Evolutionary principles for general frequency-dependent two-phenotype models in sexual populations

The evolutionary dynamics in general two-sex two-phenotype frequency-dependent selection models are studied with respect to underlying multi-allele one-locus genetic systems. Two classes of equilibria come into play: genotypic equilibria, with equilibrium allelic frequencies independent of the phenotype, and phenotypic equilibria, which are characterized by equal mean phenotypic fitnesses. The exact conditions for genotypic equilibria to exist and be stable and for phenotypic equilibria to exist and be evolutionarily attractive are examined. Using adequate definitions of mean fitnesses in general contexts of frequency-dependent selection in dioecious populations, we show that two phenotypes, when they can coexist in the population, tend to balance their fitnesses as far as is allowed by the genetic system as more alleles responsible for phenotype determination are introduced into the population.     

Frequency-dependent viability selection (a single-locus, multi-phenotype model)

Frequency-dependent natural selection models are examined where the viability of an individual in the diploid population is determined by its phenotype and the frequency of other phenotypes present. The equilibria of the multi-phenotypic system are characterized through local mean fitness functions. It is shown that stability can best be analyzed by combining the principles of maximization of population mean fitness with the evolutionary stability conditions that apply when phenotypic fitnesses relative to the genetic constraints are equal.     

Genetic and phenotypic models of natural selection

The following theorem is proposed: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness, provided that certain modes of inheritance apply; in particular, all genotypes specifying the same phenotype must have the same average fitness. If these conditions of “uniform fitness” patterns of inheritance are not met, particular genetic models of natural selection should replace an analysis of phenotypes. If the conditions are met, an analysis of the stationary conditions when the phenotypes have equal fitnesses permits quantitative statements about the outcome of selection without recourse to genetic models. Phenotypic analyses of natural selection are illustrated by models of sex ratios in plants, sexual versus asexual reproduction in plants, and parental investment by animals.     

Density dependent selection 1: A stable feasible equilibrium may not be attainable

The joint evolution of gene frequency p, and population size N is studied. It is shown that when the genotypic fitnesses are logistic functions of the population size, sets of initial states exist which lead to bizarre behavior. Even though equilibria may be locally stable, these sets of initial conditions eventually produce negative fitnesses. Alternative models are discussed as are some general properties of the mean fitness.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

Polymorphism at two loci through selection for linear metric deviation.

A model of phenotypic stabilising selection in which the fitness of an individual depends solely on its phenotype, and not directly on its genetic constitution, is explored algebraically for a system of two linked loci of unequal effect. It is found that selection for metric deviation gives rise to polymorphic gametefrequency equilibria for a variety of fitness regimes. Stability of non-trivial equilibria occurs for a wide range of parameter sets. Stability is facilitated by close linkage and inequality between gene effects. It is suggested that, in general genetic variation may be maintained under stabilising selection when the fitness of double heterozygotes exceeds that of the phenotypically intermediate homozygotes.     

THE MEASUREMENT OF SELECTION ON QUANTITATIVE TRAITS: BIASES DUE TO ENVIRONMENTAL COVARIANCES BETWEEN TRAITS AND FITNESS

The use of regression techniques for estimating the direction and magnitude of selection from measurements on phenotypes has become widespread in field studies. A potential problem with these techniques is that environmental correlations between fitness and the traits examined may produce biased estimates of selection gradients. This report demonstrates that the phenotypic covariance between fitness and a trait, used as an estimate of the selection differential in estimating selection gradients, has two components: a component induced by selection itself and a component due to the effect of environmental factors on fitness. The second component is shown to be responsible for biases in estimates of selection gradients. The use of regressions involving genotypic and breeding values instead of phenotypic values can yield estimates of selection gradients that are not biased by environmental covariances. Statistical methods for estimating the coefficients of such regressions, and for testing for biases in regressions involving phenotypic values, are described.     

Deviations of Genotypic Structures from Hardy-Weinberg Proportions under Random Mating and Differential Selection between the Sexes

Population genetic models, such as differential viability selection between the sexes and differential multiplicative fecundity contributions of the sexes, are considered for a single multiallelic locus. These selection models usually produce deviations of the zygotic genotype frequencies from Hardy-Weinberg proportions. The deviations are investigated (with special emphasis put on equilibrium states) to quantify the effect of selective asymmetry in the two sexes. For many selection regimes, the present results demonstrate a strong affinity of zygotic genotype frequencies for Hardy-Weinberg proportions after two generations, at the latest. It is shown that the deviations of genotypic equilibria from the corresponding Hardy-Weinberg proportions can be expressed and estimated by means of selection components of only that sex with the lower selection intensity. This corresponds to the well-known fact that viability selection acting in only one sex yields Hardy-Weinberg equilibria.     

