Chaos

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when protection efforts for known species are already inadequate. Given the problems of environmental deterioration and species extinction on a global scale, it might be unwise to divert a large fraction of the limited resources earmarked for conservation to molecular genetic evaluations. Nonetheless, in certain situations molecular genetic assessments are clearly a necessary and important guide to the description of biotic diversity. Furthermore, general management implications may often stem from studies funded as ‘pure’ research in molecular evolution. For example, Bermingham and Avise found that populations within each of several species of freshwater fishes in the southeastern United States exhibit marked and geographically concordant genetic differences, probably due to historical patterns of drainage isolation and These phylogenetic coalescence. subdivisions should aid in the development of a regional management program applicable to many fish species in the southeast of the

country. Molecular genetic techniques should be increasingly employed as an aid to phylogeny reconstruction, and hence biotic diversity assessment. The need for increased attention to molecular descriptions of phylogenetic differences, particu-

larly at the level of populations and species, is a special aspect of the broader call to revive the field of systematics in conservation biology26. Considerable conservation resources are devoted to the management of rare taxa; commensurate attention should be devoted to the adequacy of the biotic descriptions. Acknowledgements Brian Bowen provided helpful suggestions for the manuscript. This work has been supported by grants from the NSF.

References 1 Simberloff, D. (1988)Annu. Rev. fcol. syst. 19,473-511 2 Harris, R.B., Maguire, L.A. and Shaffer, M.L. (1987) Conserv. 8iol. 1,72-76 3 Lande, R. (1988) Science241, 1455-1460 4 Andrewartha, H.G. and Birch, L.C. (1954) The Distribution and Abundance of Animals, University of Chicago Press 5 Rails, K., Ballou. J.D. and Temofeton, A. ( 1988) Conserv. giol. 2,185-l 93’ 6 Price, M.V. and Waser, N.M. (1979) Nature 277,294-298 7 Bonnell, M.L. and Selander, R.K. (1974) Science 184,908-909 6 Lesica, P., Leary, RF., Allendorf, F.W. and Bilderback, D.E. (1988) Conserv. Biol. 2,276-282 9 O’Brien. S.J.. Roelke. M.E.. Marker. L.. Newman; A., Winkler,C.A., Meltzer,D.,’ Golly, L., Evermann, J.F., Bush, M. and Wildt, D.E. (1985) Science227,1428-1434 10 Vrijenhoek, R.C., Douglas, M.E. and

Meffe, G.K. (1985) Science 229,400-402 11 O’Brien, S.J. and Evermann. J.F. (1988) Trends Ecu/. Evol. 3,254-259 12 Wildt, D.E., Bush, M., Goodrowe, K.L., Packer, C., Pusey, A.E., Brown, J.L., Joslin, P. and O’Brien, S.J. (1987) Nature 329,328-330 13 Laerm, J., Avise, J.C., Patton, J.C. and Lansman, R.A. (1982) J. Wild/. Manage. 46,513-518 14 Avise, J.C. and Nelson, W.S. (1989) Science 243,646-648 15 Byrne, G..(1988) Science 242,190 16 James, F.C. (1983) Science 221, 184-186 17 Kornfield, I., Smith, DC., Gagnon, P.S. and Taylor, J.N. (1982) Evolution 36, 658-664 18 Meyer, A. (1987) Evolution 41, 1357-1369 19 McKitrick, M.C. and Zink, R.M. (1988) Condor 90,1-14 20 Cracraft, J. (1983) in Current Ornithology, Vol. I (Johnston, R.F., ed.), pp. 15%187, Plenum Press 21 Avise, J.C.,Arnold, J., Ball, R.M., Jr, Bermingham, E., Lamb, T., Neigel, J.E., Reeb. C.A. and Saunders, N.C. (1987) Ann;. Rev. &co/. Syst. 18; 489-522 22 Neigel, J.E. and Avise, J.C. (1986) in Evolutionary Processes and Theory (Karlin, S. and Nevo, E., eds), pp. 515-534, Academic Press 23 Pamilo, P. and Nei, M. (1988) Mol. Siol. Evol. 5,568-583 24 Federal Register (1987) Endangered and Threatened Wildlife and Plants, US Department of the Interior, Washington, DC 25 Bermingham, E. and Avise, J.C. (1986) Genetics 113,939-965 26 Wilson, E.O. (1985) Science230,1227

