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* Department of Mathematics, University of Vienna, A-1090 Vienna, Austria
Department of Genetics, Stanford University, Stanford, California 94305
1 Corresponding author: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria.
E-mail: reinhard.buerger{at}univie.ac.at
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ABSTRACT |
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 TOP ABSTRACT THE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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General and systematic treatments of genetic models with frequency- and density-dependent selection are available only for a single locus (NAGYLAKI 1979; ASMUSSEN 1983; ASMUSSEN and BASNAYAKE 1990). In these models, fitnesses are assigned directly to genotypes, though in a rather general way. These and other investigations clearly demonstrate that FDS may strongly influence the genetic structure of a population and has a much higher potential in maintaining genetic variation than frequency-independent selection. In addition, already for one-locus two-allele models in which genotypic fitnesses are linear functions of the genotype frequencies, complex dynamic behavior, such as chaos, may occur for relatively wide ranges of parameter values (ALTENBERG 1991; GAVRILETS and HASTINGS 1995).
We pursue a different approach by making specific assumptions on how selection acts on phenotypes. In doing so, we lose generality compared to some of the single-locus studies. However, it enables us to tackle questions such as, How strong must FD be relative to other selective forces, such as stabilizing selection, to have a noticeable impact on the genetic structure of a population? or, When is FD strong enough to be phenotypically detectable? To this end, we consider a quantitative trait that is subject to (frequency-independent) stabilizing selection and mediates frequency- and density-dependent selection through intraspecific competition; i.e., individuals of similar phenotypes compete. We use a functional form for fitness of the trait introduced by BULMER (1974)(1980) that does not specify the causes of stabilizing selection, i.e., whether real or induced by pleiotropic effects. Nevertheless, mathematically this model is closely related to models based on differential resource utilization and a Lotka-Volterra approach (SLATKIN 1979; CHRISTIANSEN and LOESCHCKE 1980; LOESCHCKE and CHRISTIANSEN 1984). Previous studies of this type of model assumed a single locus contributing to a normally distributed trait (BULMER 1974, 1980; SLATKIN 1979), a classical quantitative-genetics approach for a normally distributed trait (SLATKIN 1979), a single locus with multiple alleles (CHRISTIANSEN and LOESCHCKE 1980), or two recombining diallelic loci contributing additively to a quantitative trait without assumptions on its distribution (LOESCHCKE and CHRISTIANSEN 1984; BüRGER 2002a,b). Roughly, conditions were identified under which FD is strong enough to induce disruptive selection and maintain polymorphism. In the two-locus models, the amount of genetic variation and the possible equilibrium patterns and their dependence on the parameters were derived.
Studies with simple genetics (BULMER 1974, 1980; SLATKIN 1979) have shaped our intuition of the role of intraspecific competition in maintaining genetic variation by showing that if the strength of competition relative to stabilizing selection exceeds a certain threshold, the trait is under disruptive selection; hence genetic variance should be maintained. Otherwise, the trait is under stabilizing selection, which usually depletes heritable variation (see also the DISCUSSION). However, questions concerning the extent of variation maintained by FDS or if and how conclusions depend on the underlying genetic structure can be answered only on the basis of more explicit genetic models.
We complement and extend previous studies in several directions. First, we admit multiple diallelic loci to contribute (additively) to the trait. Second, the population size is allowed to change according to its demographic dynamics. Because random genetic drift is ignored, the dynamics are still deterministic. Third, stabilizing selection that is asymmetric with respect to the existing range of phenotypic values is explored. Fourth, to cover a wide range of parameters, in particular from the high-dimensional genetic parameter space, we adopt the statistical and numerical approach of BüRGER and GIMELFARB (1999)(2002). Thus, for a given number of loci, a range of recombination rates, and given ecological and demographic parameters, we calculate the quantities of interest by iterating a large number of genetic systems with randomly chosen locus effects and initial conditions until equilibrium is reached and then computing the appropriate averages. This numerical approach is complemented by analytical results and allows us to derive general inferences on the patterns of genetic variation resulting from the interaction of multilocus genetics with FDS. In particular, the strength of FD that leads to highly polymorphic genetic systems and to disruptive selection is determined. Also the evolution of mean fitness and of the equilibrium distribution of genotypic values under strong FD is examined.
This is not the first multilocus study exploring the amount of genetic variation maintained by FDS on a quantitative trait. CLARKE et al. (1988) and MANI et al. (1990) employed a model of selection that is similar to BULMER's (1980), but differs in the way competition and stabilizing selection are modeled. Moreover, they assume a finite population and admit several multiallelic loci with recurrent mutation. Their results are purely numerical and the main focus is on variation at the gene level. Therefore, direct comparison with the models discussed above is difficult. The purpose of our study is not to demonstrate that FDS can be important, but rather to identify conditions under which it has significant effects on the genetic structure of a population, what and how large these effects are, and how genetical and ecological parameters interact.
