Unpredictable selection in a structured population leads to local genetic differentiation in evolved reaction norms

Authors


G. de Jong Evolutionaire Populatiebiologie, Vakgroep Botanische Oecologie en Evolutiebiologie, Padualaan 8, NL-3584 CH Utrecht, the Netherlands. Tel: +31 30 2532246; fax: +31 30 2542219; e-mail: g.dejong@bio.uu.nl

Abstract

Unpredictability during development of the optimum phenotype under future selection leads to a compromise reaction norm with a slope that is shallower than the slope of the optimum reaction norm. Unpredictability of selection can lead to an evolved curved reaction norm when genetic variation for curvature is available even if the optimum reaction norm is linear. This requires asymmetry in the frequency distribution of the habitats of selection; at small population size, stochasticity in the number of individuals per selection habitat is sufficient to generate such asymmetry.

Unpredictability of selection in structured populations leads to local genetic differentiation of reaction norms. The mean habitat of a subpopulation is defined as the subpopulation's focal habitat. The evolved mean reaction norm of each subpopulation is anchored at the optimum genotypic value in its focal habitat. Linear reaction norms are parallel if the conditional distribution of adults around the focal habitats is the same for each subpopulation.

Adult migration and absence of zygote dispersal represents the ultimate structured population, each habitat playing the role of focal habitat. Absence of zygote dispersal requires that the flow of individuals through the habitats is used instead of the habitats’ frequencies in the prediction of the evolved reaction norm. Adult migration in absence of zygote dispersal leads to an evolved pattern of locally differentiated reaction norms with optimum genotypic value anchored in the focal habitat and, for linear reaction norms, parallel slopes.

Introduction

Scheiner (1998) presents the results of a large simulation of the evolution of reaction norms. In this simulation, Scheiner found local genetic differentiation of reaction norms between habitats. The genotypes found in a habitat differed in mean reaction norm value from the genotypes found in all other habitats. In all locally differentiated reaction norms, phenotypic plasticity was lower than the phenotypic plasticity selected for. Curvature appeared in evolved reaction norms even if the reaction norm selected for was linear. Scheiner's (1998) results differ from those of most models; this note aims at clarification and analytical prediction of his results.

In Scheiner's simulation, optimizing selection towards a habitat-specific optimum phenotypic value was present in each of many habitats. The question was whether evolution would lead to phenotypically plastic genotypes that reach the optimum phenotype in each habitat, i.e. exhibit the optimum reaction norm. The optimum reaction norm is here defined as the collection of optimum phenotypes over all habitats. If at the end of selection only one genotype were present that conformed to the optimum phenotype in each habitat, no genetic load would exist. Therefore, the collection of optimum phenotypes is here referred to as the (globally) optimum reaction norm; evolved reaction norms will be optimized, but will not necessarily be equal to the optimum reaction norm. If so, a genetic load will accompany an evolved optimized reaction norm.

In his simulation, Scheiner found a strong effect of structuring the population: limited migration led to evolved differences between the genotypes found in each habitat. Specifically, over the global environment (a collection of 50 habitats) local adaptation and local differentiation between reaction norms evolved. The evolved reaction norms for genotypes adapted to different focal habitats differed in height. If linear, the reaction norms of the different genotypes specific to different habitats were found to be more or less parallel with a slope that was more shallow than the slope given by the linear optimum reaction. Curved reaction norms evolved even if the optimum reaction norm was linear, in the presence of genetic variation for curvature. One should consult Scheiner's figure 3B–F and figure 6B–F. In each panel of Scheiner's figures 3B–F and 6B–F, the reaction norms of 10 genotypes sampled from a particular focal habitat are given. The average reaction norm over the 10 genotypes presented in each panel shows the local adaptation and local differentiation of the evolved local mean reaction norm. Evolution has led to a family of reaction norms between habitats, visible as the five mean reaction norms of the five panels. Local mating and consequent population structure had a very large effect in the evolution of phenotypic plasticity.

All evolved reaction norms differed from the optimum reaction norm. Scheiner ‘failed to find conditions that resulted in a pure plasticity strategy, unlike many previous models’ (page 318). One such previous model is the model of Via & Lande (1985). In their model, the optimum character states over habitats are always reached (apart from genetic constraints that do not concern us here). Gavrilets & Scheiner (1993), too, concluded that the optimum reaction norm will evolve. Via & Lande (1985) considered a structured population also, and concluded that structuring of the population has no effect at all: the optimum character states will evolve. The optimum reaction norm evolves in all classical quantitative genetic models of optimizing selection on phenotypic plasticity.

Why the difference in results between Scheiner's (1998) simulation and previous models?

Quite simply: the dispersal stage differs. In Scheiner (1998) adults migrate; in the models of Via & Lande (1985) and Gavrilets & Scheiner (1993) zygotes disperse. Adult migration disconnects the habitats of development and selection, and leads to less than perfect predictability of future selection.

In Via & Lande (1985), de Jong (1989, 1995), Gomulkiewicz & Kirkpatrick (1992) and Gavrilets & Scheiner (1993) a zygote pool exists. From this zygote pool zygotes disperse at random to a series of habitats. In these habitats, individuals develop and are selected. Adults leave their habitat and join a mating pool. Random mating results in a zygote pool of the next generation (Fig. 1a). Selection and development are in the same habitat. Selection is totally predictable, and the optimum reaction norm evolves.

Figure 1.

