Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ₀ (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ₀ is an increasing function of the panmictic rate and to deduce the effect on λ₀ of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ₀; bounds on λ₀ are established; and λ₀ is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.     

Clines with asymmetric migration

The consequences of asymmetric dispersion on the maintenance of an allele in a one-dimensional environmental pocket are examined. The diffusion model of migration and selection is restricted to a single diallelic locus in a monoecious population in the absence of mutation and random drift. It is further supposed that migration is homogeneous and independent of genotype, the population density is constant and uniform, and Hardy-Weinberg proportions obtain locally. If dispersion is preferentially out of an environmental pocket at the end of a very long habitat, the condition for maintaining the allele favored in the pocket becomes less stringent than for symmetric migration; dispersion preferentially into the pocket increases the severity of the condition for polymorphism. If an allele is harmful in large regions on both sides of an environmental pocket, the probability for polymorphism is decreased by asymmetric migration. The criterion for the existence of a cline is independent of the sense of the asymmetry; the cline itself is not. These phenomena are studied both analytically and numerically.—It is shown for symmetric migration and variable population density that the more densely populated parts of the habitat are more influential in determining gene frequency than the others. Thus, the higher the population density in an environmental pocket, the more easily an allele beneficial in the pocket will be maintained in the population.     

A diffusion model for geographically structured populations

Summary A diffusion model is derived for the evolution of a diploid monoecious population under the influence of migration, mutation, selection, and random genetic drift. The population occupies an unbounded linear habitat; migration is independent of genotype, symmetric, and homogeneous. The treatment is restricted to a single diallelic locus without dominance. With the customary diffusion hypotheses for migration and the assumption that the mutation rates, selection coefficient, variance of the migrational displacement, and reciprocal of the population density are all small and of the same order of magnitude, a boundary value problem is deduced for the mean gene frequency and the covariance between the gene frequencies at any two points in the habitat. Supported by the National Science Foundation (Grant No. DEB77-21494).     

Models of clines in sexual selection by female choice

Most models of sexual selection by female choice have considered discrete, homogenous populations. This paper studies the evolution of a female preference along a cline in the frequency of a preferred male trait. Single alleles control both the preference and preferred character.
Fisher's process initially causes the preference to spread to some 'maximum' frequency with a corresponding rise in character gene frequency. At this stage, the cline in preference does not necessarily mirror that of the trait. Then, however, preference frequency usually decreases, albeit very slowly, and the preference cline always comes to mirror that of the preferred character. Eventually, preference is completely lost from the cline and the character cline decays to that seen under random mating. This loss can only be prevented if the preference is initially frequent enough to push the character to fixation throughout the cline.
Consequently, a preference that arises after the preferred trait may increase very little in frequency itself and have a negligible effect on trait frequency before being lost from the population. Special conditions, such as cyclical natural selection, may be necessary to explain the spread of a preference in a cline from a low initial frequency to frequencies as high as those observed. A preference that predates the preferred trait can enter the population at a high frequency and radically alter a cline in the frequency of a preferred male trait, albeit often transiently.     

Food choice behaviour may promote habitat specificity in mixed populations of clonal and sexual Potamopyrgus antipodarum

Genetic polymorphism along an environmental gradient may be maintained if disruptive selection on habitat-specific traits leads to a correlated response in traits that reduce gene flow between habitats. We studied a short-distance cline in a population of freshwater snails Potamopyrgus antipodarum in which sexual and clonal snails coexist. Sexuals and clones show a life history cline by depth: snails reproduce at a smaller size in shallower habitats. Clones are also structured genetically across habitats and seem not to mix, even though habitats are within the dispersal distance of the snails and the opportunity for gene flow via migration must be considerable. Because habitat preference may promote divergence in both clones and sexuals along the depth gradient, we investigated whether snails show habitat-specific food choice behaviour that could reduce migration. We tested the food choice behaviour of the snails by exposing them simultaneously to food from their home and adjacent habitats. Both juvenile and adult snails from the shallow shore bank and a mid-water macrophyte habitat preferentially grazed on the vegetation of their original habitats. We suggest that the observed genetic and life history cline may be maintained by food choice behaviour that may promote a partial barrier to gene flow between the habitats. Copyright 2000 The Association for the Study of Animal Behaviour.     

