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Contributed Papers

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 kaufman@almaden.ibm.com†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 interac-tions. Are ecosystems intrinsically stable or do they naturally develop into a chaotic state where mass extinc-tion is an unavoidable consequence of the dynamics? To study this problem we developed a computer modelin which the organisms and their interactions “evolve” by a “natural selection” process. The organisms existon a multi-dimensional lattice defined both by a diverse physical landscape that does not change and by thepresence 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 evolvethat specialize or that are adapted to many niches. The particular niches that individual species are adaptedto occupy are not built into the model. These interactions develop as a consequence of the selection process. Asspecies in the model evolve, a complex food web develops. We found evidence for a “critical” level of biodiver-sity at which ecosystems are highly susceptible to extinction. Our model suggests the critical biodiversity pointis not a point of attraction in the evolutionary process. Our system naturally reaches an ordered state whereglobal perturbations are required to cause mass extinction. Reaching the ordered state beyond the criticalpoint, however, is kinetically limited because the susceptibility to extinction is so high near the critical biodi-versity. We quantify this behavior as analagous to a physical phase transition and suggest model indepen-dent measures for the susceptibility to extinction, order parameter, and effective temperature. These measuresmay also be applied to natural (real) ecosystems to study evolution and extinction on Earth as well as the in-fluence of human activity on ecosystem stability.

Biodiversidad Crítica

Resumen:

Los ecosistemas son sistemas dinámicos en los organismos sobreviven a una compleja red de in-teracciones. ¿Son intrínsecamente estables los ecosistemas o se desarrollan naturalmente hasta un estadocaótico en el que la extinción masiva es una consecuencia inevitable de la dinámica? Para estudiar este prob-lema 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 porun diverso paisaje físico que no cambia y por la presencia de otras especies que están evolucionando. Este en-rejado multidimensional define un vector de “nichos” dinámico. Los nichos posibles incluyen al paisaje físicofijo y a todas las especies. Las especies pueden evolucionar para especializarse o para adaptarse a muchosnichos. El modelo no incluye los nichos particulares a los que se adaptan especies individuales. Estas interac-ciones se desarrollan como consecuencia del proceso de selección. A medida que evolucionan las especies enel modelo, se desarrolla una compleja red alimenticia. Encontramos evidencia de un nivel “crítico” de biodi-versidad en el que los ecosistemas son altamente susceptibles de extinción. Nuestro modelo sugiere que elpunto crítico de biodiversidad no es un punto de atracción durante el proceso evolutivo. Nuestro sistema al-canza un estado ordenado naturalmente en el que se requieren perturbaciones globales para causar estin-ción masiva. Sin embargo, alcanzar el estado ordenado después del punto crítico esta limitado cinéticamenteporque la susceptibilidad de extinción es muy alta cerca del punto crítico. Cuantificamos este compor-tamiento como análogo al de fase de transición física y sugerimos parámetros independientes del modelopara medir susceptibilidad de extinción, orden y temperatura efectiva. Estas medidas también se pueden apli-car a ecosistemas naturales (reales) para estudiar la evolución y extinción sobre la Tierra, así como la influ-

encia 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 greatextinction spasms of geological history.” Although thismass extinction is due to human activity, the fossilrecord reveals several previous mass extinctions nearthe ends of the Silurian, Devonian, Permian, Triassic,and most recently in the Cretaceous periods (Wilson1992; Benton 1995). Understanding the origins and dy-namics of these extinctions is a topic of considerable in-terest both for fundamental reasons and because of theimplications for the extent of the extinction events weare now triggering (Simberloff 1986).

Two major classes of theories have been put forwardto explain these mass extinction events. The first in-vokes a major external event such as the collision of alarge asteroid or comet with the Earth (Alvarez et al.1980; Raup 1991), global climate changes (Vrba 1985),sea level changes (Newell 1952), or volcanism (Moses1989). The second class of theories postulates that eco-systems, like many other natural dynamic systems, evolvetoward 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 per-turbations may trigger macroscopic events on globallength scales (Gleick 1987). If chaotic processes were op-erating in an ecological system, then extinction events ofall sizes should be evident over an appropriate spatiotem-poral scale. Furthermore, predicting the dynamics wouldbe problematic because of the extreme sensitivity of thesystem to small perturbations. It is this sensitivity thatmakes weather prediction so difficult (Lorenz 1963).

The mathematical properties of chaotic systems havebeen shown by Mandelbrot (1983) to be fractal. The spa-tial and dynamic properties of all fractals scale (i.e., theyobey power laws). Similarly, the properties of physicalsystems that undergo second-order phase transitions ex-hibit scaling behavior at the “critical temperature” wherethe phase transition occurs (Ma 1976). This similarity ledto the term “self-organized criticality” to describe sys-tems that naturally evolve to and stay at a critical or cha-otic state (Bak & Paczuski 1995; Patterson & Fowler1996; Perry 1995; Sole & Manrubia 1995).