  Diagrams giving population life cycle. (a) Random zygote dispersal over habitats, development and selection in the same habitat, random mating of adults in a mating pool. (b) Random zygote dispersal over habitats, development, adult migration over habitats, selection in the second habitat, random mating of adults in a mating pool. (c) Zygotes in habitat of adult selection, development in focal habitat, adult migration from focal habitat over habitats, selection in habitat reached as adult, local mating per habitat, local reproduction.

Unpredictability of selection can be introduced into the model. From the zygote pool zygotes disperse at random to a series of habitats. In these habitats, individuals develop to adults. Adults migrate, to a random habitat, and are selected there. Surviving adults leave this selection habitat and join a mating pool. Random mating results in a zygote pool of the next generation (Fig. 1b). Selection and development might be in different habitats, depending upon some probability distribution. Unpredictability of selection has a strong effect on the evolved reaction norm, in general terms leading to an evolved slope of the reaction norm that is shallower than the slope of the optimum reaction norm (Gavrilets & Scheiner, 1993; Sasaki & de Jong, in press, de Jong and Sasaki, in preparation). Under the heading ‘temporally varying environment’Gavrilets & Scheiner (1993) present a model in which all development is in one environment and all selection is in another environment within each generation, but both environments are unpredictable; the important point is the lack of predictability between the habitat of development and the␣habitat of selection. It is the within-generation unpredictability not the between-generation unpredictability of the environment that causes the result.

In the simulation by Scheiner (1998), adults do not leave their habitat of selection. They mate locally, and reproduce locally; no zygote pool exists. The zygotes of the next generation develop in the habitat of selection of their parents. After development and fixing of their phenotype, adults leave, and migrate to another habitat in which they encounter selection (Fig. 1c). Selection is therefore unpredictable from their habitat of development as juveniles, the ‘focal’ habitat of their migration as adults. The lack of a zygote pool subdivides the population: a structured population results from limited adult migration. The position in this paper is that limited adult migration results in genotypic differentiation between habitats. A family of reaction norms evolves, anchored at the habitats of juvenile development and showing the slopes due to unpredictability of selection. Unpredictability of selection causes the population structure to influence the evolution of reaction norms in the simulation by Scheiner (1998).

This note aims to give an algebraic model setting the results of Scheiner's (1998) simulation in another context. First, it addresses the effect of unpredictability of selection. Second, it attempts to disentangle the effects of unpredictability of selection from the effects of a structured population in the evolution of reaction norms. Third, it shows how the probability to survive selection and migration probability become confounded with unpredictability of selection.

Model

Basics

The model considers a heterogeneous environment consisting of a number of habitats. The differences between predictable selection and unpredictable selection can be summarized in a simple model that will first be applied to one single population in a heterogeneous environment, and later to a structured population. The terminology will be that zygotes disperse and adults migrate. A zygote disperses from the zygote pool at random to any of the habitats, and develops in that habitat. Zygote dispersal decides the habitat of development. An adult migrates from its habitat of development to a second habitat, that of selection. Adult migration decides the habitat of selection. Migration breaks the direct link between development and selection.

Development occurs in discrete habitats characterized by an environmental variable x with a distribution given by fx, or in continuous habitats with a probability density p(x) for the environmental variable x. Selection occurs in discrete habitats characterized by an environmental variable y with a distribution given by fy, or in continuous habitats with a probability density p(y) for the environmental variable y. The distributions given by fx and fy or p(x) and p(y) depend upon the habitats, not on the organisms. The mean environmental value of the habitats of development is given by x, and the variance in the environmental variable x by σ2{x}. The mean environmental value of the habitats of selection is given by y, and the variance in the environmental variable y by σ2{y}. The simultaneous distribution of the environmental variables x and y for any individual developing in habitat x and being selected in habitat y is given by fxy or probability density p(x,y). The conditional probability of being selected in any habitat y if developed in habitat x is given by fy|x or p(y|x). The probability distribution implies the covariance cov{x,y} between the environmental values x and y, i.e. between habitat of development and habitat of selection. If the range of values of x and y is similar, |cov{x,y}| ≤ σ2{x}.

An individual's phenotype after development is a function of the environmental value in its habitat; what function is determined by its genotype, and is therefore the genotypic reaction norm. Only polynomial reaction norms will be considered. The genotypic reaction norm is the function of the environment defined by g=g0 + g1x + g2x2 + ···Here g0 equals the reaction norm height at x=0, g1 represents the slope and g2 represents curvature of the reaction norm, all at x=0. The genotypic value of an individual that develops in the habitat with environmental value x is given by the function value g(x)=g0 + g1x + g2x2 + ···The coefficients gi are genetically variable and differ between genotypes; a mean genotypic value of each coefficient, and a genetic variance covariance matrix of the coefficients exists in a population.

The function connecting the phenotypic optima in all␣habitats y exists; this function is given by c=c0 + c1y + c2y2 + ···, and is the optimum reaction norm. Here c0 equals the optimum reaction norm height at y=0, c1 represents the slope and c2 represents curvature of the optimum reaction norm, all at y=0. In a habitat with environmental value y optimizing selection towards c(y)=c0 + c1y + c2y2 + ···occurs. We will use quadratic optimizing selection, as that is the simplest case, but mention Gaussian selection when the difference between fitness functions becomes important (de Jong and Sasaki, in preparation). Individual fitness of an organism that develops in habitat x and is selected in habitat y is given by␣w(x,y)=1 – a[c(y) – g(x)]2 for weak selection (if c(y) – 1/√a < g(x) < c(y) + 1/√a), and by w=0 otherwise. We will assume weak selection throughout.

