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What did Trivers and Willard really predict?
ANIMAL BEHAVIOUR, 2002, 63, F1–F3 doi:10.1006/anbe.2001.1901, available online at http://www.idealibrary.com on
FORUM What did Trivers and Willard really predict? JUAN CARRANZA
Biology and Ethology Unit, University of Extremadura (Received 28 November 2000; initial acceptance 5 March 2001; final acceptance 24 May 2001; MS. number: SC-1197)
hope Trivers and Willard themselves would agree. The TW hypothesis was based on three assumptions: (1) mothers in better condition can produce young in better condition; (2) the condition of young at independence is translated into adulthood; and (3) male fitness will gain comparatively more than female fitness by slight advantages in condition. Assumption (1) means that mothers in better condition tend to have a higher budget to expend on parental care than those in poor condition. In Fig. 1a, this simply means that mothers in better condition are more likely than those in poor condition to expend r>rc on an offspring. Assumption (2) means that differences in parental investment at independence (X axis in Fig. 1a) produce differences in condition at adulthood that affect the fitness return (Y axis). Assumption (3) concerns the return curves. It requires the curves for both sexes to be different so that in a certain region of parental expenditure, such as around the critical value rc, increases in parental expenditure (hence in offspring condition) produce higher increases in fitness return for one of the sexes (usually males). Provided that the curves meet the TW assumptions, these interpretations can be assessed in the light of the graphical model. (1a) Females in better condition tend to produce bigger male than female offspring. This statement is not a prediction of the TW model but simply assumption (3). As an assumption, it also applies to females in poor condition, so the complementary statement is false. In a sexually dimorphic species, male offspring are bigger than females regardless of the condition of the mother. This statement is represented by the difference between the curves along the X axis in Fig. 1a. The smaller the difference between the curves along X, the more likely that some female offspring are bigger than males, but also the more relaxed is assumption (3) and therefore the validity of the TW predictions. (1b) Females in better condition tend to produce bigger males than females in poor condition. This is also an assumption, number (1). But mothers in better condition
he issue raised by Trivers & Willard (1973) on the ‘natural selection of parental ability to vary the sex ratio of offspring’ has produced many publications. Probably no other case exists in behavioural ecology where a couple of pages have sired so many studies. With so many descendants, however, mutations are likely to emerge. One main problem is that the Trivers–Willard (TW) hypothesis was a verbal argument and hence semantic interpretations are possible. For instance, TW wrote that ‘females in better condition tend to invest in males’. This may be interpreted in different ways, regarding either the way of allocating the investment or the nature of the comparison, as follows: (1) females in better condition tend to produce bigger (or costlier) male offspring than female offspring or they tend to produce bigger (or costlier) males than females in poor condition; (2) females in better condition tend to produce more male offspring than female offspring or they tend to produce more males than females in poor condition. Furthermore, these statements may lead to complementary ones such as: (1) females in poor condition tend to produce bigger female than male offspring or they tend to produce bigger females than other females; (2) females in poor condition tend to produce more female than male offspring or they tend to produce more females than other females. One needs only to read the original TW paper carefully (or subsequent reviews discussing it: e.g. Frank 1990; Godfray & Werren 1996; Hardy 1997) to see that some of these interpretations are not in concordance with Trivers and Willard’s ideas. I do not intend to review the TW hypothesis comprehensively here, although a simple graphical model may help to put these thoughts into a framework. Figure 1a represents the hypothetical relationship between the fitness return of a female or a male offspring and the amount of care it receives (r). I believe this represents the conditions of the TW model, and I
T
