Environmental Stress Models

In chapters 2 to 4 we discussed several models of extinction which make use of ideas drawn from the study of critical phenomena. The primary impetus for this approach was the observation of apparent power-law distributions in a variety of statistics drawn from the fossil record, as discussed in section 1.2; in other branches of science such power laws are often indicators of critical processes. However, there are also a number of other mechanisms by which power laws can arise, including random multiplicative processes (Montroll and Shlesinger 1982; Sornette and Cont 1997), extremal random processes (Sibani and Littlewood 1993), and random barrier-crossing dynamics (Sneppen 1995). Thus the existence of power-law distributions in the fossil data is not on its own sufficient to demonstrate the presence of critical phenomena in extinction processes. Critical models also assume that extinction is caused primarily by biotic effects such as competition and predation, an assumption which is in disagreement with the fossil record. As discussed in section 1.2.2.1, all the plausible causes for specific prehistoric extinctions are abiotic in nature. Therefore an obvious question to ask is whether it is possible to construct models in which extinction is caused by abiotic environmental factors, rather than by critical fluctuations arising out of biotic interactions, but which still give power-law distributions of the relevant quantities. Such models have been suggested by Newman (1996, 1997) and by Manrubia and Paczuski (1998). Interestingly, both of these models are the result of attempts at simplifying models based on critical phenomena. Newman's model is a simplification of the model of Newman and Roberts (see section 3.6), which included both biotic and abiotic effects; the simplification arises from the realization that the biotic part can be omitted without losing the power-law distributions. Manrubia and Paczuski's model was a simplification of the connection model of Solé and Manrubia (see section 4.1), but in fact all direct species-species interactions were dropped, leaving a model which one can regard as driven only by abiotic effects. We discuss these models in turn. The model proposed by Newman (1996, 1997) has a fixed number N of species which in the simplest case are noninteracting.

Self-Organized Critical Models

The models discussed in the last chapter are intriguing, but present a number of problems. In particular, most of the results about them come from computer simulations, and little is known analytically about their properties. Results such as the power-law distribution of extinction sizes and the system's evolution to the "edge of chaos" are only as accurate as the simulations in which they are observed. Moreover, it is not even clear what the mechanisms responsible for these results are, beyond the rather general arguments that we have already given. In order to address these shortcomings, Bak and Sneppen (1993; Sneppen et al. 1995; Sneppen 1995; Bak 1996) have taken Kauffman's ideas, with some modification, and used them to create a considerably simpler model of large-scale coevolution which also shows a power-law distribution of avalanche sizes and which is simple enough that its properties can, to some extent, be understood analytically. Although the model does not directly address the question of extinction, a number of authors have interpreted it, using arguments similar to those of section 1.2.2.5, as a possible model for extinction by biotic causes. The Bak-Sneppen model is one of a class of models that show "self-organized criticality," which means that regardless of the state in which they start, they always tune themselves to a critical point of the type discussed in section 2.4, where power-law behavior is seen. We describe self-organized criticality in more detail in section 3.2. First, however, we describe the Bak-Sneppen model itself. In the model of Bak and Sneppen there are no explicit fitness landscapes, as there are in NK models. Instead the model attempts to mimic the effects of landscapes in terms of "fitness barriers." Consider figure 3.1, which is a toy representation of a fitness landscape in which there is only one dimension in the genotype (or phenotype) space. If the mutation rate is low compared with the time scale on which selection takes place (as Kauffman assumed), then a population will spend most of its time localized around a peak in the landscape (labeled P in the figure).

Extinction In The Fossil Record

Of the estimated one to four billion species that have existed on the Earth since life first appeared here (Simpson 1952), less than 50 million are still alive today (May 1990). All the others became extinct, typically within about ten million years (My) of their first appearance. It is clearly a question of some interest what the causes are of this high turnover, and much research has been devoted to the topic (see, for example, Raup (1991a) and Glen (1994) and references therein). Most of this work has focussed on the causes of extinction of individual species, or on the causes of identifiable mass extinction events, such as the end-Cretaceous event. However, a recent body of work has examined instead the statistical features of the history of extinction, using mathematical models of extinction processes and comparing their predictions with global properties of the fossil record. In this book we will study these models, describing their mathematical basis, the extinction mechanisms that they incorporate, and their predictions. Before we start looking at the models however, we need to learn something about the trends in fossil and other data which they attempt to model. This is the topic of this introductory chapter. Those well versed in the large-scale patterns seen in the Phanerozoic fossil record may wish to skip or merely browse this chapter, passing on to chapter 2, where the discussion of the models begins. There are two primary colleges of thought about the causes of extinction. The traditional view, still held by most palaeontologists as well as many in other disciplines, is that extinction is the result of external stresses imposed on the ecosystem by the environment (Benton 1991; Hoffmann and Parsons 1991; Parsons 1993). There are indeed excellent arguments in favor of this viewpoint, since we have good evidence for particular exogenous causes for a number of major extinction events in the Earth's history, such as marine regression (sealevel drop) for the late-Permian event (Jablonski 1985; Hallam 1989), and bolide impact for the end-Cretaceous (Alvarez et al. 1980; Alvarez 1983, 1987).