Evolutionary Stability for Interactions among Kin under Quantitative Inheritance

The theory of evolutionarily stable strategies (ESS) predicts the long-term evolutionary outcome of frequency-dependent selection by making a number of simplifying assumptions about the genetic basis of inheritance. I use a symmetrized multilocus model of quantitative inheritance without mutation to analyze the results of interactions between pairs of related individuals and compare the equilibria to those found by ESS analysis. It is assumed that the fitness changes due to interactions can be approximated by the exponential of a quadratic surface. The major results are the following. (1) The evolutionarily stable phenotypes found by ESS analysis are always equilibria of the model studied here. (2) When relatives interact, one of the two conditions for stability of equilibria differs between the two models; this can be accounted for by positing that the inclusive fitness function for quantitative characters is slightly different from the inclusive fitness function for characters determined by a single locus. (3) The inclusion of environmental variance will in general change the equilibrium phenotype, but the equilibria of ESS analysis are changed to the same extent by environmental variance. (4) A class of genetically polymorphic equilibria occur, which in the present model are always unstable. These results expand the range of conditions under which one can validly predict the evolution of pairwise interactions using ESS analysis.     

Density-Dependent Selection Incorporating Intraspecific Competition. II. a Diploid Model

A diploid model is introduced and analyzed in which intraspecific competition is incorporated within the context of density-regulated selection. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when only that type is present. Each genotypic fitness at a single diallelic autosomal locus is a decreasing function of a distinctive effective population size perceived as a result of intraspecific competition. The resulting fitnesses are both density and frequency dependent with selective advantage determined by a balance between genotypic carrying capacity and sensitivity to intraspecific competition. A major finding is that intergenotypic interactions may allow genetic variation to be more easily maintained than in the corresponding model of purely density-dependent selection. In addition, numerical study confirms the possible existence of multiple interior equilibria and that neither overdominance in fitness nor carrying capacity is necessary for stability. The magnitude of the equilibrium population size and optimization principles are also discussed.     

On the status of mean fitness maximization as a governing principle in evolution

Mean fitness is not always maximized under natural selection. In particular, in two locus models, recombination and epistasis may combine to prevent the operation of a maximizing priciple for mean fitness. If inversion phenomena are considered, however, there exist fully polymorphic equilibria which maximize the mean fitness and moreover the initial progress of an inversion can proceed if and only if it gives rise to an increase in mean fitness. While not applicable to all models, the principle of mean fitness maximization is still useful heuristically.     

SOCIAL SELECTION AND THE EVOLUTION OF ANIMAL SIGNALS

Social selection is presented here as a parallel theory to sexual selection and is defined as a selective force that occurs when individuals change their own social behaviors, responding to signals sent by conspecifics in a way to influence the other individuals' fitness. I analyze the joint evolution of a social signal and behavioral responsiveness to the signal by a quantitative-genetic model. The equilibria of average phenotypes maintained by a balance of social selection and natural selection and their stability are examined for two alternative assumptions on behavioral responsiveness, neutral and adaptive. When behavioral responsiveness is neutral on fitness, a rapid evolution by runaway selection occurs only with enough genetic covariance between the signal and responsiveness. The condition for rapid evolution also depends on natural selection and the number of interacting individuals. When signals convey some information on signalers (e.g., fighting ability), behavioral responsiveness is adaptive such that a receiver's fitness is also influenced by the signal. Here there is a single point of equilibrium. The equilibrium point and its stability do not depend on the genetic correlation. The condition needed for evolution is that the signal is beneficial for receivers, which results from reliability of the signal. Frequency-dependent selection on responsiveness has almost no influence on the equilibrium and the rate of evolution.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

Natural selection under population regulation

The dynamical behavior of multi-allele, one-locus systems is analyzed under population regulation. Weak selection is assumed. It is shown that for sufficiently large times, t, the nth time derivative of the population number N(t) is of order n}+1 in the selection coefficients. These order relations imply there is an asymptotic “quasi-equilibrium” in which population size and mean fitness change slowly relative to changes in gene frequencies. Consistent with the results of other authors, in quasi-equilibrium the mean fitness is second-order in the selection coefficients. In an effort to understand dynamic behavior beyond the immediate neighborhood of equilibrium, the topology of mean fitness surfaces is explored. In general, population regulation leads to regions of decreasing mean fitness in which there are important changes in gene frequencies. To illustrate this and other related phenomena, I analyze models in which there is logarithmic population control, and in which genotypic fitnesses W_i(x) are linear in the allele frequencies x. Exact solutions for mean fitness W(x) are obtained for two- and three-allele systems with symmetric fertilities and mortalities.     