The column vector gRN contains the mean reaction norm coefficients: gRN = [ g0g1g2⋯]T. The square matrix GRN is the genetic variance covariance matrix of the reaction norm coefficients. The column vector cc contains the optimum reaction norm coefficients: cc= [c0 c1c2···]T. The number of elements in the vectors gRN and cc corresponds to the successive powers of x and y. The vectors gRN and cc might have different numbers of elements, but that does not influence the following argument. We will not go higher than g2, c2 for the present purpose.

The selection gradient vector βRN is defined as the column vector containing all partial derivatives of mean fitness towards the mean reaction norm coefficients, ∂w/∂gi. Finding the form of the selection gradient for several situations is what we need to do. The predicted selection response is given by

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for normally distributed reaction norm coefficients gi (Lande, 1979; Gavrilets & Scheiner, 1993) and approximately for multilocus selection on reaction norm coefficients (Barton & Turelli, 1987; de Jong, 1995, 1999).

Intermezzo: predictable selection

If selection is predictable, the habitat of selection y equals the habitat of development x (Fig. 1a, Table 1). Individual fitness is given by w(x)=1 – a[c(x) – g(x)]2. Mean fitness is given by:

Table 1.  Predicted reaction norm slope. The optimum reaction norm is linear, of the form c = c0 + c1y. The slope 1 of the evolved mean reaction norm  = 0 + 1x differs, depending upon predictability or unpredictability of selection, and quadratic or Gaussian optimizing fitness function. Thumbnail image of

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or its analogue for a discrete distribution of environmental values of habitats. Mean fitness can be rewritten in matrix form. Define the square matrix XX containing the powers of the environmental variable x:

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The matrix E[XX ] contains the expected value of the matrix XX, for discrete or continuous distributions of x:

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The number of rows and columns in the matrices XX and E[XX ] corresponds to the number of coefficients gi in the polynomial reaction norm. Mean fitness becomes equal to

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The selection gradient vector containing the ∂w/∂gi (Lande, 1979) is given by

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At the end of selection, the vector of changes in the mean genotypic values of the reaction norm coefficients, and therefore the selection gradient vector βRN, equals zero. As can be seen from eqn 1, the fact that the selection gradient vector βRN equals zero implies that the mean reaction norm coefficients all equal the coefficients of the optimum reaction norm: gRN=cc, i.e. gi=ci for all i, independent of the matrixE[XX ]. After all, βRN=0 implies βRN=[ccgRN]. All mean reaction norm coefficients become optimum under predictable selection and random zygote dispersal. This is the standard model of selection on reaction norm coefficients (Gavrilets & Scheiner, 1993; de Jong, 1995, 1999).

One should realize that structuring of the zygote population has no influence on the evolved mean reaction coefficients at all: all environmental influence is contained in the matrix E[XX ] that does not influence when the selection gradient vector βRN equals zero. Different subpopulations over different parts of the total environment will differ in the matrix E[XX ] but not in [ccgRN]. The corollary is that selection leads to identical reaction norms even in structured populations if selection is predictable (cf. Via & Lande, 1985). It is therefore impossible to obtain results as in Scheiner (1998) with zygote dispersal and predictable selection.

Unpredictable selection

Let us go back to the situation where the habitat of selection y cannot be predicted for any individual on the basis of its habitat of development. Fitness of an individual that developed in habitat x and is selected in habitat y equals w=1 – a[c(y) – g(x)]2. The habitat of selection y is only related to the habitat of development x by some probability (Fig. 1b). A possible distribution of the environmental values and of individuals over environments is given in Fig. 22a. The population is not yet structured. The predicted change in mean genotypic values of the reaction norm coefficient is given by

Figure 2.

  The effect of unpredictability of selection on the evolved reaction norm in a globally dispersing population. (a) Simultaneous distribution of habitat of development x and habitat of selection y (=0; ȳ=3.43); x and y are nearly independent (cov{x,y}/σ2{x}=0.0827); the distribution of y is asymmetric. (b) Linear optimum reaction norm and evolved reaction norms. The optimum reaction norm has the equation c(y)=10 + 0.4y, i.e. c0=10 and c1=0.4. The evolved linear reaction norm shows slope c1 cov{x,y}/σ2{x}=0.033; the presence of genetic variation for curvature together with the asymmetry in the distribution of y leads to a curved evolved reaction norm.

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but now the selection gradients refer to a different mean fitness. Mean fitness over the total environment, measured as the contribution per zygote to the zygote pool in the next generation, is given by

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or its discrete habitat analogue. Define the matrix XY of the products of powers of the environmental values x and y, and matrix YY of the products of the environmental values y, analogously to the matrix XX:

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The matrix E[XY ] represents the expected value of the matrix XY given by

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or

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i.e. the expected value of the mixed powers of the environmental variables x and y for reaction norms containing g0, g1 and g2, and optimum reaction norm containing c0, c1 and c2. The number of rows in the matrices XY and E[XY ] corresponds to the number of coefficients gi in the polynomial reaction norm. The number of columns in the matrices XY and E[XY ] correspond to the number of coefficients ci in the polynomial optimum reaction norm. Similarly, the matrix YY represents the powers of the environmental values y. The matrix E[YY ] similarly gives the expected value of the powers of y.

Mean fitness from zygote pool to zygote pool can now be written as:

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leading to the selection gradient vector

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The evolved optimized means of the reaction norm coefficients gRN are found when the selection gradient βRN=0, that is, by solving βRN=E[XY]ccE[XX]gN=0. As can be seen from eqn 2, βRN=0 cannot be simplified. In the selection gradient vector, the ranges of environmental values x and y and the probability distribution of the habitats p(x,y) as represented by E[XY] and E[XX ] remain present. The probability distributions of habitats of development and of habitats of selection influence when the selection gradient equals zero.