Chaos and Ecology Chaotic Evolution and Strange Attractors

stant. It is now increasingly realized, however, that simple and determinby David Ruelle, Cambridge Univeristic relations can produce stable sity Press, 7989. f25.00/$39.50 hbk, points, or stably cyclic oscillations, f8.95/$72.95 pbk (772 pages) ISBN or apparently random, ‘chaotic’, fluc0527368308 tuations in population density, depending on the severity of the nonlinear or density-dependent effects. Chaos Originating largely from work in population and from biology, edited by Arun V. Holden, ManchesLorentz’s studies in meteorology, ter University Press and Princeton University Press, 7986, f45/$50 hbk, ‘chaos’ is now a burgeoning subject, f 77.95679.95 pbk (324 pages) ISBN finding an increasing range of applications in the physical and bio0697 084238 logical sciences (see, for instance, Ref. I). Unfortunately, it is not yet clear Chaos in Biological Systems exactly what the implications of edited by H. Degn, A.V. Holden and chaotic dynamics and strange attracL.F. Olsen, Plenum Press, 7987. $75 tors may be for ecologists. Hassell has reviewed, with examples, ways hbk (323 pages) ISBN 0 306 42685 4 in which the presence of deterministically chaotic population fluctuUntil recently, most ecologists assumed that if a population was ations might undercut conventional methods for analysing data. Schaffer influenced only by simple densityand Kots have discussed the possidependent factors, with no environmental or other randomness, then it bility that many recorded obserwould tend to remain roughly con- vations of population densities over

time (‘time series’), which show quasi-periodic or irregular fluctuations, may be interpreted as strange including attractors; a group Schaffer, Kot, Pimm, Ellner, Gilpin, Sugihara and others (what might be called a strangely attractive collective!) is currently pursuing these ideas with as many long-term time series as they can find. The books briefly reviewed here serve, in different ways, to guide the interested novice into the unfamiliar world of deterministic chaos. In Ruelle’s title, ‘evolution’ has nothing to do with Darwin, but rather refers to the development of dynamical patterns in chaotic systems; the book has the subtitle The Statistical Analysis of Time Series for Deterministic Nonlinear Systems. The book in fact derives from notes prepared by Stefano lsola from lectures given by Ruelle, and it gives an exceptionally clear and concise account of basic mathematical ideas, with an emphasis on intuitive arguments and on what the mathematics

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means. The book is a gem; most mathematical writing in this area is a transcendent reflection of the nonlinear subject matter-too often, you have to understand all of it before you can understand any of it. The first part (entitled ‘Steps to a Deterministic Interpretation of Chaotic Signals’) uses the phenomenon of turbulence, along with specific examples such as the quadratic map and the H6non map, to motivate a discussion that includes the notion of fractal dimension and its relation to strange attractors and to chaotic series. In the second part (‘The Ergodic Theory of Chaos’) the going gets heavier; inter alia, characteristic or Lyapunov exponents are introduced, recipes for their calculation are given, and their uses as measures of the information content in a time series are discussed. Chaos, the volume edited by Holden, combines exposition of basic ideas with other chapters that sketch various applications. I think the two chapters by Lauwerier, which deal with one-dimensional and twodimensional difference equations (that is, single populations and pairs of interacting populations with discrete, nonoverlapping generations), are particularly lucid and interesting. Did you know, for example, that if a host-parasitoid interaction produces regular or irregular cycles, their period is unlikely to be less than seven generations? Mees gives a good account of general aspects of chaos in feedback systems, and Wolf explains how to ‘quantify chaos’ by using Lyapunov exponents. Among the specific applications, Schaffer and Kot make a case for the data for measles in New York City and Baltimore, 1926-1963, following a trajectory corresponding to a somewhat noisy strange attractor; they also give a more general discussion of