Numerical analysis of random drift in a cline

The equilibrium state of a diffusion model for random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The flattening of the expected cline due to random drift is very slight in natural populations. The ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a nonzero constant. The correlation between the gene frequencies at two points decreases monotonically to zero as the separation is increased with the average position fixed; the decrease is asymptotically exponential. The correlation decreases monotonically to a positive constant depending on the separation as the average position increasingly deviates from the center of the cline with the separation fixed. The correlation also decreases monotonically to zero if one of the points is fixed and the other is moved outward in the habitat, the ultimate decrease again being exponential. Some asymptotic formulae are derived analytically.——The loss of an allele favored in an environmental pocket is investigated by simulating a chain of demes exchanging migrants, the other assumptions being the same as above. For most natural populations, provided the allele would be maintained in the population deterministically, this process is too slow to have evolutionary importance.     

Clines with partial panmixia in an unbounded unidimensional habitat

In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection in an unbounded unidimensional habitat is investigated. Migration and selection are both weak. The former is homogenous and isotropic; the latter has no dominance. The population density is uniform. A simple, explicit formula is derived for the maximum value β₀ of the scaled panmictic rate β for which a cline exists. The former depends only on the asymptotic values of the scaled selection coefficient. If the two alleles have the same average selection coefficient, there exists a unique, globally asymptotically stable cline for every β≥0. Otherwise, if β≥β₀, the allele with the greater average selection coefficient is ultimately fixed. If β<β₀, there exists a unique, globally asymptotically stable cline, and some polymorphism is retained even infinitely far from its center. The gene frequencies at infinity are determined by a continuous-time, two-deme migration-selection model. An explicit expression is deduced for the monotone cline in a step-environment. These results differ fundamentally from those for the classical cline without panmixia.     

Clines with Variable Migration

The consequences of a discontinuity in the migration rate and of a geographical barrier in the habitat are studied in a diffusion model of migration and selection. The treatment is restricted to a single diallelic locus in a monoecious population in the absence of mutation and random drift. It is supposed further that migration is independent of genotype, the population density remains constant and uniform, and Hardy-Weinberg proportions obtain locally. It is shown that a discontinuity in the migration rate leads to a jump in the slope of the gene frequency, but not in the gene frequency itself, while a localized geographical barrier has precisely the opposite effect. These features of the gene frequency behavior are quantitatively related to the migration rate. The influence of the above inhomogeneities in migration on the maintenance of an allele in an environmental pocket is examined. The extent to which the critical condition for polymorphism is made less stringent by decreased migration outside the pocket and by a geographical barrier between the pocket and the rest of the habitat is evaluated.     