A straightforward solution to βRN=0 exists for when the optimum reaction norm is linear and genetic variation only exists in the two coefficients g0 and g1; that is, when the genotypic reaction norms are linear. Linear reaction norms and a linear optimum reaction norm lead to the evolved mean genotypic reaction norm coefficients:

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The mean value of the reaction norm in the habitat with x=0, g0, depends upon both y and x (Fig. 2b). The evolved mean slope of the reaction norm g1 is shallower than the optimum slope if |cov{x, y}| < σ2{x} (Fig. 2b) (Gavrilets & Scheiner, 1993; Sasaki & de Jong, in press; de Jong and Sasaki, in preparation).

The solution for gRN in the case of curved (and higher order) reaction norms can be found too; it is long and nasty looking (Appendix; de Jong and Sasaki, in preparation). In the solution for the gi, the coefficients ci of the optimum reaction norm appear accompanied by large coefficients in the expected values of the environmental variables x and y. In the evolved g0 all coefficients ci appear (Appendix, eqn A1). In the evolved g1 and g2 (and any higher gi) the optimum height c0 does not occur; but both optimum slope c1 and optimum curvature c2 do (Appendix, eqns A2 and A3).

If the optimum reaction norm is linear but genetic variation for curvature exists, the evolved curvature g2 depends upon c1 (Appendix). The evolved coefficient g2 does not in general equal zero (Appendix, eqn A4). The coefficient of c1 in g2 equals zero only if the distributions of x and y are both symmetric and x=y=0; only then will a linear reaction norm evolve. If the distribution of the environmental value y is asymmetric, i.e. E[x2y] ≠ 0, curvature will evolve given that genetic variation for curvature is available, but optimum curvature c2 equals 0 (Fig. 2b).

Selection is in fact weighed by the frequencies of the habitats of selection. Asymmetry in the distribution of the environmental values y implies that selection ‘pulls harder’ on one side of the mean environment of selection y than on another. The presence of genetic variance for curvature translates the difference in the strength of selection on the two sides of the mean in an evolved curved reaction norm, given that genetic variation for curvature is available. Symmetry in the distribution of the environmental values y implies an equal strength of selection on the two sides of the mean, and therefore no component of selection working on curvature.

Even a slight asymmetry suffices for curvature in the evolved reaction norm to appear if genetic variation for curvature is available. The distribution in Fig. 22a is fairly asymmetric: it derives from a binomial distribution with probability P=0.75. The distributions of habitats y in Fig. 33a and Fig. 44a are only slightly asymmetric. In Fig. 33a, the distribution of habitats of selection y is symmetric for individuals developing in the main habitat x=–10, and the evolved curvature depends on very subtle differences in the distribution over the habitats of selection. Even the numbers in the population become important. Low number of individuals in the population implies a realized distribution of individuals over environments that will somewhat differ from the ideal distribution, as individuals come in discrete units. The realized distribution of individuals over the habitats of selection might be asymmetric even if formally the distribution of y at each x is symmetric. Such stochasticity of the distribution over habitats might generate enough asymmetry to lead to curved optimized reaction norms even if the optimum reaction norm is linear. Simulations using individuals will be prone to showing this effect. In Scheiner's (1998) simulation the evolution of curved genotypic reaction norms when the optimum reaction norm is linear might very well be an example of this (cf.␣figure 6B–F of Scheiner, 1998).

Figure 3.

  The effect of unpredictability of selection towards a linear optimum reaction norm in a structured population, given zygote dispersal. The global environment presents habitats –25 to +25, both for development and selection. Five subpopulations, centred on the focal habitats –20, –10, 0, +10 and +20, are present. The optimum reaction norm has the equation c(y)=10 + 0.4y over the global environment. (a) Simultaneous distribution of habitat of development x and habitat of selection y for each zygote pool, for focal habitats =−10 and ȳ=10; the simultaneous distributions of the subpopulations only differ in and ȳ. (b) Locally differentiated evolved linear reaction norms for five subpopulations; c1 cov{x,y}/σ2{x}=0.196 in each locally evolved reaction norm. (c) Locally differentiated evolved curved reaction norms for five subpopulations. The presence of genetic variation for curvature together with the asymmetry in the simultaneous distribution leads to curved evolved reaction norms.

Figure 4.

  The effect of unpredictability of selection towards a linear optimum reaction norm in a population without zygote dispersal but with adult migration. Development leads to a phenotypic value that is related to the number of the habitat of development. Migration is as follows: 78% of adults stay in the focal habitat, 10% move 1 habitat left, 10% move 1 habitat right, 1% move 2 habitats to the left and 1% move 2 habitats to the right from the focal habitat. The optimum reaction norm has the equation c(y)=10 + 0.4y over the global environment. Optimizing selection of the form w=1 − 0.5* (habitat value y – habitat value x)2 exists in each habitat. The distribution is adjusted to the flow through each habitat. Limiting the number of migrating adults entering a habitat to 100 individuals causes the actual frequency distribution of habitats to differ from the expected frequency distribution, frequencies only occur in multiples of 0.01. (a) Realized simultaneous distribution of habitats x and y around focal habitat 0; the realized distribution considers 100 migrating individuals and therefore stochasticity and its resulting slight asymmetry are present. (b) Evolved linear and curved reaction norms to the distribution in (a) for focal habitat 10→0. The curvature is the consequence of the stochasticity introduced by considering migrating individuals, if genetic variation for curvature is present.

