The ancestral graph and gene genealogy under frequency-dependent selection.

Minority-advantage frequency-dependent selection has been proposed as the cause for the high level of observed polymorphism in some self/nonself-recognition systems. We present a mathematically rigorous derivation of the ancestral graph for a sample of genes that evolved according to a haploid infinite-alleles model of minority-advantage frequency-dependent selection. In the case of sufficiently weak selection, the gene genealogy can be extracted from the ancestral graph. We demonstrate that the gene genealogy under this model is identical to that obtained for a diploid model with heterozygote advantage. The case of strong selection is exemplified by a one-locus haploid self-incompatibility system; in this context, we investigate the number of alleles that can be maintained in a spatial versus a non-spatial habitat. Finally, we compare gametophytic self-incompatibility to the haploid self-incompatibility model.     

CYCLICAL ENVIRONMENTAL CHANGES AS A FACTOR MAINTAINING GENETIC POLYMORPHISM. 2. DIPLOID SELECTION FOR AN ADDITIVE TRAIT

The subject of this paper is polymorphism maintenance due to stabilizing selection with a moving optimum. It was shown that in case of two-locus additive control of the selected trait, global polymorphism is possible only when the geometric mean fitnesses of double homozygotes averaged over the period are lower than that of the single heterozygotes and of the double heterozygote (with a multiplier [1 – r]^p, which depends on recombination rate r and period length p). But local stability of polymorphism cannot be excluded even if geometric mean fitnesses of all double homozygotes are higher than that of all heterozygotes. We proved, that for logarithmically convex fitness functions, cyclical changes of the optimum cannot help in polymorphism maintenance in case of additive control of the selected trait by two equal loci. However, within the same class of fitness functions, nonequal gene action and/or dominance effect for one or both loci may lead to local polymorphism stability with large enough polymorphism attracting domain. The higher the intensity of selection and closer the linkage between selected loci the larger is this domain. Note that even simple cyclical selection could result in two forms of polymorphic limiting behavior: (a) usually expected forced cycle with a period equal to that of environmental changes; and (b) “supercycles,” nondumping auto-oscillations with a period comprising of hundreds of forced oscillation periods.     

Stable two-allele polymorphisms maintained by fluctuating fitnesses and seed banks: protecting the blues in Linanthus parryae

Motivated by data demonstrating fluctuating relative and absolute fitnesses for white- versus blue-flowered morphs of the desert annual Linanthus parryae, we present conditions under which temporally fluctuating selection and fluctuating contributions to a persistent seed bank will maintain a stable single-locus polymorphism. In L. parryae, blue flower color is determined by a single dominant allele. To disentangle the underlying diversity-maintaining mechanism from the mathematical complications associated with departures from Hardy-Weinberg genotype frequencies and dominance, we successively analyze a haploid model, a diploid model with three distinguishable genotypes, and a diploid model with complete dominance. For each model, we present conditions for the maintenance of a stable polymorphism, then use a diffusion approximation to describe the long-term fluctuations associated with these polymorphisms. Our protected polymorphism analyses show that a genotype whose arithmetic and geometric mean relative fitnesses are both less than one can persist if its relative fitness exceeds one in years that produce the most offspring. This condition is met by data from a population of L. parryae whose white morph has higher fitness (seed set) only in years of relatively heavy rain fall. The data suggest that the observed polymorphism may be explained by fluctuating selection. However, the yearly variation in flower color frequencies cannot be fully explained by our simple models, which ignore age structure and possible selection in the seed bank. We address two additional questions--one mathematical, the other biological--concerning the applicability of diffusion approximations to intense selection and the applicability of long-term predictions to datasets spanning decades for populations with long-lived seed banks.     