Correspondence: J. Carranza, Ca´tedra de Biologi´a y Etologi´a, Facultad de Veterinaria, Universidad de Extremadura, 10071 Ca´ceres, Spain (email: [email protected]) 0003–3472/02/0200F1+03 $35.00/0
F1
2002 The Association for the Study of Animal Behaviour
ANIMAL BEHAVIOUR, 63, 2
1.0 (a) 0.8 µ(r)
0.6
φ(r)
0.4 0.2 rc
0.0
0.2
1.0
0.4
0.6
0.8
1.0
Supposed cut-off line for mother's condition
(b) Fitness return
F2
0.8 µ(r)
0.6
φ(r)
0.4 0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.0 (c) 0.8 0.6
Mp Mg µ(r)
0.4
φ(r) Costs to poor-condition mothers
0.2
Costs to good-condition mothers
0.0
Fp Fg
0.2 0.4 0.6 0.8 1.0 Parental expenditure per offspring (r)
Figure 1. Hypothetical relationships between fitness returns for individual male, µ(r), and female, φ(r), offspring and the amount of parental expenditure (r) that it receives. (a) The value rc for which µ(r)=φ(r) represents the threshold for the amount of parental budget that favours either male or female offspring. (b) The cut-off lines indicate the amount of parental resources allocated to individual offspring of both sexes in relation to the mother’s condition (see text). (c) Fp and Fg indicate the optimal expenditure on female offspring by mothers in poor and good condition, respectively; Mp and Mg are the same for male offspring.
should also tend to produce bigger females than mothers in poor condition, since the assumption is that the condition of mothers affects the condition of young. In Fig. 1a, mothers can be on either side of rc. Then, the complementary statement that females in poor condition tend to produce bigger females than mothers in better condition makes no sense and it would also violate assumption (1). (2a) Females in better condition tend to produce more male than female offspring. This is the main prediction of the TW model as is the complementary statement that mothers in poor condition tend to produce more females than males. This makes sense in the light of the graphical model since expenditure above rc is more profitable in the male curve than in the female curve and the contrary is true for investments below rc.
(2b) Females in better condition tend to produce more males than females in poor condition. This is a straightforward prediction from the previous statement, as also is its complementary statement. So far, the TW model seems to be only about the sex ratio, as indicated in the title of the TW paper. Why then did Trivers and Willard mention a bias in the investment in sons and daughters? Simply because of the Fisherian equilibrium: Fisher (1930) predicted the equilibrium in the allocation of parental resources to male and female offspring at the population level. Consequently, if males cost much more than females, mothers in better condition may support the TW model without producing more males than females, by investing more resources in the production of males than in the production of females. Mothers in poor condition, in turn, should invest more resources in the production of females than in the production of males. This does not mean higher investment per individual offspring, but simply that the total parental budget of each type of mother is allocated differently to males or females. It was using this sense that Trivers and Willard claimed the differential investment, to provide a more general prediction, but not to deal with the assumptions. One might think that Trivers and Willard’s ideas are already clear and well understood by researchers in the field. However, my feeling is that they are not. Consider a recent example, by Cameron & Linklater (2000) in Animal Behaviour. Among the findings, individual mares, Equus caballus, in better condition invested more in individual male foals than in individual female foals. This supports assumption (3) of Trivers and Willard, but is not a prediction of the model. Conversely, and most surprisingly, Cameron & Linklater found that mares in poor condition invested more in individual female foals than in individual male foals. This, again, is not a prediction of the TW model. Furthermore, it contradicts assumption (3) in this species. However, it is quite easy to arrive at such interpretations. Although the majority of Trivers and Willard’s arguments were on the sex ratio and on the investment bias caused by a sex ratio bias, they also stated in their last sentence of the paper that ‘one might expect biases in parental behaviour toward offspring of different sex, according to parental condition; parents in better condition would be expected to show a bias toward male offspring’. It appears that Trivers and Willard also predicted differences in the amount of investment that an offspring of a given sex would receive depending on maternal condition. The last Trivers and Willard statement can be viewed graphically in at least two ways. If maternal budget depends on condition, one can make a cut-off line for r to separate mothers in poor and better condition (Fig. 1b). Then, mothers in better condition will produce male and female offspring at the optimal r values, but those in poor condition will be constrained to produce males with r up to the cut-off point. This means that the difference between expenditure in individuals of both sexes is higher for mothers in better condition. In the words of Trivers and Willard, ‘parents in better condition show a
