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The "Machiavellian intelligence" hypothesis (or the "social brain" hypothesis) posits that large brains and distinctive cognitive abilities of humans have evolved via intense social competition in which social competitors developed increasingly sophisticated "Machiavellian" strategies as a means to achieve higher social and reproductive success. Here we build a mathematical model aiming to explore this hypothesis. In the model, genes control brains which invent and learn strategies (memes) which are used by males to gain advantage in competition for mates. We show that the dynamics of intelligence has three distinct phases. During the dormant phase only newly invented memes are present in the population. During the cognitive explosion phase the population's meme count and the learning ability, cerebral capacity (controlling the number of different memes that the brain can learn and use), and Machiavellian fitness of individuals increase in a runaway fashion. During the saturation phase natural selection resulting from the costs of having large brains checks further increases in cognitive abilities. Overall, our results suggest that the mechanisms underlying the "Machiavellian intelligence" hypothesis can indeed result in the evolution of significant cognitive abilities on the time scale of 10 to 20 thousand generations. We show that cerebral capacity evolves faster and to a larger degree than learning ability. Our model suggests that there may be a tendency toward a reduction in cognitive abilities (driven by the costs of having a large brain) as the reproductive advantage of having a large brain decreases and the exposure to memes increases in modern societies.

Females of many species are frequently courted by promiscuous males of their own and other closely related species. Such mating interactions may impose strong selection on female mating preferences to favor trait values in conspecific males that allow females to discriminate them from their heterospecific rivals. We explore the consequences of such selection in models of the evolution of female mating preferences when females must interact with heterospecific males from which they are completely postreproductively isolated. Specifically, we allow the values of both the most preferred male trait and the tolerance of females for males that deviate from this most preferred trait to evolve. Also, we consider situations in which females base their mating decisions on multiple male traits and must interact with males of multiple species. Females will rapidly differentiate in preference when they sometimes mistake heterospecific males for suitable mates, and the differentiation of female preference will select for conspecific male traits to differentiate as well. In most circumstances, this differentiation continues indefinitely, but slows substantially once females are differentiated enough to make mistakes rare. Populations of females with broader preference functions (i.e., broader tolerance for males with trait values that deviate from females' most preferred values) will evolve further to differentiate if the shape of the function cannot evolve. Also, the magnitude of separation that evolves is larger and achieved faster when conspecific males have lower relative abundance. The direction of differentiation is also very sensitive to initial conditions if females base their mate choices on multiple male traits. We discuss how these selection pressures on female mate choice may lead to speciation by generating differentiation among populations of a progenitor species that experiences different assemblages of heterospecifics. Opportunities for differentiation increase as the number of traits involved in mate choice increase and as the number of species involved increases. We suggest that this mode of speciation may have been particularly prevalent in response to the cycles of climatic change throughout the Quaternary that forced the assembly and disassembly of entire communities on a continentwide basis.

We develop and analyze an explicit multilocus genetic model of coevolution. We assume that interactions between two species (mutualists, competitors, or victim and exploiter) are mediated by a pair of additive quantitative traits that are also subject to direct stabilizing selection toward intermediate optima. Using a weak-selection approximation, we derive analytical results for a symmetric case with equal locus effects and no mutation, and we complement these results by numerical simulations of more general cases. We show that mutualistic and competitive interactions always result in coevolution toward a stable equilibrium with no more than one polymorphic locus per species. Victim-exploiter interactions can lead to different dynamic regimes including evolution toward stable equilibria, cycles, and chaos. At equilibrium, the victim is often characterized by a very large genetic variance, whereas the exploiter is polymorphic in no more than one locus. Compared to related one-locus or quantitative genetic models, the multilocus model exhibits two major new properties. First, the equilibrium structure is considerably more complex. We derive detailed conditions for the existence and stability of various classes of equilibria and demonstrate the possibility of multiple simultaneously stable states. Second, the genetic variances change dynamically, which in turn significantly affects the dynamics of the mean trait values. In particular, the dynamics tend to be destabilized by an increase in the number of loci.

