Sasha gimelfarb died on may 11 2004
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Sasha Gimelfarb died on May 11, 2004. Reinhard Bürger Department of Mathematics, University of Vienna. A Multilocus Analysis of Frequency-Dependent Selection on a Quantitative Trait. Frequency-dependent selection. has been invoked in the explanation of evolutionary phenomena such as

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Sasha Gimelfarb

died on May 11, 2004

Reinhard Bürger

Department of Mathematics, University of Vienna

A Multilocus Analysis of Frequency-Dependent Selection on a Quantitative Trait

Frequency-dependent selection

has been invoked in the explanation of evolutionary

phenomena such as

  • Evolution of behavioral traits

  • Maintenance of high levels of genetic variation

  • Ecological character divergence

  • Reproductive isolation and sympatric speciation

Frequency-Dependent Selection Caused by Intraspecific Competition

Intraspecific competition mediated by a quantitative trait under stabilizing selection:

  • Bulmer (1974, 1980)

  • Slatkin (1979, 1984)

  • Christiansen & Loeschcke (1980), Loeschcke & Christiansen (1984)

  • Clarke et al (1988), Mani et al (1990)

  • Doebeli (1996), Dieckmann & Doebeli (1999)

  • Matessi, Gimelfarb, & Gavrilets (2001)

  • Bolnick & Doebeli (2003)

  • Bürger (2002a,b), Bürger & Gimelfarb (2004)

The General Model

  • A randomly mating, diploid population with discrete generations and equivalent sexes is considered.

  • Its size is large enough that random genetic drift can be ignored.

  • Viability selection acts on a quantitative trait.

  • Environmental effects are ignored (in particular, GxE interaction). Therefore, the genotypic value can be identified with the phenotype.

Ecological Assumptions

  • Fitness is determined by two components:

  • by stabilizing selection on a quantitative trait,

  • and

  • by competition among individuals of similar trait value,

The strength of competition experienced by

phenotype g (= genotypic value) for a given

distribution P of phenotypes is

where and VA denote the mean and (additive

genetic)variance of P.

If stabilizing selection acts independently of

competition, the fitness of an individual with

phenotype g can be written as

where F(N) describes population growth according

to N´=NF(N). (F may be as in the discrete logistic,

the Ricker, the Beverton-Holt, the Hassell, or the

Maynard Smith model.)

For weak selection ( , f = a/s constant),

this fitness function is approximated by

where is a compoundmeasure for the strength

offrequency and densitydependence relative to

stabilizing selection, i.e., .

Genetic Assumptions

  • The trait is determined by additive contributions from n diploid loci, i.e., there is neither dominance nor epistasis.

  • At each locus there are two alleles. The allelic effects at locus i are -gi/2 and gi/2. (This is general because the scaling constants can be absorbed by the position of the optimum and the strength of selection.)

Genetical and Ecological Dynamics

pi , pi´ : frequencies of gamete i in consecutive generations

Wjk :fitness of zygote consisting of gametes j, k

R(jk->i): probability that a jk-individual produces a

gamete of type i through recombination

: mean fitness

Issues and problems to be addressed

  • What aspects of genetics and what aspects of ecology are relevant, and under what conditions?

  • When does FDS have important consequences for the genetic structure of a population?

  • How does FDS affect the genetic structure?

  • How much genetic variation is maintained by this kind of FDS?

Numerical Results from a Statistical Approach (withA. Gimelfarb)

Figure, poly, th=0

Figure, poly, th=0, 0.75

Figure: poly

The Weak-Selection orLinkage-Equilibrium Approximation

  • The structure is the same as in Turelli and Barton 2004 (but ). The proofs of the results below use their results.

  • The population is assumed to be in demographic equilibrium, i.e., N and η are treated as constants.

  • All models of intraspecific competition and stabilizing selection I know have the above differential equation as their weak-selection approximation.

  • ‘Arbitrary‘ population regulation, i.e., with a unique stable carrying capacity, is admitted.

General Conclusions

  • The amount and properties of variation maintained depend in a nearly threshold-like way on , the strength of frequency and density dependence relative to stabilizing selection.

  • This critical value is independent of the number of loci and, apparently, also of the linkage map.

Weak FDS

  • If more than two loci contribute to the trait, then weak frequency dependence (< 1) can maintain significantly more genetic variance than pure stabilizing selection, but still not much. The more loci, the larger this effect.

  • FDS of such strength does not induce a qualitative change in the equilibrium structure relative to pure stabilizing selection.

  • Such FDS does not lead to disruptive selection, rather, stabilizing selection prevails.

Strong FDS

  • Strong FDS (> 1) causes a qualitative change in the genetic structure of a population by inducing highly polymorphic equilibria in positive linkage disequilibrium.

  • In parallel, such FDS induces strong disruptive selection, the fitness differences between phenotypes maintained in the population being much larger than under pure stabilizing selection.

Disruptive Selection

  • Therefore, disruptive selection should be easy to detect empirically whenever FDS is strong enough to affect the equilibrium structure qualitatively.

  • Its strength (the curvature of the fitness function at equilibrium) is s( – 1).

When Genetics Matters

  • The degree of polymorphism maintained by strong FDS depends on the number of loci and the distribution of their effects.

  • Models based on popular symmetry assumptions, such as equal locus effects or symmetric selection, are often not representative (they maintain more polymorphism).

  • Linkage becomes important only if tight. It produces clustering of the phenotypic distribution. Otherwise, the LE-approximation does a very good job.


  • Include assortative mating to study the conditions leading to divergence within a population (work in progress  K. Schneider).

  • Determine the conditions under which sympatric speciation is induced by intraspecific competition.

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