Fitness functions

Up to now, the quadratic fitness function has been used, giving fitness of an individual that develops in habitat x and is selected in habitat y as w(x,y)=1 – a[c(y) – g(x)]2. Another possibility would be to use the Gaussian fitness function, i.e. w(x,y)=exp{–a[c(y) – g(x)]2}. Mean genotypic fitness under a Gaussian fitness function does not equal an expression in mean genotypic value, as does mean genotypic fitness under a quadratic fitness function. An approximation will have to suffice (and is always used); we will not use mean genotypic fitness but the fitness at the genotypic mean: ww(g). Mean fitness over all environments becomes

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The selection gradient vector ∂w/∂gi can be computed. With predictable selection, the selection gradient vector has again the form

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but now contains a different matrix E[XX]:

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The distribution used to derive the expected values now contains fitness w(x) as well as the frequency of the environmental values over habitats:

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The distribution represents the flow of individuals through the habitats rather than the habitat frequencies. The presence of fitness itself in the flow distribution derives from the form of the derivative of the fitness function.

Unpredictable selection leads to a selection gradient vector of the form

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Again, this selection gradient involves matrices E[XX] and E[XY] that differ with the fitness function. The distribution used to derive the expected values additionally contains the fitnesses:

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with

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The frequencies f x and f ’x,y represent the relative flow of individuals through the habitats x, or x and y, within one generation, over all genotypes. These frequencies will therefore be referred to as the flow frequencies.

As with a quadratic fitness function, the selection gradient βRN equals zero at cc=gRN under predictable selection. It does not matter whether Gaussian or quadratic selection is used if selection is predictable (Table 1). However, under unpredictable selection, the matrices E[XX ] and E[XY ] form part of the solution to βRN=0. This implies that the form of the fitness function influences the evolved mean reaction norms (Table 1). The shape of the solutions to βRN=0 remains the same, but a different distribution has to be used. The value of the evolved mean reaction norm coefficients look similar to eqns 3 and 4, but in g1=c1cov{x,y}/σ2x, the distribution used includes f x,y, not fx,y.

Structured population, zygote dispersal pools

Zygote dispersal and adult migration differ in their evolutionary effects on reaction norms. Zygote dispersal from one zygote pool connects and integrates the population. Adult migration disconnects development and selection. Splitting the population into separate zygote dispersal pools tenuously linked by gene flow represents a structured population. Population structure implies we have to consider a subset of the available habitats x and habitats y for each subpopulation. Zygotes of a subpopulation disperse over a subset of the available habitats of development. Adults migrate over a corresponding subset of the habitats of selection. Adults will not migrate to habitats of selection of a different subpopulation. Selection will differentiate the gene frequencies in the surviving adults of each subpopulation. Gene frequencies, mean genotypic values, and mean reaction norms will locally differentiate.

Subpopulations will be called 1, 2, …, etc. The habitats of development and selection are specific for each subpopulation. The distributions of the environmental values x of the habitats of development and environmental values y of the habitats of selection of the subpopulations will differ in their means. The mean of the environmental values x of subpopulation 1 is x1. The habitat of development with environmental value x=x1x1 is called the ‘focal’ habitat of development for subpopulation 1. Similarly, the mean of the environmental values y of subpopulation 1 is y1. The habitat of selection with environmental value y=y1y1 is the focal habitat of selection for subpopulation 1. It is assumed that y1 equals x1, i.e. that the environmental value of the focal habitat is the same for development and selection. The probability distributions fx,y might differ between subpopulations.

Selection optimizes the reaction norm for each subpopulation separately. The above results for the evolved reaction norm of a general population over the total environment can be applied to each of the subpopulations. I will only use a linear reaction norm for optimum reaction norm, i.e. c(y)=c0 + c1y, though the argument can be used for any shape of reaction norm. This optimum reaction is identical for all subpopulations; it extends over the global environment. The evolved reaction norms for the separate subpopulations will be linear. The evolved reaction norms are easily compared by their value in habitat x=0 of the global environment, and by their slope. From eqn 3 we see that the evolved value at x=0 of the global environment becomes for subpopulation 1

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The evolved slope becomes (eqn 4)

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where cov{x,y} and σ2{x} are conditional upon focal habitat x1. The evolved mean of the genotypic reaction norms in subpopulation 1 is therefore given by

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Substituting eqns 5 and 6 into eqn 7a yields the equation for the evolved mean reaction norm of subpopulation 1:

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The mean reaction norm value in the focal habitat x1 is found by substitution of the environmental value of the focal habitat x1 for x and simplification:

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The evolved mean reaction norm value of subpopulation 1 in its focal habitat lies on the optimum reaction norm. Similar reasoning shows that the evolved mean reaction norm values in all focal habitats per subpopulation lie on the optimum reaction norm. The reaction norm values in the focal habitats follow the optimum reaction norm, and differ between subpopulations.

How much do the reaction norm values in the focal environments differ? If the optimum reaction norm is linear, the difference in height at x=0 between the reaction norms of subpopulations 1 and 2 is given by c1(1 cov{x,y})(x2x1) (from eqn 7b). The evolved mean slopes g1(j) of the subpopulations are identical if the conditional distributions of x and y are identical around the focal habitats of each subpopulation. The slopes of the evolved mean reaction norms are all shallower than the optimal slope. The consequence is a family of parallel reaction norms, one per subpopulation. Population structure leads to local differentiation in reaction norms.