Several authors have constructed models for evolutionin an “ecosystem” (Plotnick & Gardner 1993; Drake1990; Drake et al. 1992; Durrett & Levin 1994). Some ofthese models lead to chaotic behavior. For example, Plot-nick & McKinney (1993) applied percolation theory tostudy the extinction process. In the Plotnick–McKinneymodel, depending on the relative rates of species cre-ation and species death, the system may be tuned to apoint where extinction events of all sizes may occur.When tuned to this critical state, the death of a singlespecies may trigger a mass extinction.

Flyvbjerg et al. (1993) developed a model based on anevolutionary fitness landscape, the “shifting balance the-

ory” described by Wright (Wright 1982; Jongeling 1996).Over time, they observed extinction events (equal to thenumber of transformations) of all sizes (Gould & Eldredge1993). Quiescent periods are characterized by most spe-cies having similar “fitness,” and avalanches of extinc-tions 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 theecology as a whole is “most fit” at the critical point. Inhis book

The Origins of Order

, Kauffman (1993) postu-lates that natural selection drives biological “adaption tothe edge of chaos.”

These models suggest that natural ecosystems couldbe in a critical state. The question is, how can one tell ifchaos is an inevitable consequence of the dynamics? Ifthe behavior of ecosystems is analogous to the behaviorof other dynamic systems that evolve to or through acritical state, then a mathematical framework already ex-ists to describe their dynamics. To apply this framework,one must find measures of ecosystems that are analo-gous to the variables used to describe the state of otherdynamic systems: an effective temperature, susceptibil-ity, and order parameter (Ma 1976). It is not at all obvi-ous, a priori, what these measures would be.

We set out to develop a model in which both organ-isms and their interactions evolve into a complex web ofinterdependency. In allowing the interactions to de-velop as a consequence of the model, we sought toavoid building into the web of interdependency featuresthat would predetermine the dynamic behavior of themodel. Our goal is neither to reproduce the richness ofnatural biological interactions nor to prove that the ex-tinction dynamics exhibited by our model are the sameas natural extinction dynamics. Rather, we hope to usethis model to develop measures and to show how theymight be applied to understand extinction dynamics inreal biological systems.

The Model

Our model world is a square lattice (100

3

100). The lat-tice, which does not use periodic boundary conditions,is analogous to an isolated island in a lifeless sea. Eachlattice site is randomly assigned one of six local environ-ment types; therefore, no one environment type occu-pies an unbroken path spanning the lattice (Plotnick &Gardner 1993). At every location or site (x,y), the envi-ronments 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 condi-tions. 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 groupor species it belongs to. For example, at some instant intime at a particular site (i,j) there may exist at the bot-

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tom of the chain of ecological dependence an organismof a particular species, call it s(17). The existence of spe-cies s(17) on site (i,j) may provide a niche for anotherspecies on the same site. This second species also has alabel which might be s(3). The species s(3) also had tocompete for a niche, but the niche that makes it possi-ble for s(3) to exist on site (i,j) is not the environmente(i,j); it is s(17). Higher on the chain another organismmay occupy the niche created by the presence of s(3).

Instantaneously at a particular site the species are or-ganized in this hierarchical fashion. Over a longer periodof time, and over the world as a whole, organisms com-pete in a more complex food web. The same speciess(3) living on s(17) at site (i,j) and time

t

may be livingon a different species (occupying a different niche) atsome other site (m,n) at time

t

. The model is dynamic sothe chain of interdependency at any site and time

t

maybe different at time

t

9

. The organisms exist on this multi-dimensional lattice with local conditions defined bothby the physical landscape (e(x,y)), which does notchange, and by the presence of other species. This mul-tidimensional lattice defines a dynamic vector of avail-able “niches” (Plotnick & Mckinney 1993). The possibleniches include all of the local environment types and allof the species themselves. The particular niches individ-ual species are adapted to live on are not built into themodel. Adaptation occurs through a natural selectionand ultimate speciation process. The only restriction webuilt into the model regarding the food web is that nospecies may live on itself. This is our “conservation ofenergy” rule. This is not a predator-prey model. Organ-isms in the model are never consumed by other organ-isms. In dynamic balance the presence of prey makes itpossible for a predator to survive in the same sense thatthe presence of a tree makes it possible for an epiphyteto survive, but the mathematical model only tracks theexistence of the relationship or dependency. A speciesbecomes extinct only if there is no niche available for itto live on or if it cannot successfully compete for anavailable niche somewhere on the lattice, at which timeevery individual organism of that species disappears.

