1. Sex allocation theory Dominique Allainé UMR-CNRS 5558 « Biométrie et Biologie Evolutive » Université Lyon 1 France allaine@biomserv.univ-lyon1.fr

2. Allocation to sex Sex allocation is the allocation of resources to male versus female reproductive function Definition Charnov, E.L. 1982. The theory of sex allocation

3. Introduction Two types of reproduction concerned : Dioecy: individuals produce only one type of gamete during their lifetime Hermaphrodism: individuals produce the two types of gametes during their lifetime sequential hermaphrodism simultaneous hermaphrodism

4. Introduction Problematic 4.In what condition hermaphrodism or dioecy is evolutionarily stable? 1.In dioecious species, what is the sex ratio maintained by natural selection? 2.In sequential hermaphrodites, what is the order of sexes and the time of sex change ? 3.In simultaneous hermaphrodites, what is, at equilibrium, the resource allocated to males and females at each reproductive event ? 5.In what condition, natural selection favors the ability of individuals to modify their allocation to sexes ?

5. Introduction An old problem « … I formerly thought that when a tendency to produce the two sexes in equal numbers was advantageous to the species, it would follow from natural selection, but I now see that the whole problem is so intricate that it is safer to leave its solution for the future. » Charles Darwin, 1871. The descent of man, and selection in relation to sex. 2nd Edition 1874

6. Fisher’s model The first solution « If we consider the aggregate of an entire generation of such offspring it is clear that the total reproductive value of the males in this group is exactly equal to the total value of all the females, because each sex supply half the ancestry of all future generations of the species. From this it follows that the sex ratio will so adjust itself, under the influence of Natural Selection, that the total expenditure incurred in respect of children of each sex, shall be equal » R.A. Fisher. 1930. The genetical theory of natural selection

7. Two comments 1. It is a frequency-dependant model 2. It is a verbal model How to demonstrate the Fisher’s equal allocation principle? Fisher’s model

8. Concept adapted to ecology by J. Maynard-Smith An ESS is a strategy, noted r* that, if played in the population, cannot be invaded by an alternative strategy s, played by a mutant individual introduced in the population Fisher’s model ESS approach The fitness of an individual playing the strategy s in a population where individuals play the strategy r is noted W(s,r)

9. Fisher’s model ESS approach W(r*,r*) > W(s,r*)  r* is the unique best response to r* r* is an ESS if: Or: W(r*,r*) = W(s,r*) r* is not the unique best response to r*  and W(r*,r) > W(r,r) but r* is a better response to r than r

10. Formalization of the Fisher’s model Consider a population of N females of a dioecious species with discrete generations Each female produces C offspring at each reproductive event Consider S1 and S2 the proportions of males and females that survive to the age of first reproduction Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons We consider: 1. a continuous variable, for example the proportion of males produced 2. a strategy r adopted by the females of the population 3. a strategy s adopted by a mutant female 4. an optimal strategy r* When applied to sex allocation :

11. The offspring of the N+1 females, produce as a whole K offspring The relative contribution of the mutant female to grandchildren through her sons is: (1) (2) Formalization of the Fisher’s model 1. Formalization by Shaw and Mohler (1953) The relative contribution of the mutant female to grandchildren through her daughters is:

12. The relative contribution of the mutant female to genes of grandchildren, that is her relative fitness, is the sum of (1) and (2) (3) If N is large, (3) can be approximated by: (4) This is the Shaw and Mohler equation Formalization of the Fisher’s model 1. Formalization by Shaw and Mohler (1953)

13. Then (4) becomes : (5) This is a generalization of the Shaw and Mohler’s equation Formalization of the Fisher’s model 1. Formalization by Shaw and Mohler (1953)

14. 2. The marginal value criterion Consider a population of N females of a dioecious species with discrete generations Each female allocates an optimal proportion M* of resources to males and an optimal proportion F* = 1-M* of resources to females Consider a mutant female that allocates a proportion M of resources to males and a proportion F = 1-M of resources to females Formalization of the Fisher’s model

15. The relative fitness of the mutant female in a population allocating M* is:  (M) is the competitive ability of males having received an allocation M  (F) is the competitive ability of females having received an allocation F is the genetic profit through males is the genetic profit through females 2. The marginal value criterion Formalization of the Fisher’s model

16. In the Fisher’s model, competitive abilities (  (M) and  (F)) are linear:  (M) = aM and  (F) = bF a and b being constants for example:  (M) = CsS1 and  (F) = C(1-s)S2 with a = CS1 and b = CS2 and allocation is measured by sex ratio (M = s and F = (1-s)) These competitive abilities are often measured by the number of males and females offspring surviving to the age of reproduction 2. The marginal value criterion Formalization of the Fisher’s model

