A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait

The equilibrium structure of an additive, diallelic multilocus model of a quantitative trait under frequency- and density-dependent selection is derived. The trait is under stabilizing selection and mediates intraspecific competition as induced, for instance, by differential resource utilization. It is assumed that stabilizing selection is weak, but the strength of competition may be arbitrary relative to it. Density dependence is caused by population regulation, which may be of a very general kind. The number and effects of loci are arbitrary, and stabilizing selection is not necessarily symmetric with respect to the range of phenotypic values. All previously studied models of intraspecific competition for a continuum of resources known to the author reduce to a special case of the present model if overall selection is weak. Therefore, in this case our results are applicable as approximations to all these models. Our central result is the (nearly) complete characterization of the equilibrium and stability structure in terms of all parameters. It is derived under the sole assumption that selection is weak enough relative to recombination to ignore linkage disequilibrium. In particular, necessary and sufficient conditions on the strength of competition relative to stabilizing selection are found that ensure the maintenance of multilocus polymorphism and the occurrence of disruptive selection. In this case, explicit formulas for the number of polymorphic loci at equilibrium, the allele frequencies, the genetic variance, and the strength of disruptive selection are obtained. For two loci, the effects of linkage are investigated analytically; for several loci, they are studied numerically.     

Long-term evolution of polygenic traits under frequency-dependent intraspecific competition

We analytically investigate the long-term evolution of a continuously varying quantitative character in a diploid population that is determined additively by a finite number of loci. The trait is under a mixture of frequency-dependent disruptive selection induced by intraspecific competition and frequency-independent stabilizing selection. Moreover, the trait is restricted to a finite range by constraints on the particular loci. Our investigations are based on explicit analytical results (provided by Bürger [2005. A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait. J. Math. Biol. 50, 355-396]; Schneider [2006. A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition. J. Math. Biol. 52, 483-523]) on the short-term dynamics under the assumption of linkage equilibrium. We show that the population always reaches a long-term equilibrium (LTE), i.e., an equilibrium that is resistant against perturbations of mutations of sufficiently small effect. In general, several LTEs can coexist. They can be calculated explicitly, and we provide necessary and sufficient conditions for their existence. In the case that more than one LTE exists, we exemplify numerically that the evolutionary outcome depends crucially on the initial genetic architecture, on the joint distribution of mutational effects across loci, and on the particular realization of the mutation process. Therefore, long-term evolution cannot be predicted from the ecology alone. We further show that a partial order exists for the LTEs. The set of LTEs has a 'largest' element, an LTE which is reached during long-term evolution if the effects of the occurring mutant alleles are sufficiently large.     

The conditions for speciation through intraspecific competition

Abstract It has been shown theoretically that sympatric speciation can occur if intraspecific competition is strong enough to induce disruptive selection. However, the plausibility of the involved processes is under debate, and many questions on the conditions for speciation remain unresolved. For instance, is strong disruptive selection sufficient for speciation? Which roles do genetic architecture and initial composition of the population play? How strong must assortative mating be before a population can split in two? These are some of the issues we address here. We investigate a diploid multilocus model of a quantitative trait that is under frequency‐dependent selection caused by a balance of intraspecific competition and frequency‐independent stabilizing selection. This trait also acts as mating character for assortment. It has been established previously that speciation can occur only if competition is strong enough to induce disruptive selection. We find that speciation becomes more difficult for very strong competition, because then extremely strong assortment is required. Thus, speciation is most likely for intermediate strengths of competition, where it requires strong, but not extremely strong, assortment. For this range of parameters, however, it is not obvious how assortment can evolve from low to high levels, because with moderately strong assortment less genetic variation is maintained than under weak or strong assortment sometimes none at all. In addition to the strength of frequency‐dependent competition and assortative mating, the roles of the number of loci, the distribution of allelic effects, the initial conditions, costs to being choosy, the strength of stabilizing selection, and the particular choice of the fitness function are explored. A multitude of possible evolutionary outcomes is observed, including loss of all genetic variation, splitting in two to five species, as well as very short and extremely long stable limit cycles. On the methodological side, we propose quantitative measures for deciding whether a given distribution reflects two (or more) reproductively isolated clusters.     