Nonrandom larval dispersal can steepen marine clines

Sharp and stable clinal variation is enigmatic when found in species with high gene flow. Classical population genetic models treat gene flow as a random homogenizing force countering local adaptation across habitat discontinuities. Under this view, dispersal over large spatial scales will lower the effectiveness of adaptation by natural selection at finer spatial scales. Thus, random gene flow will create a shallow phenotypic cline across an ecotone in response to a steep selection gradient. In sedentary marine species that disperse primarily as larvae, nonrandom dispersal patterns are expected due to coastal hydrodynamics. Surprisingly sharp phenotypic and genotypic clines have been documented in marine species with high gene flow. We are interested in the extent to which nonrandom dispersal could accentuate such clines. We model a linear species range in which populations have stable and uniform densities along a selection gradient; in contrast to random dispersal, convergent advection of larvae can amplify phenotypic differentiation if coupled with a semipermeable dispersal barrier in the convergence zone. The migration load caused by directional dispersal pushes the phenotypic mean away from the local trait optimum in downstream populations, that is, near the convergence zone. A dispersal barrier is possible as a result of colliding currents if the water and larvae are mostly displaced offshore, away from suitable settlement habitat. Disjunctions in a quantitative trait were enlarged in the convergence zone by faster current flows or a more complete dispersal barrier. With advection of larvae per generation one-third as far as the average dispersal distance by diffusion, convergence on a dispersal barrier with 40% permeability generated a trait disjunction across the convergence zone of two phenotypic standard deviations. Without directional dispersal, similar clines also developed across a habitat gap, where population density was low, or across dispersal barriers with less than 1% permeability. These findings suggest that the types of hydrographic phenomena often associated with marine transition zones can strongly affect the balance between gene flow and selection and generate surprisingly steep clines given the large-scale gene flow expected from larvae.     

Genetic drift widens the expected cline but narrows the expected cline width

Random genetic drift shifts clines in space, alters their width, and distorts their shape. Such random fluctuations complicate inferences from cline width and position. Notably, the effect of genetic drift on the expected shape of the cline is opposite to the naive (but quite common) misinterpretation of classic results on the expected cline. While random drift on average broadens the overall cline in expected allele frequency, it narrows the width of any particular cline. The opposing effects arise because locally, drift drives alleles to fixation--but fluctuations in position widen the expected cline. The effect of genetic drift can be predicted from standardized variance in allele frequencies, averaged across the habitat: . A cline maintained by spatially varying selection (step change) is expected to be narrower by a factor of √1- relative to the cline in the absence of drift. The expected cline is broader by the inverse of this factor. In a tension zone maintained by underdominance, the expected cline width is narrower by about 1- relative to the width in the absence of drift. Individual clines can differ substantially from the expectation, and we give quantitative predictions for the variance in cline position and width. The predictions apply to clines in almost one-dimensional circumstances such as hybrid zones in rivers, deep valleys, or along a coast line and give a guide to what patterns to expect in two dimensions.     

Group selection on the boundary of a stable population

This article presents a model of group selection via differential extinction acting on small boundary populations of a large, fixed population. We restrict consideration to extinction of populations at or near habitat carrying capacity, thus modeling the kind of situation envisioned by Wynne-Edwards (1962). Under the assumption that the extinction rate is large relative to individual genetic parameters, we discuss the effect of differential extinction on the distribution of gene frequencies within boundary micropopulations. In these circumstances, it is shown that differential extinction is most likely to produce a bimodal distribution of the gene frequencies if the form of the extinction operator approximates a step function, with a critical allele frequency at which the extinction rate shifts from high to low. The need for a rather close approximation to such discontinuous behavior limits the possible conditions under which differential extinction might be important. We conclude with a comparison of our model with that of Levins (1970a) and suggest some technically feasible generalizations of our approach.     

Numerical analysis of weak random drift in a cline

The equilibrium state of a diffusion model for weak random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric, and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a non-zero constant. The correlation between the gene frequencies at two points decreases monotonically to zero as the separation is increased with the average position fixed; the decrease is asymptotically exponential. The correlation decreases monotonically to a positive constant depending on the separation as the average position increasingly deviates from the center of the cline with the separation fixed. The correlation also decreases monotonically to zero if one of the points is fixed and the other is moved outward in the habitat, the ultimate decrease again being exponential. All the results are parameter free. Some asymptotic formulae are derived analytically.     