In female heterogamous (ZW) species, sex-linked genes coding for maternal products that are packaged into the egg open a unique arena for genetic conflict that does not occur in male heterogamous (XY) species. Z-linked maternal-effect alleles that help sons and harm daughters are expected to go to fixation, as are W-linked alleles that help daughters and harm sons. This conflict differs from known cases of meiotic drive, because sex-specific ontogeny, physiology, and gene expression greatly simplify the genetic interactions that lead to sexual conflict. Selection on maternal-effect genes may substantially alter the evolution of ZW compared with XY systems.

The paper entitled "Sympatric speciation," which was published by John Maynard Smith in 1966, initiated the development of mathematical models aiming to identify the conditions for sympatric speciation. A part of that paper was devoted to a specific two-locus, two-allele model of sympatric speciation in a population occupying a two-niche system. Maynard Smith provided some initial numerical results on this model. Later, Dickinson and Antonovics (1973) and Caisse and Antonovics (1978) performed more extensive numerical studies on the model. Here, I report analytical results on the haploid version of the Maynard Smith model. I show how the conditions for sympatric and parapatric speciation and the levels of resulting genetic divergence and reproductive isolation are affected by the strength of disruptive selection and nonrandom mating, recombination rate, and the rates of male and female dispersal between the niches.

We consider mathematical models describing the evolutionary consequences of antagonistic interactions between male offence, male defence and female reproductive tract and physiology in controlling female mating rate. Overall, the models support previous verbal arguments about the possibility of continuous coevolutionary chase between the sexes driven by two-way (e.g. between male offence and female traits) and three-way (e.g. between male offence, male defence and female traits) inter-sexual antagonistic interactions. At the same time, the models clarify these arguments by identifying various additional potential evolutionary dynamics and important parameters (e.g. genetic variances, female optimum mating rates, strength of selection in females and the relative contributions of first and second males into offspring) and emphasizing the importance of initial conditions. Models also show that sexual conflict can result in the evolution of monandry in an initially polyandrous species and in the evolution of random mating in a population initially exhibiting non-random mating.

Adaptive radiation is defined as the evolution of ecological and phenotypic diversity within a rapidly multiplying lineage. When it occurs, adaptive radiation typically follows the colonization of a new environment or the establishment of a "key innovation," which opens new ecological niches and/or new paths for evolution. Here, we take advantage of recent developments in speciation theory and modern computing power to build and explore a large-scale, stochastic, spatially explicit, individual-based model of adaptive radiation driven by adaptation to multidimensional ecological niches. We are able to model evolutionary dynamics of populations with hundreds of thousands of sexual diploid individuals over a time span of 100,000 generations assuming realistic mutation rates and allowing for genetic variation in a large number of both selected and neutral loci. Our results provide theoretical support and explanation for a number of empirical patterns including "area effect," "overshooting effect," and "least action effect," as well as for the idea of a "porous genome." Our findings suggest that the genetic architecture of traits involved in the most spectacular radiations might be rather simple. We show that a great majority of speciation events are concentrated early in the phylogeny. Our results emphasize the importance of ecological opportunity and genetic constraints in controlling the dynamics of adaptive radiation.

Theoretical studies of speciation have been dominated by numerical simulations aiming to demonstrate that speciation in a certain scenario may occur. What is needed now is a shift in focus to identifying more general rules and patterns in the dynamics of speciation. The crucial step in achieving this goal is the development of simple and general dynamical models that can be studied not only numerically but analytically as well. I review some of the existing analytical results on speciation. I first show why the classical theories of speciation by peak shifts across adaptive valleys driven by random genetic drift run into trouble (and into what kind of trouble). Then I describe the Bateson-Dobzhansky-Muller (BDM) model of speciation that does not require overcoming selection. I describe exactly how the probability of speciation, the average waiting time to speciation, and the average duration of speciation depend on the mutation and migration rates, population size, and selection for local adaptation. The BDM model postulates a rather specific genetic architecture of reproductive isolation. I then show exactly why the genetic architecture required by the BDM model should be common in general. Next I consider the multilocus generalizations of the BDM model again concentrating on the qualitative characteristics of speciation such as the average waiting time to speciation and the average duration of speciation. Finally, I consider two models of sympatric speciation in which the conditions for sympatric speciation were found analytically. A number of important conclusions have emerged from analytical studies. Unless the population size is small and the adaptive valley is shallow, the waiting time to a stochastic transition between the adaptive peaks is extremely long. However, if transition does happen, it is very quick. Speciation can occur by mutation and random drift alone with no contribution from selection as different populations accumulate incompatible genes. The importance of mutations and drift in speciation is augmented by the general structure of adaptive landscapes. Speciation can be understood as the divergence along nearly neutral networks and holey adaptive landscapes (driven by mutation, drift, and selection for adaptation to a local biotic and/or abiotic environment) accompanied by the accumulation of reproductive isolation as a by-product. The waiting time to speciation driven by mutation and drift is typically very long. Selection for local adaptation (either acting directly on the loci underlying reproductive isolation via their pleiotropic effects or acting indirectly via establishing a genetic barrier to gene flow) can significantly decrease the waiting time to speciation. In the parapatric case the average actual duration of speciation is much shorter than the average waiting time to speciation. Speciation is expected to be triggered by changes in the environment. Once genetic changes underlying speciation start, they go to completion very rapidly. Sympatric speciation is possible if disruptive selection and/or assortativeness in mating are strong enough. Sympatric speciation is promoted if costs of being choosy are small (or absent) and if linkage between the loci experiencing disruptive selection and those controlling assortative mating is strong.

