.”
There may not have been time for life on these islands to
evolve past the critical biodiversity (critical slowing
down). Third, perhaps the observations have not yet
been made at the maximum length (it may exist but may
not yet have been observed). Finally, the ascent of humans may already have changed the global environment
enough to drive biodiversity to the critical point. Reanalysis of existing paleobiographic data may help determine the order parameter, susceptibility to extinction,
and critical biodiversity for life on Earth. The world is in
the midst of another mass extinction. If our theory is
partially correct, we should all heed the advice of the
conservation biologists. Preserving the remaining diversity may be the best strategy to optimize the near-term
survival probability of all species including our own.
Acknowledgments
The authors would like to acknowledge the helpful criticisms of J. Drake, D. Gessler, D. Perry, R. Plotnick, and
M. Stanton. The time they took to look at this work and
their clarification of biological concepts was greatly appreciated.
Literature Cited
Alvarez, L. W., F. Asaro, H. V. Michael, and W. Alvarez. 1980. Extraterrestrial cause for the Cretaceous-territory extension. Science 208:
1095–1108.
Aronson, R. 1994. Scale-independent biological processes in the marine environment. Oceanography and Marine Biology: an Annual
Review 32:435–460.
Bak, P., C. Tang, and K. Wiesenfeld. 1988. Self-organized criticality.
Physical Review A 38:364–374.
Bak, P., and M. Paczuski. 1995. Complexity, contingency, and criticality. Proceedings of the National Academy of Sciences of the USA
92:6689–6696.
Benton, M. J. 1995. Diversification and extinction in the history of life.
Science 268:52–58.
Boecklen, W. J., and D. Simberloff. 1986. Area-based extinction models
in conservation. Pages 247–277 in D. K. Elliott, editor. Dynamics of
extinction. John Wiley and Sons, New York.
Darlington, P. J., Jr. 1957. Zoogeography: The geographical distribution of animals. John Wiley and Sons, Inc., New York.
Diamond, J. M. 1984. “Normal” extinctions of isolated populations.
Pages 191–245 in M. H. Nitecki, editor. Extinctions. The University
of Chicago Press, Chicago.
Drake, J. A. 1990. Communities as assembled structures: do rules govern pattern? Trends in Ecology and Evolution 5:159–163.
Critical Biodiversity
531
Drake, J. 1990. The mechanics of community assembly and succession. Journal of Theoretical Biology 147:213–233.
Drake, J. A., G. J. Whitteman, and G. R. Huxel. 1992. Development of
biological structure: critical states, and approaches to alternative
levels of organization. Pages 457–463 in J. Eisenfield, D. S. Levine,
and M. Witten, editors. Journal of biomedical modeling and simulation. Elsevier Science Publishers, Amsterdam.
Durrett, R., and S. Levin. 1994. Stochastic spatial models: a user’s guide
to ecological applications. Philosophical Transactions of the Royal
Society of London, B 343:329–350.
Flyvbjerg, H., K. Sneppen, and P. Bak. 1993. Mean field theory for a
simple model of evolution. Physical Review Letters 71:4087.
Gleick, J. 1987. Chaos, making of a new science. Viking, New York.
Gould, S. J., and N. Eldredge. 1993. Punctuated equilibrium comes of
age. Nature 366:223–227.
Jongeling, T. B. 1996. Self-organization and competition in evolution: a
conceptual problem in the use of fitness landscapes. Journal of
Theoretical Biology 178:369–373.
Kauffman, S. A., and S. Johnson. 1991. Coevolution to the edge of
chaos coupled fitness landscapes poised states and coevolutionary
avalanches. Journal of Theoretical Biology 149:467–506.
Kauffman, S. A. 1993. The origins of order. Oxford University Press,
New York.
Lorenz, E. N. 1963. The mechanics of vacillation. Journal of the Atmospheric Sciences 20:448–464.