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THE MODEL |
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TOPABSTRACTTHE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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Ecological assumptions:
The first fitness component is frequency independent and may reflect some sort of direct selection on the trait, for example, through differential supply of a resource whose utilization efficiency is phenotype dependent. However, frequency-independent stabilizing selection could as well be caused by indirect selection through pleiotropic side effects of alleles that contribute primarily to fitness-related traits (e.g., ROBERTSON 1967; KEIGHTLEY and HILL 1990; BüRGER 2000, chap. VII). We ignore environmental variation and deal directly with the fitnesses of genotypic values. In the absence of genotype-environment interaction, this is no restriction because in the present model the only effect of including environmental variance was a deflation of the selection intensity. For simplicity, we sometimes use the words genotypic value and phenotype synonymously.
Stabilizing selection is modeled by the quadratic function
 | (1) |
is the position of the optimum. Of course, S(g) is assumed to be positive on the range of possible phenotypes, thus restricting the admissible values of Vs. We use a scale on which the range of possible phenotypes is the interval [0, 1] (see Genetic assumptions below). Hence, if 
0.5, then Vs must satisfy Vs 1/22. We exclude pure directional selection by assuming 0 < < 1.
The second component of fitness is FD. We assume that competition between phenotypes g and h can be described by
 | (2) |
. For our scaling this means V
0.5. Equation 2 implies that competition between individuals of similar phenotypes is much stronger than that between individuals of very different phenotypes, as will be the case if different phenotypes preferentially utilize different food resources. Small V implies a strong frequency-dependent effect of competition, whereas in the limit V 
, FD vanishes. Let P(h) denote the relative frequency of individuals with phenotype h. Then the intraspecific competition function 
P
, which measures the strength of competition experienced by phenotype g if the population distribution is P, is given by
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 | (3) |
and VA denote the mean and (additive genetic) variance, respectively, of the distribution P of genotypic values.
Similar to BULMER's (1974, 1980) model, we assume that the absolute fitness of an individual with genotypic value (phenotype) g is given by
 | (4) |
and 
are positive parameters and N denotes the total population size. For notational simplicity, the dependence of W(g) on N and P is omitted. In the context of density-dependent growth models, the parameter in (4) is related to the growth rate of the population and is proportional to the carrying capacity. More precisely, in a monomorphic population in which 
and VA = 0, the fitness (of the population) becomes – N/; hence the carrying capacity is
 | (5) |
Our model of FDS is closely related to models that are based on Lotka-Volterra equations (SLATKIN 1979; CHRISTIANSEN and LOESCHCKE 1980; LOESCHCKE and CHRISTIANSEN 1984). The relation between these models and ours is worked out in BüRGER (2002b), where it is shown that all these ecological models lead to fitness functions that are mathematically equivalent to first order in 1/Vs and 1/V. Thus, for sufficiently weak overall selection, the differences vanish. As explained in BüRGER (2002b), the present choice of the fitness function makes the model more easily amenable to mathematical analysis and does not lead to certain special effects that a Gaussian fitness function causes under strong selection. If stabilizing selection is modeled by a Gaussian fitness function, the equilibrium structure is much more complex than with quadratic fitness. This is so in the absence of FD (NAGYLAKI 1989; WILLENSDORFER and BüRGER 2003) as well as for very strong FD (LOESCHCKE and CHRISTIANSEN 1984). Our fitness function is also closely related to that of MATESSI et al. (2001); if higher-order terms are omitted, the induced dynamics and equilibrium structure become equivalent.
Genetic assumptions:
The trait values g are determined additively by n diallelic loci. There is no dominance or epistasis. The contribution of one allele at each locus 
is zero, and the contribution, ß, of the other allele is a random number between zero and one. It is assumed that the minimum and maximum genotypic values are always zero and one. Therefore, the actual contribution by the second allele at locus is scaled to be 
It follows that the genotypic value of the total heterozygote is always 1/2, and the average allelic effect among the n loci controlling the trait is 
. This normalization has the advantage that the strength of selection on genotypes can be compared for different numbers of contributing loci. The effect of an increasing number of loci is a finer resolution of possible phenotypes through genotypic values.
