Scaling of Size and Dimorphism in Primates I: Microevolution

I used a new quantitative genetics model to predict relationships between sex-specific body size and sex-specific relative variability when populations experience differences in relative intensity of sex-specific selection pressures—stronger selection on males or females—and direction of selection: increase or decrease in size. I combined Lande's (Evolution 34: 292–305) model for the response of sex-specific means to selection with a newly derived generalization of Bulmer's (Am. Nat. 105: 201–211) model for the response of relative variability to selection. I used this combined response model to predict correlations of sex-specific size and relative variability under various starting conditions, which one can compare to correlations between closely related primate populations. One can then compare predicted patterns of sex-specific selection pressures to social and ecological variables pertaining to those populations to identify likely forces producing microevolutionary change in sexual size dimorphism (SSD). I provide examples of this approach for populations representing three taxa: Papio anubis, Saguinus mystax, and Cercopithecus aethiops pygerythrus. Model results suggest that microevolutionary changes in SSD can result from greater selection acting on males or females, and that natural selection or natural and sexual selection combined, rather than sexual selection alone, may sometimes explain sex-specific selection differentials.

Appendices

APPENDIX A: DERIVATION OF THE GENERALIZED BULMER EFFECT

Based on an analysis of Lande's quantitative genetic model for sexually dimorphic polygenic traits (Lande, 1980), Leutenegger and Cheverud (1982, 1985) present the expressions:

$$ \overline{\rm R}_{\rm M} = \left( \frac{1}{2} \right)\left( {h_{\rm M}^2 s_{\rm M} i_{\rm M} + h_{\rm M} h_{\rm F} r_{\rm a} s_{\rm M} i_{\rm F} } \right)$$

(A.1)

and

$$\overline{\rm R}_{\rm F} = \left(\frac {1}{2} \right)\left( {h_{\rm F}^2 s_{\rm F} i_{\rm F} + h_{\rm M} h_{\rm F} r_{\rm a} s_{\rm F} i_{\rm M} } \right)$$

(A.2)

where $\overline{\rm R}$ is the mean response of the trait to selection, h ² is the heritability of the trait, s is the phenotypic standard deviation, i is the selection intensity, and r _a is genetic correlation between males and females for that trait. Subscripts indicate the sex to which the parameter refers (_M, males; _F, females). Heritability, selection intensity, and genetic correlation are all unitless values; thus the response to selection is measured in units of the standard deviation, i.e., the units used to measure the phenotypic trait.

The response (R) is the change in value of the continuous trait from one generation to the next; for example:

$$M^* = M + R_{\rm M} ,$$

(A.3)

where M is the variable for the continuous trait in the parental generation of males and M ^* is the variable for the trait in the offspring generation of males.

What follows is a derivation of the equations describing the sex-specific changes in variance between parental and offspring generations due to selection. Bulmer (1971) derived equations for the case when males and females do not differ in the expression of a continuous trait; here I derive a generalization that one can apply to all conditions ranging from absence of sex-linkage in the trait, i.e., Bulmer's model, where r _a = 1, to complete sex-linkage of a trait (r _a = 0). The male equation is derived here; the female equation is obtained simply by replacing M with F every time it appears, and vice versa.

The variable R _M (distinct from the mean of this variable, $\overline{\rm R}_{\rm M}$) is the sum of two independent variables: the change in males due to selection on males in the parental generation (R _Mm), and the change in males due to selection on females in the parental generation (R _Mf). One can re-express Eq. (A.3) as follows:

$$M^* = M + R_{{\rm Mm}} + R_{{\rm Mf}} ,$$

(A.4)

This figure represents the sex-specific distribution of body mass for a population of males. The pre-selection distribution fills the entire area under the heavy black curve with mean $\overline {\rm M}$ (heavy solid vertical line) and standard deviation s (light solid vertical line). Males that fall into the crosshatched portion of the distribution are selected against and do not reproduce. The post-selection distribution excludes the crosshatched area and has mean $\overline {\rm M} ^\prime$ (heavy dashed line) and standard deviation s′ (light dashed line).

where

$$\overline{\rm R}_{{\rm Mm}} = \left(\frac{1}{2}\right)(h_{\rm M}^2 s_{\rm M} i_{\rm M} )$$

(A.5)

and

$$\overline{\rm R}_{{\rm Mf}} = \left(\frac{1}{2}\right)(h_{\rm M} h_{\rm F} r_{\rm a} s_{\rm M} i_{\rm F} )$$

(A.6)

Next, I follow Bulmer in modeling selection as the truncation of a distribution at a particular value (Fig. A.1), which allows the post-selection distribution to be described as

$${\rm M}' = {\rm M} + d{\rm M},$$

(A.7)

where M′ is the variable for the trait in post-selection males. The other variable in Eq. (A.7), dM, describes the difference between pre- and post-selection parental distributions. It is independent of M and has a mean of $\overline {dM}$ and variance $ds_M^2$, where

$$\overline {d{\rm M}} = \overline {{\rm M}'} - \overline {\rm M}$$

(A.8)

and

$$ds_{\rm M}^2 = s{'}_{\rm M}^2 - s_{\rm M}^2 .$$

(A.9)

Fig. A.1.

