1. Polymorphism in Time-Varying Environments Claus Rueffler 1,2, Hannes Svardal 1, Peter A. Abrams 2 & Joachim Hermisson 1 MaBS Mathematics and Biosciences Group 1 Department of Mathematics, University of Vienna 2 Dept. of Ecology and Evolutionary Biol., Univ. of Toronto

2. Overview Protected polymorphism in time-varying environments –the classical view –the generalized Lottery-model –the storage effect Evolutionary stability of protected polymorphism Evolutionary stability of protected polymorphism against the invasion of mixed strategies (bet-hedging) –the quick answer –a closer look the role of initial proportions the role of genetic correlations Phenotypic plasticity in time-varying environments –an integrative model-framework –bet-hedging and pure plasticity as limiting cases –full model Summary

3. Maintenance of Genetic Polymorphism in Time-Varying Environments The traditional view: -> temporally fluctuating selection does generally not promote maintenance of genetic variation (review: Hedrick 1986) -> quantitative genetics: random temporal variation in phenotypic fitness increases genetic variance only slightly or not at all (Lande, Barton, Turelli) Based on models with discrete and non-overlapping generations

4. The Storage Effect Chesson and Warner (1981, 1985) Random temporal variation can promote coexistence of different genotypes in models with overlapping generations: the storage effect prerequisites -> fluctuating genotype specific recruitment rates -> long-lived life-history state (adults or seeds/diapausing eggs)

5. The Generalized Lottery Model N : generation overlap H : probability to reproduce s r : survival of reproducing individuals s d : survival of “dormant” individuals : # surviving offspring in environment e t Two special cases 1) Long-lived Adults: 2) Long-lived Seeds, annual Adults:

6. -> K sites (patches) exist -> each season (1-γ)K sites are cleared -> all offspring compete for empty sites -> offspring produced by genotype z i given environment e t Assume two coexisting types z 1 and z 2 -> per capita contribution of persisting individuals -> population dynamics of phenotype z 1 that coexists with type z 2 Lottery Competition for Empty Sites

7. Long-term average per capita growth rate of a rare mutant with phenotype z m in a resident population with phenotype z r where Invasion Fitness

8. Invasion fitness: mutant doomed to extinctionmutant able to invade resident cannot invade mutant no mutual invasion mutant goes to fixation mutual invasion mutant and resident can coexist in protected dimorphism What invasion fitness can tell us

9. The Storage Effect: a Numerical Example I Example (long-lived adults): s r =0 ( ⇒ γ=0: no generation overlap) The genotype with the highest expected offspring number wins no coexistence

10. The Storage Effect: a Numerical Example II Example (long-lived adults): s r =0.5 ( ⇒ γ=0: overlapping generations), K=102 -> per capita growth rate under favorable conditions is highest for a type at low density -> maximal loss under bad conditions is set by adult death rate -> offspring from years with good recruitment are “stored” for many years -> rare types protected from extinction protected polymorphism possible genotype z 1 genotype z 2 density at t=0 density at t=1 given environment e 1 environment e 2 density after adult mortality 100 2 50 1 101 1 50 52 For simplicity we assume: - only two environmental conditions exist - each genotype fails completely to reproduce in the “wrong” environment

11. Protected Polymorphism does not Imply Evolutionary Stability Evolutionary stability: no mutant genotype exists that can invade the protected polymorphism and thereby drive both resident types to extinction We answer this question by asking the opposite question: When does invasion of a resident genotype result in a genetic polymorphism? When is a protected polymorphism evolutionary stable?

12. Environmental Variation and the Environment-Phenotype Match continuous range of environmental conditions (e.g. date of first frost) two environmental conditions (e.g. early vs. late frost) strength of stabilizing selection : selection variance # surviving offspring decreases with increasing mismatch of phenotype z and environment e t probability distribution of environmental conditions : environmental variance c=2 c=1/2 c=1 trade-off curvature determines performance of a generalist c<1 : convex (strong) trade-off c=1 : linear trade-off c>1 : concave trade-off environmental variation imposes trade-off

13. Relationship Between the Models with Continuous and Discrete Environmental Conditions In the following conditions for convergence and invadability will be given for the model with continuous environments in terms of and. The equivalent conditions for the model with two resources can be obtained by replacing: In the second part of the talk (Hannes) results will be given for the model with two environmental conditions.