Dynamics:
Gametes are designated by i, their frequencies among zygotes in consecutive generations by pi and p'i, and the fitness of a zygote consisting of gametes j and k by Wjk (we do not indicate the frequency and density dependence). Let R(jk i) denote the probability that a randomly chosen gamete produced by a jk individual is i. The function R is determined by the pattern of recombination between loci. Since random mating is assumed and gamete frequencies are measured after reproduction and before selection, Hardy-Weinberg proportions obtain and the genetic dynamics can be described in terms of gamete frequencies by the well-known system of recursion relations
 | (6) |
is the mean fitness (e.g., BüRGER 2000).
The ecological dynamics follow the standard recursion
 | (7) |
and VA = 0, the classical discrete logistic equation is obtained, N' = N( – N/). As is well known, monotone convergence to the carrying capacity K = ( – 1) occurs if and only if 
2 and oscillatory convergence occurs if 2 < 3 (e.g., BüRGER 2000). Throughout this study, we are concerned only with sufficiently small values so that the ecological dynamics are simple; i.e., convergence to a unique equilibrium population size occurs. Also the initial N0 was chosen sufficiently small (except for exploratory reasons, N0 = K) so that population size always remained positive. Because this study is devoted to the equilibrium structure and the equilibrium population size is uniquely determined if 3, this choice of N0 is no restriction. |
THE STATISTICAL APPROACH |
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TOPABSTRACTTHE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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We proceeded as follows. For a given set of ecological parameters (, , V, , Vs), a given number n of loci, and a given range of recombination rates, we constructed 1000 of what we call genetic parameter sets (allelic effects of loci and recombination rates between adjacent loci from the given range). For each genetic parameter set, allelic effects were obtained by generating values ß ( = 1, 2, ... , n) as independent random variables, uniformly distributed between 0 and 1, and transforming them into the actual allelic effects, 
= 1/2ß/
kßk. The additivity assumption yields the genotypic values, and from Equations 1, 3, and 4, the genotypic fitnesses Wjk are calculated in each generation. Recombination rates between adjacent loci, r,+1 ( = 1, ... , n – 1), were either assumed to be all 1/2 or obtained as independent random variables, uniformly distributed between 0 and 0.01. We assumed absence of interference and refer to these two scenarios as free recombination and tight linkage, respectively.
For each of such constructed genetic parameter sets, the recursion relations (6) and (7) were numerically iterated starting from 10 different randomly chosen initial gamete distributions. To make the initial distributions more evenly distributed in the gamete state space, they were chosen such that the (Euclidean) distance between any two of them was no less than a predetermined value (0.25, 0.30, and 0.35 for two, three, and four loci, respectively). An iteration was stopped after generation t when either an equilibrium was reached (in the sense that the geometric distance between gamete distributions in two consecutive generations was <10–12), or t exceeded 106 generations (in some cases even 107). If equilibrium was not reached, the parameter combination was excluded from the analysis. Usually, the proportion of excluded runs was small enough not to introduce a bias. In a small region of ecological parameters (the so-called transitory region, see next section), extremely slow convergence was the rule because of very flat fitness landscapes. Since for some parameter combinations in this region only 
20% of the runs converged within 106 generations, we stopped runs that had not converged only after 107 generations. At that time 90% had fulfilled our convergence criterion. If convergence within the specified maximum number of generations did not occur, it was because of extremely slow convergence of allele frequencies. Apart from a very flat fitness function, the main reason for slow convergence was the presence of alleles of extremely small effect. No instance of complicated dynamic behavior was detected.
For each parameter combination with n 4 loci, all statistics are based on 1000 genetic parameter sets that led to equilibration; for n = 5 loci they are based on 500 such genetic parameter sets. For each parameter set we calculated the number of different equilibria, the gamete frequencies at each equilibrium, and the number of trajectories (initial distributions) converging to each equilibrium. Using this database, the equilibrium properties were analyzed. Whenever we use the term equilibrium without qualification, we mean a (locally asymptotically) stable equilibrium, unless otherwise mentioned.
Most of our numerical results are for two, three, and four loci; only a few are for five or more because some four- and most five-locus parameter combinations are extremely time consuming. With five loci, for instance, 1600 generations take 1 sec on an AMD Athlon processor with 1.3 GHz. Since with five loci, typical runs for weak or very strong FD equilibrate only after several hundred thousand generations (for intermediate FD more than 10 times as many), 10 initial conditions for each parameter combination are taken, and average statistics are performed over 500 genetic parameter sets, the computing time was correspondingly long. The total computer time for the project was equivalent to 2 CPU years, but we used several computers.