Environmental fluctuations and the maintenance of genetic diversity in age or stage-structured populations

The ability of random fluctuations in selection to maintain genetic diversity is greatly increased when generations overlap. This result has been derived previously using genetic models with very special assumptions about the population age structure. Here we explore its robustness in more realistic population models, with very general age structure or physiological structure. For a range of genetic models (haploid, diploid, single and multilocus) we find that the condition for maintaining genetic diversity generalizes almost without change. Genetic diversity is maintained by selection if a product of the form (generation overlap)×(selection intensity)×(variability in the selection regime) is sufficiently large, where the generation overlap is measured in units of Fisher's reproductive value. This conclusion is based on a local evolutionary stability analysis, which differs from the standard “protected polymorphism” criterion for the maintenance of genetic diversity. Simulation results match the predictions from the local stability analysis, but not those from the protected polymorphism criterion. The condition obtained here for maintaining genetic diversity requires fitness fluctuations that are substantial but well within the range observed in many studies of natural populations.     

Evolutionary dynamics in frequency-dependent two-phenotype models

General frequency-dependent selection models based on two phenotypic classes are analyzed with underlying one-locus multiallele phenotypic determination systems in diploid populations. It is proved that the mean phenotypic fitnesses tend to equality over discrete generations and genetic mutations if a phenotypic polymorphism is to be maintained. The exact conditions are examined. The present results are valid for a wide class of models whenever random groupings or assortative patterns based on phenotype and affecting fitness, linearly or not, are independent of sex, mating preferences, or kinship. They can also be applied to two-sex haploid models.     

Hard selection in haploid species

The formulation of hard selection is reviwed in the context of haploid viabilities and the criteria for stability of the fixation states are given. In contrast to soft selection, both fixation states can be simultaneously stable. However, this is not possible if the migration matrix is positive definite. In the case of only two demes there is at most one polymorphic equilibrium as occurs under soft selection, but the internal equilibrium may be unstable in contrast to the soft selection case. The question of hard versus soft protection is posed in the context of haploid viabilities and the principle hard protection implies soft protection holds with a similar degree of generality as in the diploid case.     

Competitive divergence in non-random mating populations

A haploid model of frequency-dependent selection and assortative mating is introduced and analyzed for the case of a single multiallelic autosomal locus. Frequency-dependent selection is due to intraspecific competition mediated by a quantitative character under stabilizing or directional selection. Assortment is induced by the same trait. We analyze the equilibrium structure and the local stability properties of all possible equilibria. In the limit of weak selection we obtain global stability properties by finding a Lyapunov function. We provide necessary and sufficient conditions for the maintenance of polymorphism in terms of the strength of stabilizing selection, intraspecific competition and assortment. Our results also include criteria for the ability of extreme types to invade the population. Furthermore, we study the occurrence of disruptive selection and provide necessary and sufficient conditions for intraspecific divergence to occur.     

Selection and the Evolution of Genetic Life Cycles

The evolution of haploid and diploid phases of the life cycle is investigated theoretically, using a model where the relative length of haploid and diploid phases is under genetic control. The model assumes that selection occurs in both phases and that fitness in each phase is a function of the time spent in that phase. The equilibrium and stability conditions that allow for all-haploid, all-diploid, or polyphasic life cycles are considered for general survivorship functions. Types of stable life cycles possible depend on the form of the viability selection. If mortality rates are constant, either haploidy or diploidy is the only stable life cycle possible. Departures from constant mortality can give qualitatively different results. For example, when survivorship in each phase is a linear, decreasing function of the time spent in the phase, stable haploid, diploid or polyphasic life cycles are possible. The addition of genetic variation at a coevolving viability locus does not qualitatively affect the outcome with respect to the maintenance of polyphasic cycles but can lead to situations where more than one life cycle is concurrently stable. These results show that trade-offs between the advantages of being diploid and of being haploid may help explain the patterns of life cycles found in nature and that the type of selection may be critical to determining the results.     

