A theory of natural selection incorporating interaction among individuals. II. Use of related groups

The present series of studies attempts to accommodate interaction among individuals in evolutionary theory. The interaction phenomenon is characterized by two dimensions (direct and associate) of gene activity. For optimal selection results, a balance between the two dimensions must occur. In the first paper of the series, it was shown that random interactions resulted in an unbalanced selection response. The expression for gene frequency change involved direct, but not associate, effects. The next three papers of the series (II, III and IV) are designed to explore the possibility that restricting interactions to certain non-random patterns may ameliorate the problem of selection balance.In the present study the interactions are restricted to related individuals in a population that is in Hardy-Weinberg equilibrium. A preliminary analysis in which interactions are restricted to full-sibs is made. This analysis is extended to the more general case in which interactions occur among related individuals of any class whose coefficient of relationship is measured by ‘r’. The classical pairwise interaction results of Hamilton are verified and extended to include interactions among individuals in groups of arbitrary size, n.Restricting interactions to related individuals tends to improve the condition of selection balance. It does this by introducing associate effects into the expression for gene frequency change. The extent to which this is accomplished is a function of the coefficient of relationship (r), and the number of interacting genotypes.     

A theory of natural selection incorporating interaction among individuals. III. Use of random groups of inbred individuals

The present series of studies attempts to accommodate interaction among individuals in evolutionary theory. The interaction phenomenon is characterized by two dimensions (direct and associate) of gene activity. For optimal selection results, a balance between the two dimensions must occur. In the first paper of the series, it was shown that random interactions resulted in an unbalanced selection response in that the direct, but not associate, effects were included in the expression for gene frequency change. The next three papers of the series (II, III and IV) were designed to determine whether or not selection with life-history models that involved non-random interactions would be useful in ameliorating the problem of selection balance.In the present study, non-randomness is generated by restricting interactions to inbred individuals. It is demonstrated that this form of non-random gene association within interacting genotypes does not improve selection balance. Thus restricting interaction to groups of inbred individuals does not result in the introduction of associate effects into the expression for gene frequency change. It is shown that inbreeding in the base population merely accelerates the unbalanced response normally occurring when selection operates on random, non-inbred individuals.     

A theory of natural selection incorporating interaction among individuals. IV. Use of related groups of inbred individuals

The present series of papers attempts to accommodate interaction among individuals in evolutionary theory. The interaction phenomenon is genetically characterized by two dimensions (direct and associate) of gene activity. For optimal selection results, a balance between the two dimensions must occur. In the first paper of the series, it was shown that random interactions resulted in an unbalanced selection response in that the direct, but not associate, effects were included in the expression for gene frequency change. The next three papers of the series (II, III and IV) were designed to determine whether or not selection with life-history models that involved non-random interactions would be useful in ameliorating the problem of selection balance.In the present study, two kinds of non-random gene association are analyzed jointly by restricting interactions to related individuals that are derived from inbred base populations. The analyses are generalized to accommodate heterogeneous as well as homogeneous groups of interacting individuals. The joint contributions of inbreeding and consanguinity to selection response are analysed by use of the nine gene-identity parameters devised by Harris.It is demonstrated that consanguinity alone or in conjunction with inbreeding does improve selection balance. However, inbreeding alone does not. Also, the influence of inbreeding is not dependent on group size, whereas the influence of consanguinity is conditioned by the size of the group. Thus, by introducing associate effects into the selection process, the use of related groups can provide the genetical bases for the evolution of social behavior phenomena such as altruism.     

A theory of natural selection incorporating interaction among individuals. V. Use of random synchronized groups

In the first paper of the current series, (I), a complex interaction model capable of describing any kind of interaction among individuals was developed. However, selection operating on random groups with regard to this model yielded short- and long-term results which were unbalanced. In subsequent kin-selection papers (II, III, and IV), a systematic analysis demonstrated that use of non-random groups could partially solve the balance problem.The present study is the first of several to employ a different approach to the problem of accommodating interaction. This approach involves changing the life-history model itself in such a way that the fitness components of individuals within groups are synchronized. Synchronization of fitness components produces total fitness values which are symmetrical. In the present study, selection operating on random groups for a model having symmetrical fitness values is evaluated for both balance and efficiency. It is demonstrated that the selection response is balanced and yields short- and long-term optimum results, but under a variety of conditions the efficiency can be low.     