EVOLUTION OF DOMINANCE UNDER FREQUENCY-DEPENDENT INTRASPECIFIC COMPETITION IN AN ASSORTATIVELY MATING POPULATION

We study the evolution of higher levels of dominance as a response to negative frequency-dependent selection. In contrast to previous studies, we focus on the effect of assortative mating on the evolution of dominance under frequency-dependent intraspecific competition. We analyze a two-locus two-allele model, in which the primary locus has a major effect on a quantitative trait that is under a mixture of frequency-independent stabilizing selection, density-dependent selection, and frequency-dependent selection caused by intraspecific competition for a continuum of resources. The second (modifier) locus determines the degree of dominance at the trait level. Additionally, the population mates assortatively with respect to similarities in the ecological trait. Our analysis shows that the parameter region in which dominance can be established decreases if small levels of assortment are introduced. In addition, the degree of dominance that can be established also decreases. In contrast, if assortment is intermediate, sexual selection for extreme types can be established, which leads to evolution of higher levels of dominance than under random mating. For modifiers with large effects, intermediate levels of assortative mating are most favorable for the evolution of dominance. For large modifiers, the speed of fixation can even be higher for intermediate levels of assortative mating than for random mating.     

Competition can maintain genetic but not environmental variance in the presence of stabilizing selection

A population in which there is stabilizing selection acting on quantitative traits toward an intermediate optimum becomes monomorphic in the absence of mutation. Further, genotypes that show least environmental variation are also favored, such that selection is likely to reduce both genetic and environmental components of phenotypic variance. In contrast, intraspecific competition for resources is more severe between phenotypically similar individuals, such that those deviating from prevailing phenotypes have a selective advantage. It has been shown previously that polymorphism and phenotypic variance can be maintained if competition between individuals is "effectively" stronger than stabilizing selection. Environmental variance is generally observed in quantitative traits, so mechanisms to explain its maintenance are sought, but the impact of competition on its magnitude has not previously been studied. Here we assume that a quantitative trait is subject to selection for an optimal value and to selection due to competition. Further, we assume that both the mean and variance of the phenotypic value depend on genotype, such that both may be affected by selection. Theoretical analysis and numerical simulations reveal that environmental variance can be maintained only when the genetic variance (in mean phenotypic value) is constrained to a very low level. Environmental variance will be replaced entirely by genotypic variance if a range of genotypes that vary widely in mean phenotype are present or become so by mutation. The distribution of mean phenotypic values is discrete when competition is strong relative to stabilizing selection; but more genotypes segregate and the distribution can approach continuity as competition becomes extremely strong. If the magnitude of the environmental variance is not under genetic control, there is a complementary relationship between the levels of environmental and genetic variance such that the level of phenotypic variance is little affected.     

Does competitive divergence occur if assortative mating is costly?

Most models of sympatric speciation have assumed that assortative mating has no costs. A few studies, however, have shown that the costs for being choosy can prevent such speciation. Here, we investigate the role of the strength of assortment and of the costs for being choosy for a simple genetic model of a single ('magic') trait that mediates both intraspecific competition for a continuum of resources and assortative mating, which is induced by choosy females who preferentially mate with males of similar phenotype. Choosiness may be costly if it is difficult to find a mating partner. Such magic trait models are considered to be most conducive of sympatric speciation. We consider a sexually reproducing population of haploid individuals that is density regulated. The trait is determined by a single locus with multiple alleles. The strength of stabilizing selection (caused by a unimodal resource distribution), the strength of competition, the degree of assortment and the costs for being choosy are independent parameters. We investigate analytically and numerically how these parameters determine the equilibrium and stability structure. In particular, we identify conditions under which no polymorphism at all is maintained as well as conditions under which strong competitive divergence occurs, or the population even splits into two reproductively isolated classes of highly diverse phenotypes. If costs are absent or moderate, genetic variability tends to be minimized at intermediate strengths of assortment, and reproductively isolated classes of phenotypes are a likely result of evolution only for intermediate or strong competition and for very strong assortment. The likelihood of divergence depends relatively weakly on the costs as long as they are not high. With high costs, however, increasingly strong assortment rapidly depletes all genetic variation, and strong competitive divergence is prevented.     