Non-Equilibrium Models

Sibani and co-workers have proposed a model of the extinction process, which they call the "reset model" (Sibani et al. 1995,1998), which differs from those discussed in the preceding chapters in a fundamental way; it allows for, and indeed relies upon, nonstationarity in the extinction process. That is, it acknowledges that the extinction record is not uniform in time, is not in any sense in equilibrium, as it is assumed to be in the other models we have considered. In fact, extinction intensity has declined on average over time from the beginning of the Phanerozoic until the Recent. Within the model of Sibani et al., the distributions of section 1.2 are all the result of this decline, and the challenge is then to explain the decline, rather than the distributions themselves. In figure 1.9 we showed the number of known families as a function of time over the last 600 My. On the logarithmic scale of the figure, this number appears to increase fairly steadily and although, as we pointed out, some of this increase can be accounted for by the bias known as the "pull of the recent," there is probably a real trend present as well. It is less clear that there is a similar trend in extinction intensity. The extinctions represented by the points in figure 1.1 certainly vary in intensity, but on average they appear fairly constant. Recall however, that figure 1.1 shows the number of families becoming extinct in each stage, and that the lengths of the stages are not uniform. In figure 6.1 we show the extinction intensity normalized by the lengths of the stages—the extinction rate in families per million years—and on this figure it is much clearer that there is an overall decline in extinction towards the Recent. In order to quantify the decline in extinction rate, we consider the cumulative extinction intensity c(t) as a function of time. The cumulative extinction at time t is defined to be the number of taxa which have become extinct up to that time.

Fitness Landscape Models

Kauffman (1993, 1995; Kauffman and Levin 1987; Kauffman and Johnsen 1991) has proposed and studied in depth a class of models referred to as NK models, which are models of random fitness landscapes on which one can implement a variety of types of evolutionary dynamics and study the development and interaction of species. (The letters N and K do not stand for anything; they are the names of parameters in the model.) Based on the results of extensive simulations of NK models, Kauffman and co-workers have suggested a number of possible connections between the dynamics of evolution and the extinction rate. To a large extent it is this work which has sparked recent interest in biotic mechanisms for mass extinction. In this chapter we review Kauffman's work in detail. An NK model is a model of a single rugged landscape, which is similar in construction to the spin-glass models of statistical physics (Fischer and Hertz 1991), particularly p-spin models (Derrida 1980) and random energy models (Derrida 1981). Used as a model of species fitness the NK model maps the states of a model genome onto a scalar fitness W. This is a simplification of what happens in real life, where the genotype is first mapped onto phenotype and only then onto fitness. However, it is a useful simplification which makes simulation of the model for large systems tractable. As long as we bear in mind that this simplification has been made, the model can still teach us many useful things. The NK model is a model of a genome with N genes. Each gene has A alleles. In most of Kauffman's studies of the model he used A = 2, a binary genetic code, but his results are not limited to this case. The model also includes epistatic interactions between genes—interactions whereby the state of one gene affects the contribution of another to the overall fitness of the species. In fact, it is these epistatic interactions which are responsible for the ruggedness of the fitness landscape.

Summary

In this book we have studied a large number of recent quantitative models aimed at explaining a variety of large-scale trends seen in the fossil record. These trends include the occurrence of mass extinctions, the distribution of the sizes of extinction events, the distribution of the lifetimes of taxa, the distribution of the numbers of species per genus, and the apparent decline in the average extinction rate. None of the models presented match all the fossil data perfectly, but all of them offer some suggestion of possible mechanisms which may be important to the processes of extinction and origination. In this chapter we conclude our review by briefly summarizing the properties and predictions of each of the models once more. Much of the interest in these models has focused on their ability (or lack of ability) to predict the observed values of exponents governing distributions of a number of quantities. In Table 7.1 we summarize the values of these exponents for each of the models. Most of the models we have described attempt to provide possible explanations for a few specific observations. (1) The fossil record appears to have a power-law (i.e., scale-free) distribution of the sizes of extinction events, with an exponent close to 2 (section 1.2.2.1). (2) The distribution of the lifetimes of genera also appears to follow a power law, with exponent about 1.7 (section 1.2.2.4). (3) The number of species per genus appears to follow a power law with exponent about 1.5 (section 1.2.3.1). One of the first models to attempt an explanation of these observations was the NK model of Kauffman and co-workers. In this model, extinction is driven by revolutionary avalanches. When tuned to the critical point between chaotic and frozen regimes, the model displays a power-law distribution of avalanche sizes with an exponent of about 1. It has been suggested that this could in turn lead to a power-law distribution of the sizes of extinction events, although the value of 1 for the exponent is not in agreement with the value 2 measured in the fossil extinction record.

Interspecies Connection Models

In the Bak-Sneppen model studied in the previous chapter there is no explicit notion of an interaction strength between two different species. It is true that if two species are closer together on the lattice, then there is a higher chance of their participating in the same avalanche. But beyond this there is no variation in the magnitude of the influence of one species on another. Real ecosystems, on the other hand, have a wide range of possible interactions between species, and as a result the extinction of one species can have a wide variety of effects on other species. These effects may be helpful or harmful, as well as strong or weak, and there is in general no symmetry between the effect of A on B and B on A. For example, if species A is prey for species B, then A's demise would make B less able to survive, perhaps driving it also to extinction, whereas B's demise would aid A's survival. On the other hand, if A and B compete for a common resource, then either's extinction would help the other. Or if A and B are in a mutually supportive or symbiotic relationship, then each would be hurt by the other's removal. A number of authors have constructed models involving specific speciesspecies interactions, or "connections." If species i depends on species j , then the extinction of j may also lead to the extinction of i, and possibly give rise to cascading avalanches of extinction. Most of these connection models neither introduce nor have need of a fitness measure, barrier, viability, or tolerance for the survival of individual species; the extinction pressure on one species comes from the extinction of other species. Such a system still needs some underlying driving force to keep its dynamics from stagnating, but this can be introduced by making changes to the connections in the model, without requiring the introduction of any extra parameters.