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PROPERTIES OF GENETIC VARIATION AT EQUILIBRIUM |
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TOPABSTRACTTHE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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For each parameter combination, we recorded the equilibria to which at least 1 of the 10 trajectories converged, their number, the number of trajectories converging to a given equilibrium, the number of polymorphic loci at each equilibrium, the (additive) genetic variance, VA, and the genic variance, VLE (i.e., the variance that would be observed under linkage equilibrium), at equilibrium. As a measure for global linkage disequilibrium, we use the ratio VA/VLE. To facilitate the comparison of genetic systems with different numbers of loci, we calculated the ratio Vr = VA/Vmax of the genetic variance and the maximum possible variance, Vmax, in the given genetic system under the assumption of linkage equilibrium.
These values were then averaged over all 1000 genetic parameter sets, and standard deviations were calculated. This yielded our "quantities of interest" for each set of ecological parameters, number of loci, and recombination scenario. The data presented in the figures and tables are such averages. We denote the average over Vr by 
r and refer to it as the relative genetic variance. Its use is preferable when comparing systems with different numbers of loci, because the variance itself is strongly dependent on the average effect among loci, which decreases in proportion to 1/(2n). For a given number of loci, the relative genetic variance r and the (average) genetic variance A behave very similarly (results not shown). Because 
, the expectation (and in principle the whole distribution) of Vmax can be calculated for each n. For n = 2, 3, 4, we have 
, and 
0.041, respectively. Multiplying r by E[Vmax] yields an estimate of A that typically is within 10% of the "true" value (results not shown). The polymorphism displayed in the figures is the average number of polymorphic loci at a stable equilibrium, the average being taken over all 10 trajectories in all 1000 genetic parameter sets.
For later use we note that for any number of loci we have Vmax 
, where the maximum is attained if the alleles at one locus have effects 0 and 1/2 and those at other loci have no effect, and VA 1/8, where the maximum is attained if there are only the two gametes with effects 0 and 1/2 in the population, each with frequency 1/2, because zygotes are in Hardy-Weinberg proportions.
For a given number of loci and given recombination scenario, the genetic parameter sets as well as the initial conditions are the same for all ecological parameter sets. Therefore, variation among quantities of interest comes almost exclusively from variation in the ecological parameters. Only the exclusion of slow runs leads to slight variation among the genetic parameter sets used for different ecological parameter combinations.
An important role in this article is played by the quantity
 | (8) |
separately; however, to leading order in 1/Vs (the strength of stabilizing selection) and 1/V (the strength of competition), they depend only on f. In addition, we introduce the compound demographic parameter
 | (9) |
The symmetric case:
We begin by discussing numerical results for the symmetric case in which the optimum coincides with the genotypic value of the completely heterozygote genotypes, i.e., = 1/2. The demographic parameters here are = 2 and = 10,000. This value of ensures that rapid convergence of N to the carrying capacity occurs. The role of and is studied further below.
Figure 1 visualizes some of the most important general findings. It shows a distinct, threshold-like dependence of the (average) relative genetic variance and the (average) amount of polymorphism on f as f increases from values <1.0 to values > 1.2. This threshold is more pronounced the larger the number of loci is. The genetic variance is slowly, but steadily, increasing with f if f < 1. Then a rapid increase occurs in a narrow interval, and for free recombination the genetic variance nearly levels off as f exceeds 1.2. For larger values of f and free recombination, the genetic variance is very slightly higher than Vmax (actually, very slowly increasing as f increases further) and nearly independent of the number of loci. The average amount of linkage disequilibrium is very small in this case, even for large values of f; i.e., the average VA/VLE is always <1.03 (results not shown). For tight linkage, the genetic variance increases to values much higher than Vmax as f increases above 1.0. The increase beyond Vmax is almost solely due to the build-up of strong positive linkage disequilibrium; the more loci, the higher the linkage disequilibrium. If f 1.25, we have VLE = Vmax under both recombination scenarios because then a unique stable, fully polymorphic equilibrium is maintained (see Tables 1 and 2) at which all allele frequencies are 1/2. Therefore, our measure VA/VLE for linkage disequilibrium equals Vr if FD is strong. For weak or moderate FD (f < 1), linkage has only a minor, in general diminishing, effect on the genetic variance (the more loci, the weaker the effect). As under pure stabilizing selection (BüRGER and GIMELFARB 1999), not only the (absolute) genetic variance but also the relative variance decreases with increasing number of loci if FD is weak (for some five-locus data see Table 3).