the way chaotic trajectories can easily arise in prey-predator systems. Chaos in Biological Systems, edited by Degn et a/., is a collection of conference papers showing how ideas about strange attractors and chaotic dynamics are finding specific applications to laboratory data concerning cardiac physiology, neurobiology, cycles in glycolytic reactions, and various other topics. Olsen extends the analysis of Schaffer and Kot (also represented here in a chapter by Schaffer) to time series for the incidence of measles, mumps, rubella, chickenpox, pertussis and scarlet fever in Copenhagen, and for the first three of these six finds similar evidence that the dynamics may be governed by a strange attractor. Markus et al. give a good discussion of the population dynamics that can arise when intrinsically chaotic behaviour is further complicated by periodically or randomly varying growth conditions; in collaboration with a different set of co-authors, Markus also explores how order and disorder in spatial patterns may be characterized. The book edited by Degn et al. overlaps to some extent with another recent conference volume4, which ranges more widely over nonlinear processes in biology. The latter volume contains a particularly interesting series of chapters dealing with various explicitly nonlinear aspects of the transmission dynamics of HIV/AIDS. Here, several different authors make the point that once heterogeneities are present - be they geographical, social (as in rates of acquiring new sexual partners), genetic, or other-they can interact with nonlinearities in such a way that simple calculations based on average values can be misleading; the chapter by Anderson et al. combines these epidemiological consid-

1989

erations with conventional human demography, to make a tentative assessment of the likely impact of HIV/AIDS upon long-term patterns of human population growth in Africa. There are some ecologists who feel that the ‘flute music’ (Robert Paine’s phrase) of mathematicians has little to do with understanding real populations. Some, indeed, seem to see the music as coming from the pipes of Pan, distracting and confusing those who are so unwise as to listen. If, for these or other reasons, you are unfamiliar with the ideas discussed above, I urge you to consider the recursion relation x~+~= ax, (1 - xt), to set the constant a at, say, a=3.9, and to iterate the sequence on a hand calculator or a PC [this relation represents exponential growth, modified by a simple ‘crowding factor’ (1 - xt), which causes the effective growth rate to fall as population density increases]. The result may surprise you, and I hope it will prompt you to return to the review articles in TREE by Hassell and by Schaffer and Kot referred to above. For a longer excursion into the subject, the books reviewed here are, in their different ways, excellent guides. Robert M. May ZOOlOgY Dept,Oxford University, Oxford OX1 35 UK

References 1 Gleick, J. (1987) Chaos: Making a New Science, Viking Press 2 Hassell, M.P. (1988) Trends Ecol. Evol. 1,90-93 3 Schaffer, W.M. and Kot, M. (1986) Trends Ecol. Evol. 1,58-63 4 Perelson, AS., Goldstein, B., Dembo, M. and Jacquez, J.A., eds (1988) Nonlinearity in Biology and Medicine, Elsevier

Violence Aggressionand War:TheirBiologicaland SocialBases edited by Jo Groebel and Robert A. Hinde, Cambridge University Press, 1989. f25.00/$49.50 hbk, f8.95/ $14.95 pbk (xvi + 237 pages) ISBN 0521353564 The world has lost Niko Tinbergen, Konrad Lorenz and Luigi Valzelli all major influences on public perceptions of aggression. Lorenz is associated with the view that human aggression is a basic instinct inherited from our animal ancestors. 282

Tinbergen carried out elegant experiments with fishes and birds, clearly illustrating aggression’s roles in territoriality and mating preferences. Valzellil investigated the impact of drugs and neural circuitry on aggression. It seems appropriate to evaluate the relevance of animal studies to our understanding of phenomena such as ‘war’. Aggression and War is certainly timely. The book‘s rationale is contained in the Seville Statement on Violence (quoted in the Preface), which attempts to counter some erroneous impressions credited (not

always fairly) to Lorenz, Tinbergen and Valzelli. These include the opinions that (a) humans inherited a propensity to make war from their animal ancestors, (b) war and other violent behaviours are genetically programmed into human nature, (c) selection for increased aggressiveness has occurred in human evolution, (d) humans have a ‘violent brain’, and (e) war is caused by an ‘instinct’. The volume includes thoughtful essays by diverse scientists, providing valuable crossdisciplinary material to support the view that these are oversimplifica-