FORUM
bias toward male offspring’. But this does not mean that mothers in poor condition show a bias towards female offspring (see Fig. 1b). Alternatively, reproductive costs may increase at an accelerating rate with parental expenditure, and the cost functions may differ for mothers in good and poor condition. Then, optimal expenditure in male and female offspring may differ according to the mother’s condition (Fig. 1c). Regardless of whether one considers the difference or the ratio between fitness return and reproductive costs as the optimization criterion, the higher the rate at which costs for mothers in poor condition increase with respect to costs for those in better condition, the greater the difference between the two types of mothers in the optimal expenditure on individual offspring according to sex. However, again, mothers in poor condition should not show a bias to individual female offspring, but at most a smaller bias to individual male offspring than mothers in better condition. Furthermore, is the last sentence of Trivers and Willard’s paper really a prediction of their model? Surely it is partially implicitly included in assumption (1), since the condition of the young (of both sexes) at independence should be related to the mother’s condition during the period of parental investment. However, if differences between the expenditure on both sexes are not the same for the two types of mothers, which depends on the shape of the curves (e.g. Fg Fp
Smith 1980; Bull 1981; Charnov 1982; Frank 1987; Leimar 1996; Lessels 1998), I agree with some reviews (e.g. Clutton-Brock 1991; Godfray & Werren 1996) that some of the implications of Trivers and Willard’s ideas may lack the appropriate theoretical background, and that greater integration of empirical and theoretical work on the TW hypothesis is needed. Ideas contained in this paper have benefited from discussions with C. Mateos, J. Valencia and by comments from two anonymous referees. Research by J.C. was supported by project 1FD97-1504 during the writing of the paper. References Bull, J. J. 1981. Sex ratio evolution when fitness varies. Heredity, 46, 9–26. Cameron, E. Z. & Linklater, W. L. 2000. Individual mares bias investment in sons and daughters in relation to their condition. Animal Behaviour, 60, 359–367. Charnov, E. L. 1982. The Theory of Sex Allocation. Princeton, New Jersey: Princeton University Press. Clutton-Brock, T. H. 1991. The Evolution of Parental Care. Princeton, New Jersey: Princeton University Press. Ferna´ndez-Llario, P., Carranza, J. & Mateos-Quesada, P. 1999. Sex allocation in a polygynous mammal with large litters: the wild boar. Animal Behaviour, 58, 1079–1084. Festa-Bianchet, M. 1996. Offspring sex ratio studies of mammals: does publication depend upon the quality of the research or the direction of the results? Ecoscience, 3, 42–44. Fisher, R. A. 1930. The Genetical Theory of Natural Selection. Oxford: Oxford University Press. Frank, S. A. 1987. Individual and population sex allocation patterns. Theoretical Population Biology, 31, 47–74. Frank, S. A. 1990. Sex allocation theory for birds and mammals. Anual Review of Ecology and Systematics, 21, 13–55. Godfray, H. C. J. & Werren, J. H. 1996. Recent development in sex ratio studies. Trends in Ecology and Evolution, 11, 59–63. Hardy, I. C. W. 1997. Possible factors influencing vertebrate sex ratios: an introductory overview. Applied Animal Behaviour Science, 51, 217–241. Leimar, O. 1996. Life history analysis of the Trivers and Willard sex-ratio problem. Behavioral Ecology, 7, 316–325. Lessells, K. 1998. A theoretical framework for sex-biased parental care. Animal Behaviour, 56, 395–407. Maynard Smith, J. 1980. A new theory of sexual investment. Behavioural Ecology and Sociobiology, 7, 247–251. Trivers, R. L. & Willard, D. E. 1973. Natural selection of parental ability to vary the sex ratio of offspring. Science, 179, 90–92.