It is well established that sexual conflict can drive an endless coevolutionary chase between the sexes potentially leading to genetic divergence of isolated populations and allopatric speciation. We present a simple mathematical model that shows that sexual conflict over mating rate can result in two other general regimes. First, rather than "running away" from males, females can diversify genetically into separate groups, effectively "trapping" the males in the middle at a state characterized by reduced mating success. Female diversification brings coevolutionary chase to the end. Second, under certain conditions, males respond to female diversification by diversifying themselves. This response results in the formation of reproductively isolated clusters of genotypes that emerge sympatrically.

Although empirical evidence has shown that many male traits have evolved via sexual selection by female mate choice, our understanding of the adaptive value of female mating preferences is still very incomplete. It has recently been suggested that female mate choice may result from females evolving resistance rather than attraction to males, but this has been disputed. Here, we develop a quantitative genetic model showing that sexual conflict over mating indeed results in the joint evolution of costly female mate choice and exaggerated male traits under a wide range of circumstances. In contrast to tradition explanations of costly female mate choice, which rely on indirect genetic benefits, our model shows that mate choice can be generated as a side-effect of females evolving to reduce the direct costs of mating.

Using a weak migration and weak mutation approximation, I studied the average waiting time to parapatric speciation. The description of reproductive isolation used is based on the classical Dobzhansky model and its recently proposed multilocus generalizations. The dynamics of parapatric speciation are modelled as a biased random walk performed by the average genetic distance between the residents and immigrants. If a small number of genetic changes is sufficient for complete reproductive isolation, mutation and random genetic drift alone can cause speciation on the time-scale of ten to 1,000 times the inverse of the mutation rate over a set of loci underlying reproductive isolation. Even relatively weak selection for local adaptation can dramatically decrease the waiting time to speciation. The actual duration of the parapatric speciation process (that is the duration of intermediate forms in the actual transition to a state of complete reproductive isolation) is shorter by orders of magnitude than the overall waiting time to speciation. For a wide range of parameter values, the actual duration of parapatric speciation is of the order of one over the mutation rate. In general, parapatric speciation is expected to be triggered by changes in the environment.

We study genetic variation in phenotypic plasticity maintained by a balance between mutation and weak stabilizing selection. We consider linear reaction norms allowing for spatial and/or temporal variation in the environments of development and selection. We show that the overall genetic variation maintained does not depend on whether the trait is plastic or not. The genetic variances in height and slope of a linear reaction norm, and their covariance, are predicted to decrease with the variation in the environment. Non-pleiotropic loci influencing either height or slope are expected to decrease the genetic variance in slope relative to that in height. Decrease in the ratio of genetic variance in slope to genetic variance in height with increasing variation in the environment presents a test for the presence of loci that only influence the slope, and not the height. We use data on Drosophila to test the theory. In seven of eight pair-wise comparisons genetic variation in reaction norm is higher in a less variable environment than in a more variable environment, which is in accord with the model's predictions.