None of the species-species interactions are built intothe model. The food web evolves by “mutation” or,more precisely, speciation. What is required in themodel is a “data structure” that keeps track of the rela-tionships between all existing species and all environ-ment types. Mutation will be accomplished by small ran-dom changes to this data structure. The data structure isnot a “genetic” code. It is an accounting scheme thattracks how well every species is adapted to every exist-ing 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 theexisting environment types and to live on the other ex-isting species, respectively. The use of two separate vec-tors is for convenience because (1) the environments do

not change in our model and (2) the species vector dy-namically changes length as new species come into ex-istence (through mutation and speciation) and as oldspecies die.

Data Structure

The numbers or adaptive weights stored in these twovectors reflect the relative “fitness” of species for differ-ent possible niches and are used to determine the out-come of competition between individuals. Consideragain species s(3). Call it an orchid. The available nichesin the world include the local environment types and allof the existing species. The orchid is not adapted tomost of these niches. It may be adapted to only one typeof local environment. Suppose the label for this environ-ment 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 ex-cept for the fourth element. The magnitude of that ele-ment reflects how strongly the orchid is adapted to theenvironment. These adaptive weights are relative andthe orchid may be adapted to environment type fourwith a weight of two. If there are six different local envi-ronment types, the environment vector for the orchid isthen (0,0,0,2,0,0). As before, the orchid is also adaptedto live on species s(17), which we will call a fig tree. Forsimplicity in this example, suppose s(17) is the onlyother species on which the orchid is adapted to live.Then the species vector for the orchid is zero except forelement 17 which measures how strongly the orchid isadapted to the fig tree. That element might have a value(weight) of nine. The orchid may speciate into a new or-ganism with new adaptations or different adaptive weights.

Competition Process

Given the vectors of relative competitive weights for ev-ery species, the competition process between individualorganisms is straightforward. At every time cycle of theprogram, each individual on the grid competes to existagain on the sites where it existed in the previous timecycle and competes for sites that are nearest neighborsto the sites where it existed. The nearest neighbors toany site on a square lattice are, by definition, the foursites offset by

6

1 unit in x or y. The competition takesplace simultaneously at all sites (x,y) on the map. Con-sider a specific example of organisms competing to liveon a particular site (i,j) at time

t

. The individuals of vari-ous species at site (i,j) and at the four nearest neighborsites to (i,j) are put on a list. Even if a species is repre-sented at more than one of the nearest neighbor sites, itis only added to the list once. A competition process fol-lows, first for the spot at the bottom of the chain of eco-

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logical dependence (the environment e(i,j)). Organismsthat exist higher on the chain of ecological dependenceon a neighboring site may also be able to compete forthe bottom niche on site (i,j). Species not adapted toe(i,j) have zero adaptive weight for that element of theirenvironment vectors and do not enter this round ofcompetition. The organism that wins the competitionfor the first niche at time

t

, is chosen probabilisticallybased on the relative adaptive weights. The single win-ner is not necessarily representative of the most welladapted species at a particular site and time, but that is amore likely event than selection of a relatively less fitcompetitor. Having selected the organism that will fillthe bottom niche, the organism is removed from the listof competitors

for that site

and added to a map matrixm(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 onewhen it is present. Species that lost the competition forthe first niche then seek to occupy the niche created bythe 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 respectivespecies vectors. Each species selected in this manner be-comes, itself, the next available niche. The species notyet selected compete to fill the next niche. The processcontinues until either all represented species have beenchosen or until none of the organisms awaiting a nicheare adapted to the last available niche. Any species re-maining have then lost the competition for (i,j) at time

t

.

Creation Event

At the beginning of the simulation, the world (grid) isseeded with a single individual of a single species, s(0),which is adapted to live at a single type of environmentwith an adaptive weight of one. A specific site, (i,j), forthis creation event is chosen at random on the edge ofthe grid. Because no environment e(i,j) provides a con-tinuous path across the map (i.e., it does not percolate),the seed species s(0) will spread to at most a few neigh-boring sites. To spread further, the seed organism mustspeciate.

Speciation

In our model every mutation event creates a new spe-cies from the parent organism. We have included twotypes of mutations. The first is an increase in fitness, byone, of one element of the environment or species vec-tor. The element selected for this net increase in fitnessis random. It is equally likely to occur in any possibleniche (environment or species vector element). The sec-ond type of mutation involves a shift of adaptive weight.In this shift, the fitness (adaptive weight) for one niche

is decreased by one, and the competitiveness for a differ-ent niche is increased by one. These niches are chosenat random with one restriction. The niche in which thespecies becomes less competitive must initially havenonzero weight. The second mutation process allowsnew species to “explore” the space of possible interac-tions without increasing overall fitness. Of course the ac-tual fitness of individuals is a relative quantity dependingboth on the characteristics of the species and on the lo-cal conditions in the multidimensional lattice. Theseconditions vary from site to site and as a function of timewithin the model.

