Ecosystems are dynamic systems in which the survival of organisms is subject to a complex web of interactions. Indeed, since the fossil record indicates a series of mass extinctions, some scientists wonder if ecosystems inevitably become intrinsically stable, or if they naturally develop to a chaotic state where mass extinction is an unavoidable consequence of complex dynamics. This question may be important both in understanding mass extinction events, and in planning successful nature reserves.

Two major classes of theories have been put forward to explain the known mass extinctions. The first class invokes an external global catastrophe such as the collision of a large asteroid or comet with the Earth [1], global climate changes [2], sea level changes [3], or vulcanism [4].

The second class of theories postulates that ecosystems, like many other natural dynamic systems, evolve toward a chaotic or "critical" state. Evolution to a chaotic state is called "self-organized criticality" [5]. In a chaotic system, the smallest change or local perturbation may trigger a catastrophic change that affects the entire system [6]. If ecosystems exhibit self-organized criticality, they will evolve to a state that is extremely sensitive to small perturbations.

Is there zoological evidence for ecosystems existing near a "critical" or chaotic state? One characteristic of such a state is that the variables that describe a system at criticality exhibit a power law relationship of the form y = x^D, where the exponent D is a constant. When such a power law is observed, the system is said to be fractal, or to exhibit "scaling" behavior [7]. There is abundant evidence of such scaling behavior in natural ecosystems, due to the meticulous work of many conservation biologists [8]. In his book Zoogeography: The Geographical Distribution of Animals, Philip Darlington compiles data from several authors who measured how species diversity in the Antilles Islands increased with island area [9].

MacArthur and Wilson later recognized the scaling or fractal behavior in this data. In An Equilibrium Theory of Insular Zoogeography, they showed that the number of species (S) scales with island area (A) with a power (z), S = CA^z (where C is a constant). This relationship has been verified by several investigators. The power law holds for numerous types of species over numerous island groups [10]. Experimentally, z ranges from 0.1 to 0.5, often with the exponent constant over many decades in area ranging from square meters to many square kilometers [11]. The power law is observed for different island groups, but the value of the exponent, z, varies from island group to island group, depending on how far a given island group is from a continental landmass. More isolated island groups seem to exhibit larger exponents in the species-area relationship. Groups closer to continental landmasses experience a higher rate of new species introduction and exhibit smaller exponents [12].

Lovejoy et al. measured species diversity in Amazon forest areas before and after they were isolated by the clear-cutting and burning of neighboring lands to create cattle ranches [13]. We analyzed Lovejoy's data and found that the species-area relation holds for these reserves as well. For example, over a range of isolated areas from 1 to 1,000 hectares, the number of butterfly species scales with the area to a power of 0.18.

However, the presence of a power law relationship between species number and area does not prove self-organized critical behavior. Many systems, when "near" a critical point, exhibit scaling over some range of length scales, but exhibit a cutoff length scale beyond which the power law breaks down. Furthermore, the fact that a dynamic system is found to be in a chaotic state at some instant in time does not prove that such a state represents the ultimate or final dynamics (or the so-called point of attraction).

How then can one determine if ecosystems become dynamically stable or if they naturally evolve to a state of perpetual chaos? Theoretical models are often of little use, as the interactions assumed in a model will often predetermine the dynamics. In a recent paper published in Conservation Biology [14], we reported the results of a computer model of the evolution of life designed to discriminate between these two scenarios.

The model was structured so that the interactions and relationships between organisms (as well as the "species" themselves) evolved by a natural selection process. Our study revealed an intriguing third possible explanation for the observed scaling: the model indicated a "critical" level of biodiversity at which point ecosystems become susceptible to mass extinctions. However, this state of critical biodiversity was not the endpoint or point of attraction in the evolution process. Our "ecosystems" eventually reached a dynamically stable state. Because the system is quite susceptible to mass extinction near the critical biodiversity, and because mass extinction lowers diversity, the system typically approaches the critical point many times before it finally evolves to a new dynamically stable state.

Primitive State

We present animations of the three dynamic regimes that develop in our simulation. In these animations, each species is denoted by a different color. (Click on each "State" link for thumbnails and explanations; animations can be downloaded.) The "world" is an island of 100 x 100 microenvironments, each connected to four nearest neighbor microenvironments. Early in the evolution process, the system is in a primitive state characterized with fairly low biodiversity. Each "species" is represented by a color. In the primitive state, colors seem to be uniformly distributed across the grid (there is an absence of persistent spatial structure). This indicates that species, including possible keystone species, are distributed almost uniformly throughout the world.

Critical State

At the critical state, species are still connected throughout the grid. The pattern instantaneously appears similar to the primitive state. Over time, this critical state is most susceptible to mass extinction. During a mass extinction event, more than half the existing species may be wiped out - this dynamic process is evident in the animation. A mass extinction event lowers the biodiversity and may send the system back to the primitive state for a long period of time. Hence the critical state must act as a kinetic barrier to increased diversity.

Ordered State

Eventually, the system may evolve to what we call the ordered state. In the ordered state, species on the grid organized themselves into separate subecosystems. Specialization occurs, creating well-defined communities. In this state, the species become specialized to compete within these communities. The separate communities appear as well-defined spatial patterns on the map. These patterns are uncorrelated with the local environments. The species themselves define the large-scale patterns or environments, which persist over long periods of time. Each community undergoes changes, and species are replaced by more fit species at the same rate as before. However, the changes 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 have never observed a collapse back to the primitive state.

The model is described in full in the Conservation Biology paper by Kaufman et al. [15]. Independent of the model, this notion of a state of critical biodiversity as a kinetic transition state may be more general in nature. Indeed, in condensed matter physics it is common to observe critical slowing down when a dynamic system passes through a critical point [16]. For example, the relaxation time in a ferromagnet diverges near the Curie temperature where spontaneous magnetization first becomes possible.

In a recent letter to Nature, Huisman and Weissing [17]) report the results of a completely different dynamic model of biodiversity in plankton, and also find evidence for a system that passes through a regime of "competitive chaos" before evolving to a dynamically stable state. This behavior is exactly that predicted by our study.

If the evolutions of natural ecosystems are also subject to a kinetic barrier or "critical biodiversity," then the observation of the MacArthur-Wilson scaling relationship for various island groups suggests the biodiversity of these groups may be near the critical point. Understanding the origins of this scaling in real ecosystems may be important for several reasons. Since the diversity supported in a reserve will scale with the area of that reserve, knowledge of both the actual diversity in the reserve and the critical diversity will provide information on how large an area is required to preserve the existing diversity.

Conversely, that same information will help predict how the diversity will collapse if the reserve areas are reduced. Since the critical diversity itself depends on area, a great deal of experimental data is required to develop a predictive theory of ecosystem diversity. In our Conservation Biology article, we suggest measures that can be applied to experimental data to obtain the order parameter - a species's susceptibility to extinction - and a quantitative indication of the critical biodiversity point for an ecosystem. The measure may be defined by any consistent measure of diversity (e.g., number of species, number of families, number of genera). The analysis requires not only a measure of the diversity as a function of time, but also information about the rate of introduction of new species, and the size of extinction events as a function of time. Such data may be obtained from both living ecosystems and from the fossil record. If there exists a critical biodiversity, and if we could measure it as a function of area over a large enough scale, we could then deduce the critical biodiversity for life on earth.