# Mathematical Evolutionary Theory

Edited by Marcus W. Feldman
Pages: 352
https://www.jstor.org/stable/j.ctt7ztkct

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. PREFACE
(pp. ix-2)
4. ### Part I Stochastic and Deterministic Genetic Theory

• [Part I Introduction]
(pp. 5-8)

The foundation stone of population genetic theory with finite population size is the Wright-Fisher sampling model. It is described in the paper by Ewens and underlies much of the discussion in the papers by Gillespie, Watterson, and Kaplan and Hudson. It also forms the background for part of Tavaré’s paper. The binomial (or multinomial) sampling scheme produces a Markov chain that describes the change in the genetic constitution of the population over time. The eigenvectors of this Markov chain have not been found in a useful form, and Wright in 1931 used a diffusion approximation to the discrete time stochastic...

• CHAPTER ONE The Effective Population Sizes in the Presence of Catastrophes
(pp. 9-25)
Warren J. Ewens

My long association with Sam Karlin through our work in mathematical population genetics has been a most memorable and enjoyable experience for me. It started in 1964 when I was a postdoctoral student at Stanford, has continued to this day, and my aim in this paper is to continue it even further. A simplified description of one aspect of our association is as follows: I would become interested in some problem and partially develop its mathematical properties, but would eventually be defeated by some aspect of the mathematical analysis, or not see the full generality of the theory, whereupon I...

• CHAPTER TWO The Neutral Alleles Model with Bottlenecks
(pp. 26-40)
Geoffrey A. Watterson

If a sample ofngenes is chosen at random from a large population (say of sizeNdiploids) then Ewens’ (1972) sampling distribution describes the probability that the sample will contain a certain number of alleles, at various frequencies for those alleles. Perhaps the neatest way to describe the distribution is first to introduce the sample “frequency spectrum”β1,β2,…,βn, where

βi= number of alleles havingirepresentative genes in the sample

Then Ewens’ formula for the probability of getting a particular set of values for the spectrum is$P({{\beta }_{1}},{{\beta }_{2}},\ldots ,{{\beta }_{n}})=\frac{n!}{{{(\theta )}_{(n)}}}\prod\limits_{i=1}^{n}{\left[ {{(\theta /i)}^{{{\beta }_{i}}}}/{{\beta }_{i}}! \right]}$ (2.1.1)whereβ1,β2,…,βnare nonnegative integers such that...

• CHAPTER THREE The Genealogy of the Birth, Death, and Immigration Process
(pp. 41-56)
Simon Tavaré

It is indeed a pleasure for me to contribute to this dedicatory volume for Professor Samuel Karlin. Among Karlin’s many contributions that address mathematical or statistical issues in the broad area of biology is a collection devoted to the analysis of a variety of stochastic processes that arise in the mathematical theory of population genetics. This theory is the most developed (and the most elegant) in the setting of the infinitely many neutral alleles models, and it is to such problems that this paper is addressed.

In a seminal paper, Karlin and McGregor (1967) describe the following model. Imagine families...

• CHAPTER FOUR When Not to Use Diffusion Processes in Population Genetics
(pp. 57-70)
John H. Gillespie

The use of diffusion processes to approximate the dynamics of population genetics models has yielded insights that would be unapproachable by exact methods. Wright (1931, 1945) laid the foundations with techniques that he developedde novoto obtain stationary densities and leading eigenvalues of processes that arise in population genetics. Building on Wright’s results, Kimura (1955, 1964) pioneered work on the transient properties of diffusions through the use of the forward equation to obtain transient densities and through the backward equation to obtain fixation probabilities. The Australian group (Moran 1962; Watterson 1962; Ewens 1964, 1965) did the most important early...

• CHAPTER FIVE The Effect of Population Subdivision on Multiple Loci without Selection
(pp. 71-85)
Freddy Bugge Christiansen

When previously isolated populations are mixed, the genotypic proportions at a locus with variable gene frequencies show an excess of homozygotes as compared to the Hardy-Weinberg proportions corresponding to the mean gene frequency in the mixed population (Wahlund 1928). This well-known signature of population mixing disappears after just a single breeding by random mating in a population with nonoverlapping generations. A more profound and long-lasting effect of population mixing is the creation of distributional interactions between alleles of a pair of loci with variable gene frequencies (Sinnock and Singh 1972; Prout 1973; Nei and Li 1973; Feldman and Christiansen 1975)....

• CHAPTER SIX Complete Characterization of Disequilibrium at Two Loci
(pp. 86-110)
Bruce S. Weir and C. Clark Cockerham

All possible disequilibrium coefficients involving one or two genes at each of two loci are defined. The behavior over time of the various classes of coefficients—four digenic, two trigenic, and one quadrigenic—is expressed in terms of two-locus descent measures. Within populations, maximum likelihood estimates of the disequilibrium coefficients are presented, and so are expected values of their sampling variances. Simulations confirm that hypothesis tests can be based on the asymptotic normality of the estimates. It is recommended that all disequilibrium coefficients be considered in the statistical analysis of genotypic data.

The transmission of genetic material between generations in...

• CHAPTER SEVEN The Reduction Principle for Genetic Modifiers of the Migration Rate
(pp. 111-138)
Uri Liberman and Marcus W. Feldman

The extent to which the demes in a system of subpopulation are separated is one of the key parameters in the genetic evolution of each deme, and of the system considered as a whole. The interaction between migration and selection in such systems has recently been surveyed by Karlin (1982), especially with regard to conditions for the maintenance of genetic polymorphism. In the present paper we shall take the equilibrium structure engendered by the interaction of migration and selection as the point of departure from which to examine the evolution of a gene that controls the rate of migration in...