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1.25 all equilibria are completely polymorphic, irrespective of recombination. They are also unique (Tables 1–3) and, as already noted, symmetric in the sense that all allele frequencies are 1/2. They differ, of course, in their linkage disequilibria. If f < 1 and recombination is free, then the average degree of polymorphism is <1 but steadily increasing and nearly independent of the number of loci. As f increases beyond 1, a marked increase of polymorphism to the maximum possible amount occurs. For tight linkage and f < 1, on average equilibria exhibit more polymorphism than for free recombination, and for a range of values f the amount of polymorphism maintained may even decrease as f increases. This latter phenomenon is more pronounced if stabilizing selection is stronger (results not shown) and was demonstrated analytically for two loci (BüRGER 2002a,b). The bars in different shades of gray between the top and bottom of Figure 1 indicate the shape of the fitness function at equilibrium for the corresponding value f (see DISRUPTIVE vs. STABILIZING SELECTION).
Tables 1 and 2 provide detailed insights into the effects of increasing FD on the equilibrium structure. For free recombination (Table 1), no stable equilibria with two or more polymorphic loci were found if f < 1. As Table 1 shows, the introduction of FD leads to a steady increase of single-locus polymorphisms at the expense of stable monomorphic equilibria until a complete restructuring of the equilibrium pattern occurs between f = 1 and f = 1.25. If f > 1, stable monomorphic equilibria cease to exist and as soon as f 1.25, there is a unique stable, fully polymorphic equilibrium. Table 2 shows that similar behavior occurs with linked loci, the difference being that with linkage and weak or moderate FD, stable equilibria may exist at which two or more loci are polymorphic. But again, single-locus polymorphisms are increasing with increasing f at the expense of stable equilibria with no or more than one polymorphic locus. As in the case of free recombination, if f > 1, stable equilibria cannot be monomorphic, and if f 1.25, there is a unique fully polymorphic equilibrium with all allele frequencies equaling 1/2. We call the region in which the degree of polymorphism and genetic variance increase rapidly the transitory region.
Table 2 shows that for f < 1, multilocus polymorphisms, provided they exist, always have negative linkage disequilibrium. In the transitory region, polymorphisms with negative, zero, and positive linkage disequilibrium may exist, whereas for large enough f all equilibria exhibit positive linkage disequilibrium. In the transitory region, the equilibrium structure may be rather complex and several stable equilibria with different degrees of polymorphism may coexist for certain parameter combinations. For instance, with four loci, free recombination, Vs = 1.25, and f = 1.04 (as in Figure 1 and Table 1), an unsymmetric four-locus polymorphism coexisting with six two-locus polymorphisms was found. A detailed analytical study of the two-locus model with = 1/2 was performed by BüRGER (2002a)(b). It provides a good guide for shaping the intuition on how the equilibrium structure changes under increasingly strong FD.
The occurrence of multiple stable equilibria for weak FD is consistent with previous results of pure stabilizing selection (e.g., BARTON 1986; BüRGER and GIMELFARB 1999). The reason is that the phenotypes of several homozygous genotypes may closely match the optimum of the stabilizing-selection fitness function and, hence, may be locally stable. Similarly, single-locus polymorphisms may be stable if both homozygous genotypes code for phenotypes close to the optimum. The equilibrium configurations for strong FD are discussed below in PHENOTYPIC DISTRIBUTIONS AT EQUILIBRIUM.
Without showing the results, we remark that the average deviation of the equilibrium mean genotypic value from the optimum is always very small. It is maximal in the absence of FD, reaching 0.39 
for two loci and 
for four loci (the maximum possible deviation is 0.5). It decreases with increasing number of loci and increasing strength of FD. If f > 1, then all deviations are <0.005; if f 1.25, they are zero because the allele frequencies are 1/2 at equilibrium.
Results analogous to Figure 1 and Tables 1 and 2 were obtained by fixing V (= 0.5, the strongest possible competition in our model) and varying Vs across its full range (not shown). The main difference is that for small Vs tight linkage has a somewhat stronger effect (in the direction predicted from the above results). Results for five loci were obtained only for some parameter combinations. Therefore, they are summarized separately in Table 3.
Analytical estimates of the transitory region:
The multilocus model is too complex to admit a complete mathematical analysis. However, the stability of the monomorphic equilibria can be determined, and this provides analytical insight into the dependence of the transitory region on the various parameters in the model. Let 
denote the value of the compound demographic parameter 
(9) at the equilibrium population size 
. In the APPENDIX, we prove that there is a critical value 
S, which can be written as
 | (10) |
S no stable monomorphic equilibrium can exist, whatever the allelic effects and the recombination rates are. If f < S, stable monomorphic equilibria may exist for appropriate allelic effects and recombination rates. Importantly, the critical value S is independent of the number of loci and typically only slightly larger than . For the parameters used in Figure 1 and Tables 1 and 2, i.e., = 1/2, Vs = 1.25, (10) yields 
. Because for this range of values f (i.e., f near one) we have 10,000 10,200 (Table 4), Equation 9 gives 1 0.96, which yields a S between 1.03 and 0.99, respectively, in accordance with our numerical findings.