Adaptation to a steep environmental gradient and an associated barrier to gene exchange in Littorina saxatilis

Steep environmental gradients offer important opportunities to study the interaction between natural selection and gene flow. Allele frequency clines are expected to form at loci under selection, but unlinked neutral alleles may pass easily across these clines unless a generalized barrier evolves. Here we consider the distribution of forms of the intertidal gastropod Littorina saxatilis, analyzing shell shape and amplified fragment length polymorphism (AFLP) loci on two rocky shores in Britain. On the basis of previous work, the AFLP loci were divided into differentiated and undifferentiated groups. On both shores, we have shown a sharp cline in allele frequencies between the two morphs for differentiated AFLP loci. This is coincident with a habitat transition on the shore where the two habitats (cliff and boulder field) are immediately contiguous. The allele frequency clines coincide with a cline in shell morphology. In the middle of the cline, linkage disequilibrium for the differentiated loci rises in accordance with expectation. The clines are extremely narrow relative to dispersal, probably as a result of both strong selection and habitat choice. An increase in F(ST) for undifferentiated AFLPs between morphs, relative to within-morph comparisons, is consistent with there being a general barrier to gene flow across the contact zone. These features are consistent either with an episode of allopatric divergence followed by secondary contact or with primary, nonallopatric divergence. Further data will be needed to distinguish between these alternatives.     

Habitat selection reduces extinction of populations subject to Allee effects

Theoretical studies indicate that a single population under an Allee effect will decline to extinction if reduced below a particular threshold, but the existence of multiple local populations connected by random dispersal improves persistence of the global population. An additional process that can facilitate persistence is the existence of habitat selection by dispersers. Using analytic and simulation models of population change, I found that when habitat patches exhibiting Allee effects are connected by dispersing individuals, habitat selection by these dispersers increases the likelihood that patches persist at high densities, relative to results expected by random settlement. Populations exhibiting habitat selection also attain equilibrium more quickly than randomly dispersing populations. These effects are particularly important when Allee effects are large and more than two patches exist. Integrating habitat selection into population dynamics may help address why some studies have failed to find extinction thresholds in populations, despite well-known Allee effects in many species.     

Stable partial migration under a genetic threshold model of migratory behaviour

Many species show migratory behaviour in response to seasonal changes in environmental conditions. A peculiar, yet widespread phenomenon is partial migration, when a single population consists of both migratory and non‐migratory individuals. There are still many open questions regarding the stability and evolutionary significance of such populations. For passerines the inheritance of migratory activity is best described by the threshold model of quantitative genetics. Such a model has not yet been employed in theoretical studies, in which stability of partially migratory populations is usually linked to group differences in survival or reproduction. Here we develop a parsimonious model featuring a conditional genetic threshold for passerine migratory behaviour under which stable partial migration can be observed, and we explore the resulting selection landscape. Our model results show a cline in migratory behaviour across the landscape, from fully migratory populations to fully residential populations, with a fairly wide zone of partially migratory populations, which is stable in both time and space under a wide range of parameter settings. Temporal stability of the zone is linked with the yearly variance in both migration survival and resident winter survival. In contrast to other theoretical studies, we show that density dependence in winter survival is not essential for observing partially migratory populations. In addition, we observe that selection on the genetic threshold value occurs mainly at the borders of the zone of partial migration. This result suggests that fully migratory and fully residential populations in areas far from the zone of partial migration can harbour genetic diversity that allows the appearance of the alternative phenotype under (a wide range of) different conditions.     

Geographical Variation in a Quantitative Character

A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.     

Change and stability in a steep morph‐frequency cline in the snail Cepaea nemoralis (L.) over 43 years