We develop a simple framework for modeling speciation and diversification as a continuous process of accumulation of genetic (or morphological) differences accompanied by species and subpopulation extinction and/or range expansion. This framework can be used to approach a number of questions such as species-area distribution, species-range size distribution, the rate of ecological turnover, asymmetries of range division between sister species, waiting time until speciation and extinction, the relationship between the geographic range size and the probability of speciation, the relationships between subpopulation-level parameters and metapopulation-level parameters, and the effects of taxonomic level on these rates, distributions, and parameters. We illustrate some of these applications using numerical simulations. We develop approximations describing the dependence of the number of different taxonomic units, their average range size, and the rate of their turnover on the system size, the rate of fixation of genetic (or morphological) changes in local demes, and the rate of local extinction and colonization.

Geographic variation may ultimately lead to the splitting of a subdivided population into reproductively isolated units in spite of migration. Here, we consider how the waiting time until the first split and its location depend on different evolutionary factors including mutation, migration, random genetic drift, genetic architecture, and the geometric structure of the habitat. We perform large-scale, individual-based simulations using a simple model of reproductive isolation based on a classical view that reproductive isolation evolves as a by-product of genetic divergence. We show that rapid parapatric speciation on the time scale of a few hundred to a few thousand generations is plausible even when neighboring subpopulations exchange several individuals each generation. Divergent selection for local adaptation is not required for rapid speciation. Our results substantiates the claims that species with smaller range sizes (which are characterized by smaller local densities and reduced dispersal ability) should have higher speciation rates. If mutation rate is small, local abundances are low, or substantial genetic changes are required for reproductive isolation, then central populations should be the place where most splits take place. With high mutation rates, high local densities, or with moderate genetic changes sufficient for reproductive isolation, speciation events are expected to involve mainly peripheral populations.

A growing amount of experimental data indicates extremely rapid evolution of traits and proteins related to fertilization in many diverging taxa. These data come from studies of sperm or pollen competition between closely related species, and from molecular studies of fertilization proteins. The positive selection for evolutionary novelty that appears to be acting on fertilization systems seems paradoxical because successful reproduction requires the close matching of female and male traits. It has been suggested that perpetual coevolution between the sexes can result from sexual conflict in mating. Sexual conflict occurs when characteristics that enhance the reproductive success of one sex reduce the fitness of the other sex. Numerous examples of sexual conflict resulting from sensory exploitation, polyspermy and the cost of mating have been discussed in detail. The potential for coevolution due to such conflict has been evaluated experimentally. Here I develop a simple mathematical model describing coevolutionary dynamics of male and female traits involved in reproduction. The model shows that continual change in such traits at a constant speed is expected whenever females (or eggs) experience fitness loss from having too many compatible males (or sperms). The plausibility of runaway coevolution increases with increasing population size. Rapid evolution of reproductive barriers driven by sexual conflict may explain increased speciation rates after colonization of new habitats ('adaptive radiation') and high species richness in resource-rich environments.

Metapopulation models and stepping-stone models in genetics are based on very different underlying dispersal structures, yet it can be difficult to distinguish the behaviour of the two kinds of models. We demonstrate a striking qualitative difference in the equilibrium behaviour possible with these two kinds of dispersal. If, in a local patch, there are multiple stable equilibria (and consequently an unstable equilibrium), we demonstrate that, for the spatial system with a metapopulation structure, at equilibrium every patch has to be near one of the stable equilibria. This contrasts with the clinal structure possible with a stepping-stone or continuous space model; thus the result can be used to deduce qualitative information about the form of dispersal from observations of allele frequencies.

A holey adaptive landscape is an adaptive landscape where relatively infrequent well-fit combinations of genes form a contiguous set that expands throughout the genotype space. I formulate and study a series of simple models describing the dynamics of speciation on holey adaptive landscapes driven by mutation and random genetic drift. Unlike most previous models that concentrate only on some stages of speciation, the models studied here describe the complete process of speciation from initiation until completion. The evolutionary factors included are selection (reproductive isolation), random genetic drift, mutation, recombination, and migration. In these models, pre- and postmating reproductive isolation is a consequence of cumulative genetic change. I study possibilities for speciation according to allopatric, parapatric, peripatric, and vicariance scenarios. The results presented here, together with earlier numerical simulations, strongly suggest that rapid speciation, including simultaneous emergence of several new species, is a plausible outcome of the evolutionary dynamics of subdivided populations. Rapid speciation is most likely for populations that are subdivided into a large number of small subpopulations. Speciation is possible even when subpopulations exchange several individuals per generation. Selection for local adaptation is not necessary for rapid speciation. I briefly discuss implications of the dynamics on holey adaptive landscapes for the nearly neutral theory of molecular evolution and for the theory of evolution of genetic canalization.