Lovejoy, T. E., J. E. Rankin, R. O. Bierregaard Jr., K. S. Brown Jr., L. H.
Emmons, and M. E. Van der Voort. 1984. Ecosystem decay of Amazon forest remnants. Pages 295–325 in M. H. Nitecki, editor. Extinctions. The University of Chicago Press, Chicago.
Lovejoy, T. E., R. O. Bierregaard Jr., A. B. Rylands, J. R. Malcolm, C. E.
Quintela, L. H. Harper, K. S. Brown Jr., et al. 1986. Edge and other
effects of isolation on Amazon forest fragments. Pages 257–285 in
M. E. Soulé, editor. Conservation biology: the science of scarcity
and diversity. Sinauer Associates, Sunderland, Massachusetts.
Ma, S. K. 1976. Modern theory of critical phenomena. Benjamin-Cummings, Reading, Massachusetts.
MacArthur, R. H., and E. O. Wilson. 1963. An equilibrium theory of insular zoogeography. Evolution 17:373–387.
Mandelbrot, B. B. 1983. The fractal geometry of nature. Freeman, San
Francisco.
Moses, C. O. 1989. A geochemical perspective on the causes and periodicity of mass extinctions. Ecology 70:812–823.
Newell, N.D. 1952. Periodicity in invertebrate evolution. Journal of Paleontology 26:371–385.
Parker, V. T., and R. L. Simpson, editors. 1989. Ecology of soil seed
banks. Academic Press, San Diego.
Patterson, R., and A. Fowler. 1996. Evidence of self organization in
planktic foraminiferal evolution: implications for interconnectedness of paleoecosystesm. Geology 24:215–218.
Perry, D. 1995. Self organizing systems across scales. Trends in Ecology and Evolution 10:241–244.
Plotnick, R., and R. Gardner. 1993. Lattices and landscapes. Lectures
on Mathematics in the Life Sciences 23:129–157.
Plotnick, R., and M. McKinney. 1993. Ecosystem organization and extinction dynamics. PALAIOS 8:202–212.
Raup, D. M. 1991. Extinction: bad genes or bad luck? W. W. Norton,
New York.
Simberloff, D. 1972. Models in biogeography. Pages 160–191 in T. J. M.
Schopf, editor. Models in paleobiology. Freeman, Cooper and Company, San Francisco, California.
Simberloff, D. 1986. Are we on the verge of a mass extinction in tropical rain forests? Pages 165–180 in D. K. Elliott. Dynamics of extinction. John Wiley and Sons, New York.
Soulé, M. E., B. A. Wilcox, and C. Holtby. 1979. Benign neglect: a
model of faunal collapse in the game reserves of East Africa. Biological Conservation 15:259–272.
Conservation Biology
Volume 12, No. 3, June 1998
532
Critical Biodiversity
Sole, R. V., and S. C. Manrubia. 1995. Are rainforests self-organized in a
critical state? Journal of Theoretical Biology 173:31–40.
Stauffer, D. 1985. Introduction to percolation theory. Taylor and
Frances, London.
Terborgh, J. 1974. Preservation of natural diversity: the problem of extinction prone species. Bioscience 24:715–722.
Vrba, E. S. 1985. Environment and evolution: alternative causes of the
temporal distribution of evolutionary events. Suid-Afrikaanse Tydskrif vir Wetenskap 81:229–236.
Conservation Biology
Volume 12, No. 3, June 1998
Kaufman et al.
Wilcox, B. A. 1978. Super-saturated island faunas - species-age relationship for lizards on post-pleistocene land-bridge islands. Science
199:996–998.
Williamson, M. 1989. Natural extinction on islands. Philosophical
Transactions of the Royal Society of London, B 325:457–468.
Wilson, E. O. 1992. The diversity of life. W. W. Norton and Co., New
York.
Wright, S. 1982. Character change speciation and the higher taxa. Evolution 36:427–443.