Figure 33a presents a distribution for the combination of habitats of development and habitats of selection that an individual in a subpopulation focused on habitat −10 might find. Some asymmetry is present in this distribution. The distribution of adult migration used within each subpopulation is assumed to be identical for all subpopulations; the subpopulations are characterized by their focal habitats. In Fig. 33b, genetic variation is restricted to height g0 and slope g1. In Fig. 33b, the evolved mean reaction norms for five subpopulations are plotted, given the distribution over habitats of Fig. 33a but with different focal habitats. In Fig. 33c, genetic variation for curvature g2 is present too though the optimum reaction norm is linear. Some slight asymmetry is present in the distribution of y. The presence of genetic variation for curvature, together with the presence of asymmetry in the distribution of the environmental values of the habitats of selection leads to an evolved family of curved reaction norms.

Structured population, no zygote dispersal

In the presence of a zygote pool, zygotes are independent individuals, and each individual experiences development and selection as described by its probability distribution over habitats. The distributions p(x) or fx and p(y) or fy refer to the actual distribution of environmental values over habitats as given by habitat frequency. The simultaneous distribution p(x,y) or fxy refers to the probability that individuals experience a combination of environmental values x and y (Fig. 1b). Note that the probability distribution occurred totally before selection in the life cycle of a generation (Fig. 1b). The probability distributions of individuals over habitats and selection are therefore separate phenomena that can be modelled separately. This independence of selection and distribution of individuals over habitats is specific to a life cycle containing a zygote pool.

No zygote dispersal pool is present at all if parents mate locally and development of the juveniles of the next generation occurs in the habitat of selection of the parents. In a life cycle that contains only adult migration and no zygote dispersal (Fig. 1c), development of the offspring of a mating is no longer independent of the parents and independent between the offspring of a mating (Table 1). The highest possible amount of population structure is present when no zygote dispersal occurs at all, but only adult migration (Fig. 1c). The absence of a zygote pool and the connection between selection in the parents and development of the offspring complicates a direct analysis. It is no longer immediately clear how to write a recurrence equation on which to base a prediction of the evolution of the reaction norm.

The only possibility is to base the recurrence equation on the one stage where individuals are not grouped, at adult migration. The starting habitat of migration is now the focal habitat of a subpopulation. The presence of adults as the migrating stage in the life cycle implies that the probability distribution of development and selection no longer refers to one set of individuals within one generation. From migration to migration, a probability distribution of habitats of selection is joined to a probability of survival in these habitats. The probability distribution of habitats need no longer be independent of selection. The flow of the individuals through the habitats has to be considered.

Flow of individuals through the habitats

We will focus on descendants of generation 0 juveniles from a focal habitat. Any habitat can be taken as focal habitat: the reasoning will be the same for all habitats. Local mating, for instance in habitat x=0, leads to juveniles of generation 0 born in this habitat. We will follow descendants of generation 0 through two generations. Juveniles of generation 0 develop a habitat-specific genotypic value given by their genotypic reaction norm. As adults, these individuals migrate from their habitat of development x0 to their habitats of selection y0, where the subscript now indicates generation. The habitat of selection y0 is measured in distance z0 from x0: y0=x0 + z0. The probability distribution of migration distance is given by fz0. Mean migration distance z equals zero. In the habitats y0, optimizing selection occurs according to w0, where again the subscript specifies the generation. Mating is local in the habitat of selection, at random between adults originating from any habitat. Local offspring belong to generation 1, and develop a habitat-specific phenotypic value given by their reaction norm. Note that the same habitat that was a habitat y0 to the parents is a habitat x1 to their offspring (Table 1).

The offspring of the juveniles from the original and focal habitat, that is, the adults of generation 1, emerge from their habitats of development x1 according to the following number:

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Fitness of a migrating generation 0 adult is measured in its migrating offspring. Mean fitness of generation 0 adults migrating from the focal habitat is given by

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Maximum fitness is found at ∂w(x)/∂gi=0. For a quadratic fitness function, this implies fitness is maximum if each individual in the focal habitat developed towards the optimum phenotypic value of the habitat of selection of their parents. The best prediction by each individual is that it will encounter the same selection environment as its parents did, as mean dispersal is zero, and the best option for each individual is to develop accordingly. This is slightly modified if the fitness function is Gaussian. For a Gaussian fitness function, fitness itself enters in the weighing of the selection strength over habitats (see above); the best option for a developing individual slightly differs from the optimum phenotype of its parents.

Over one generation, only height of the reaction norm is directly selected in this life cycle, as we consider migrating adults from one environment. The slope of the reaction norm only becomes important with the juveniles of the migrating adults of generation 0, as these are the first individuals to experience different habitats of development. A second round of migration is necessary to assess the selection on slope. First-generation adult offspring migrate according to the same migration distribution z, now called z1 referring to generation 1, from their habitat of development x1 to new habitats of selection y1. Their phenotypic value depends upon their habitat of development x1. Optimizing selection occurs in the habitats of selection y1, according to a local fitness w1. Surviving adults mate locally, and their generation 2 offspring develop locally. The number of the emerging (grand)offspring is again weighted by selection:

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for any given habitat x1 of emerging adults of generation 1 and habitat y1 of selection of adults in generation 1. The flow through the habitats is therefore migration probability weighted by selection. To arrive at the evolved reaction norm for each focal habitat, this flow that compounds migration and selection has to be used.

The flow defines fitness of the original emerging adults of generation 0, counted in grand-offspring of generation 2. Fitness is given by

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The immediate selection result for a focal habitat can be predicted by the first two adult generations after emerging adults of generation 0 left the focal habitat. Migration leads to percolation of genes, but the selection structure around the focal habitat is stable. The highest fitness value as a function of the reaction coefficients can be found. It implies we use another form of selection gradient: βRN = ∂w/∂gi instead of βRN=∂/∂gi. We are now primarily interested in the slope of the reaction norm, and will assume that reaction norm height has evolved towards its optimum value (see above at eqn 8b). That is, in the focal habitat individuals reach the optimum reaction norm value for that focal habitat, for the local genotype.

