The Genetics of Altruism. by S. A. Boorman; P. R. LeavittReview by: Donald LudwigSIAM Review, Vol. 24, No. 3 (Jul., 1982), pp. 366-368Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030205 .

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366 BOOK REVIEWS

provides most of the link between local and global theory in this book. The presentation of several important theorems and their discussions in this chapter are particularly adapted to more advanced and general discussions. For example, in discussing orientation of surfaces, a theorem is stated without proof: "Every compact surface [without boundary] in E3 is orientable." (The parenthetical expression is added by this reviewer in order to avoid confusion by the students.) Clearly this theorem can be generalized to hypersur- faces of codimension 1 in En. In discussing the Gauss-Bonnet theorem, the Euler characteristic is introduced via triangulation, and it is stated without proof that every regular region of a surface admits a triangulation. The theorem is then stated and proven in the general form with many special cases as corollaries and many applications. The Poincare theorem relating the sum of indices of a vector field with only isolated singular points on a compact surface S to the Euler characteristic of S is proven. Last but not least, the theorem of Hoft and Rinor on completeness properties is proven for complete surfaces in a form that can easily be generalized.

This is a well written text on traditional topics of differential geometry. A carefully seleced set of exercises, some of which supplement the text, are given at the end of most sections. This book is recommended as a text for an undergraduate course in differential geometry, and is also suited for background reading for graduate courses in geometry. A more expanded bibliography would be beneficial.

K. K. LEE University of Rochester

The Genetics of Altruism. By S. A. BOORMAN and P. R. LEAVITT. Academic Press, New York, 1980. xx + 459 pp. $29.50. Boorman and Leavitt have made a major contribution to the central theoretical

problem of sociobiology. This problem was defined by Edward 0. Wilson (1975, p. 3) as follows:

In a Darwinist sense the organism does not live for itself. Its primary function is not even to reproduce other organisms; it reproduces genes, and it serves as their temporary carrier. Each organism generated by sexual reproduction is a unique, accidental subset of all the genes constituting the species. Natural selection is the process whereby certain genes gain representation in the following generations superior to that of other genes located at the same chromosome positions. When new sex cells are manufactured in each generation, the winning genes are pulled apart and reassembled to manufacture new organisms that, on the average, contain a higher proportion of the same genes. But the individual organism is only their vehicle, part of an elaborate device to preserve and spread them with the least possible biochemical perturbation. Samuel Butler's famous aphorism, that the chicken is only an egg's way of making another egg, has been modernized: the organism is only DNA's way of making more DNA....

In the process of natural selection, then, any device that can insert a higher proportion of certain genes into subsequent generations will come to characterize the species.... As more complex social behavior by the organism is added to the genes' techniques for replicating themselves, altruism becomes increasingly prevalent and eventually appears in exaggerated forms. This brings us to the central theoretical problem of sociobiology: how can altruism, which by definition reduces personal fitness, possibly evolve by natural selection?

The most extreme forms of social behavior occur in the Hymenoptera (ants, bees and wasps), and in the Isoptera (termites). Within the order Hymenoptera, eusociality (characterized by cooperative care for the young, more or less sterile castes, and offspring assisting their parent) has arisen independently perhaps 11 times. Only one occurrence is known outside the Hymenoptera, in the Isoptera. What is special about the Hymenop- tera? W. D. Hamilton (1964) pointed out the peculiarities of relationship which are caused by the haplodiploid mode of sex determination.

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In the Hymenoptera and a few other groups of organisms, fertilized eggs develop into females, and unfertilized eggs develop into males. Consequently, female offspring of the same parents have exactly the same genes inherited from their father, and an average of l/2 of their genes in common from their mother. Their coefficient of relationship is therefore l/2 * 1 + '/2 * /2 = 3/4. On the other hand, females have only l/2 of their genes in common with their daughters. Loosely speaking, if a gene tries to perpetuate itself, there is a greater return from investing in sisters than in offspring. Hence we should expect most females to be sterile, etc. This remarkable piece of reasoning seems to answer a major problem at a single stroke. It has stimulated a vigorous development of social and evolutionary theory. Outstanding expositions are Wilson (1975) and Dawkins (1976).

In view of the connection between population genetics and social behavior, Boorman and Leavitt have undertaken a systematic investigation of the evolution of sociality. Their book is divided into three parts, which are concerned with three possible mechanisms for the evolution of altruistic behavior. Altruism is defined as behavior on the part of individual I which tends to increase the number of offspring of individual II, at the expense of a reduction of the number of offspring of individual I. As was indicated in the quotation from Wilson, one would expect that natural selection would reduce altruism.

The first mechanism considered by Boorman and Leavitt is "reciprocity selection." In this model, individuals I and II are not necessarily related, and altruism is governed by a recessive allele at a simple locus. It turns out that altruism will be favored if the percentage of altruists in the population is high enough, but will otherwise be at a disadvantage. How can an altruistic gene spread from an initial low frequence? The authors propose that altruism could establish itself in a small isolated population by chance, and then spread into a larger, partially isolated population. The feasibility of such a "cascade" process is demonstrated by computer simulations, and some approximations. The mechanism of reciprocity selection seem to be most appropriately applied to higher vertebrates. Reference is made to cooperative hunting in lions.

