Mathematical Topics in Population Genetics by Ken-ichi KojimaReview by: Milena RoudnáFolia Geobotanica & Phytotaxonomica, Vol. 7, No. 1 (1972), pp. 95-97Published by: SpringerStable URL: http://www.jstor.org/stable/4179662 .

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REPORTS AND REVIES 95

sphare. Beispiele einer solchen Zirkulation einzelner Elemente (Stickstoff und Kohlenstoff) sind durch ihre Gehalte in einzelnen Reservoiren und durch die Austauschverhaltnisse zwischen diesen gekennzeichnet. Die Faihigkeit der Okosysteme, einzelne Elemente oder Verbindungen zu kon- zentrieren kann sich negativ durch Konzentrierung schadlicher Stoffe, wie z. B. der Radioisotope und Pestizide bis zur schaidlichen Konzentration in Nahrungsmitteln des Menschen, geltend machen. Diese Erkenntnis muss vor massenhafter Anwendung verschiedener organischer Be- kampfungsmittel in der Agrikultur und auch der ungeregelten Ausnutzung der Atomenergie dringend warnen.

Die Biosphiire - oder genauer gesagt - die Sphiare der Okosysteme bildet die Umwelt des Menschen. Die Okosysteme produzieren die Nahrung fur die explosiv sich vermehrende mensch- liche Population, sie bauen die schadlichen Produkte der Zivilisation ab, sie konnen aber einige Stoffe in einer dem Menschen schiidlicher Menge konzentieren. Der Mensch wird in der Zukunft gegen zwei grosse Probleme 6kologischer Natur gestellt werden: Regelung und Ernahrung der menschlichen Population und Beibehaltung der Umwelt in einem fuir ihn gunstigem Zustand. Mit einem kurzen Blick auf diese Problematik schliesst WHITTAKER sein aufschlussreiches Buch.

JAROSLAV MORAVEC Botanisches Institut der Tschechoslowakischen Akademie der Wissenschaften, Pr&honice bei Praha

KEN-ICHI KOJIMA (Ed.)

MATHEMATICAL TOPICS IN POPULATION GENETICS

Springer-Verlag, Berlin, Heidelberg et New York 1970, 400 pp., 55 figures.

Probabilistics and statistical analyses are very important methods in contemporary genetics. An interdisciplinary area between mathematics and biology, now called Biometrics, Biometry, or Applied Statistics, was established by Galtonian geneticists as early as the late 19th century and at the beginning of the 20th century.

Most works in the field of population genetics are based on the Mendelian gene theory. R. A. FISHER and J. B. S. HALDANE in England and S. WRIGHT in the United States were the founders of population genetics. (Their most famous works were published in the period 1916-1932.) There have been very capable investigators with an excellent mathematical back- ground, whose papers some of are included in the present book, the aim of which is "to document major issues and ideas that appeared during the development of population genetics in the past half century" (editor). Some papers are the results of original research, others are reviews of some aspects of population genetics. The book is the first volume in a series of monographs and reports - Biomathematics.

The first paper, by S. WRIGHT, deals with "Random drift and the shifting balance theory of evolution". The author explains the term "drift" and demands the adjective "steady" or "random" according to the case. In random drift the shifting balance theory of evolution and its three phases (local random drift, local mass selection, interdemic selection) and their various complications are analyzed. The effects of fluctuating environmental conditions, of persistent local differences, of secular changes and the ultimate control by ecological opportunity are dis- cussed. A mathematical definition for random drift is suggested. The shifting balance theory is contrasted with other Mendelian theories of evolutionary dynamics.

In the paper "Changes in mean fitness under natural selection" J. R. G. TURNER discusses the very interesting question of genetic variances in fitness, introduced by FISHER. He gives equations describing populations and compares FISHER's theory with KIMURA'S new theorem which corrects some of FIsHER's errors. But the experimental evidence shows that the changes in mean population growth, size, and biomass are not covered by any adequate theory and that the theory would need to be elaborated.

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96 FOLIA GEOBOTANICA ET PHYTOTAXONOMICA, 7, 1972

R. H. RICHARDSON discusses "Models and analyses of dispersal patterns". He distinguishes two general types of dispersal models, gives their mathematical function and application by popu- lation and ecological geneticists. Several examples of spread or distribution from nature are analyzed. In the author's opinion there are three stages of genetical application. Certain difficulties and shortcomings which may occur during biological study and some errors of investigators are pointed out.

C. C. COCKERHAM analyzes mathematically "Avoidance and rate of inbreeding". He determined a general formula for the inbreeding coefficient and observed the consequences of subdivisions of the population. In the systems of mating of close relatives he distinguishes two classes according to the type of ancestors and draws conclusions for the rate of inbreeding.

J. F. CROW discusses "Genetic loads and the cost of natural selection". He reviews briefly the development of the meaning of the word "load", which is closely connected with fitness in population. The author gives a list of some of the factors that may produce a genetic load and analyzes mathematically in detail some kinds of load under certain conditions in the population, especially mutational and segregational. The development of the principle of the cost of natural selection and its several modifications are mentioned.

M. KIMURA contributed to the book reviewed here the paper "Stochastic processes in population genetics, with special reference to distribution of gene frequencies and probability of gene fixation". One of the central problems in the mathematical theory of population genetics is the change of gene frequencies by mutation, natural selection, migration and random sampling of gametes. According to the author "the sampling of gametes introduces an element of chance into the change of gene frequencies so that the process must be treated as a stochastic one". Several examples of the application of this process in population genetics are given.