The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

The maintenance (or not) of polygenic variation by soft selection in heterogeneous environments

On the basis of single-locus models, spatial heterogeneity of the environment coupled with strong population regulation within each habitat (soft selection) is considered an important mechanism maintaining genetic variation. We studied the capacity of soft selection to maintain polygenic variation for a trait determined by several additive loci, selected in opposite directions in two habitats connected by dispersal. We found three main types of stable equilibria. Extreme equilibria are characterized by extreme specialization to one habitat and loss of polymorphism. They are analogous to monomorphic equilibria in singe-locus models and are favored by similar factors: high dispersal, weak selection, and low marginal average fitness of intermediate genotypes. At the remaining two types of equilibria the population mean is intermediate but variance is very different. At fully polymorphic equilibria all loci are polymorphic, whereas at low-variance equilibria at most one locus remains polymorphic. For most parameters only one type of equilibrium is stable; the transition between the domains of fully polymorphic and low-variance equilibria is typically sharp. Low-variance equilibria are favored by high marginal average fitness of intermediate genotypes, in contrast to single-locus models, in which marginal overdominance is particularly favorable for maintenance of polymorphism. The capacity of soft selection to maintain polygenic variation is thus more limited than extrapolation from single-locus models would suggest, in particular if dispersal is high and selection weak. This is because in a polygenic model, variance can evolve independently of the mean, whereas in the single-locus two-allele case, selection for an intermediate mean automatically leads to maintenance of polymorphism.     

Similarity selection and the evolution of sex: revisiting the red queen

For over 25 years, many evolutionary ecologists have believed that sexual reproduction occurs because it allows hosts to change genotypes each generation and thereby evade their coevolving parasites. However, recent influential theoretical analyses suggest that, though parasites can select for sex under some conditions, they often select against it. These models assume that encounters between hosts and parasites are completely random. Because of this assumption, the fitness of a host depends only on its own genotype (“genotypic selection”). If a host is even slightly more likely to encounter a parasite transmitted by its mother than expected by random chance, then the fitness of a host also depends on its genetic similarity to its mother (“similarity selection”). A population genetic model is presented here that includes both genotypic and similarity selection, allowing them to be directly compared in the same framework. It is shown that similarity selection is a much more potent force with respect to the evolution of sex than is genotypic selection. Consequently, similarity selection can drive the evolution of sex even if it is much weaker than genotypic selection with respect to fitness. Examination of explicit coevolutionary models reveals that even a small degree of mother–offspring parasite transmission can cause parasites to favor sex rather than oppose it. In contrast to previous predictions, the model shows that weakly virulent parasites are more likely to favor sex than are highly virulent ones. Parasites have figured prominently in discussions of the evolution of sex, but recent models suggest that parasites often select against sex rather than for it. With the inclusion of small and realistic exposure biases, parasites are much more likely to favor sex. Though parasites alone may not provide a complete explanation for sex, the results presented here expand the potential for parasites to contribute to the maintenance of sex rather than act against it.     

On the number of stable equilibria and the simultaneous stability of fixation and polymorphism in two-locus models

It is shown that in simple symmetric two-locus, two-allele constant fitness models the bound of four simultaneously stable equilibria previously accepted for general two-locus, two-allele models is exceeded. Situations with five and six stable equilibria are exhibited. These involve four chromosomal fixations and either one or two polymorphic stable equilibria.     

Why kin and group selection models may not be enough to explain human other-regarding behaviour

Models of kin or group selection usually feature only one possible fitness transfer. The phenotypes are either to make this transfer or not to make it and for any given fitness transfer, Hamilton's rule predicts which of the two phenotypes will spread. In this article we allow for the possibility that different individuals or different generations face similar, but not necessarily identical possibilities for fitness transfers. In this setting, phenotypes are preference relations, which concisely specify behaviour for a range of possible fitness transfers (rather than being a specification for only one particular situation an animal or human can be in). For this more general set-up, we find that only preference relations that are linear in fitnesses can be explained using models of kin selection and that the same applies to a large class of group selection models. This provides a new implication of hierarchical selection models that could in principle falsify them, even if relatedness--or a parameter for assortativeness--is unknown. The empirical evidence for humans suggests that hierarchical selection models alone are not enough to explain their other-regarding or altruistic behaviour.