Models of Extinction A Review M. E. J. Newman and R. G. Palmer Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501. U.S.A. Abstract We review recent work aimed at modelling species extinction over geological time. We discuss a number of models which, rather than dealing with the direct causes of particular extinction events, attempt to predict overall statistical trends, such as the relative frequencies of large and small extinctions, or the distribution of the lifetimes of species, genera or higher taxa. We also describe the available fossil and other data, and compare the trends visible in these data with the predictions of the models. arXiv:adap-org/9908002v1 6 Aug 1999 Contents 1 Causes of extinction 3 2 The data 4 2.1 Fossildata........................................ .......... 4 2.1.1 Biases in the fossil data . ........ 4 2.2 Trendsinthefossildata .............................. ............ 6 2.2.1 Extinctionrates ................................... ....... 6 2.2.2 Extinction periodicity . ....... 8 2.2.3 Origination and diversity . ....... 9 2.2.4 Taxon lifetimes . ...... 10 2.2.5 Pseudoextinction and paraphyly . .......... 11 2.3 Otherformsofdata ................................. ........... 12 2.3.1 Taxonomicdata .................................... ...... 12 2.3.2 Artificial life . ..... 13 3 Early models of extinction 14 4 Fitness landscape models 15 4.1 TheNKmodel ....................................... ........ 15 4.2 EvolutiononNKlandscapes............................. ........... 17 4.3 Coevolvingfitnesslandscapes . ............. 18 4.4 Coevolutionaryavalanches. ............. 19 4.5 Competitivereplacement.. .. .. .. .. .. .. .. .. .. .. .. .. ............ 20 5 The Bak–Sneppen model and variations 22 5.1 TheBak–Sneppenmodel.............................. ............ 22 5.2 Self-organized criticality . ............ 23 5.3 Time-scales for crossing barriers . ............... 26 5.4 The exactly solvable multi-trait model . ............. 26 5.5 Models incorporating speciation . ............. 27 5.6 Modelincorporatingexternalstress. ................. 29 6 Inter-species connection models 30 6.1 TheSol´e–Manrubiamodel. ............. 31 6.2 VariationsontheSol´e–Manrubiamodel . ............... 33 6.3 AmaralandMeyer’sfoodchainmodel . ............. 33 6.4 Abramson’sfoodchainmodel . ............ 34 7 Environmental stress models 35 7.1 Newman’smodel ..................................... ......... 35 7.2 Shortcomingsofthemodel. .. .. .. .. .. .. .. .. .. .. .. ............. 38 7.3 The multi-trait version of the model . ............. 38 7.4 Thefinite-growthversionofthemodel . ............... 39 7.5 ThemodelofManrubiaandPaczuski . ............. 39 8 Sibani’s reset model 40 8.1 Extinction rate decline . ........... 41 8.2 Theresetmodel .................................... .......... 42 8.3 Extinctionmechanisms................................ ........... 43 9 Conclusions 43 References 45 1 Causes of extinction 3 Of the estimated one to four billion species which accepted (Glen 1994), but almost all of the alterna- have existed on the Earth since life first appeared here tives are also exogenous in nature, ranging from the (Simpson 1952), less than 50 million are still alive to- mundane (climate change (Stanley 1984, 1988), ocean day (May 1990). All the others became extinct, typi- anoxia (Wilde and Berry 1984)) to the exotic (volcan- cally within about ten million years (My) of their first ism (Duncan and Pyle 1988, Courtillot et al. 1988), appearance. It is clearly a question of some interest tidal waves (Bourgeois et al. 1988), magnetic field re- what the causes are of this high turnover, and much versal (Raup 1985, Loper et al. 1988), supernovae (El- research has been devoted to the topic (see for ex- lis and Schramm 1995)). There seems to be little dis- ample Raup (1991a) and Glen (1994) and references agreement that, whatever the causes of these mass ex- therein). Most of this work has focussed on the causes tinction events, they are the result of some change in of extinction of individual species, or on the causes of the environment. However, the mass extinction events identifiable mass extinction events, such as the end- account for only about 35% of the total extinction ev- Cretaceous event. However, a recent body of work has ident in the fossil record at the family level, and for examined instead the statistical features of the history the remaining 65% we have no firm evidence favouring of extinction, using mathematical models of extinction one cause over another. Many believe, nonetheless, processes and comparing their predictions with global that all extinction can be accounted for by environ- properties of the fossil record. In this paper we review mental stress on the ecosystem. The extreme point of