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D, in the present notation given by
 | (11) |
D a single, globally stable, fully polymorphic equilibrium exists at which all allele frequencies are equal to 1/2. Although we cannot prove this for more than two loci, our numerical results suggest that it continues to be true. The two-locus result implies that f > D is necessary for the existence of a uniquely determined, globally stable, fully polymorphic equilibrium for all possible locus effects. Straightforward manipulations of the conditions f = S and f = D yield simple formulas for the corresponding critical values of V and Vs in terms of the remaining parameters (see, e.g., the last paragraph of the APPENDIX). For the parameters used in Figure 1 and Tables 1 and 2, (11) yields D = 1.33. Because in this range of values f, is between 10,200 and 10,500 (Table 4), this gives a D between 1.28 and 1.21, respectively, which again is in accordance with our numerical findings.
The role of the demographic parameters:
Before investigating asymmetric selection, we examine how the properties of our model depend on the demographic parameters and and derive an approximation for the population size expected at equilibrium. As indicated by Equations 10 and 11, a crucial role is played by the compound demographic quantity .
An estimate for the equilibrium population size can be obtained from (7) by the condition 
. From (4) it is straightforward to compute an explicit expression for 
in terms of the first four moments of the phenotypic distribution at equilibrium. If the mean coincides with the optimum, 
, as is almost always satisfied to a close approximation in our numerical results (in particular, if more than two loci are involved), the following representation is obtained,
 | (12) |
Vs, and equating the resulting expression to one, we obtain for the equilibrium population size the approximation
 | (13) |
K = ( – 1) if and only if VA = 0 or
 | (14) |
1/8 , V 1/2, and Vs 1/8 if = 1/2). Table 4 shows that (14) predicts the value f at which = K nearly correctly (although 
is assumed in its derivation), namely f = 0.5 if = 2. Second, numerical evaluation of (13) shows that it indeed provides a very accurate approximation of the numerically observed average (compare the last two columns in Table 4). This is also the case for all other parameter combinations of Table 4, as well as for many others (e.g., for other values of and ; results not shown).
Differentiation of (13) with respect to V shows that is decreasing in V if and only if > 2Vs/(2Vs – VA), which is only slightly larger than one. Hence, increases with f unless is very small. Table 4 confirms this. This is in accordance with ASMUSSEN (1983), who showed for rather general one-locus models of FDS that the population size at a stable interior equilibrium can exceed the carrying capacity if intraspecific competition between like genotypes is stronger than that between unlike genotypes. In SLATKIN's (1979) quantitative-genetic model of Lande's kind, the population size at the equilibrium with nonzero genetic variance is proportional to 1/
.
Equation 13 predicts that the equilibrium population size is always proportional to , and this has been verified numerically (results not shown). Another important point is (Table 4) that for a wide range of values f, the equilibrium population size deviates from K by only a few percent [cf. (13) and recall that on average VA is very small]. Therefore, we have
 | (15) |
= 2. Table 4 also shows that the deviation of from K decreases with an increasing number of loci. This is confirmed by (13) because (unless linkage disequilibrium is extremely high) in our model the average VA decreases with increasing number of loci since the average effect of a locus decreases in proportion to 1/(2n).
Numerical iterations for various ecological parameter sets in which has been changed by a factor of 10 show that the only statistically significant effect on the properties of the system is that the carrying capacity and, hence, the equilibrium population size are changed by the corresponding factor. Similarly, different values of induce exactly the transformation indicated by (10), (11), and (13), but have no statistically significant effect on the relative genetic variance, the number of equilibria maintained, or their degree of polymorphism (results not shown). For > 3, the dynamics of N become complicated and were not studied in detail. However, in the few runs performed the dynamics of gamete frequencies seemed to be nearly unaffected by the fluctuations of the population size. A similar observation was reported by CLARKE and BEAUMONT (1992).
If density dependence is ignored and the population size is assumed constant and equal to K, then up to a scale transformation of f by the factor 
, the same results as for a changing population size are obtained. For instance, for a fixed population size of 10,000, the relative genetic variance and the polymorphism at some value f are about the same as those for variable N at the value f (results not shown). Thus, for fixed population size, the critical values S and D are somewhat larger than those under density dependence because, by (14), > K if f > 1/2( – 1)–1. Hence, FD must be slightly stronger to dominate pure stabilizing selection.