Selection at the haploid and diploid phases: cyclical variation

This paper considers a deterministic model, where selection acts at both the diploid and haploid phases of development. The stability criteria for the maintenance of a protected polymorphism are derived. The implications of the model are examined through several examples, with emphasis on the comparison of these conditions to those of models that view selection as acting only at the diploid or haploid phases. It is found that the consideration of both phases broadens the conditions over such models.     

The evolution of genetic diversity.

The existence within natural populations of large amounts of genetic variation in molecules and morphology presents an evolutionary problem. The 'neutralist' solution to this problem, that the variation is usually unimportant to the organism displaying it, has now lost much of its strength. Interpretations that assume widespread heterozygous advantage also face serious difficulties. A resolution is possible in terms of frequency-dependent selection by predators, parasites and competitors. The evidence for pervasive frequency-dependent selection is now very strong. It appears to follow naturally from the behaviour of predators, from the evolutionary lability of parasites, from the ecology of competition and, at the molecular level, from the phenomena of enzyme kinetics. Such selection can explain the maintenance not only of conventional polymorphism but also of continuous variation in both molecular and morphological characters. It can account for the occurrence of diversity within groups of haploid and self-fertilizing organisms, and for the evolution of differences between individuals in their systems of genetic control.     

Preferential mating in symmetric multilocus systems: stability conditions of the central equilibrium

Several multilocus models that incorporate both preferential mating and viability selection are studied. Specifically, a class of symmetric heterozygosity models are considered that assign individuals to phenotypic classes according to which loci are in heterozygous state regardless of the actual allelic content. Otherwise, an arbitrary number of loci, number of alleles per locus, and arbitrary recombination scheme, viability parameters and preferential mating pattern based on phenotypes are allowed. The conditions for the stability of a central polymorphism are indicated and interpreted. The effects of viability and preference selection may be summarized in a single quantity for each phenotypic class, a generalized fitness. Preferential assortative mating alone can produce stability for a central polymorphism as in the case of viability selection when sexual attractiveness or general fitness increases with higher levels of heterozygosity. The situation is more complex with sexual selection.     

Some circumstances assuring monomorphism in subdivided populations

It is well known that in a subdivided population subject to soft selection with two alleles at one locus, instability of both fixation states (a “protected polymorphism”) entails at least one stable polymorphic equilibrium. Although stable polymorphic and monomorphic equilibria can coexist in general, a stable fixation state (monomorphic equilibrium) precludes the existence of any polymorphic equilibrium under the circumstances of haploid or submultiplicative diploid viabilities. This provides that a stable monomorphism is robust against random fluctuations in allele frequencies. It also increases the known circumstances where there is a unique globally attracting stable equilibrium, i.e., where allele frequencies are determined by the selection-migration structure independent of the history of the system.     

The statistical inevitability of stability-diversity relationships in community ecology

In this article, we explain an often overlooked process that may significantly contribute to positive correlations between measures of species diversity and community stability. Empirical studies showing positive stability-diversity relationships have, for the most part, used a single class of stability (or, more accurately, instability) measures: the temporal variation in aggregate community properties such as biomass or productivity. We show that for these measures, stability will essentially always rise with species diversity because of the statistical averaging of the fluctuations in species' abundances. This simple probabilistic process will operate in the absence of any strong species interactions, although its strength is driven by the relative abundances of species, as well as by the existence of positive or negative correlations in the fluctuations of species. To explore the possible importance of this effect in real communities, we fit a simple simulation model to Tilman's grassland community. Our results indicate that statistical averaging might play a substantial role in explaining stability-diversity correlations for this and other systems. Models of statistical averaging can serve as a useful baseline for predictions of community stability, to which the influences of both negative and positive species interactions may then be added and tested.     