17. It follows that:  2. The marginal value criterion Formalization of the Fisher’s model This is the Shaw and Mohler equation

18. 3. Model of inclusive fitness Formalization of the Fisher’s model Consider a population of N females of a dioecious species with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider S1 and S2 the proportions of males and females that survive to the age of first reproduction Consider a mutant female that produces a proportion s of sons

19. the fitness W of a female is measured by: W = number of adult daughters + number of females inseminated by her sons 3. Model of inclusive fitness Formalization of the Fisher’s model

20. Then, the relative fitness of a mutant female is : 3. Model of inclusive fitness Formalization of the Fisher’s model This is the Shaw and Mohler equation

21. The relative fitness of a mutant female is : Formalization of the Fisher’s model The Shaw and Mohler (1953)’ equation

22. Solution with equal costs of production The question is : Does a mutant female contribute more to the next generation than a non mutant female? In other words : Is the fitness of the mutant female greater than the fitness of a non mutant female? Or, does the « mutant » allele will invade the population ? Or, is the relative fitness of the mutant female W(s,r) greater than 1 ? Fisher’s model

23. What is the optimal sex ratio (allocation)? Solution with equal costs of production Fisher’s model Two conditions are needed: To have an extremum To have a maximum r* is the value such that the fitness W is maximised for s=r=r*

24. r* = 0.5 is an ESS Solution with equal costs of production Fisher’s model

25. 1.The derived does not depend on s 2.If r = r* = 0.5, W’ = 0 whatever the value of s thus W(s,r*) = cte 3.If r = r*, s = r* is the best but not the unique best response to r* 4.If r 0 and s = 1 is the best response to r 5.If r > 0.5, W’ < 0 and s = 0 is the best response to r Solution with equal costs of production Fisher’s model Comments:

26. s ≠ r s = r best response (s) Sex ratio in the population (r) Solution with equal costs of production Fisher’s model

27. The first model assumed that the energetic cost of production of both sexes was the same. Allocation was measured directly by the sex ratio. Fisher (1930) « From this it follows that the sex ratio will so adjust itself, under the influence of Natural Selection, that the total expenditure incurred in respect of children of each sex, shall be equal » What happens if the costs of production of the two sexes differ ? Fisher’s model Model with different costs of production

28. Consider a population of N females of a dioecious species with discrete generations Each female has a quantity R of resources to allocate at each reproductive event Each female allocates a proportion q* of resources to the production of males Consider a mutant female allocating a proportion q of resources to the production of males Fisher’s model Model with different costs of production

29. and then Same form as the model with equal costs Fisher’s model Model with different costs of production

30. The optimal strategy is an equal allocation to males and females This is the Fisher’s prediction ! Fisher’s model Model with different costs of production

31. Because each female has a quantity R of resources to allocate at each reproductive event and because young of the two sexes are not equally costly to produce, q* = 0.5 implies that the Fisher’s equal allocation principle can be written as: n ♂ x C ♂ = n ♀ x C ♀ Fisher’s model Model with different costs of production

32. Fisher’s model does not predict a sex ratio equal to 0.5 in the population if costs of production of the two sexes differ. Costs of production should be used sensu Trivers (1972) that is to say in term of fitness cost and not only in term of energetic cost (cf. Charnov 1979). Fisher’s principle should be rephrased in terms of equal investment rather than of equal allocation Conclusion Fisher’s model

33. Biased sex ratio Local Mate Competition (LMC) Patches of habitat In some species of parasitoids, the environment is made of patches, each patch being occupied by fertilized females

34. Biased sex ratio LMC laying mating laying Offspring born on a patch mate on the patch Then, males die and fertilized females disperse to vacant patches ♀ ♂ ♀ ♀ ♀ ♂ ♂ ♀ ♀ ♂ ♂ ♀ ♀ ♀ ♀ ♂ ♂ ♀ ♀ ♂ ♀ ♂ ♀ ♂ ♀ ♀ ♀

35. Biased sex ratio In this kind of species, there is a local competition between males to fertilize females on the birth patch Males are then the costly sex and a female-biased sex ratio is expected LMC

36. Biased sex ratio To predict the sex ratio in this situation, Hamilton has relaxed one assumption of the Fisher’s model: the hypothesis of a panmictic reproduction Consider a population of n females of a diploid species, dioecious with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons LMC : diploid species (Hamilton 1967)

37. LMC Biased sex ratio It comes :

38. r* is an ESS Sex ratio in the population (r) s best response to r Unique best response n = 3 LMC : diploid species Biased sex ratio