A potentially important conclusion resulting from Equations 10, 11, and 15 is that for small growth rates , much larger f, hence stronger competition, is needed to induce a qualitative change in the equilibrium structure because D > S > –1. This is not immediately obvious for models based on the Lotka-Volterra functional form (e.g., SLATKIN 1979), in which the location of this region would depend on 
2k/V, where 2k is the width of the available resources. However, since Equation 19 of BüRGER (2002b) informs us that 2k is proportional to Vs with proportionality constant /N – 1, the models indeed lead to equivalent predictions.
These multilocus results confirm and generalize the conclusion of BüRGER (2002a)(b) that, compared with pure stabilizing selection, FD leads to a quantitative, but not to a qualitative change, in the equilibrium properties of genetic variation as long as f < – 1. Only larger f induces a qualitative change. Notably, the critical value at which this occurs is independent of the number of loci.
The asymmetric case:
Now we turn to the asymmetric case and explore the consequences of a shifted optimum. Even in the absence of FD, i.e., for pure stabilizing selection, very little work on the asymmetric case has been done and no simple, general results are available (HASTINGS and HOM 1990; GAVRILETS and HASTINGS 1993; BüRGER 2000, pp. 213–216). Our numerical results are based on the assumption of free recombination and that the position of the optimum is at = 0.75. The main findings are summarized in Figure 2 and Tables 5 and 3, which compare the asymmetric case with the corresponding symmetric case from Figure 1 and Table 1. Figure 2 clearly demonstrates the existence of a transitory region and shows that its lower bound is nearly unaffected by the shift in the optimum, as predicted by our theoretical estimate (10) (for = 0.75, we obtain 
instead of the previous 1.03, and is nearly identical in both cases). However, although there is a distinctive increase of both the relative genetic variance and the polymorphism in a small range of values f, there is no clear upper bound of the transitory region. Instead, both quantities increase very slowly until f = 2.5 is reached. For the given strength of stabilizing selection (Vs = 1.25), this is the largest possible value in this model and attained if V = 0.5.
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(compare Table 5 with Table 1, and see Table 3) and, except for two loci, asymmetry leads only to a slight loss of genetic variance. For five loci, the difference nearly vanishes. If FD dominates (f > 1.1), then a shifted optimum leads to substantially reduced genetic variance and polymorphism relative to a symmetric one. Under the strongest possible FD in our model, the relative genetic variance and the average amount of polymorphism still are diminished by 20% relative to the symmetric case. Even for such strong FD, a substantial fraction of stable equilibria exists at which not all loci are polymorphic. Actually, the more loci there are, the lower the fraction of fully polymorphic equilibria. The relative genetic variance depends only very weakly on the number of loci. The deviation of the mean from the optimum is about as small as in the symmetric case (results not shown).
Thus, for weak or moderate FD (f S), free recombination, and three or more loci, a shift in the optimum of small or moderate size (i.e., 0.25 0.75 so that is closer to the middle of the phenotypic range than to its boundaries) has only a minor effect on the amount and structure of the genetic variation at equilibrium. For two loci, the effect is much larger because a shifted optimum destroys the inherent symmetries of the two-locus model that maintain much more relative genetic variance than that in models with three or more loci (see BüRGER and GIMELFARB 1999).
For some ecological parameter combinations, results were also obtained for an optimum at the upper boundary of phenotypic values, i.e., for = 1. Then the trait is under pure directional selection if FD is absent. We found that for free recombination, four loci, Vs = 1.25, and the strongest possible FD, i.e., f = 2.5, the relative genetic variance is only 0.4, the average polymorphism is 2.0, and four-locus polymorphisms have a frequency of <0.02. For weak FD (f = 0.83), no polymorphism at all is maintained. This is consistent with the finding of LOESCHCKE and CHRISTIANSEN (1984) that for a single locus with multiple alleles, even under strong FD, no variation is maintained if the trait is under directional selection.
We also computed standard deviations of our quantities of interest. For three, four, and five loci, the standard deviation of the relative genetic variance is nearly twice the mean in the absence of FD and decreases to a value close to the mean as f increases to S. This holds for both recombination scenarios and for = 0.5 and = 0.75. Hence, genetic variation between genetic systems varies greatly for weak to moderate FD. Under free recombination, the standard deviation decreases further in absolute terms as f increases and is always <2% of the mean if f > D. Hence, the genetic variance maintained under strong FD is nearly independent of the locus effects. For tight linkage, the standard deviation of the variance is between 15 and 20% of the mean if f > D; thus in absolute terms it increases for large f.