So far we have defined no tuning parameters in thismodel. Because we want the system to select the “favor-able” interactions while increasing fitness quasi-stati-cally, we set 60% of mutations to result in a shift in com-petitiveness (“neutral” speciation) and 40% to an increasein competitiveness (“advantageous” speciation). In bothtypes of mutation processes, the parent species is a“possible” niche for the child species. Recall that no spe-cies can “live on itself.”

Finally, one must decide what absolute rate of muta-tion to allow. The speciation process is intended to in-troduce small local changes. One can then measure anyresponses or “avalanches” of mass extinction that occurin response to a new species. For this reason we adopt aquasi-static approximation. On average, we assume thereis enough time between speciation events so that newspecies that

can

spread across the grid will before an-other speciation event occurs. If speciation occurs at anoverall higher average rate, then this system will neverbe in steady state and an arbitrary length scale would bebuilt into model (namely how far a new species canspread before it speciates again). To avoid this we dy-namically reduce the mutation rate as the population ofthe grid increases. Speciation occurs just before eachcompetition, or spreading, event. A “child” species thencompetes for the site on which it was created as well asthe four nearest neighbors. The parent species is repre-sented in this competition as well. The mutation proba-bility per competition event per species per site is de-fined as

,

where L, the width of the grid, equals 100. In reality, ge-netic mutation occurs with constant probability for ev-ery reproductive event (independent of population).The quasi-static approximation implies the actual rate ofmutation is always small compared to the rate speciescan spread by competition. In this regime it is possibleto study the response of a system in steady state to sin-gle, local perturbations.

Our goal in choosing mutation rules is to incorporatein the model a means for the “ecosystem” to explore dif-ferent interaction spaces through a series of small per-

R0.75

L----------

1

population--------------------------

×=

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turbations. By running the program many times one cansample the most likely possible food web organizations.The absolute mutation rate was made sufficiently smallso that it is an “irrelevant tuning parameter.” In thequasi-static approximation the absolute frequency ofmutation is low enough to allow new species to spreadto their steady state range. Lowering the mutation ratefurther would only increase the time in the simulationwhen 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 organ-isms on the grid? Because the particular speciation pro-cess we chose will slowly increase overall fitness, newspecies are likely to replace parent species over time.Whether or not new keystone species support the hier-archical chain of ecological dependence determines thestability in this model. The question of whether an exist-ing species provides a niche for a new mutant definesthe most important parameter in our model. To resolvethe question we turn to Darwin. We assume speciationarises from a mutation process in a population of organ-isms of the same species present on a given site. As wediscussed, one can consider that a species that providesa niche for another species is being preyed upon andthat the populations are in dynamic balance. With re-gard to the “prey,” survival of the fittest implies thatthose individuals that evolve to resist predation are morelikely to survive and reproduce. If there are no predatorspresent, however, there is no evolutionary advantage tothose individuals that resist predation. To incorporatethis phenomenon in the model, we established the fol-lowing rules.

(1) If an existing species could “prey” upon the parentof a new mutant

and

if the existing species is

not

present on the site when and where the mutationoccurred, then it can survive on the niche createdby the child species with probability P1.

(2) If an existing species could prey upon the parentof a new species

and

if the existing species

is

present on the site when and where the speciationoccurred, then it can survive on the niche createdby the child species with probability P2

,

P1. It isless likely to survive on the niche created by thechild because the new species may have evolvedto resist predation.

(3) If an existing species could not prey upon the par-ent, then it cannot prey upon the child. In this lastrule we treat the mutation as a small perturbationfrom the parent species (which was not a viableniche for the predator).

These three conditions cover all possibilities. If P1

5

P2, species would never evolve to resist predation. Inthis case the model never produced specialization andall species evolved to live everywhere. Because we don’tknow a priori how to define P1 and P2, we studied bothP1

5

1.0, P2

5

0.0 and P1

5

0.75, P2

5

0.25. Theseconditions produce the same dynamics. On first inspec-tion this rule seems quite destabilizing, making it ex-tremely likely that a keystone species would be replacedby a variant that would not support existing predators.One might expect this instability to be the source of ex-tinctions on all scales, and perhaps the cause of self or-ganized critical behavior. The actual behavior of themodel was a surprise. Even for P1

5

1.0 and P2

5

0.0(the parameters used for the data reported here) wefound evolution to a dynamically stable state.

Results and Discussion

We ran the simulation 14 times (using different initialrandom number seeds) for 200,000 cycles. A cycle, ortime step, is defined by the competition and spreadingprocess. In one time step an organism may spread froma site it occupies to a nearest neighbor site. The mini-mum time required for an organism to spread across thelattice is

t

5

L cycles where L (

5

100) is the linear size ofthe grid. The data were analyzed after 100,000 and200,000 cycles (typical program, Figs. 1 & 2).