Populations of the polymorphic land snail Cepaea nemoralis (L.) from Deepdale, Derbyshire, UK, sampled in 1965–67, showed a pattern of area effects, with steep clines among groups of populations differing in shell colour and banding morph frequencies. In 2010, most of these populations were resampled. In particular, a continuous transect made in 1967 of 42 quadrats (18.34 × 18.34 m) across a steep cline in several morph frequencies was completely resampled. In the dale as a whole, yellow shells had increased in frequency. In the transect, the frequencies of banding morphs showed no significant changes, although colour morphs showed some changes. Pink shells had increased in frequency in a section in which scrub had developed, and brown shells had increased in frequency in the area in which they had originally been at the highest frequency. In each case, the selection coefficients were of the order of 4%. Yellow had increased elsewhere. Nevertheless, both in the dale as a whole and in the transect, the pattern of geographical change in morph frequencies had remained essentially the same. Estimates of migration based on previous studies of marked snails and on modelling of the effect of drift and migration suggest that, regardless of whether the cline is a product of differential selection or of the gradual merging of previously separate founding populations, it has been in existence for a long time, and that migration occurs over greater distances than estimated from direct observation on marked snails. Although we can demonstrate that selection has occurred, the origin and maintenance of the cline and others similar to it remain in doubt; the development and maintenance of polymorphism in this species may require consideration of several processes operating on different time scales. © 2012 The Linnean Society of London     

The role of natural selection in genetic differentiation of worldwide populations of Drosophila ananassae

The main evolutionary forces leading to genetic differentiation between populations are generally considered to be natural selection, random genetic drift, and limited migration. However, little empirical evidence exists to help explain the extent, mechanism, and relative role of these forces. In this study, we make use of the differential migration behavior of genes located in regions of low and high recombination to infer the role and demographic distribution of natural selection in Drosophila ananassae. Sequence data were obtained from 13 populations, representing almost the entire range of cosmopolitan D. ananassae. The pattern of variation at a 5.1-kb fragment of the furrowed gene, located in a region of very low recombination, appears strikingly different from that of 10 noncoding DNA fragments (introns) in regions of normal to high recombination. Most interestingly, two main haplotypes are present at furrowed, one being fixed in northern populations and the other being fixed or in high frequency in more southern populations. A cline in the frequency of one of these haplotypes occurs in parallel latitudinal transects. Taken together, significant clinal variation and a test against alternative models of natural selection provide evidence of two independent selective sweeps restricted to specific regions of the species range.     

Differential gene exchange between parapatric morphs of Littorina saxatilis detected using AFLP markers

Speciation requires the acquisition of reproductive isolation, and the circumstances under which this could evolve are of great interest. Are new species formed after the acquisition of generalized incompatibility arising between physically separated populations, or may they arise as a result of the action of disruptive selection beginning with the divergence of a rather restricted set of gene loci? Here we apply the technique of amplified fragment length polymorphism (AFLP) analysis to an intertidal snail whose populations display a cline in shell shape across vertical gradients on rocky shores. We compare the F_ST values for 306 AFLP loci with the distribution of F_ST estimated from a simulation model using values of mutation and migration derived from the data. We find that about 5% of these loci show greater differentiation than expected, providing evidence of the effects of selection across the cline, either direct or indirect through linkage. This is consistent with expectations from nonallopatric speciation models that propose an initial divergence of a small part of the genome driven by strong disruptive selection while divergence at other loci is prevented by gene flow. However, the pattern could also be the result of differential introgression after secondary contact.     

The diffusion model for migration and selection in a dioecious population

The diffusion approximation is derived for migration and selection at a multiallelic locus in a dioecious population subdivided into a lattice of panmictic colonies. Generations are discrete and nonoverlapping; autosomal and X-linked loci are analyzed. The relation between juvenile and adult subpopulation numbers is very general and includes both soft and hard selection; the zygotic sex ratio is the same in every colony. All the results hold for both adult and juvenile migration. If ploidy-weighted average selection, drift, and diffusion coefficients are used, then the ploidy-weighted average allelic frequencies satisfy the corresponding partial differential equation for a monoecious population. The boundary conditions and the unidimensional transition conditions for coincident discontinuities in the carrying capacity and migration rate extend identically. The previous unidimensional formulation and analysis of symmetric, nearest-neighbor migration of a monoecious population across a geographical barrier is generalized to symmetric migration of arbitrary finite range, and the transition conditions are shown to hold for a dioecious population. Thus, the entire theory of clines and of the wave of advance of favorable alleles is applicable to dioecious populations.This work was supported by National Science Foundation grant BSR-9006285