Quantitative variability and multilocus polymorphism under epistatic selection

We study multilocus polymorphism under selection, using a class of fitness functions that account for additive, dominant, and pairwise additive-by-additive epistatic interactions. The dynamic equations are derived in terms of allele frequencies and disequilibria, using the notions of marginal systems and marginal fitnesses, without any approximations. Stationary values of allele frequencies and pairwise disequilibria under weak selection are calculated by regular perturbation techniques. We derive conditions for existence and stability of the multilocus polymorphic states. Using these results, we then analyze a number of models describing stabilizing selection on additive characters, with some other factors, and determine the conditions under which genetic quantitative variability is maintained.     

ADAPTIVE DYNAMICS IN ALLELE SPACE: EVOLUTION OF GENETIC POLYMORPHISM BY SMALL MUTATIONS IN A HETEROGENEOUS ENVIRONMENT

We demonstrate how a genetic polymorphism of distinctly different alleles can develop during long-term frequency-dependent evolution in an initially monomorphic diploid population, if mutations have only small phenotypic effect. As a specific example, we use a version of Levene's (1953) soft selection model, where stabilizing selection acts on a continuous trait within each of two habitats. If the optimal phenotypes within the habitats are sufficiently different, then two distinctly different alleles evolve gradually from a single ancestral allele. In a wide range of parameter values, the two locally optimal phenotypes will be realized by one of the homozygotes and the heterozygote, rather than by the two homozygotes. Unlike in the haploid analogue of the model, there can be multiple polymorphic evolutionary attractors with different probabilities of convergence. Our results differ from the population genetic models of short-term evolution in two aspects: (1) a polymorphism that is population genetically stable may be invaded by a new mutant allele and, as a consequence, the population may fall back to monomorphism, (2) long-term evolution by allele substitutions may lead from a population where polymorphism is not possible into one where polymorphism is possible.     

Evolution in fine-grained environments. II. Habitat selection as a homeostatic mechanism

A model of genotype specific habitat selection is developed for an organism subject to within-lifetime environmental fluctuations. Habitat selection is first overlaid upon both hard and soft selection Levene models with either discrete or continuous habitats. It is shown that even if all genotypes have identical physiological and fitness responses within a habitat, habitat selection can still maintain a polymorphism. In other words, physiological divergence is not a necessary prerequisite for divergence in habitat preferences. Within-lifetime environmental variability is then assumed to occur within each chosen habitat. It is shown that habitat selection acts as an evolutionary filter that can enhance the fitness impact of some niches and effectively eliminate the impact of others such that it generally increases the chances for a polymorphism under soft selection. However, density-dependent effects obscure the relationship between physiological fitness and evolutionary outcome. Indeed, it is possible for selection to favor an allele causing its bearers to preferentially go to the niche to which they are least physiologically adapted. Hence, changes in habitat preference can evolve before an organism has completely adapted physiologically to a new habitat. The fitness impact of habitat selection interacts with both homeostatic avoidance mechanisms (i.e., short-term buffering) and with tolerance (long-term) mechanisms. In general, habitat selection will be most favored in those organisms deficient in long-term tolerance. Moreover, habitat selection tends to accentuate selection favoring short-term avoidance mechanisms. Thus, organisms displaying much habitat selection should have poor physiological long-term tolerances but effective physiological short-term avoidance mechanisms. Finally, if the fitness costs associated with habitat selection are too large to be ignored and are comparable for all genotypes, habitat selection directs the selective pressures back onto the physiological homeostatic capabilities. Hence, the very existence and extent of habitat selection depends critically upon the physiological capabilities of the organism.     

On the coevolution of pathogen and host: II. Selfing hosts and haploid pathogens

The theory of discrete time coevolution is applied to the problem of maintenance of genetic polymorphism with selfing hosts and haploid pathogens. It is shown that the usual simplifying assumption, discrete synchroized generations with no intraspecific frequency-dependent selection, precludes stability. This situation is not corrected by the incorporation of special features such as mutation, alternate hosts, partial outcrossing of the hosts, or genetic recombination in the pathogen population.     