If optimizing selection is quadratic, the selection gradient ∂w/∂g1 equals zero if

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The solution resembles the form c1 cov{x,y}/σ2{x}, but includes w0. Independence of the migration distance per generation, i.e. independence of z0 and z1, reduces eqn 10a, however, to g1=c1.

Gaussian optimizing selection leads a modification of the flow frequencies, as we have seen above. With Gaussian selection, the selection gradient ∂w/∂g1 equals zero if

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The solution has the form c1 cov{x,y}/σ2{x}, using the flow frequencies f ′=fzw instead of the habitat frequency fz for the migration distribution. Independence of the migration distance per generation, i.e. independence of z0 and z1, does not reduce eqn 10b to g1=c1. The reaction norm slope at which the fitness of individuals in the focal environment is maximal deviates from the optimum slope.

A family of reaction norms

Let us first look at linear reaction norms selected towards a linear optimum reaction norm. At any focal habitat X, the predicted value of the mean reaction norm slope g1 becomes

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where cov{x,y} and σ2{x} might be conditional upon focal habitat X. The ratio of covariance between the habitats from which the second generation emerges and the habitats from which the first generation emerges determines the predicted reaction norm slope. The predicted value of the mean reaction norm coefficient g0 (giving the value at x=0 of the overall environment) becomes

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The predicted value of the mean reaction norm for focal habitat X becomes over all habitats x:

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The mean reaction norm value in the focal habitat X becomes

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That is, the predicted mean reaction norm value in the focal habitats follows the optimum reaction norm. The mean reaction norm slopes g1(X) specific to focal habitats x are more shallow than the slope of the optimum reaction norm (eqn 4). The consequence is a family of parallel reaction norms anchored at the optimum values in the focal habitats.

The flow after two migration periods of individuals originating from the focal habitat 0 is given in Fig. 44a. The evolved reaction norm values in the focal habitats follow the optimum reaction norm (Fig. 4b). In Fig. 44b, the slopes of the evolved mean reaction norms are identical as migration patterns from each focal habitat and selection patterns over habitats were assumed identical. The result is a family of reaction norms, one for each habitat, and a series of locally differentiated populations. Population structure in the form of absence of zygote dispersal leads to strong genetic differentiation, and local subpopulations. Shallow reaction norm slopes cause migrating individuals to have a low fitness at a relatively small distance from their focal environment.

Genetic variation in the reaction norm coefficient g2 might again lead to the prediction of curved reaction norms if some asymmetry in the distribution of individuals over habitats is present, i.e. if some asymmetry in the flow exists. Symmetric migration patterns and a symmetric selection pattern around the focal habitat would, however, lead to a symmetric flow of individuals from any focal habitat. But, that symmetric flow is the expected flow. Low numbers of individuals might not realize the expected flow, but lead to stochasticity in the actual flow. The actual flow may easily be asymmetric due to stochasticity. Asymmetry due to stochasticity in flow is sufficient for curved reaction norms to evolve in the presence of a linear optimal reaction norm over the global environment and genetic variation for curvature. The realized flow is shown in Fig. 44a and used to predict the curved reaction norm of Fig. 44b.

Model and simulation

The simulations by Scheiner (1998) showed that local genetic differentiation will prevail over diffusion of genotypes. Only adult migration diffuses genotypes through the population. Selection counteracts the decay of local genotype frequencies through migration, and tightens population structure. Given that the evolved reaction norms have less than optimal slope, one can easily find that the range of effective migration of individuals and the range of effective gene flow is far less than the total environmental range.

The present model does not show that the genetic differentiation due to selection overrules gene flow and diffusion of genes over the total environment. The model is aimed to predict which mean reaction norms are implied by a given migration structure. The predicted evolved mean reaction norms are vertically staggered, following the optimum reaction norm in elevation, and have a predicted slope that is shallower than optimum. This pattern implied by the model emerges in the simulation.

Discussion

The simulation by Scheiner (1998) showed that local genetic differentiation of reaction norms is possible in a structured population. Local genetic differentiation of reaction norms took a special form, of reaction norms anchored at the optimum phenotypic value in their focal environment accompanied by a shallower than optimal slope. The model here implies that the major condition for this to happen is unpredictability of selection, due to adult migration.

Scheiner suggests that genetic architecture is the cause of his results: the results in his figures 3B– and 6B–F are found in a model Scheiner calls ‘epistatic’. Looking accurately at Scheiner's model shows that though the model is written in a different parameterization, it is the same model as the one used here (de Jong, 1999). Scheiner's ‘epistatic’ model is fully additive, as is the model here; no epistasis in its technical sense is present. Genetic architecture is not the cause of any differences between Scheiner's simulation and previous models, such as Via & Lande (1985) and Gavrilets & Scheiner (1993).

Adults are the only migration stage in Scheiner's model. The consequence of the absence of zygote dispersal in Scheiner (1998) is a highly structured population; in fact, this is highest degree of structuring of the population that is possible with a given number of habitats. The persistence of genetic differentiation in Scheiner's simulation of a structured population is a major result. Migration of 11% in each direction in each generation does not stop population differentiation. Local genotypes have highly differentiated reaction norms; the genotypic reaction norms in Scheiner's figures 3B–F and 6B–F show both phenotypic plasticity of genotypes adapted to a particular focal habitat and a cline due to genetic differentiation between genotypes.