It is useful to discuss, qualitatively, the dynamics ob-served in a typical run, or evolution time series, on thegrid. In a typical simulation, there is a period of timewhere evolution takes place in a localized region (upperleft-hand corner of Fig. 1 at

t

5

2090). Eventually, one ortwo species acquire, through mutation, the ability tospread throughout the grid (first frame in Fig. 1, initialspreading event). Often we found that two speciesspread together. In this example (Fig. 1), 15 species ex-ist and two of them are in the process of spreadingacross the grid together. In frame 3, over 10,000 timesteps later, some of the primitive children of the initialseed are still evident in the upper left corner. This framealso captures the almost coincident birth of two newspecies capable of spreading throughout the grid. This isa rare event given the quasi-static mutation rate (one mu-tation per 133 time steps on average). The last twoframes of Figure 1 are representative of a persistent stateof the ecosystem. We call it the “primitive” state. Thereis a fair degree of diversity on much of the grid area, butthere is no overall spatial structure or organization. Thisstate may persist for long periods of time. It is highly un-stable and susceptible to sudden periods of extinctionactivity where one or more keystone species are re-placed, making large numbers of species unfit. In thisstate two types of global changes can occur: mass ex-tinction where large numbers of species die, and species

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replacement where dominant species on the grid are re-placed. The primitive state is characterized by long peri-ods of quiescence when the total number of speciesdoes not change and by sudden mass extinctions wherelarge numbers of species die. This is exactly the inter-mittent behavior discussed by Flyvbjerg et al. (1993).Relative fitness of many species on the grid may be af-fected by a single mutation effect. With few local ex-ceptions, no species is isolated or protected. In thisprimitive state, many of the species percolate through-out the grid.

In the “ordered” state species on the grid have orga-nized themselves into separate subecosystems. Special-ization has occurred creating well defined communities.This is evident both in Figure 2 and in detailed examina-tion of the species and environment vectors. Not all evo-lutionary 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 forcompetition within well defined communities. The col-lection of these separate communities forms well de-fined spatial patterns on the map. These patterns are un-correlated with the local environments. The speciesthemselves define these large-scale patterns or environ-ments which persist over long periods of time (Fig. 2).Each community undergoes changes, and species are re-placed by more fit species at the same rate as before.These changes, however, are localized to the clustersthat contain the communities and do not lead to mas-sive species loss. Once a system evolves to this orderedstate, we never observe a collapse back to the primitivestate.

For each run of the simulation it is possible to measurethe number of species versus time (Fig. 3). From thesetime series it is possible to make some general observa-tions about the dynamics. Extinction events of different

Figure 1. The early stages of evolution in one ecosys-tem 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 long- range 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 ex-tinctions 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 envi-ronment types. It reflects specialization and formation of separate “communities” of species. Mutation and extinction continue as before but the spatial organiza-tion persists.

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sizes occur. Time series C (Fig. 3) corresponds to therun depicted in Figs. 1 and 2. It exhibits three mass ex-tinctions before 100,000 cycles where almost half thespecies on the island die. Large extinctions are evidentin each of these time series, especially early in the emer-gence of each ecosystem. Over longer periods of timethe fluctuations grow more slowly than the overall diver-sity. The fraction of species extinguished, however, de-creases as the diversity of life increases. The model pro-duces ecosystems that seem to pass through a statecharacterized by fluctuations on all scales. Is this state acritical point? It seems not to be a point of attraction forthe ecosystems. Eventually, diversity and specializationproduce dynamic stability.

It is desirable to eliminate the time variable in analysisof this problem. What are meaningful time-independentmeasures? For these we look to standard problems incondensed matter physics. We are interested in deter-mining if there is a critical point associated with this dy-namic system and if that critical point is the point of at-traction. Critical points are associated with second-orderphase transitions (Ma 1976). Some textbook examplesare ferromagnetism and percolation (Ma 1976; Stauffer1985). Formally, these concepts apply to systems inequilibrium. Critical behavior, however, can often be ob-served in dynamic systems in “steady state” as they are“tuned” through or near a critical point. In a second-order phase transition, the order parameter varies smoothlyfrom zero to a finite value with a power law:

,

where T is an effective temperature and T

c

is the criticalpoint. In percolation the effective temperature is the

M T Tc– β=

density of points on the lattice and the critical tempera-ture is the percolation threshold (that density where thelargest cluster first spans the system). At any criticalpoint many properties exhibit power law behavior or“scaling.” Often the observation of scaling is used as evi-dence that a dynamic system is at a critical point. Manyproperties, however, exhibit scaling over some lengthscale even when T is not tuned to T

c

. This length scaleover which the power law behavior is observed is thecorrelation length, which diverges at T

c

and decreaseswith a Boltzman Law as the system is tuned away fromthe critical point.