A theory of natural selection incorporating interaction among individuals. VI. Use of non-random synchronized groups

The previous study, (V), in this series approached the problem of accommodating interaction among individuals in a population by considering a life-history model which resulted in the synchronization of fitness components (viability and fecundity) for members within groups of interacting individuals. It was shown that such synchronization resulted in symmetric fitness values, and that selection operating on symmetric fitness values produced optimum short- and long-term results. However, it was also shown that selection operating on the random groups of the model could be inefficient.The present paper demonstrates that use of non-random (related) groups can increase the efficiency of symmetric selection without destroying its short-term balance. The consequences of long-term selection are more complicated and depend on the complexity of the genetic model.     

A theory of natural selection incorporating interaction among individuals. VII. Use of groups consisting of one sire and one dam

A previous study, (V), in this series dealt with the problem of interaction among individuals in a population by utilizing a conceptual life-history model that assumed the synchronization of fitness components (viability and fecundity) for members within groups of interacting individuals. The model also assumed that the group members were not differentiated in any way. It was shown that these assumptions of synchronization and homogeneity of group members resulted in overall symmetric fitness values, and that selection operating on these symmetric values produced optimum short- and long-term results.Since the idealized model of paper (V) yields optimum selection results, it is of interest to consider specific biological mechanisms which can be used to fulfill the conceptual assumptions involved. The solution adopted in this study is to, first, require survival to occur at the group level. This forces the viability parameters to be identical for all group members.Second, the group composition is restricted to a single pair mating. This forces the fecundity parameters to be identical for members (sire and dam) of the same group. However, this solution causes a complication. Since each individual is identified as belonging to one of the two sexual types, it is necessary to extend the previous analysis in paper (V), to accommodate differentiated group members. It is shown that this differentiation leads to a non-symmetric fitness matrix. However, it is further shown that the selection results can be obtained by use of a ‘combined’ fitness matrix that is symmetric. Therefore, the important result is demonstrated that selection involving differentiated sire-dam groups possesses all of the optimum short- and long-term properties inherent in the original non-differentiated selection procedure developed in paper (V). It is speculated that these results could be of some significance with regard to the evolution of a monogamous, pair-bonding family structure.     

A theory of natural selection incorporating interaction among individuals. VIII. Use of groups consisting of a sire and several dams

The previous paper, (VII), in this series dealt with a group structure that consisted of a single mating pair. It was demonstrated that selection operating on such groups produced optimum short- and long-term results. The present study extends this group structure to include a single sire and several, (n ? 1), dams. The objective of the present study is to determine whether or not the optimum evolutionary results inherent with groups consisting of a single mating pair extend to groups consisting of multiple matings.It is demonstrated that extending the group from a single mating pair to include multiple matings converts the strictly symmetric into a modified-symmetric selection procedure that combines symmetric and non-symmetric properties. Thus the optimum evolutionary results of groups consisting of a single mating pair do not extend completely to groups consisting of multiple matings.     

A theory of natural selection incorporating interaction among individuals. IX. Use of groups consisting of a mating pair and sterile diploid caste members

This paper and the next member of the series, deal with genetical mechanisms responsible for the evolution of eusociality (a level of social organization that includes differentiated sterile castes) among the “social” insects. Eusociality has evolved in a number of different species. Two different types of genetic systems are represented among these species: diplodiploidy (both sexes diploid) and haplodiploidy (haploid males and diploid females). The present paper examines the evolution of a sterile caste system in the context of diplodiploidy, and the next paper considers the evolution of eusociality in the context of haplodiploidy.The present study demonstrates that if the sterile diploid caste members are related to the reproductive members of the group, eusociality can evolve. This is true because the caste associate gene effects are included in the function determining gene frequency change (i.e. Δp_i). Also, since the caste gene effects are expressed only through the associate dimension of gene activity, they can cause morphological and behavioral adaptations to occur which are peculiar to the caste members, and need not be expressed in the reproducing members of the group. Thus caste differentiation is possible.     