39. r* n LMC : diploid species Biased sex ratio

40. LMC : general solution Biased sex ratio More generally (case of haplodiploid species): Where  depends on inbreeding

41. Many studies on parasitoid species give evidence that the sex ratio may be extremely biased towards females in these species. Biased sex ratio LMC: test in parasitoid wasps

42. Models derived from the LMC: dispersal of sons out of the patch (Nunney et Luck 1988) variable number of females in space and time (Luck et al 1993) model of host quality (Charnov 1982) model of « constrained females» (Godfray 1990) Werren’s model (1980) … LMC Biased sex ratio

43. Local Resource Competition (LRC) Clark (1978) Biased sex ratio tt+1 Male dispersal

44. LRC In some primate species, males disperse early while females stay with their mother beyond sexual maturity. Daughters compete with each other (and with their mother if alive) for resources. There is a local competition for resources between related females Biased sex ratio Females are then the costly sex and a male-biased sex ratio is expected

45. Consider a population of N females of a diploid species, dioecious with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons Competition for resources affects females’ survival. Then the survival of daughters will depend on sex ratio [  (r) or  (s)] LRC Biased sex ratio

46. LRC However,  ’(r*) > 0 => r* > 0.5 Biased sex ratio

47. LRC: test in primates (Clark 1978) Biased sex ratio Galago crassicaudatus From Clark (1978)

48. LRC: test in birds (Gowaty 1993) Biased sex ratio Dispersal female biased Sex ratio female biased Dispersal male biased Sex ratio male biased %males

49. Local Resource Enhancement (LRE) Emlen et al. (1986) Biased sex ratio In cooperative breeders, the sex ratio seems biased towards the helping sex Initially, we thought that helpers help because they are in excess in the population. Being in excess for an unknown reason, individuals of the helping sex do not find mate and they can increase their fitness by helping. However, Gowaty and Lennartz (1985) proposed an alternative interpretation. because they help that helpers are produced in excess They argued that it is because they help that helpers are produced in excess This hypothesis was formalized by Emlen et al. in 1986

50. LRE In cooperatively breeding species, offspring of one sex generally stay in the family group and help parents in raising young. For example, helpers are provisioning food for young (local resource enhancement). The helping sex is less costly in fitness term because it provides a fitness benefit to parents by increasing reproductive success or decreasing the workload of parents. So, helpers reimburse parental investment. Helper repayment model Biased sex ratio ➨

51. LRE Consider a population of N females of a diploid species, dioecious with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons Helpers effect is expressed by a multiplicative coefficient H in the production of offspring Biased sex ratio

52. LRE Helpers’ effects are assumed to be additive !!! Helpers’ effect depends on the sex ratio produced by the mother Biased sex ratio

53. LRE Biased sex ratio Pen & Weissing demonstrated that:

54. LRE In the great majority of cooperatively breeding species, only one sex helps so: The ESS depends only on 2 parameters : 1.Mean number of helpers 2.Contribution of each helper ! Remember that the model assumes that contributions are additive ! Biased sex ratio

55. LRE Biased sex ratio Example : the alpine marmot Marmota marmota Territorial family groups One monogamous dominant pair Subordinates of both sexes (  2 year olds) Yearlings Juveniles: one litter per year (mean = 4) Dominants Subordinates Yearling s Juveniles

56. LRE: test in the alpine marmot Biased sex ratio La Grande Sassière National Reserve French Alps (2340 m a.s.l) Open alpine meadow

57. Biased sex ratio From 1990 to 2002, 20 family groups with 499 marked individuals number, age, sex and social status For each group: number, age, sex and social status of each individuals LRE: test in the alpine marmot

58. Biased sex ratio Test of the Helper Repayment Hypothesis number of subordinate males number of subordinate females juvenile survival We determined the winter survival of 198 juveniles from 53 litters Winter survival of juveniles = 0.78 (95% confidence interval : 0.72-0.84) Subordinate males may be considered as helpers and may reimburse parental investment by warming juveniles during winter LRE: test in the alpine marmot

59. Biased sex ratio Complete sex composition at emergence was determined for 53 litters representing a total of 207 juveniles The overall sex ratio was 0.578 and significantly departed from 0.5 (95% confidence interval [0.511; 0.643]) Sex ratio at emergence Sex ratio at birth Five females in captivity gave birth to 22 sexed neonates 13 were males giving an overall sex ratio of 0.59 LRE: test in the alpine marmot

60. Biased sex ratio Test of the Helper Repayment Hypothesis LRE: test in the alpine marmot Mean number of helpers = 0,836 Mean effect of a helper = mean percentage of increase in survival b = 0,107 Sex ratio predicted = 0,541 Observed sex ratio = 0.578 [0.511; 0.643]

61. Biased sex ratio What individual strategy should be ? Individual level Should all females have the same strategy ? Or not ?