14. Genetic polymorphism evolutionary stable (z* is branching point): environmental variance, discounted by generation overlap, larger than strength of stabilizing selection Analysis of the candidate ESS z* : selection variance : environmental variance In the limit of complete generation overlap (γ→1): time-varying environments equivalent to MacArthur-Roughgarden resource-competition model

15. Evolutionary Stability Against Mixed-Strategy Bet-Hedging  canalized genotype-phenotype map: one genotype ⇒ one phenotype  Alternative: one genotype produces different phenotypes in some fixed proportion  Mixed strategy two phenotypes z 1 and z 2 are produced with probability q and 1-q Genotype: (z 1,z 2,q) Idea: Bet-Hedging - insurance against environmental uncertainty - geometric mean increases with decreasing between-year variance Question Can a protected and evolutionary stable genetic polymorphism be invaded by a mixed strategy?

16. Heterotheca latifolia (Asteracae) disk achene ray achene disk achene: -immediate germination -wind dispersed ray achene: -delayed germination -poor dispersal Seed heteromorphism (different seeds on same plant) (common in some families) Aegilops speltoides (Poaceae) speltoides typeligustica type dispersal polymorphism one-locus with two alleles ligustica dominant over speltoides Within-population genetic polymorphism (few examples)

17. Only if mixed strategies are not possible genetic polymorphism are evolutionary stable. With this perspective genetic polymorphism are the result of constraints on the genotype-phenotype map. Evolutionary Stability Against Mixed Strategies Closer look: How does a mixed strategy evolve de novo? IN OUR MODEL Any evolutionarily stable polymorphism can be invaded by a mixed strategy, where the phenotypes in the mixture equal the phenotypes in the polymorphism and where q is arbitrary. (follows immediately from Jensen’s Inequality)

18. Accommodating Mixed Strategies:Higher-Dimensional Trait Spaces q unspecified Evolutionary properties of the singular strategy Convergence Stability Jacobian of the selection gradient (derivative of fitness gradient) Invadability Selection Hessian of invasion fitness (2nd derivative of invasion fitness) Evolution in higer-dimensional trait spaces ⇒ selection (gradient) ⇒ covariance between traits (variance-covariance matrix) on the diagonal: q is neutral

19. singular point in the (z 1, z 2 )-plane (here (z*,z*)=(0.5,0.5)) Invadability of Singular Points in the (z 1,z 2 )-plane Maximum uninvadable Saddle Point invadable in some directions Minimum invadable stabilizing selection weaker than environmental variance stabilizing selection stronger than discounted environmental variance intermediate strength of stabilizing selection

20. Convergence Stability of Singular Points in the (z 1,z 2 )-plane Putting invadability and convergence together : uninvadable + convergence stable => CSS whereby convergence stability and invadability (almost) exclude each other evolution of a single bet-hedging genotype : convergence stable and invadable in some directions : convergence stable in some, invadable in all directions

21. Convergence Stable in Some, Invadable in all Directions direction of the steepest slope of the fitness landscape (dominant eigenvector) direction with convergence => mutants in this region coexist with singular strategy q=0.5q=0.9q=0.7 Observation: evolutionary properties of singular point strongly depend on q, the initial proportion of the two types in a mixed strategy

22. Constraints for the Evolution of Mixed Strategies I initial q-value direction of the steepest slope of the fitness landscape (dominant eigenvector) direction with convergence => mutants in this region coexist with singular strategy direction through the singular point For q 0.75 the steepest increase in the fitness landscape points into a direction resulting in the coexistence of mixed strategies

23. Full model: Phenotypic Plasticity vs Bet-hedging vs Genetic Polymorphism only most specialized phenotypes are produced : both environmental cue and internal noise influence phenotype determination! : developmental noise contributes equally to and : time-varying environments with imperfect cues favors a mixture of bet-hedging and plasticity no branching points => no genetic polymorphism Single convergence stable and uninvadable strategy:

24. (1-r) r

25. Long-Term Dynamics of Two Mixed Strategies II Hypothesis ⇒ initially constraints may limit the independent evolution of z 1 and z 2 ⇒ a mutation changing z 1 is likely to change z 2 in the same direction positive correlation between z 1 and z 2

26. Genetic polymorphism favoured by ⇒ asymmetric q ⇒ positive covariance between z 1 and z 2 q covariance Long-Term Dynamics of Two Mixed Strategies II