We present a new simple model for the evolution of premating reproductive isolation. Using this model we first analyze the level of genetic variability maintained by mutation in a large stable population. Then we consider the plausibility of the evolution of strong premating reproductive isolation after a founder event. We demonstrate that after a founder event a new adaptive combination of genes may rise to high frequencies in the presence of an old combination of genes. We compare the probabilities of speciation after a founder event with those in a stable population and with those when reproductive isolation is due to viability selection against hybrids. We argue that premating reproductive isolation is more efficient than postmating reproductive isolation in maintaining the integrity of sympatric species. This might have contributed to the pattern of stronger premating isolation than postmating isolation between closely related pairs of sympatric species.

We study the strength of the genetic barrier to neutral gene flow in a general one-locus, two-allele model that includes viability selection as well as fertility selection and premating isolation. We have separately considered adult migration and pollen migration. Our theoretical results suggest that selection against hybrid formation in the form of fertility selection or assortative mating is more effective in preventing introgression of neutral alleles than is reduction in hybrid viability. We argue that in experimental studies of introgression testing of F 1 hybrids is as important as testing of parental forms. To illustrate the utility of this approach, we estimate the strength of the barrier to neutral gene flow between Piriqueta caroliniana and P. viridis, between Iris hexagona and I. fulva, and between Chorthippus brunneus and C. biguttulus., (© 1998 The Society for the Study of Evolution.)

A classical view of speciation is that reproductive isolation arises as a by-product of genetic divergence. Here, individual-based simulations are used to evaluate whether the mechanisms implied by this view may result in rapid speciation if the only source of genetic divergence are mutation and random genetic drift. Distinctive features of the simulations are the consideration of the complete process of speciation (from initiation until completion), and of a large number of loci, which was only one order of magnitude smaller than that of bacteria. It is demonstrated that rapid speciation on the time-scale of hundreds of generations is plausible without the need for extreme founder events, complete geographic isolation, the existence of distinct adaptive peaks or selection for local adaptation. The plausibility of speciation is enhanced by population subdivision. Simultaneous emergence of more than two new species from a subdivided population is highly probable. Numerical examples relevant to the theory of centrifugal speciation and to the conjectures about the fate of 'ring species' and 'sexual continuums' are presented.

I formulate and study a series of simple one-locus two-allele models for maternal (parental) selection. I show that maternal (parental) selection can result in simultaneous stability of equilibria of different types. Thus, in the presence of maternal (parental) selection the outcome of population evolution can significantly depend on initial conditions. With maternal selection, genetic variability can be maintained in the population even if none of the offspring of heterozygous mothers survive. I demonstrate that interactions of maternal and paternal selection can result in stable oscillations of genotype frequencies. A necessary condition for cycling is strong selection.

We study a general dynamical model describing coevolution of two haploid populations with two alleles at a single locus under weak linear symmetric frequency-dependent selection. A novel and more realistic element of our modeling approach is that both species are allowed to evolve. We analyse conditions for "evolutionary chase" between two phenotypically similar species in which one species evolves to decrease its resemblance with the other species while this other species evolves to increase its resemblance with the first species. We apply our results to a series of simple population genetics models describing classical Müllerian and Batesian mimicries as well as intermediate cases. We show that one of the most important factors influencing the plausibility of non-equilibrium dynamics in systems of mimicry is the relationship between the strength of between-species and within-species interactions. This indicates that this relationship should be the focus of both experimental and theoretical work. Our results suggest that systematic studies of frequencies of different mimicry morphs through time may be very useful.