Selection modeled as the truncation of a distribution after Bulmer (1971).

Selection intensity (i) is defined as the difference of the pre- and post-selection means, divided by the pre-selection standard deviation, so

$$i_{\rm M} = \frac{{\overline {d{\rm M}} }}{{s_{\rm M} }}$$

(A.10)

and

$$i_{\rm F} = \frac{{\overline {d{\rm F}} }}{{s_{\rm F} }}.$$

(A.11)

Replacing i _M and i _F in Eqs. (A.5) and (A.6) yields

$$\overline{\rm R}_{{\rm Mm}} = \left(\frac{1}{2}\right)\left( {h_{\rm M}^2 } \right)\left( {\overline {dM} } \right)$$

(A.12)

and

$$\overline{\rm R}_{{\rm Mf}} = \left(\frac{1}{2}\right)\left( {h_{\rm M} h_{\rm F} r_{\rm a} \frac{{s_{\rm M} }}{{s_{\rm F} }}} \right)\left( {\overline {dF} } \right).$$

(A.13)

One can now represent the independent variables R _Mm and R _Mf as

$$R_{{\rm Mm}} = \left[\left(\frac{1}{2}\right )h_{\rm M}^2\right]d{\rm M}$$

(A.14)

and

$$R_{{\rm Mf}} = \left( {\left(\frac{1}{2}\right)h_{\rm M} h_{\rm F} r_{a} \frac{{s_{\rm M} }}{{s_{\rm F} }}} \right)dF,$$

(A.15)

where R _Mm and R _Mf are expressed as linear transformations of the variables dM and dF, respectively. Their variances are as shown below:

$${\rm Var[}R_{{\rm Mm}} {\rm ]} = \left[\left( \frac{1}{4} \right)h_{\rm M}^4\right] ds_{\rm M}^2$$

(A.16)

and

$${\rm Var}\,[R_{{\rm Mf}} ] = \left[\left( \frac{1}{4}\right) h_{\rm M}^2 h_{\rm F}^2 r_{a}^2 \frac{{s_{\rm M}^2 }}{{s_{\rm F}^2 }}\right] ds_{\rm F}^2 .$$

(A.17)

Because the value of the continuous trait in the preselection parental males (M) is independent of the responses due to selection in the parental males and females (dM and dF, respectively), variance in offspring males is expressed as

$${\rm Var}\,{\rm [}{\rm M}^* {\rm ]} = {\rm Var}\,[{\rm M}] + {\rm Var}\,[{\rm R}_{{\rm Mm}} ] + {\rm Var}\,{\rm [}{\rm R}_{{\rm Mf}} {\rm ],}$$

(A.18)

which expands to

$$s_{\rm M}^{*2} = s_{\rm M}^2 + \left[\left( \frac{1}{4}\right) h_{\rm M}^4\right] ds_{\rm M}^2 + \left[\left( \frac{1}{4}\right) h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm M}^2 }}{{s_{\rm F}^2 }}\right] ds_{\rm F}^2 ,$$

(A.19)

where$s_{\rm M}^{*2}$ is the variance of M ^*. The equivalent expression for female offspring is as follows:

$$s_{\rm F}^{*2} = s_{\rm F}^2 + \left[\left( \frac{1}{4}\right) h_{\rm F}^4\right] ds_{\rm F}^2 + \left[\left( \frac{1}{4}\right) h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm F}^2 }}{{s_{\rm M}^2 }}\right] ds_{\rm F}^2 .$$

(A.20)

Equations (A.19) and (A.20) describe the change in variance for a continuous trait in male and female offspring due to selection on the parental generation. If there is no sex-linkage in the trait and selection acts equally on both sexes, then males and females form a single distribution. In that case, r _a equals 1, and one can remove the subscripts from all other parameters in Eqs. (A.19) and (A.20) because there is no distinction between female and male parameters. For such a situation, both equations reduce to

$$s^{*2} = s^2 + \left( {1/2h^4 } \right)ds^2 .$$

(A.21)

Equation (A.21) is exactly the equation Bulmer (1971) derived to describe the effect of selection on variance in a monomorphic population.

