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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

a multilocus analysis of frequency dependent selection on a quantitative trait
Reinhard Bürger

Department of Mathematics, University of Vienna

A Multilocus Analysis of Frequency-Dependent Selection on a Quantitative Trait
frequency dependent selection
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
intraspecific competition mediated by a quantitative trait under stabilizing selection
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
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
Ecological Assumptions
  • Fitness is determined by two components:
  • by stabilizing selection on a quantitative trait,
  • and
  • by competition among individuals of similar trait value,
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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.

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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.)

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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
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
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
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?
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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
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
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
  • 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
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
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.
outlook
Outlook
  • 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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