Unpredictability of selection is the major cause of Scheiner's result. Many features have no influence on the evolution of reaction norms when selection is predictable. The same features have a large influence on the evolution of reaction norms when selection is unpredictable from the habitat of development. At least five features make themselves felt with unpredictability of selection. First, the habitat distributions influence the evolved reaction norm if selection is unpredictable from development, causing the evolved reaction norm to differ from the optimum reaction norm. The evolved phenotype in any habitat of development is an average over the selected optima in the habitats of selection. The average is weighted by the frequencies of the habitats of selection, and the selection strengths. Second, as the selection strengths in the separate habitats now influence the outcome of selection, the exact form of the fitness function matters, given unpredictability of selection (de Jong and Sasaki, in preparation). Third, an evolved reaction norm is shallower than the optimum reaction norm. Obviously, if the habitat of selection is independent of the habitat of development, the evolved optimized reaction norm will be equal for all habitats of development, whatever the optimum reaction norm and whatever the genetic variation that is available. In this extreme case, all reaction norm coefficients gi except the height coefficient g0 will equal zero. An evolved reaction norm that is shallower than the optimum reaction norm has been demonstrated for a linear reaction norm; the same is true for curved reaction norms. Fourth, evolved reaction norms that curve are possible even when the optimum reaction norm is linear, given the appropriate genetic variation. The condition is asymmetry in the distribution of the habitats of selection. Selection pulls harder at the side of the mean that occurs with higher frequency, and the consequence is bending of the reaction norm. Fifth, population structure surfaces with unpredictability of selection. The existence of subpopulations has no effect if development and selection occur in the same habitat (Via & Lande, 1985; de Jong, 1999).

Predictability of selection from development causes the optimum reaction norm to evolve. Evolution of the optimum reaction norm implies that no genetic load is present at the end of selection. Unpredictability leads to a reaction norm that differs from the optimum reaction norm and is shallower. The difference between the evolved reaction norm and the optimum reaction norm implies that at the end of evolution, selection goes on in many habitats: a genetic load remains present. If optimum and evolved reaction norms are both linear, genetic load is lowest in the mean habitat of selection. Genetic load will increase with distance of habitat from the mean habitat. The pattern that will be found over habitats is stabilizing selection. Anholt (1991) found a pattern resembling stabilizing selection over habitats in damselflies. If optimum and evolved reaction norms are both curved, the two curves will intersect at two environmental values of the habitats of selection. Genetic load is zero in these habitats. The pattern found over habitats will be disruptive selection. A prediction of the present model is therefore that curved reaction norms and unpredictability of selection from development will be accompanied by apparent disruptive selection over habitats in field data.

Adult migration without zygote dispersal leads to an ESS reaction norm, rather than to a reaction norm that can be straightforwardly predicted from quantitative genetics theory. In the literature, one finds optimization models of the evolution of phenotypic plasticity next to quantitative genetics models. Houston & McNamara (1992) predict phenotypic plasticity in life-history decisions by an optimization model. The description of the life-history in Houston and McNamara's model of the evolution of phenotypic plasticity in clutch size strongly suggests that the case they consider is similar to the case of adult migration in the absence of a zygote dispersal pool.

Unpredictability of selection is of great importance in the evolution of phenotypic plasticity in natural populations (Kingsolver & Huey, 1998). A reaction norm pattern as in this model (cf. Fig. 3b) has been found in experimental studies of the reaction norms for wing length in Drosophila melanogaster sampled from different populations along the East Coast of the USA (Coyne & Beecham, 1987). The linear reaction norms for wing length are parallel between populations. If we assume that the phenotypic value of wing size found in a northern population at a northern temperature is its optimal value, and that the phenotypic value of wing size found in a southern population at a southern temperature is its optimal value, the line connecting such points is steeper than the reaction norm slopes. The life history of D. melanogaster corresponds to the life history of a structured population with local zygote dispersal and adult migration. Selection on adults will be largely independent of larval development. Larval development depends upon temperature, and the reaction norm of adult size follows. Of the phenotypically plastic adult size characters, wing/thorax ratio is the least genetically variable. Adult size might be selected according to wing/thorax ratio (Morin et al., 1996; Moreteau et al., 1997) giving temperature-dependent flight capabilities. The life history of D. melanogaster is therefore compatible with the model: selection on wing/thorax ratio need not be predictable from the temperature of the rotting fruit of development. Evolved wing/thorax ratio will have a shallower slope than optimum wing thorax ratio. Field study and model give similar results.

Acknowledgment

Sam Scheiner confirmed his simulation uses adult migration without zygote dispersal. I thank Sam Scheiner for comments on several versions of the manuscript.

Appendix

The evolved optimized reaction norm coefficients RN are found when the selection gradient βRN=0, that is, by solving

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For linear reaction norms and linear optimum reaction norms the solution is found in the main text. The solution for RN in the case of curved reaction norms equals

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The Determinant Det of E[XX] equals

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The solutions of the mean reaction norm coefficients are:

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If genetic variation for g2 is present but the optimum reaction norm is linear so that c2=0,

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showing that in general 2 has a value other than zero; therefore, the evolved reaction norm is not linear. In Scheiner (1998) migration is symmetric and systematic, while zygotes do not disperse. Using the migration distribution itself would give =ȳ,E[xy]=E[x2] while the symmetry of both the distribution of x and the distribution of y gives E[x3]=0 and E[x2y]=0. In Scheiner (1998) the asymmetry in the migration distributions seems totally the consequence of individual modelling and the consequent stochasticity.

If the distribution of environmental values x and y over the habitats of development and of selection is bivariate normal, with means =ȳ=0, variances σ2{x} and σ2{y} and correlation ρ, the moments are:

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Therefore, the evolved coefficients become

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Ancillary