Sasha Gimelfarb died on May 11, 2004

2. Sasha Gimelfarb died on May 11, 2004

3. Reinhard Bürger Department of Mathematics, University of Vienna A Multilocus Analysis of Frequency-Dependent Selection on a Quantitative Trait

4. 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

5. Frequency-Dependent Selection Caused by Intraspecific Competition

6. 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)

7. 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.

8. Ecological Assumptions • Fitness is determined by two components: • by stabilizing selection on a quantitative trait, • and • by competition among individuals of similar trait value,

9. 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.

10. 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.)

11. 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., .

12. 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.)

13. 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

14. 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?

15. Numerical Results from a Statistical Approach (withA. Gimelfarb)

16. Figure, poly, th=0

17. Figure, poly, th=0, 0.75

18. Figure: poly

19. The Weak-Selection orLinkage-Equilibrium Approximation

21. 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.

29. 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.

30. 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.

31. 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.

32. 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).

33. 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.

34. 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.