To determine if extinction dynamics in ecosystems is atrue critical phenomenon, it is important not only tolook for scaling, but to find an order parameter, suscep-tibility, and effective temperature for the system. Mea-sures of these quantities need not be unique (severalquantities can often be used to measure the degree of or-der in a system, for example). The measures should begeneral and apply to real ecosystems as well as differentmodels. Because diversity determines the overall organi-zation (or lack thereof) in an ecosystem, we use as an ef-fective temperature the total diversity, S (number of spe-cies living). This is a quantity we can measure inanalyzing time series (Fig. 3) to study how susceptibilityto extinction varies with biodiversity. To measure sus-ceptibility to extinction,

x

(S), we measure

average

ex-tinction size as a function of diversity S. This is an inte-gral, the sum of total number of extinctions (speciesdeaths)

E

(S

9

) for S

9

,

S for each S. This sum for eachparticular diversity, S, cannot begin until the system hasfirst evolved to diversity S:

.

This quantity is not necessarily finite. In fact one ex-pects susceptibility to be singular at a critical point. Fora finite simulation it typically exhibits a cusp. Becausethe integral cannot practically be carried out to t

5

`

weinvoke the law of large numbers and plot

x

(S) averagedover 14 runs of the simulation computed at

t

5

100,000and

t

5

200,000 (Fig. 4). A cusp is observed near S

5

18where the susceptibility to extinction peaks. The peakdoes not move as the simulation time is doubled.

To define the order parameter we need first to under-stand what we mean by the ordered state. We are inter-ested in the intrinsic relationship between diversity andextinction. The ordered state is the state with the small-est (relative) extinction probability per species per per-turbation event. The perturbation events are the specia-tion events. We define E(S) as the number of extinctionsthat occur at diversity S integrated over all time. C(S) is

χ S( ) 1S--- E t S ′ t( ),( )θ S ′ S,( ) td

t0

s' t0( ) s>

∞

∫≡ θ 0= : S ′ S≥

θ 1= : S ′ S<{

Figure 3. Three typical time series showing the biodi-versity as a function of time. The curve C is the ecosys-tem evolution depicted in Figs. 1 and 2. Note the ab-sence of mass extinction above a diversity of about 20 species. The fluctuation or extinction size does not grow as fast as the overall diversity.

528

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Figure 4. The susceptibility to extinction as a function of diversity has a cusp or maximum at a critical biodi-versity near 18 species. This critical point does not shift as the simulation time is doubled.

the number of creation or mutation events as a functionof diversity. The probability of an extinction event at di-versity S per creation event per species is

,

where the normalization constant is

.

Similarly, the probability of an extinction event at diver-sity S per creation event is

,

where the normalization constant is

.

The probability of surviving C(S) perturbation events isthen

,

where

r

1,2

5

r

1

or

r

2

. This then defines two differentmeasures for an order parameter, M: one measures theaverage survival probability at diversity S and one mea-sures the survival probability per species at diversity S.

,

and

.

Neither order parameter is a total survival probability forindividual species. If a species survives at diversity S,there is still the probability it will become extinct at di-

ρ1 S( ) E S( ) γSC S( )⁄=

γ E S( )SC S( )---------------

S∑=

ρ2 S( ) E S( ) αC S( )⁄=

α E S( )C S( )------------

S∑=

P C S( ) S,( ) 1 ρ1 2, S( )–{ } C S( )=

M1 1 E S( )γSC S( )------------------–

C S( )

=

M2 1 E S( )αC S( )----------------–

C S( )

=

versity S 1 1. All species eventually become extinct. Thesensitivity to fluctuations that is lost in the averagingprocess that defines the order parameter is contained inthe definition of susceptibility.

The order parameters defined above are finite. Weplotted M1(S) and M2(S) versus S, for 14 trials run to t 5200,000 (Fig. 5). As a further check that we were closeto the steady state value for M1,2, we also plotted a thirdmeasure of the order parameter, M3(S) (Fig. 5). The M3 isdefined as the integrated extinctions that occur beforethe system first evolves to diversity S. This measure is al-ways 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 soas S→`, M2→1. Normalization at finite diversity triviallysets M3(Smax) 5 1.

The three measures of the order parameter are consis-tent. The M1,2 peaks near S 5 0 because the initial seed-ing is a singular process. All of these possible order pa-rameters suggest a critical point near S 5 18. The tail inthe order parameter, M3, below S 5 18 is a finite size ef-fect intrinsic to a finite system. To understand this effectconsider the example of percolation. As points are addedto a lattice, the first “infinite” cluster of adjacent pointsforms at the critical concentration pc. For a finite lattice,there is a nonzero probability of forming a finite clusterthat 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 be-low which the species survival probability falls expo-nentially (Figs. 4 & 5). The particular value of Sc (near 18species) is not universal and scales with system area.What is special about this point? Does the extinctionrate increase when the diversity is near Sc? The averagenumber of deaths per time cycle is constant (about 133deaths/cycle) and just below the constant mutation rate.