The second general mechanism of evolution of altruism is kin selection, where individuals I and II are related. The major intended application is haplodiploid genetics, where individuals I and II have the same mother.

A total of nine models are considered. They correspond to diploid vs. haplodiploid inheritance, single or multiple insemination of the female, and dominant vs. recessive character of the altruistic gene. Conditions for stability and instability of the altruistic trait are determined, and they are compared with Hamilton's theory. Briefly stated, it is shown that Hamilton's quantitative conclusions hold only in very special cases, that they hold qualitatively in a wider class, but they are not a reliable guide in general. In particular, it is shown that very effective altruism encourages "cheating," i.e. produces a final state where a stable fraction of non altruists are present.

A third mechanism for the evolution of altruism is group selection, where certain subpopulations are more successful on average, by avoiding extinction or through superior colonization capabilities. The theory of group selection has a troubled history, as do most plausible theories of population biology. Here the major emphasis is upon work of Levins (1970a), (1970b). Levins focussed upon differential extinction rates for altruistic and nonaltruistic populations. A model based upon classical population genetics leads to a singular nonlinear diffusion equation, which is double trouble. The authors respectfully point out weaknesses and errors in Levins' theory, and further point out that a general solution of the problem would be analogous to finding a general solution of the Boltzmann equation. Finally Boorman and Leavitt consider the influence of altruism upon colonizing success. The results are diverse and will not be reviewed here.

This work is an outstanding example of modern applied mathematics. The authors

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368 BOOK REVIEWS

are familiar with the relevant biological literature, terminology and problems. Although they sometimes have recourse to intricate or abstract mathematical methods, the motivation is always clear, and the results are presented in a useful way. This includes a careful statement of assumptions and restrictions. The authors are free of the customary arrogance of mathematicians when they discover logical lapses in the work of scientists. They realize that the pioneering ideas are the most important thing, and mathematical rigor always has a subordinate role. For all of these reasons, the work of Boorman and Levitt has had a major influence on the field of sociobiology. Such influence is, after all, the decisive test of applied mathematics.

The success of Boorman and Levitt raises additional questions, which would not be appropriate for a lesser work. Does this work qualify as science? Is the theory subject to disproof? Have parameters been estimated on the basis of data and the accuracy of the estimates been assessed? The answer to all of these questions is, "not yet." The authors are troubled by this situation, and they include a lengthy discussion of the limitations of the data. By its very nature, evolution covers its own tracks. Therefore it is doubtful whether proper data will ever be available. If it were not so fundamental and important, one would declare the subject to be hopeless.

The authors have used some of their material in model-building courses. It is eminently suited for such purposes. However, the lack of confrontation with data is a very serious handicap. Model-building courses which neglect data perpetuate the gap between applied mathematics and statistics. This gap, in turn, ensures that much "applied" mathematics will have negligible scientific impact.

REFERENCES

RICHARD DAWKINS (1976), The Selfish Gene, Oxford Univ. Press, London, paperback version 1978 by Granada Press.

W. D. HAMILTON (1964), The genetical evolution of social behavior, I, II, J. Theoret. Biol. 7, pp. 1-16, 17-52.

R. LEVINS (1970a), Extinction, Some Mathematical Problems in Biology; M. Gerstenhaber, ed., Lectures on Mathematics in the Life Sciences, vol. 2, American Mathematical Society, Providence, RI, pp. 75-108.

(1970b), Fitness and optimization, in Mathematical Topics in Population Genetics, K. Kojima, ed., Springer-Verlag, Berlin and New York, pp. 389-400.

EDWARD 0. WILSON (1975), Sociobiology, Harvard Univ. Press, Cambridge, MA.

DONALD LUDWIG University of British Columbia

Optimization with Disjunctive Constraints, by HANIF D. SHERALI and C. M. SHETTY. Springer-Verlag, Berlin, 1980. viii + 156 pp. $15.00, paper. Many optimization problems have restrictions of an "either-or" nature. Disjunctive

programming is an attempt to deal with such situations directly instead of by using zero-one variables. This book presents various aspects of the theory of such problems.

Two situations are studied in detail. The first involves the requirement that the feasible point lie inside one of a finite set

of polyhedra, each of which is defined by hyperplanes. An inequality (or cutting-plane) is valid for the feasible region (the union of the polyhedra) if and only if the inequality is valid for each polyhedron separately. Chapter III reports methods of computing "deep" cutting-planes under various conditions. It follows Sherali and Shetty [3] closely. Chapters IV-VI are primarily devoted to descriptions of work of other authors, emphasiz- ing examples rather than proofs.

The second type of disjunctive program consists of a single polytope P, together with

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