Sewall Wright's powerful metaphor of rugged adaptive landscapes has formed the basis for discussing evolution and speciation for more than 60 years. However, this metaphor, with its emphasis on adaptive peaks and valleys, is to a large degree a reflection of our three-dimensional experience. Both genotypes and phenotypes of biological organisms differ in numerous characteristics, and, thus, the dimension of 'real' adaptive landscapes is much larger than three. Properties of multidimensional adaptive landscapes are very different from those of low dimension. Consequently, something that is seen as a theoretical challenge in a low-dimensional case might be a trivial problem in a multidimensional context and vice versa. In particular, the problem of how a population crosses an adaptive valley on its way from one adaptive peak to another, which Wright attempted to solve with his shifting balance theory, may be non-existent. A new framework is emerging for deepening our understanding of evolution and speciation, which provides a plausible multidimensional alternative to the conventional view of rugged adaptive landscapes.

Dobzhansky's model of epistatic selection assumes that viable genotypes form "clusters" in genotype space so that populations can evolve from one state to a reproductively isolated state following a "ridge" of well-fit genotypes without crossing any deep adaptive valleys. Recently, the importance of Dobzhansky-type models in evolutionary studies has been reemphasized by Gavrilets (1997a) and Gavrilets and Gravner (1997) who argue that the existence of "ridges" of well-fit genotypes connecting reproductively isolated genotypes is actually a general property of multidimensional adaptive landscapes. Using rigorous techniques and numerical simulations, I analyze clines in the frequencies of selected and neutral alleles maintained by a balance of migration and Dobzhansky-type epistatic selection acting on two diallelic loci. I show that Dobzhansky-type epistatic selection can build up a very strong barrier to neutral gene flow. I describe properties of clines that are indicative of Dobzhansky-type selection., (© 1997 The Society for the Study of Evolution.)

I study the dynamics of a simple quantitative genetic model describing coevolution of two antagonistic species of the victim-exploiter type. In this model, individuals are different with respect to an additive polygenic character that is under direct stabilizing selection and which also determines the strength of within and between species interactions. The model assumes that between species interactions are most intense when the victim's and exploiter's phenotypes match. I show that a cyclic coevolutionary chase is possible under a broad range of conditions. In most cases, the system cycles if the "victim" has a stronger incentive to win and/or a larger genetic variance, and is under stronger stabilizing selection than the "exploiter". The results presented here provide counter-examples to recent studies that (1) question the applicability of "Red Queen" and "arms race" metaphors for continuously varying traits; (2) argue for the existence of crucial differences between major and minor loci dynamics; and (3) attribute a stabilizing role to coevolution.

We study the structure and properties of adaptive landscapes arising from the assumption that genotype fitness can only be 0 (inviable genotype) or 1 (viable genotype). An appropriate image of resulting ("holey") fitness landscapes is a (multidimensional) flat surface with many holes. We have demonstrated that in the genotype space there are clusters of viable genotypes whose members can evolve from any member by single substitutions and that there are "species" defined according to the biological species concept. Assuming that the number of genes, n, is very large while the proportion of viable genotypes among all possible genotypes, p, is very small, we have deduced many qualitative and quantitative properties of holey adaptive landscapes which may be related to the patterns of speciation. Relationship between p and n determines two qualitatively different regimes: subcritical and supercritical. The subcritical regime takes place if p is extremely small. In this case, the largest clusters of viable genotypes in the genotype space have size of order n and there are many of such size; typical members of a cluster are connected by a single ("evolutionary") path; the number of different (biological) species in the cluster has order n; the expected number of different species in the cluster within k viable substitutions from any its member is of order k. The supercritical regime takes place if p is small but not extremely small. In this case, there exists a cluster of viable genotypes (a "giant" component) that has size of order 2n/n; the giant component comes "near" every point of the genotype space; typical members of the giant component are connected by many evolutionary paths; the number of different (biological) species on the "giant" component has at least order n2; the expected number of different species on the "giant" component within k viable substitution from any its member is at least of order kn. At the boundary of two regimes all properties of adaptive landscapes undergo dramatic changes, a physical analogy of which is a phase transition. We have considered the most probable (within the present framework) scenario of biological evolution on holey landscapes assuming that it starts on a genotype from the largest connected component and proceeds along it by mutation and genetic drift. In this scenario, there is no need to cross any "adaptive valleys"; reproductive isolation between populations evolves as a side effect of accumulating different mutations. The rate of divergence is very fast: a few substitutions are sufficient to result in a new biological species. We argue that macroevolution and speciation on "rugged" fitness landscapes proceed according to the properties of the corresponding holey landscapes.