A theory of natural selection incorporating interaction among individuate. X. Use of groups consisting of a mating pair together with haploid and diploid caste members

This paper and the previous member of the series, deal with genetical mechanisms responsible for the evolution of eusociality (a level of social organization that includes differentiated sterile castes) among the “social” insects. Eusociality has evolved in a number of different species. Two different types of genetic systems are represented among these species: diplodiploidy (both sexes diploid) and haplodiploidy (haploid males and diploid females). The previous paper examined the evolution of a sterile caste system in the context of diplodiploidy, and the present paper considers the evolution of eusociality in the context of haplodiploidy.The present study demonstrates that selection operating with regard to random groups within the haplodiploid inheritance system cannot result in the evolution of a sterile caste system. Thus haplodiploidy, in itself, is not sufficient for the evolution of eusociality. However, if the sterile caste members are related to the reproductive members of the group, the appropriate caste associate gene effects are included in the function determining gene frequency change (i.e. Δp_i), and therefore, eusociality can evolve. This is true for both haploid and diploid castes.In comparing the two modes of inheritance, it is demonstrated that haplodiploidy provides up to 37·5% increased selection efficiency relative to diplodiploidy in evolving a social caste system in the absence of inbreeding.     

A mathematical theory of natural and artificial selection—I

Mathematical expressions are found for the effect of selection on simple Mendelian populations mating at random. Selection of a given intensity is most effective when amphimixis does not affect the character selected, e.g. in complete inbreeding or homogamy. Selection is very ineffective on autosomal recessive characters so long as they are rare. Reprinted fromTransactions of the Cambridge Philosophical Society, Vol 23, pp. 19–41 (1924) with the permission of Cambridge University Press.     

A porous-medium approach for modeling heart mechanics. I. theory

A method is developed for evaluating the distribution of the blood pressure, stresses and strains of the muscle fibers, and motion of the cardiac wall due to the cyclic contractions of the heart. The cardiac system is subdivided into two media: the chambers and the wall; the latter is enclosed by two impermeable surfaces (with one interface separating the two media and the other confining the wall). The momentum balance equation for the blood in the (two) cardiac ventricles is averaged, yielding a modification of Forchheimer's law, namely inclusion of the time derivative of the flux. The contracting muscle provides the driving force for the blood flow, and the endocardial velocity is thus taken as identical to that of the blood in the cavity next to the wall. Translation of the endocardium is governed by the blood pressure gradients in the ventricles. The blood pressure and stress-strain pattern in the myocardium are analyzed by applying concepts of the theory of mixtures to the blood and to the saturated solid matrix. With the blood pressure simulated by a modified Darcy law with relative fluid-solid velocity, fiber stresses and strains can be assessed with the aid of appropriate constitutive and compatibility laws.     

A theory of follicle selection: I. Hypotheses and examples

A theory of follicle selection is developed in which all follicles, ovulatory and atretic, inherit the same developmental plan for responding to circulating concentrations of estradiol and gonadotropins. In the model, this plan is represented by a maturation surface that determines the rate of follicle growth as a function of follicle maturity and circulating hormone concentration. Examples of maturation surfaces are constructed for single and multiple spontaneous ovulators. When model follicles are activated from the reserve pool at random times, spontaneous and coordinated cycles of maturation develop in which the number of ovulatory follicles is controlled within a small and predictable range. For a given maturation surface, this range is nearly independent of the number of follicles in the interacting population, their activation times, and their initial maturities. The theory is therefore consistent with Lipschütz's Law of Follicular Constancy. The theory can account for certain statistical effects that occur with age on the timing of ovulation and the control of ovulation number. Maturation surfaces are also constructed that exhibit spontaneous transitions from cyclic to steady anovulatory behavior. The theory predicts an important regulatory role for atretic follicles in controlling both the timing of maturation and the number of follicles that reach ovulatory maturity.     

Some population genetic models combining artificial and natural selection pressures: I. One-locus theory

A series of one-locus genetic models involving the combined effects of artificial and natural selection forces are analyzed. Other factors incorporated in this work include the influence of the imposed or inherent mating system, the relevance of timing in application of the two types of selection forces, the importance of multiallelism and dominance relations.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.     

A macro-equation of natural selection

The deterministic dynamical theory of biological populations, developed widely on the basis of the classical work of Fisher and Volterra, in most cases deals with characteristics which cannot be measured directly, e.g. frequencies of various genotypes within a population, their fitness values, competition coefficients, etc. Thus, a theory dealing with a small number of simple averaged macro-characteristics, easily accessible to a direct measurement, would be of great importance. The present paper contains an attempt to establish an equation contributing to such a would-be macrotheory. It is a relationship begween the average fitness of a population (the Malthusian growth parameter), the selective delay (a new concept, introduction in section 3) and the entropy of the equilibrium structure which the population tends to under the natural selection process. A possible method of checking the proposed relationship experimentally is indicated.