27. Developmental Switch “Development is phenotypic change in a responsive phenotype due to inputs from the environment and the genome.” (West-Eberhard 2003) Four evolving traits z 1, z 2 : specialization coefficients of alternative phenotypes a : weighting factor for environmental cue e i t : threshold-value of developmental switch Four different model variants (1) z 1 =z 2 Canalized Genotype-Phenotype Map (a and t neutral) (2) a=0: Bet-hedging vs Genetic Polymorphism (3) a=1: Phenotypic Plasticity vs Genetic Polym. (4) Full model

28. Summary: Polymorphism in Time-Varying Environments coexistence of different genotypes is only possible in life-historieswith overlapping generations mixed strategy bet-hedgers can invade genetic polymorphism various constraints on mixed strategies favour genetic polymorphism of pure strategies asymmetric proportions of phenotypes in a mixed strategy positive correlation between the phenotypes in a mixed strategy

29. Summary: Polymorphism in Time-Varying Environments coexistence of different genotypes is only possible in life-historieswith overlapping generations mixed strategy bet-hedgers can invade genetic polymorphism various constraints on mixed strategies favour genetic polymorphism of pure strategies asymmetric proportions of phenotypes in a mixed strategy positive correlation between the phenotypes in a mixed strategy

30. Summary: Polymorphism in Time-Varying Environments coexistence of different genotypes is only possible in life-historieswith overlapping generations mixed strategy bet-hedgers can invade genetic polymorphism various constraints on mixed strategies favour genetic polymorphism of pure strategies asymmetric proportions of phenotypes in a mixed strategy positive correlation between the phenotypes in a mixed strategy

31. Summary: Polymorphism in Time-Varying Environments coexistence of different genotypes is only possible in life-historieswith overlapping generations mixed strategy bet-hedgers can invade genetic polymorphism various constraints on mixed strategies favour genetic polymorphism of pure strategies asymmetric proportions of phenotypes in a mixed strategy positive correlation between the phenotypes in a mixed strategy

32. Summary: Polymorphism in Time-Varying Environments coexistence of different genotypes is only possible in life-historieswith overlapping generations mixed strategy bet-hedgers can invade genetic polymorphism various constraints on mixed strategies favour genetic polymorphism of pure strategies asymmetric proportions of phenotypes in a mixed strategy positive correlation between the phenotypes in a mixed strategy

33. 1 Mendelian genes Within species polymorphism Disruptive Selection And Then What? (Rueffler et al. 2006, TREE) Sexual dimorphism Sympatric species Mixed Strategies

34. Developmental Switch “Development is phenotypic change in a responsive phenotype due to inputs from the environment and the genome.” (West-Eberhard 2003)

35. Developmental Switch “Development is phenotypic change in a responsive phenotype due to inputs from the environment and the genome.” (West-Eberhard 2003) Four evolving traits z 1, z 2 : specialization coefficients of alternative phenotypes a : weighting factor for environmental cue e i t : threshold-value of developmental switch

37. A new perspective on mixed strategies (Leimar 2005) one-dimensional trait space with fitness minimum: genetic polymorphism can evolve trait space of a genotype with mixed strategies one-dimensional trait space embedded as diagonal The curvature of the fitness landscape in the -45° direction measures the strength of selection for a for a genotype corresponding a mixed strategy with and Disruptive selection in the -45°-direction stronger than in the +45°-direction!

39. This Talk  protected genetic polymorphism in time-varying environments consequences of overlapping generation: the storage effect a model family  evolutionary stability of genetic polymorphism invasion analysis condition for evolutionary stability  evolutionary stability and mixed strategies  evolution in higher-dimensional trait spaces constraint I: asymmetric proportions in mixed strategies constraint II: genetic correlations  Summery  the broader context: Disruptive selection and then what?  Extensions

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46. Alternative evolutionary attractors ⇒ one mixed strategy ⇒ genetic polymorphism of two pure strategies Long-Term Dynamics of Two Mixed Strategies I Initial dynamics determine long-term fate

47. Specifying Reproduction: Environmental Variation and the Environment-Phenotype Match Probability distribution of environmental conditions e t (e.g. date of first frost) : environmental variance Strength of stabilizing selection: # surviving offspring decreases with increasing mismatch of phenotype z and environment e t : strength of stabilizing selection (variance of the the selection kernel) : mean of the selection curve determines phenotype

48. Finding a Candidate ESS : strength of stabilizing selection : environmental variance (e t ) Directional selection acts on the genotype until. z* cannot be invaded by any z m ≠z*: fitness maximum ⇒ no evolutionary stable protected polymorphism exists z* can be invaded by some z m ≠z*: fitness minimum ⇒ evolutionary stable protected polymorphism are possible