a number of these models, describing their mathemat- view has been put forward (though not entirely seri- ical basis, the extinction mechanisms which they in- ously) by Raup (1992), who used statistical analyses corporate, and their predictions. We also discuss the of fossil extinction and of the effects of asteroid impact trends in fossil and other data which they attempt to to show that, within the accuracy of our present data, predict and ask how well they achieve that goal. As it is conceivable that all terrestrial extinction has been we will see, a number of them give results which are in caused by meteors and comets. This however is more a reasonable agreement with the general features of the demonstration of the uncertainty in our present knowl- data. edge of the frequency of impacts and their biotic effects The outline of the paper is as follows. In Section 1 than a realistic theory. we give a brief synopsis of the current debate over the At the other end of the scale, an increasing num- causes of extinction. In Section 2 we describe the fos- ber of biologists and ecologists are supporting the idea sil record as it pertains to the models we will be dis- that extinction has biotic causes—that extinction is a cussing, as well as a number of other types of data natural part of the dynamics of ecosystems and would which have been cited in support of these models. In take place regardless of any stresses arising from the Sections 3 to 8 we describe in detail the modelling environment. There is evidence in favour of this view- work which is the principal topic of this review, start- point also, although it is to a large extent anecdotal. ing with early work such as that of Willis (1922) and Maynard Smith (1989) has given a variety of different van Valen (1973), but concentrating mainly on new re- examples of modern-day extinctions caused entirely by sults from the last five years or so. In Section 9 we give species interactions, such as the effects of overzealous our conclusions. predators, or the introduction of new competitors into formerly stable systems. The problem is that extinc- tion events of this nature usually involve no more than 1 Causes of extinction a handful of species at the most, and are therefore too small to be picked out over the “background” level of There are two primary colleges of thought about the extinction in the fossil data, making it difficult to say causes of extinction. The traditional view, still held with any certainty whether they constitute an impor- by most palaeontologists as well as many in other tant part of this background extinction. (The distinc- disciplines, is that extinction is the result of exter- tion between mass and background extinction events nal stresses imposed on the ecosystem by the environ- is discussed in more detail in Section 2.2.1.) The re- ment (Benton 1991, Hoffmann and Parsons 1991, Par- cent modelling work which is the primary focus of this sons 1993). There are indeed excellent arguments in review attempts to address this question by looking in- favour of this viewpoint, since we have good evidence stead at statistical trends in the extinction record, such for particular exogenous causes for a number of major as the relative frequencies of large and small extinction extinction events in the Earth’s history, such as marine events. Using models which make predictions about regression (sea-level drop) for the late-Permian event these trends and comparing the results against fossil (Jablonski 1985, Hallam 1989), and bolide impact for and other data, we can judge whether the assump- the end-Cretaceous (Alvarez et al. 1980, Alvarez 1983, tions which go into the models are plausible. Some 1987). These explanations are by no means universally 4 2 The data of the models which we discuss are based on purely morously referred to as the “Lipps–Signor” effect, is biotic extinction mechanisms, others on abiotic ones, seen in the origination times of taxa, which in general and still others on some mixture of the two. Whilst the err on the late side in poorly represented taxa. By results of this work are by no means conclusive yet— grouping fossil species into higher taxa, we can work there are a number of models based on different extinc- with denser data sets which give more accurate esti- tion mechanisms which agree moderately well with the mates of origination and extinction dates, at the ex- data—there has been some encouraging progress, and pense of throwing out any information which is spe- it seems a promising line of research. cific to the lower taxonomic levels (Raup and Boya- jian 1988). (Higher taxa do, however, suffer from a greater tendency to paraphyly—see the discussion of 2 The data pseudoextinction in Section 2.2.5.) In this section we review the