F3
FORUM What did Trivers and Willard really predict? JUAN CARRANZA
Biology and Ethology Unit, University of Extremadura (Received 28 November 2000; initial acceptance 5 March 2001; final acceptance 24 May 2001; MS. number: SC-1197)
hope Trivers and Willard themselves would agree. The TW hypothesis was based on three assumptions: (1) mothers in better condition can produce young in better condition; (2) the condition of young at independence is translated into adulthood; and (3) male fitness will gain comparatively more than female fitness by slight advantages in condition. Assumption (1) means that mothers in better condition tend to have a higher budget to expend on parental care than those in poor condition. In Fig. 1a, this simply means that mothers in better condition are more likely than those in poor condition to expend r>rc on an offspring. Assumption (2) means that differences in parental investment at independence (X axis in Fig. 1a) produce differences in condition at adulthood that affect the fitness return (Y axis). Assumption (3) concerns the return curves. It requires the curves for both sexes to be different so that in a certain region of parental expenditure, such as around the critical value rc, increases in parental expenditure (hence in offspring condition) produce higher increases in fitness return for one of the sexes (usually males). Provided that the curves meet the TW assumptions, these interpretations can be assessed in the light of the graphical model. (1a) Females in better condition tend to produce bigger male than female offspring. This statement is not a prediction of the TW model but simply assumption (3). As an assumption, it also applies to females in poor condition, so the complementary statement is false. In a sexually dimorphic species, male offspring are bigger than females regardless of the condition of the mother. This statement is represented by the difference between the curves along the X axis in Fig. 1a. The smaller the difference between the curves along X, the more likely that some female offspring are bigger than males, but also the more relaxed is assumption (3) and therefore the validity of the TW predictions. (1b) Females in better condition tend to produce bigger males than females in poor condition. This is also an assumption, number (1). But mothers in better condition
he issue raised by Trivers & Willard (1973) on the ‘natural selection of parental ability to vary the sex ratio of offspring’ has produced many publications. Probably no other case exists in behavioural ecology where a couple of pages have sired so many studies. With so many descendants, however, mutations are likely to emerge. One main problem is that the Trivers–Willard (TW) hypothesis was a verbal argument and hence semantic interpretations are possible. For instance, TW wrote that ‘females in better condition tend to invest in males’. This may be interpreted in different ways, regarding either the way of allocating the investment or the nature of the comparison, as follows: (1) females in better condition tend to produce bigger (or costlier) male offspring than female offspring or they tend to produce bigger (or costlier) males than females in poor condition; (2) females in better condition tend to produce more male offspring than female offspring or they tend to produce more males than females in poor condition. Furthermore, these statements may lead to complementary ones such as: (1) females in poor condition tend to produce bigger female than male offspring or they tend to produce bigger females than other females; (2) females in poor condition tend to produce more female than male offspring or they tend to produce more females than other females. One needs only to read the original TW paper carefully (or subsequent reviews discussing it: e.g. Frank 1990; Godfray & Werren 1996; Hardy 1997) to see that some of these interpretations are not in concordance with Trivers and Willard’s ideas. I do not intend to review the TW hypothesis comprehensively here, although a simple graphical model may help to put these thoughts into a framework. Figure 1a represents the hypothetical relationship between the fitness return of a female or a male offspring and the amount of care it receives (r). I believe this represents the conditions of the TW model, and I
T
Correspondence: J. Carranza, Ca´tedra de Biologi´a y Etologi´a, Facultad de Veterinaria, Universidad de Extremadura, 10071 Ca´ceres, Spain (email: [email protected]) 0003–3472/02/0200F1+03 $35.00/0
F1
2002 The Association for the Study of Animal Behaviour
ANIMAL BEHAVIOUR, 63, 2
1.0 (a) 0.8 µ(r)
0.6
φ(r)
0.4 0.2 rc
0.0
0.2
1.0
0.4
0.6
0.8
1.0
Supposed cut-off line for mother's condition
(b) Fitness return
F2
0.8 µ(r)
0.6
φ(r)
0.4 0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.0 (c) 0.8 0.6
Mp Mg µ(r)
0.4
φ(r) Costs to poor-condition mothers
0.2
Costs to good-condition mothers
0.0
Fp Fg
0.2 0.4 0.6 0.8 1.0 Parental expenditure per offspring (r)
Figure 1. Hypothetical relationships between fitness returns for individual male, µ(r), and female, φ(r), offspring and the amount of parental expenditure (r) that it receives. (a) The value rc for which µ(r)=φ(r) represents the threshold for the amount of parental budget that favours either male or female offspring. (b) The cut-off lines indicate the amount of parental resources allocated to individual offspring of both sexes in relation to the mother’s condition (see text). (c) Fp and Fg indicate the optimal expenditure on female offspring by mothers in poor and good condition, respectively; Mp and Mg are the same for male offspring.