Figure 5. Three measures of the order parameter,M: M1 measures the average survival probability at di-versity S, M2 measures the survival probability per spe-cies at diversity S, and M3 the integrated extinctions that occur before the system first evolves to diversity S.

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Kaufman et al. Critical Biodiversity 529

There is no change in death rate or mutation rate nearthe critical diversity. What, then, causes the susceptibil-ity to extinction to change with diversity? We plottedthe average time the system spends in a state with diver-sity S and the average number of species deaths as afunction of diversity (Fig. 6). This data also seem to besingular near Sc. Near the critical biodiversity evolutionslows down. Mutation and extinction rates do not slowdown; the system simply “gets stuck” in the primitivestate depicted in Figure 1. Evolution to a diversity higherthan that of the primitive state is kinetically limited. Evo-lution slows down because the ecosystem is subject tomass extinctions and collapses back to lower diversity.Hence the measured E(S) and C(S) also peak near S 5 18and define the critical point as evidenced in the orderparameter and susceptibility.

Critical Slowing Down

This kinetic barrier may be a manifestation of “criticalslowing down” (Ma 1976). As a system evolves to a criti-cal point, it takes longer and longer to come to equilib-rium. Near Tc the system is very sensitive to fluctuations.A system in such a state could be mistaken for an exam-ple of a self-organized critical phenomenon as it couldspend exponentially long times in a state that exhibitsscaling and “avalanches” of all sizes. If stuck near Sc anecosystem would truly be perched on the edge of chaos.However, that is not necessarily the point of attractionor ultimate ordered state of the system. In our modelevolution eventually produces diversity greater than Sc

leading to specialization and true dynamic stability.There are still extinction events where S decreases butthese do not grow with S. Note that the ordered state isnot characterized by an explosion in population withmore and more species living per site. The diversity persite increases more slowly than S. As the number of spe-

cies on the island increases, specialization leads to theformation of smaller and smaller clusters or communi-ties with separate local food webs. Local mutationevents or species loss are limited to these smaller clus-ters. Massive extinction would require global changes af-fecting the entire grid.

Comparison with Zoogeographical Evidence

The conclusion that life evolves through a critical stateto a dynamically stable state may seem natural to biolo-gists and naturalists. The generality of this conclusion isthe key question. A computer model can never repro-duce all of the dynamic behavior observed in nature.Nevertheless, physical systems driven by mathematicallysimilar forces and interactions often fall into the same“universality class.” That is, the power laws that governthe scaling behavior of the system variables near the crit-ical point have common “critical exponents.” Some-times even very simple models will produce the sameexponents as the more complex natural systems. Tocompare our model to real ecosystems evolving throughnatural selection, we can compare the behavior of themodel with experimental observation of real ecosys-tems. These data are abundant due to the hard work ofconservation biologists (Darlington 1957; Williamson1989).

Darlington (1957) compiled data from several authorswho measured the increase in species diversity with is-land area in the Antilles Islands. MacArthur and Wilson(1963) first recognized the scaling behavior of speciesnumber with island area. This species-area relationship,S 5 CAz, has been found by several investigators to holdfor numerous types of species over numerous islandgroups (Darlington 1957; Boecklen & Simberloff 1986).Experimentally, z ranges from 0.1 to 0.5, often with thepower law valid over many decades in area (Wilson1992; MacArthur & Wilson 1963; Diamond 1984). Thefact that the scaling law holds but the exponent variesfrom island group to island group has been attributed tohow far various island groups are from continentalmasses. This separation distance influences the introduc-tion rate of new species to the different island groupsand perhaps determines how far the ecosystems ofthose groups are from steady state or dynamic equilib-rium. More isolated island groups seem to exhibit largeexponents in the species-area relationship. Regions far-ther from equilibrium exhibit smaller exponents (Wil-son 1992; Diamond 1984).

The species-area relation is also obeyed in reserves.Lovejoy et al. (1984, 1986) selected reserves ranging insize from 1 to 1000 ha for preservation from develop-ment north of Manaus. Forest areas around the reserveswere clearcut or burned for establishment of cattleranches. Lovejoy et al. measured species diversity before

Figure 6. The average rate at which species die is inde-pendent of time. The critical point occurs because the system spends more time near the critical biodiversity.

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and after isolation of these reserves. We find from thesedata that the species-area relation seems to hold forthese reserves. For example, the number of butterflyspecies scales with an exponent of 0.18.