A common conclusion in several recent publications devoted to the deterministic analysis of the third phase of Wright's shifting-balance theory is that under reasonable conditions phase three should proceed easily. I argue that the mathematical equations analyzed in these papers do not correspond to the biological situation they were meant to describe. I present a more appropriate study of the third phase of the shifting balance. My results show that the third phase can proceed only under much more restricted conditions than the previous studies suggested. Migration should be neither too strong not too weak relative to selection. The higher peak should be sufficiently dominant over the lower peak. Recombination can greatly reduce the plausibility of this phase or completely preclude peak shifts. A very important determinant of the ultimate outcome of the competition between different peaks is the topological structure of the network of demes. Peak shifts in two-dimensional networks of demes are more difficult than in one-dimensional networks. Phase three can be accomplished easiest if it is initiated in one of the peripheral demes., (© 1996 The Society for the Study of Evolution.)

Frequency-dependent selection is an important determinant of the evolution of gametophytic self-incompatibility systems in plants, aposematic (warning) and cryptic coloration, systems of mimicry, competitive interactions among members of a population, mating preferences, predator-prey and host-parasite interactions, aggression and other behavioural traits. Past theoretical studies of frequency-dependent selection have shown it to be a plausible mechanism for the maintenance of genetic variability in natural populations. Here, through an analysis of a simple deterministic model for frequency-dependent selection, we demonstrate that complex dynamic behaviour is possible under a broad range of parameter values. In particular we show that the model exhibits not only cycles and chaos but also, for a more restricted set of parameters, transient chaos and intermittency: alterations between an apparently deterministic behaviour and apparently chaotic fluctuations. This behaviour, which has not been stressed within the population genetics literature, provides an explanation for erratic dynamics of gene frequencies.

We study the transient dynamics of the genotypic variance of an additive trait under stabilizing selection, recombination and random drift. We show how interaction of these factors determines the form and the rates of change of different components of the genotypic variance. Let Vg be the genic variance of the trait and CL be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that the dynamics of the system on the plane (Vg, CL) are typically characterized by a quick approach to a straight line with slow evolution along this line afterwards. We show that the number of loci, n, and the population size, N, affect the expected dynamics of Vg mainly through the ratio N/n. We use our analytical and numerical results in interpreting the published results of artificial stabilizing selection experiments. The analysis suggests that it is drift and not selection that most likely led to the reduction of genetic variability in most of these experiments. Even very strong stabilizing selection only slowly removes polygenic variability from populations.

We propose a simple model for analyzing the effects of microenvironmental variation in quantitative genetics. Our model assumes that the sensitivity of the phenotype to fluctuations in microenvironment has a genetic basis and allows for genetic correlation between trait value and microenvironmental sensitivity. We analyze the effects of short-term stabilizing and directional selection on the genotypic and microenvironmental components of phenotypic variance. Our model predicts that stabilizing selection on a quantitative trait increases developmental canalization. We show that stabilizing selection can result in an increase in the heritability. Our findings may provide an explanation for the results of selection experiments in which artificial stabilizing selection did not change the heritability coefficient or increased it., (© 1994 The Society for the Study of Evolution.)

We present exact conditions for stability of monomorphic equilibria in a general multilocus multiallele system and of specific polymorphic equilibria in general one- and two-locus multiallele systems. We show how these exact results on one- and two-locus systems can be used in approximate analysis of polymorphic equilibria in multilocus systems under selection strong relative to recombination. We determine conditions for existence and stability of polymorphic equilibria in specific models of quadratic stabilizing selection on additive polygenic traits.

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.

An approach to the investigation of the evolution of quantitative traits on the basis of analysis of two-locus marginal systems dynamics has been developed. It has been shown that under stabilizing selection the "quasi-stationary" state is quickly reached and maintained continuously. The "quasi-stationary" state is characterized by small changes in allele frequencies and by linkage disequilibrium that significantly decreases genotypic variance. Equations defining the role of linkage disequilibrium in the stationary state of mutation-selection balance are derived.