should also tend to produce bigger females than mothers in poor condition, since the assumption is that the condition of mothers affects the condition of young. In Fig. 1a, mothers can be on either side of rc. Then, the complementary statement that females in poor condition tend to produce bigger females than mothers in better condition makes no sense and it would also violate assumption (1). (2a) Females in better condition tend to produce more male than female offspring. This is the main prediction of the TW model as is the complementary statement that mothers in poor condition tend to produce more females than males. This makes sense in the light of the graphical model since expenditure above rc is more profitable in the male curve than in the female curve and the contrary is true for investments below rc.
(2b) Females in better condition tend to produce more males than females in poor condition. This is a straightforward prediction from the previous statement, as also is its complementary statement. So far, the TW model seems to be only about the sex ratio, as indicated in the title of the TW paper. Why then did Trivers and Willard mention a bias in the investment in sons and daughters? Simply because of the Fisherian equilibrium: Fisher (1930) predicted the equilibrium in the allocation of parental resources to male and female offspring at the population level. Consequently, if males cost much more than females, mothers in better condition may support the TW model without producing more males than females, by investing more resources in the production of males than in the production of females. Mothers in poor condition, in turn, should invest more resources in the production of females than in the production of males. This does not mean higher investment per individual offspring, but simply that the total parental budget of each type of mother is allocated differently to males or females. It was using this sense that Trivers and Willard claimed the differential investment, to provide a more general prediction, but not to deal with the assumptions. One might think that Trivers and Willard’s ideas are already clear and well understood by researchers in the field. However, my feeling is that they are not. Consider a recent example, by Cameron & Linklater (2000) in Animal Behaviour. Among the findings, individual mares, Equus caballus, in better condition invested more in individual male foals than in individual female foals. This supports assumption (3) of Trivers and Willard, but is not a prediction of the model. Conversely, and most surprisingly, Cameron & Linklater found that mares in poor condition invested more in individual female foals than in individual male foals. This, again, is not a prediction of the TW model. Furthermore, it contradicts assumption (3) in this species. However, it is quite easy to arrive at such interpretations. Although the majority of Trivers and Willard’s arguments were on the sex ratio and on the investment bias caused by a sex ratio bias, they also stated in their last sentence of the paper that ‘one might expect biases in parental behaviour toward offspring of different sex, according to parental condition; parents in better condition would be expected to show a bias toward male offspring’. It appears that Trivers and Willard also predicted differences in the amount of investment that an offspring of a given sex would receive depending on maternal condition. The last Trivers and Willard statement can be viewed graphically in at least two ways. If maternal budget depends on condition, one can make a cut-off line for r to separate mothers in poor and better condition (Fig. 1b). Then, mothers in better condition will produce male and female offspring at the optimal r values, but those in poor condition will be constrained to produce males with r up to the cut-off point. This means that the difference between expenditure in individuals of both sexes is higher for mothers in better condition. In the words of Trivers and Willard, ‘parents in better condition show a