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DISRUPTIVE VS. STABILIZING SELECTION |
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TOPABSTRACTTHE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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-shaped (concave with a local maximum), resulting in stabilizing selection, 
-shaped (convex with a local minimum), resulting in disruptive selection, or else, which we call complicated selection. The fitness function was calculated at each stable equilibrium and classified according to one of the three categories. For a symmetric optimum ( = 0.5), a fitness function classified as complicated is M-shaped in all cases examined; i.e., there are two local optima (not at the extreme genotypic values) separated by a local minimum. The shape of the M, however, can be rather variable because the minimum can be very shallow or very deep, and the fitness of the extreme genotypic values can be very low or just slightly lower than the maximum. The M is asymmetric if 
. For an asymmetric optimum or at unstable equilibria in the symmetric case, other shapes may occur, such as monotone decreasing or increasing fitness, or one local maximum and one local minimum. In natural populations, the shape of a fitness function is difficult to determine exactly and often approximated by a quadratic polynomial through regression. Therefore, we also approximated our fitness function by a quadratic polynomial and classified it as stabilizing selection if it was concave (curved downward) and as disruptive otherwise.
In Figures 1 and 2, the horizontal bars below the top section display the dependence of the shape of the fitness function on f. Except for the region in which selection is complicated, the same type of selection occurs at all equilibria and for all genetic parameter combinations. The coincidence between the type of selection acting and the properties of genetic variation maintained at equilibrium is remarkable. The figures clearly show that complicated selection occurs in the transitory region, i.e., if S < f < D. Under the assumption that the mean coincides with the optimum, (as is satisfied to a close approximation for the vast majority of parameter combinations if more than two loci contribute to the trait—the more loci, the better the approximation), it is straightforward to show that W(g) is -shaped if and only if
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, (16) is satisfied for all possible genetic parameter combinations if and only if f < .
Another straightforward calculation shows that if 
(as is approximately the case in our numerical results) disruptive selection occurs if and only if the derivative of W(g) at g = 0 is positive. This is the case if and only if
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= 1/2 and VA = 1/8, which is the largest possible value of VA in Hardy-Weinberg equilibrium, the right-hand side of (17) coincides with D (11). Therefore, disruptive selection occurs for all possible genetic systems in this model if f > D. These results suggest that in the symmetric case, disruptive selection should be almost always detectable if FD is strong enough to maintain a unique fully polymorphic equilibrium.
For the asymmetric optimum, the region with complicated selection is substantially extended and much stronger FD is needed to produce disruptive selection (Figure 2). This is also predicted by (17), which gives a critical value of f between 1.7 and 1.9 if VA = 1/8 and the numerically obtained 's (in this region between 10,300 and 10,800) are plugged in. In the transitory region (and only there), the fitness function is often not only complicated, but also very flat near equilibrium, thus leading to extremely slow convergence (cf. THE STATISTICAL APPROACH).
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EVOLUTION OF MEAN FITNESS |
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TOPABSTRACTTHE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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For the present multilocus model we investigated such and related issues using our statistical approach. For every trajectory, we calculated the ratio 
/W0 of the mean fitness at equilibrium (which, of course, is one) and the initial mean fitness and averaged /W0 over all 1000 genetic parameter sets pertaining to a given ecological parameter combination. Because initial points are randomly chosen, the average ratio 
provides a measure for the net change of population mean fitness during evolution. We also calculated the percentage of trajectories, %RF, for which mean fitness at equilibrium was lower than that initially. Some of these results are summarized in Table 4.
A number of noteworthy features are observed. As f increases from zero, the average ratio decreases from a value > 1 to 1.000 as f reaches 1.0 ( S). In parallel, the proportion of trajectories for which fitness is reduced at equilibrium increases from nearly zero (except for two loci, when this proportion is a few percent) to 10% or more. This percentage reaches a maximum of 40% or more in the transitory region and declines as f increases beyond D. For free recombination this decline is very slow, but for tightly linked loci it is rapid; the more loci, the more rapid the decline. decreases slightly below one for f > D if recombination is free, but increases quite substantially if linkage is tight. Therefore, under moderate or strong FD (f > S) the dynamics are "on average" nonadaptive if the loci are unlinked. However, if the loci are tightly linked and FD is very strong, then mean fitness decreases only with low probability and, on average, increases substantially above that of randomly chosen initial distributions. As we see by way of example in the next section, this is because with tightly linked loci, the phenotypic distribution shows signs of clustering at its extreme phenotypic values.
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PHENOTYPIC DISTRIBUTIONS AT EQUILIBRIUM |
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TOPABSTRACTTHE MODELTHE STATISTICAL APPROACHPROPERTIES OF GENETIC VARIATION...DISRUPTIVE VS. STABILIZING...EVOLUTION OF MEAN FITNESSPHENOTYPIC DISTRIBUTIONS AT...DISCUSSIONAPPENDIX: ACKNOWLEDGEMENTSLITERATURE CITED |
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