Sexually selected dichromatism in the hihi Notiomystis cincta: multiple colours for multiple receivers

Why do some bird species show dramatic sexual dichromatism in their plumage? Sexual selection is the most common answer to this question. However, other competing explanations mean it is unwise to assume that all sexual dichromatism has evolved by this mechanism. Even if sexual selection is involved, further work is necessary to determine whether dichromatism results from competition amongst rival males, or by female choice for attractive traits, or both. Here, we test whether sexually dichromatic hihi (Notiomystis cincta) plumage is currently under sexual selection, with detailed behavioural and genetic analyses of a free‐living island population. Bateman gradients measured for males and females reveal the potential for sexual selection, whilst selection gradients, relating reproductive success to specific colourful traits, show that there is stabilizing selection on white ear tuft length in males. By correlating colourful male plumage with different components of reproductive success, we show that properties of yellow plumage are most likely a product of male–male competition, whilst properties of the black and white plumage are an outcome of both male–male competition and female choice. Male plumage therefore potentially signals to multiple receivers (rival males and potential mates), and this may explain the multicoloured appearance of one of the most strikingly dichromatic species in New Zealand.     

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.     

Competition and evolution along environmental gradients: patterns, boundaries and sympatric divergence

The present study examined how competitive interactions and environmental conditions generate species boundaries and determine species distributions. A spatially explicit, quantitative genetic, two-species competition model was used to manipulate the strengths of competition, gene flow and local adaptation along environmental gradients. This allowed us to assess the long-term persistence of each species and whether the ranges they inhabited had boundaries in space or were unlimited. We found that a species boundary arises along less steep environmental gradients when the strength of stabilizing selection and diversifying selection are similar. We also found that a species boundary may arise along shallow environmental gradients if interspecific competition is more intense than intraspecific, which relaxes previous requirements for steep gradients for generating range limits. We determined an analytical form for the critical environmental gradient as a function of ecological and genetic parameters at which a species boundary is expected to arise by competition. Results suggest an alternative to resource competition as an explanation for phenotypic divergence between sympatric competitors. Competitors sharing a trait that is under stabilizing selection along an environmental gradient may segregate spatially and evolve in different regions, with phenotypic sympatric divergence reflecting the resulting clines. Along various types of environmental gradients, variation in stabilizing selection intensities could lead to contrasting patterns in the distribution of species. For stabilizing selection strengths in accord with field data estimates, this study predicts that the level of sympatric character divergence would be limited along environmental gradients.     

Optimization under frequency-dependent selection

We consider a model of frequency-dependent selection, which we refer to as the Wildcard Model. A variety of more specific models, representing quite diverse biological situations, are covered by the Wildcard Model as particular cases. Two very different particular models that are subsumed by the Wildcard Model are the game theoretically motivated two-phenotype model of Lessard [Lessard, S.,1984. Evolutionary dynamics in frequency-dependent two-phenotype models, Theor. Popul. Biol. 25, 210-234], and the model of selection on a continuous trait due to intraspecific competition of Bürger [Bürger, R., 2005. A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait. J. Math. Biol. 50 (4), 355-396] and Schneider [Schneider, K.A., 2006. A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition. J. Math. Biol. 52 (4), 483-523]. Both these models have been shown in the past to have a global Lyapunov function (LF) under appropriate genetic assumptions. We show that (i) the Wildcard Model in continuous time for a single multiallelic locus, or for multiple multiallelic loci in linkage equilibrium, has a global LF, of which the Lessard and Bürger-Scheneider LF are special cases in spite of their widely different biological interpretations; (ii) the LF of the Wildcard Model can be derived from an LF previously identified for a model of density- and frequency-dependent selection due to Lotka-Volterra competition, with one locus, multiple alleles, multiple species and continuous-time dynamics [Matessi, C., Jayakar, S.D., 1981. Coevolution of species in competition: A theoretical study. Proc. Natl. Acad. Sci. USA, 78 (2, part2), 1081-1084]. We extend the LF with density and frequency dependence to the multilocus case with linkage-equilibrium dynamics. As a possible application of our results, the optimization principle we established can be used as a tool in the study of long-term evolution of various models subsumed by the Wildcard Model based on explicit short-term dynamics.     

Maximization principles for frequency-dependent selection I: the one-locus two-allele case

In this article we study the one-locus two-allele version of the pairwise-interaction model of frequency-dependent selection in discrete and continuous time. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles. We provide a systematic approach that covers all possible facets of the dynamical behavior of the model, and we illustrate our results by concrete examples. We show that the mean fitness of the population is nondecreasing if the interaction coefficients are symmetric and positive. Moreover, monotonic convergence to the set of equilibria always occurs, which is not true if we also consider negative interaction coefficients. For asymmetric interaction, we provide necessary conditions when the mean fitness is nondecreasing and sufficient conditions when it is not. Furthermore, in discrete time, we show that limit cycles cannot occur, unless some interaction coefficients are negative.     