In his paper "Theory of limits to selection with line crossing" W. G. HILL makes a theoretical comparison of alternative breeding systems which utilize only selection within lines to improve a cross between two strains. A pure line selection and selection on cross performance - recurrent selection to a tester and reciprocal recurrent selection - are considered. The author gives the results obtained under conditions of complete dominance, partial dominance, overdominance and initial equilibrium, overdominance and initial disequilibrium, assuming in each case the same value of N8 (N is the effective population size for a cycle of progeny testing and selection, 8 is the standardized selection differential).

A. ROBERTSON presents "A theory of limits in artificial selection with many linked loci". Certain types of selections were studied. Both intermediate initial gene frequencies and extreme gene frequencies were taken into account. Several varieties of the simulation programme, carried out on the Edinburgh University computer, are given. The optimum intensity of selection, the variance between replicate lines at fixation, the half life of the selection process and the effect of simultaneous segregation of several chromosomes are also discussed.

J. A. SVED and 0. MAYO review the theories of "The evolution of dominance". They start with FISHER'S theory, show some of its weak points and present the opinions of PLUNIKETT,

MULLER and WRIGHT. showing a tendency to improve the above theory. The authors conclude, that there is nothing to determine which theory is to be accepted. Crucial evidence could hardly be obtained whether modifiers have been selected solely for their effect on an unfavourable and rare heterozygote, or for their effect on a wider class of genotypes.

H. E. SCHAFFER discusses "Survival of mutant genes as a branching process". A fundamental assumption in the theory of the branching process is for genes or chromosomes to act independ- ently in their survival and reproduction. The application of this theory is therefore restricted to the study of survival of mutations in very large or infinite populations. In a finite population the process is conditioned by a given finite population size and can be studied by means of the theory of Markov chains.

C. C. LI in his paper "The incomplete binomial distribution" reviews the methods of estimating the parameter p of a binomial distribution whose first term is missing. He distinguishes in this respect three periods and analyzes in detail the individual methods. But a method which is superior to all others in both simplicity and efficiency has been developed recently (MANTEL, 1951). It involves only direct counting and there is no equation to solve.

At the end of his study of "Fitness and optimization" R. LEVINS comes to the conclusion that in cases of simple Mendelian selection populations will differ in nature in the same direction as their optima. But the optimization principle is not justified in a fluctuating environment. In all cases examined by the author the appropriate measure of fitness depends on the ecological parameters and the variances of these parameters over environments.

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REPORTS AND REVIEWS 97

The editor K. KOJIMA and R. C. LEWONTIN contributed the paper "Evolutionary significance of linkage and epistasis". They discuss the rate of change in population fitness, the question of unstability of genotype and the rate of changes in various parameters in multilocus systems. According to the authors "a population in actual environments may never come to the point of equilibrium due to various minor changes, including random changes, in the system of fitness values".

It is really very difficult to review this book unless one can study thoroughly both sides (the mathematical and the genetical) of the problems in question. But it is beyond any doubt that the given interdisciplinary area has developed a great deal since its beginning. It may be supposed that it will develop in the future even more rapidly thanks to the use of computers for solving very complex problems. There are still many interesting questions to be investigated. The book appears at the beginning of a new phase in the study of genetio populations in which research workers recognize various factors such as biochemical aspects, ecological conditions for a popu- lation, and so forth.

The comprehensive list of literature on genetics and mathematical application in genetics (see references at the end of individual papers) is praiseworthy.

MILENA ROUDNA

Botanical Institute Czechoslovak Academy of Sciences, Prfihonice near Praha

D. WALKER and R. G. WEST (Ed.)

STUDIES IN THE VEGETATIONAL HISTORY OF THE BRITISH ISLES

Cambridge University Press, London 1970, 266 pp.; ? 8.-, $ 25.-

When Professor H. GODWIJN left the Chair of Botany at Cambridge, his pupils wanted to honour him with a collection of papers. These papers edited by D. WALKER and R. G. WEST, are now published by the Cambridge University Press. The connecting link between the miscellaneous writings in the collection is the authors' endeavour to present a survey (contributing to a better, understanding) of the vegetational history of the British Isles during the Quaternary period, whether by means of pollen analyses, macroscopic analyses of plant remains, Quaternary geology, archaeology, or palaeophytogeography. Professor GODWIN himself is interested in all these disciplines, and owing to the versatility of both himself and of his pupils Great Britain is one of the few countries which can boast of an almost perfect and complete survey of the history of its flora and vegetation.

The miscellany opens with an essay by R. G. WEST on glacial, periglacial and non-glacial environment, on the pleistocene flora and its subdivisions in the British Isles (pp. 1-11). The author characterizes the vegetation of the temperate stages (interglacials), paying special attention to the behaviour and occurrence of the main forest trees in these periods. The paper ends with a survey of the oldest interglacial finds of representative plants belonging to the main phytogeographic elements of the British flora.

G. F. MITCHELL and W. A. WATTS (pp. 12-21) present a comparative study on the history, pleistocene occurrence and spreading of the representatives of the family Ericaceae. The study is especially concerned with Ireland, but it also takes a wider view of the entire area of the British Isles and also of a part of western Europe.

Changes of land and sea level in Scotland during the last 10,000 years studied on the basis of pollenanalytical and stratigraphical research and of C14 dating are discussed in an essay by J. J. DONNER (pp. 23-37). At the same time he draws a comparison with Flandrian (Holocene) land/sea level changes in England, Holland, Norway and the Baltic sea.

A regional synthesis of vegetational history for NW. England is offered by WINIFRED PENNINGTON (pp. 41-79). It is a survey showing clearly the differences between the vegetational changes during the Late Glacial, early Post-glacial and later Post-glacial periods in Cumberland Lowland, Lake District and the lowlands round Morecambe Bay. As the author reprints the

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