62. Biased sex ratio Individual level Since the selection is only for the total expenditure, only the mean sex ratio is fixed and there is no effect on the variance, that is, a population can have any degree of heterogeneity so long as the totals expended on the production of each sexes are equal (Kolman 1960) Individuals producing offspring in sex ratios that deviate from 50/50 are not selected against as long as these deviations exactly cancel out and result in a sex ratio at conception of 50/50 for the local breeding population (Trivers and Willard 1973)

63. Biased sex ratio Individual level Parents should overproduce offspring of the most profitable sex in term of fitness return (Trivers and Willard 1973)  Facultative sex ratio adjustment

64. Biased sex ratio Individual level: Trivers and Willard (1973) Assumptions of the Trivers and Willard’s hypothesis 1.The condition of the young at the end of PI depends on the condition of the mother during PI 2. Differences in condition of young at the end of PI endure into adulthood 3. A slight advantage in condition has disproportionate effects on male reproductive success compared to the effects on female RS 3. => especially designed for polygynous species

65. Biased sex ratio Predictions of the Trivers and Willard’s hypothesis Females in relatively better condition tend to produce males and females in relatively poor condition tend to produce females Many studies aimed to test the TW model especially in ungulates => inconsistent results probably because: 1. assumptions not respected 2. predictions not clear (Leimar 1996) Individual level: Trivers and Willard (1973)

66. Biased sex ratio Individual level: LRC Prediction of the LRC hypothesis Females in a low quality environment should produce more offspring of the dispersing sex The prediction may be the opposite of TW prediction: for example, in many primates, daughters are philopatric. So, dominant females (in good situation) should overproduce daughters and this is the opposite prediction of TW. => inconsistent results probably because :

67. Biased sex ratio Individual level: Burley (1981) In species where some males are more attractive to females than others, thereby leading to variation in male mating and reproductive success, and where male attractiveness has a genetic basis, females mated to attractive males should produce male-biased litters. Assumptions and prediction

68. Biased sex ratio Individual level: Burley (1981) From Griffith et al. (2003) On the blue tit Parus caeruleus Assume the heritability of UV coloration

69. Biased sex ratio Individual level: Burley (1981) Heritability confirmed but lowFrom Kölliker et al. (1999) Test at the individual level: Great tit Parus major

70. Biased sex ratio Individual level: Burley (1981) In many species (especially in mammals), the attractiveness is hard to define But if EPP occurs, and assuming that EPM are more attractive to females than cuckolded males, we predict that the sex ratio should increase with the proportion of EPY in the litter and that EPY should be more often males than their half-sib WPY Most studies in birds failed to show that the sex ratio of EPY was more male-biased than the sex ratio of their half-sib WPY

71. Biased sex ratio Individual level: Burley (1981) In the alpine marmot we found that the sex ratio in mixed litters was more male-biased as the proportion of EPY increased EPY were more likely males (SR = 0.62 ± 0.09) than their half-sib WPY (0.44 ± 0.08) but the difference was not significant (p = 0.2) => lack of power ? Test at the individual level: alpine marmots

72. Biased sex ratio Individual level: LRE Prediction of the LRE hypothesis Females should produce more offspring of the helping sex when helpers are absent in the family group

73. Biased sex ratio Test at the individual level Prediction: offspring sex ratio should be biased towards the helping sex when helpers are absent (Gowaty & Lennartz 1985, Pen & Weissing 2000) Test: use of generalized linear mixed models with a binomial error, the mother nested to the territory as a random term, and the presence of helpers as a fixed term. From 1992 to 2002, a litter occurred in 82 group-years (66%). All juveniles were sexed for 63 litters. LRE: test in the alpine marmot

74. Biased sex ratio Test across females in the population sr = 0.66 sr = 0.49 Only the presence of helpers had a significant effect on sr (  2 = 8.74, df =1, p = 0.003) Test in individual females across multiple years Ten mothers remained several years in their territory They produced a sex ratio according to their social environment (p = 0.002): Helpers absent: sr = 0.65 [0.54;0.74]helpers present: sr = 0.46 [0.36; 0.56] LRE: test in the alpine marmot

75. Biased sex ratio LRE: test in the alpine marmot These results suggest that mothers are able to facultatively adjust the sex ratio facultatively adjust the sex ratio of their offspring Mechanism ???