The effects of intraspecific competition and stabilizing selection on a polygenic trait

The equilibrium properties of an additive multilocus model of a quantitative trait under frequency- and density-dependent selection are investigated. Two opposing evolutionary forces are assumed to act: (i) stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and (ii) 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 equilibrium structure, in particular, number, degree of polymorphism, and genetic variance of stable equilibria, is affected by the strength of frequency dependence, and what role the number of loci, the amount of recombination, and the demographic parameters play. To this end, we employ a statistical and numerical approach, complemented by analytical results, and explore how the equilibrium properties averaged over a large number of genetic systems with a given number of loci and average amount of recombination depend on the ecological and demographic parameters. We identify two parameter regions with a transitory region in between, in which the equilibrium properties of genetic systems are distinctively different. These regions depend on the strength of frequency dependence relative to pure stabilizing selection and on the demographic parameters, but not on the number of loci or the amount of recombination. We further study the shape of the fitness function observed at equilibrium and the extent to which the dynamics in this model are adaptive, and we present examples of equilibrium distributions of genotypic values under strong frequency dependence. Consequences for the maintenance of genetic variation, the detection of disruptive selection, and models of sympatric speciation are discussed.     

Field and experimental evidence for competition's role in phenotypic divergence

Resource competition has long been viewed as a major cause of phenotypic divergence within and between species. Theory predicts that divergence arises because natural selection favors individuals that are phenotypically dissimilar from their competitors. Yet, there are few conclusive tests of this key prediction. Drawing on data from both natural populations and a controlled experiment, this paper presents such a test in tadpoles of two species of spadefoot toads (Spea bombifrons and S. multiplicata). These two species show exaggerated divergence in trophic morphology where they are found together (mixed-species ponds) but not where each is found alone (pure-species ponds), suggesting that they have undergone ecological character displacement. Moreover, in pure-species ponds, both species exhibit resource polymorphism. Using body size as a proxy for fitness, we found that in pure-species ponds disruptive selection favors extreme trophic phenotypes in both species, suggesting that intraspecific competition for food promotes resource polymorphism. In mixed-species ponds, by contrast, we found that trophic morphology was subject to stabilizing selection in S. multiplicata and directional selection in S. bombifrons. A controlled experiment revealed that the more similar an S. multiplicata was to its S. bombifrons tankmate in resource use, the worse was its performance. These results indicate that S. multiplicata individuals that differ from S. bombifrons would be selectively favored in competition. Our data therefore demonstrate how resource competition between phenotypically similar individuals can drive divergence between them. Moreover, our results indicate that how competition contributes to such divergence may be influenced not only by the degree to which competitors overlap in resource use, but also by the abundance and quality of resources. Finally, our finding that competitively mediated disruptive selection may promote resource polymorphism has potentially important implications for understanding how populations evolve in response to heterospecific competitors. In particular, once a population evolves resource polymorphism, it may be more prone to undergo ecological character displacement.     

Intraspecific variation in sperm length is negatively related to sperm competition in passerine birds

Spermatozoa are among the most diversified cells in the animal kingdom, but the underlying evolutionary forces affecting intraspecific variation in sperm morphology are poorly understood. It has been hypothesized that sperm competition is a potent selection pressure on sperm variation within species. Here, we examine intraspecific variation in total sperm length of 22 wild passerine bird species (21 genera, 11 families) in relation to the risk of sperm competition, as expressed by the frequency of extrapair paternity and relative testis size. We demonstrate, by using phylogenetic comparative methods, that between-male variation in sperm length within species is closely and negatively linked to the risk of sperm competition. This relationship was even stronger when only considering species in which data on sperm length and extrapair paternity originated from the same populations. Intramale variation in sperm length within species was also negatively, although nonsignificantly, related to sperm competition risk. Our findings suggest that postcopulatory sexual selection is a powerful evolutionary force reducing the intraspecific phenotypic variation in sperm-size traits, potentially driving the diversification of sperm morphology across populations and species.     

Gene duplications contribute to the overrepresentation of interactions between proteins of a similar age

Background

Disruptive selection has been documented in a growing number of natural populations. Yet, its prevalence within individual systems remains unclear. Furthermore, few studies have sought to identify the ecological factors that promote disruptive selection in the wild. To address these issues, we surveyed 15 populations of Mexican spadefoot toad tadpoles, Spea multiplicata, and measured the prevalence of disruptive selection acting on resource-use phenotypes. We also evaluated the relationship between the strength of disruptive selection and the intensity of intraspecific competition??an ecological agent hypothesized to be an important driver of disruptive selection.