5. ### Part II Behavior, Ecology, and Evolutionary Genetics

• [Part II Introduction]
(pp. 141-144)

The papers of this section are somewhat more applied in their approach than those of Part I. The first two address the evolution of behavior but from very different points of view, one in terms of cultural evolution, the other in terms of kin selection. In Chapter 8 Feldman and Cavalli-Sforza continue their theoretical studies on the coevolution of genes and culture with their model for the simultaneous evolution of lactose absorption and milk use. Lactose absorption and malabsorption is a genetically based phenotypic distinction manifest in the presence of a cultural distinction, namely, use or nonuse of milk. In...

• CHAPTER EIGHT On the Theory of Evolution under Genetic and Cultural Transmission, with Application to the Lactose Absorption Problem
(pp. 145-173)
Marcus W. Feldman and Luigi L. Cavalli-Sforza

It is perhaps surprising that the evolutionary theory of discrete valued phenotypic variation which does not obey simple Mendelian rules of inheritance has received less quantitative study than has continuous or quantitative variation. The reason probably is the wide acceptance of a genetic basis for continuous variation in terms of many loci, each of small effect, whose aggregate produces a continuous distribution of phenotypes (Fisher 1918). This hypothesis may lead to the distribution most amenable to dynamic analysis, which is the Gaussian. For discrete valued or for qualitative traits, the most common assumption is that the variants are themselves different...

• CHAPTER NINE Two-Locus Models of Kin Selection among Haplodiploids: Effects of Recombination and Epistasis on Relatedness
(pp. 174-206)
Marcy K. Uyenoyama

Hamilton’s (1964a,b) theory of kin selection proposes that natural selection favors genes that promote altruism among genetically related individuals, provided that relatedness between donor and recipient exceeds the ratio of the cost to the benefit associated with the behavior. Studies of dynamic recursion systems that explicitly model the processes of genetic transmission, zygote formation, and interactions among relatives have restricted attention for the most part to single loci that modify the rate of performance of altruism (see reviews in Charlesworth 1980; Uyenoyama and Feldman 1980; Michod 1982). The multilocus models of kin selection that have been analyzed to date fall...

• CHAPTER TEN Resource Allocation in Mendelian Populations: Further in ESS Theory
(pp. 207-246)
Sabin Lessard

Modern ESS (Evolutionarily Stable Strategy) theory began with Maynard Smith and Price’s (1973) paper. But its origins can be traced in early frequency-dependent selection models (dealing, e.g., with mimicry or sexual selection) and in sex ratio evolution principles (see, e.g., Fisher 1930). In this respect, Shaw and Mohler (1953) shed some light onto a basic notion, that of reproductive value in sexual populations, with the formula$\frac{{{m}_{0}}}{2m}+\frac{{{f}_{0}}}{2f},$ (10.1.1)wherem0andf0refer to individual fitnesses through male and female functions (or progenies) with mean valuesmandfin the population. Given a (convex) fitness set for all possible (m0,...

• CHAPTER ELEVEN Sexual Selection Models and the Evolution of Melanism in Ladybirds
(pp. 247-269)
Peter O’Donald and Michael E. N. Majerus

Sexual selection by female preference is expected to give rise to a “rare male advantage” (O’Donald 1977, 1980). This arises because, at a low frequency, a preferred male phenotype will be preferred by a relatively greater proportion of females than at a high frequency. Stable polymorphisms will then be maintained, either when more than one male phenotype is the object of female preference, or when the sexual selection is balanced by natural selection (O’Donald 1973, 1980).

Karlin and O’Donald (1981) analyzed multiallelic models in which either (1) females exercised separate preferences for distinct phenotypes; or (2) females increasingly preferred males...

• CHAPTER TWELVE The Evolution of Marine Life Cycles
(pp. 270-300)
Jonathan Roughgarden

Although the marine environment harbors animal species with a great diversity of life cycles, some generalizations have long been known. By 1900 two cycles had been identified for the species that have their adult phase in a benthic habitat (i.e., the adult lives on, or in, a rocky or muddy substrate). Either the entire life cycle is spent in the benthic habitat, or the life cycle begins with a morphologically distinctive larval phase that feeds in the water column (a pelagic larva) before metamorphosis into the benthic adult phase. Furthermore, in the entirely benthic life cycle, either the adult gives...

• CHAPTER THIRTEEN An Evolutionary Model for Highly Repeated Interspersed DNA Sequences
(pp. 301-314)
Norman L. Kaplan and Richard R. Hudson

Within the genomes of mammalian species there are many distinct families of repetitive DNA elements. The lengths of the repeated DNA sequences vary from a few nucleotides to several thousand, and their copy number from several hundred to several million. For some families the repeats occur tandemly, while for others the individual copies are dispersed throughout the genome. In this paper we are interested in studying evolutionary models for highly repeated short interspersed families (SINEs as denoted by Singer 1982). Examples of such families are the Alu family in humans, the B1 family in mice, the Monomer family in Galagos,...

• CHAPTER FOURTEEN Statistics and Population Genetics of the HLA System
(pp. 315-334)
Walter F. Bodmer and Julia G. Bodmer

Simple statistical 2 × 2χ2analysis of reactions of groups of sera on a panel of cells from different individuals led to the definition of the HLA system. Positive associations between pairs of sera indicate common determinants, and so the first antigens were simply defined by the consensus of reactions of a set of sera, all or most pairs of which were significantly positively associated. This led Van Rood to describe the two-allele system he called 4a and 4b, later to become related to the HLA-B locus (Van Rood and Van Leeuwen 1963). Recognizing that alleles at a locus...

6. LIST OF CONTRIBUTORS
(pp. 335-336)
7. INDEX
(pp. 337-341)