Selection in diplo-haplonts

A deterministic selection model is considered for diplohaplontic populations which reproduce in non-overlapping generations and form zygotes by random fusion of gametes. The model allows for vegetative propagation as well as for viability and fertility selection in both the haploid and diploid phases. Fertility selection in the haploid phases is permitted to differ according to sex, and all selection coefficients are assumed to be constant over the generations. The main result obtained is that this model, including all of the above selective forces, is formally and analytically equivalent to and thus shares all of the properties of the “classical” viability selection model. This result is essentially due to the fact that when haplogenotypic frequencies in the sporophytes are considered in consecutive generations, all of the different selection forces can be expressed with the help of a single selection coefficient for each diplogenotype. The benefits from this simplification are demonstrated with the help of two examples for a single, multiallelic locus. Restricting selection to the haplophases only, it is shown that fertility selection acting in opposing directions in the sexes can lead to stable multiallelic polymorphisms without assuming spatial heterogeneity of the environment. However, it is unlikely that the conditions for this will be realized in actual populations for more than two alleles. The second example is concerned with the problem of maintaining several sexual types (males, females and bisexuals) in a population, with special emphasis on the conditions for the establishment of dioecious or monoecious systems of sexuality or a mixture of these.     

Simulating the maintenance of a rare fish morph experiencing negative frequency dependent selection

Empirical work assessing the maintenance of rare genotypes in natural populations is difficult over very long time scales. Skirting this problematic issue is possible with theory and simulations. Major theoretical constructs, including mutation-selection balance and balancing selection, explain the theoretical maintenance of rare genotypes, and the occurrence of multiple, rare genotypes over time. Additionally, numerical simulations are valuable tools for assessing evolving biological systems because they allow for monitoring systems over long time scales, as well as for controlling model parameters, thus contributing to the exploration of system dynamics that cannot be assessed in nature. Here we employed numerical simulations to explore the importance of several biological factors that contribute to the maintenance of a fish color-pattern polymorphism. We present a numerical model of a two-morph fish polymorphism that allowed us to test the sensitivity of the rare morph's persistence and the population's stability to multiple parameters. Our simulations ran over 10,000 years (where one year is approximately one generation) and demonstrated the maintenance of a stable polymorphism with a rare morph which persisted at a frequency of ∼10⁻², which is in-fact the frequency of the rare, mottled black mosquitofish morph in natural populations. This pigmentation polymorphism is stable, independent of changes in population size, but can be destabilized with very high predation when coupled with very low birth rates. Employing models with empirical fitness estimates is a valuable tool for monitoring rare vertebrate morphs in nature, however few studies exist that have accomplished this task. Our approach can be adapted for modeling rare morphs (particularly in additional live-bearing fishes like sailfin mollies) that also harbor rare, pigmentation morphs within large populations.     

Roles of maternal effects in maintaining genetic variation: Maternal storage effect

Understanding the maintenance of genetic variation remains a central challenge in evolutionary biology. Recent empirical studies suggest the importance of temporally varying selection, as allele frequencies have been found to fluctuate substantially in the wild. However, previous theory suggests that the conditions for the maintenance of genetic variation under temporally fluctuating selection are quite restrictive. Using mathematical models, we demonstrate that maternal genetic effects, whereby maternal genotypes affect offspring phenotypes, can facilitate the maintenance of polymorphism in temporally varying environments. Maternal effects result in mismatches between genotypes and phenotypes, thereby buffering the influence of selection on allele frequency. This decreases the magnitude of allele‐frequency fluctuations and creates conditions for the maintenance of variation when selection causes fluctuations. Therefore, maternal effects may result in a temporal storage effect (“maternal storage effect”). On the other hand, when selection does not cause fluctuations (e.g., linear negative frequency‐dependent selection), maternal genetic effects moderate the relative importance of selection compared to genetic drift and promote stochastic allele extinction in finite populations. Thus, maternal effects can play an important role in the maintenance of polymorphism, but the direction of the effect depends on the nature of selection.