Results

Disruptive selection was the predominant mode of quadratic selection across all populations. However, a directional component of selection favoring an extreme ecomorph??a distinctive carnivore morph??was also common. Disruptive selection was strongest in populations experiencing the most intense intraspecific competition, whereas stabilizing selection was only found in populations experiencing relatively weak intraspecific competition.

Conclusions

Disruptive selection can be common in natural populations. Intraspecific competition for resources may be a key driver of such selection.     

Additive genetic variation under intraspecific competition and stabilizing selection: a two-locus study

A diallelic two-locus model is investigated in which the loci determine the genotypic value of a quantitative trait additively. Fitness has two components: stabilizing selection on the trait and a frequency-dependent component, as induced, for instance, if the ability to utilize different food resources depends on this trait. Since intraspecific competition induces disruptive selection, this model leads to a conflict of selective forces. We study how the underlying genetics (recombination rate and allelic effects) interacts with the selective forces, and explore the resulting equilibrium structure. For the special case of equal effects, global stability results are proved. Unless the locus effects are sufficiently different, the genetic variance maintained at equilibrium displays a threshold-like dependence on the strength of competition. For loci with equal effects, the equilibrium fitnesses of genotypic values exhibit disruptive selection if and only if competition is strong enough to maintain a stable two-locus polymorphism. For unequal effects, disruptive selection can be observed for weaker competition and in the absence of a stable polymorphism.     

Polyandry reduces sperm length variation in social insects

Postcopulatory sexual selection, either in the form of sperm competition or cryptic female choice, is an important selective force that is thought to have generated the enormous variation in sperm morphology observed interspecifically. However, the evolutionary significance of intraspecific variation in sperm morphology, and the role that postcopulatory sexual selection plays in influencing this variation, remains poorly investigated in invertebrates. Here, we tested the hypothesis that postcopulatory sexual selection reduces variation in sperm morphology, both between and within males, in 27 species of eusocial ants and bees. These eusocial species offer an unusual opportunity to assess how selection acts on variance in sperm morphology, as haploid males produce clonal, haploid sperm that does not experience haploid-diploid conflict. We provide solid evidence that males of polyandrous ant and bee species indeed produce less-variable sperm, indicating that sperm competition selected for sperm of superior quality. Our results offer a mechanistic explanation for the evolution of high-quality sperm and provide comprehensive evidence that sperm morphology of social insects is influenced by sexual selection.     

Experimental evidence that competition and habitat use shape the individual fitness surface

A key prediction made by theories of density‐dependent competition is that resource overlap should increase the intensity of competition. By extension, we can predict that competition should lead to density‐dependent natural selection. I studied natural selection on limb length and body size in a total of seven populations of Anolis sagrei over 3 years in the Bahamas. Experimental manipulations of population density on small off‐shore cays revealed that the strength of natural selection on body size increased with density, suggesting that density‐dependent intraspecific competition drives natural selection. At low density, reduced competition revealed significant selection on limb length driven by changes in perch diameter, indicating that selection favoured a match between morphology and habitat. The role habitat played in shaping selection was further illuminated by inter‐annual changes in vegetation structure stemming from variation in precipitation among years. Thus, changes in both the intensity of competition across spatial replicates, and in resource availability through time, revealed changes in the targets of natural selection. Results provide empirical support for the long‐standing hypothesis that density‐dependent natural selection shapes the fitness surface of Greater Antilles anoles.     

Species coexistence under resource competition with intraspecific and interspecific direct competition in a chemostat

Competition theory has developed separately for direct competition and for exploitative competition. However, the combined effects of the two types of competition on species coexistence remain unclear. To examine how intraspecific and interspecific direct competition contributes to the coexistence of species competing for a single resource, we constructed a chemostat-type resource competition model. With general functions for intraspecific and interspecific direct competition, we derived necessary and sufficient conditions (except for a critical case that rarely occurs in a biological sense) that determine the number of stably coexisting species. From these conditions, we found that the number of coexisting species is determined just by the invasibility of each species into subcommunities with a smaller number of species. In addition, using a combination of rigorous mathematical theory and a simple graphical method, we can demonstrate how the stronger intraspecific direct competition facilitates species invasion, leading to a larger number of coexisting species.