FORUM
bias toward male offspring’. But this does not mean that mothers in poor condition show a bias towards female offspring (see Fig. 1b). Alternatively, reproductive costs may increase at an accelerating rate with parental expenditure, and the cost functions may differ for mothers in good and poor condition. Then, optimal expenditure in male and female offspring may differ according to the mother’s condition (Fig. 1c). Regardless of whether one considers the difference or the ratio between fitness return and reproductive costs as the optimization criterion, the higher the rate at which costs for mothers in poor condition increase with respect to costs for those in better condition, the greater the difference between the two types of mothers in the optimal expenditure on individual offspring according to sex. However, again, mothers in poor condition should not show a bias to individual female offspring, but at most a smaller bias to individual male offspring than mothers in better condition. Furthermore, is the last sentence of Trivers and Willard’s paper really a prediction of their model? Surely it is partially implicitly included in assumption (1), since the condition of the young (of both sexes) at independence should be related to the mother’s condition during the period of parental investment. However, if differences between the expenditure on both sexes are not the same for the two types of mothers, which depends on the shape of the curves (e.g. Fg Fp
Smith 1980; Bull 1981; Charnov 1982; Frank 1987; Leimar 1996; Lessels 1998), I agree with some reviews (e.g. Clutton-Brock 1991; Godfray & Werren 1996) that some of the implications of Trivers and Willard’s ideas may lack the appropriate theoretical background, and that greater integration of empirical and theoretical work on the TW hypothesis is needed. Ideas contained in this paper have benefited from discussions with C. Mateos, J. Valencia and by comments from two anonymous referees. Research by J.C. was supported by project 1FD97-1504 during the writing of the paper. References Bull, J. J. 1981. Sex ratio evolution when fitness varies. Heredity, 46, 9–26. Cameron, E. Z. & Linklater, W. L. 2000. Individual mares bias investment in sons and daughters in relation to their condition. Animal Behaviour, 60, 359–367. Charnov, E. L. 1982. The Theory of Sex Allocation. Princeton, New Jersey: Princeton University Press. Clutton-Brock, T. H. 1991. The Evolution of Parental Care. Princeton, New Jersey: Princeton University Press. Ferna´ndez-Llario, P., Carranza, J. & Mateos-Quesada, P. 1999. Sex allocation in a polygynous mammal with large litters: the wild boar. Animal Behaviour, 58, 1079–1084. Festa-Bianchet, M. 1996. Offspring sex ratio studies of mammals: does publication depend upon the quality of the research or the direction of the results? Ecoscience, 3, 42–44. Fisher, R. A. 1930. The Genetical Theory of Natural Selection. Oxford: Oxford University Press. Frank, S. A. 1987. Individual and population sex allocation patterns. Theoretical Population Biology, 31, 47–74. Frank, S. A. 1990. Sex allocation theory for birds and mammals. Anual Review of Ecology and Systematics, 21, 13–55. Godfray, H. C. J. & Werren, J. H. 1996. Recent development in sex ratio studies. Trends in Ecology and Evolution, 11, 59–63. Hardy, I. C. W. 1997. Possible factors influencing vertebrate sex ratios: an introductory overview. Applied Animal Behaviour Science, 51, 217–241. Leimar, O. 1996. Life history analysis of the Trivers and Willard sex-ratio problem. Behavioral Ecology, 7, 316–325. Lessells, K. 1998. A theoretical framework for sex-biased parental care. Animal Behaviour, 56, 395–407. Maynard Smith, J. 1980. A new theory of sexual investment. Behavioural Ecology and Sociobiology, 7, 247–251. Trivers, R. L. & Willard, D. E. 1973. Natural selection of parental ability to vary the sex ratio of offspring. Science, 179, 90–92.
F3