It is trivial to numerically simulate the Lovejoy experi-ment with our computer model. The same grids or “is-lands” that were allowed to develop to 200,000 timesteps were used as input. Smaller circular areas (rangingin radius from 1 to 50) were isolated, and the remainingarea outside this “reserve” was numerically “poisoned”by setting the local environment type to a new valuethat no species could live on. The mutation rate was setto zero and the system allowed to relax. We found thatthe species-area relation is obeyed by our simulationwith 0.14 , z , 0.28. The average value is 0.19 (Fig. 7).Where the species diversity of the “pre-isolate” islandwas near Sc, the power law held for over three decadesin area (up to the size of the simulation). After isolationthe collapse to the steady state number of species oc-curred very fast (in one to three time steps). This is notsurprising given the quasi-static mutation rate of theoriginal simulation, which guarantees the “pre-isolate”islands to be near steady state. Given the range of islandarea exponents observed in nature it is not possible toconclude based on an average z 5 0.19 that our model isin the same universality class as real ecosystems (thoughit is consistent). We can, however, take the numericalexperiment one step further. If the spread in natural val-ues for z is not random but actually a reflection of howfar the island ecosystems are from dynamic equilibrium,it may be useful to study the collapse in species diversityin numerical models deliberately driven out of equilib-rium. In nature this question is important because spe-cies will of necessity migrate to reserves from clearcutareas, but those reserves may not be able to support all

those species looking for refuge. Simberloff (1972),Boeklin and Simberloff (1986), Diamond (1984), Ter-borgh (1974), Wilcox (1978), and Soulé (1979) have allstudied relaxation models and faunal collapse models tolend some predictability to the problem of how biodi-versity will decay in reserves of restricted area. Depend-ing on initial assumptions, collapse dynamics were foundto be exponential or algebraic functions in time.

Again we used as input the data from three “islands”evolved over 200,000 cycles. The “evolution” processwas then continued with seven mutation rates up to 50times the quasi-static rate for up to 500 time steps. Eachisland was thereby driven far from equilibrium. At thishigh rate of speciation, the diversity increases linearly intime. The mutation rate was then “turned off” and eitherthe entire island or some isolated area within the islandwas allowed to relax back to a steady state. The collapsedynamics always had the functional form

in agreement with Diamond (1984). The time constantof collapse, t, scaled linearly with isolate area and wasindependent of initial speciation rate. Using the fittedvalues for So (the peak diversity) and Seq, it was possibleto determine how each scaled with isolate area. Thedata suggest power law behavior only down to a lowerlength scale, set by the high mutation rate. The expo-nents were still in the range of natural species-area data:So z A0.3 and Seq z A0.5. The peak diversity (system fur-thest from steady state) scales with area with a larger ex-ponent than the steady state diversity. This is consistentwith the experimental observation of smaller exponentsfor continental isolates that experience higher speciesmigration rates.

Conclusions

We set out to discriminate between two classes of theo-ries explaining mass extinction: one predicting that eco-systems evolve to a dynamically stable state with massextinction triggered only by global catastrophe, anotherpredicting that mass extinction is an intrinsic propertyof natural dynamic systems. Our model yields yet a thirdpossibility. Although natural systems may in fact evolveto an ordered state with a high degree of stability, attain-ing this level of diversity may require evolution througha critical point susceptible to fluctuations of all sizes. Be-cause of critical slowing down, a real ecosystem may getstuck near a point of “critical biodiversity” for long peri-ods of time before evolving to an ordered state.

In the absence of global disruption, ecosystems maynaturally evolve toward stability and order. Some of thezoogeographical data, however, present a sobering pos-sibility. Whereas the observation of scaling is not proof

S t( ) So Seq–( )e t τ⁄– Seq+=

Figure 7. The Wilson-MacArthur island area relation-ship 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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Kaufman et al. Critical Biodiversity 531

that a system is at a critical point, one expects a systemthat has evolved to the ordered state to exhibit an upperlength-scale where scaling breaks down. If no cutoff ormaximum correlation length is observed, the islands inquestion may be very close to criticality. There are fourobvious possibilities. First, our work may lack a keyproperty of natural selection and perhaps these islandshave evolved to the critical state. Second, the absence ofa cutoff length may arise if the islands are “too young.”There may not have been time for life on these islands toevolve past the critical biodiversity (critical slowingdown). Third, perhaps the observations have not yetbeen made at the maximum length (it may exist but maynot yet have been observed). Finally, the ascent of hu-mans may already have changed the global environmentenough to drive biodiversity to the critical point. Reanal-ysis of existing paleobiographic data may help deter-mine the order parameter, susceptibility to extinction,and critical biodiversity for life on Earth. The world is inthe midst of another mass extinction. If our theory ispartially correct, we should all heed the advice of theconservation biologists. Preserving the remaining diver-sity may be the best strategy to optimize the near-termsurvival probability of all species including our own.

Acknowledgments

The authors would like to acknowledge the helpful criti-cisms of J. Drake, D. Gessler, D. Perry, R. Plotnick, andM. Stanton. The time they took to look at this work andtheir clarification of biological concepts was greatly ap-preciated.

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