APPENDIX B: RELATIONSHIP OF MEAN RESPONSES TO RAW DATA

One can log-transform data for a continuous trait such as body size to produce scale-free variables, such that

$$M = \log [X_{\rm M} ]$$

(B.1)

and

$$F = \log [X_{\rm F} ],$$

(B.2)

where M and F are variables for male and female distributions as defined in Appendix A, and X is the size variable as originally measured, e.g., in kg. When the log transformation is performed, one can rewrite the mean sex-specific responses to selection ($\overline{\rm R}_{\rm M}$ and$\overline{\rm R}_{\rm F}$ from Eqs. (A.1) and (A.2)) as

$$\overline{\rm R}_{\rm M} = \overline {\log [X_{\rm M}^* ]} - \overline {\log [X_{\rm M} ]}$$

(B.3)

and

$$\overline{\rm R}_{\rm F} = \overline {\log [X_{\rm F}^* ]} - \overline {\log [X_{\rm F} ]} ,$$

(B.4)

where the asterisk refers to the offspring generation. Eqs. (B.3) and (B.4) are equivalent to

$$\overline{\rm R}_{\rm M} = \log \left[ {{\rm GM}_{{\rm XM}}^* } \right] - \log \left[ {{\rm GM}_{{\rm XM}} } \right]$$

(B.5)

and

$$\overline{\rm R}_{\rm F} = \log [{\rm GM}_{{\rm XF}}^* ] - \log [{\rm GM}_{{\rm XF}} ],$$

(B.6)

where GM_XM and GM_XF are the geometric means of the raw size measurements for males and females, respectively. Using the log rules, one can also express Eqs. (B.5) and (B.6) as

$$\overline{\rm R}_{\rm M} = \log \left[ {\frac{{{\rm GM}_{{\rm XM}}^* }}{{{\rm GM}_{{\rm XM}} }}} \right]$$

(B.7)

and

$$\overline{\rm R}_{\rm F} = \log \left[ {\frac{{{\rm GM}_{{\rm XF}}^* }}{{{\rm GM}_{{\rm XF}} }}} \right].$$

(B.8)

In Eqs. (B.7) and (B.8), $\overline{\rm R}_{\rm M}$ and $\overline{\rm R}_{\rm F}$ are shown to be the logarithms of ratios of offspring mean size divided by parent mean size. Replacing these ratios with the symbols p _XM and p _XF yields

$$\overline{\rm R}_{\rm M} = \log [p_{{\rm XM}} ]$$

(B.9)

and

$$\overline{\rm R}_{\rm F} = \log [p_{{\rm XF}} ]$$

(B.10)

where

$${\rm GM}_{{\rm XM}}^* = {\rm GM}_{{\rm XM}} \times p_{{\rm XM}}$$

(B.11)

and

$${\rm GM}_{{\rm XF}}^* = {\rm GM}_{{\rm XF}} \times p_{{\rm XF}} .$$

(B.12)

Thus the response variables$\overline{\rm R}_{\rm M}$ and$\overline{\rm R}_{\rm F}$ are shown to be the logarithms of scalars that describe the proportional change in sex-specific mean size between parent and offspring generations.

Leutenegger and Cheverud (1982, 1985) define the response of sexual dimorphism to sexual selection as the difference between the male and female responses, namely

$$\overline{\rm R}_{{\rm SD}} = \overline{\rm R}_{\rm M} - \overline{\rm R}_{\rm F} .$$

(B.13)

Substitution of Eqs. (B.7) and (B.8) into (B.13) yields

$$\overline{\rm R}_{{\rm SD}} = \log \left[ {\frac{{{\rm GM}_{XM}^* }}{{{\rm GM}_{XM} }}} \right] - \log \left[ {\frac{{{\rm GM}_{XF}^* }}{{{\rm GM}_{XF} }}} \right]$$

(B.14)

which can be shown to be equal to

$$\overline{\rm R}_{{\rm SD}} = \log \left[ {\frac{{{\rm (GM}_{{\rm XM}}^* /({\rm GM}_{{\rm XM}} ))}}{{{\rm (GM}_{{\rm XF}}^* /({\rm GM}_{{\rm XF}} ))}}} \right] = \log \left[ {\frac{{{\rm (GM}_{{\rm XM}}^* /({\rm GM}_{{\rm XF}}^* ))}}{{{\rm (GM}_{{\rm XM}} /({\rm GM}_{{\rm XF}} ))}}} \right].$$

(B.15)

One can calculate an index of sexual size dimorphism,$\overline {{\rm SD}}$, which is the ratio of mean male size divided by mean female size:

$$\overline {{\rm SD}} = \frac{{{\rm GM}_{{\rm XM}} }}{{{\rm GM}_{{\rm XF}} }},$$

(B.16)

which allows one to express Eq. (B.15) as

$$\overline{\rm R}_{{\rm SD}} = \log \left[ {\frac{{\overline {{\rm SD}} ^* }}{{\overline {{\rm SD}} }}} \right];$$

(B.17)

i.e., the logged ratio of sexual size dimorphism in the offspring generation divided by sexual size dimorphism in the parental generation. One can replace the ratio with the symbol$p_{\overline {{\rm SD}} }$, yielding

$$\overline{\rm R}_{{\rm SD}} = \log [p_{\overline {{\rm SD}} } ],$$

(B.18)

where

$$\overline {{\rm SD}} ^* = \overline {{\rm SD}} \times p_{\overline {{\rm SD}} } .$$

(B.19)

The response of sexual size dimorphism to selection ($\overline {R_{{\rm SD}} }$) is therefore the logarithm of a scalar that describes the proportional change in sexual size dimorphism between parent and offspring generations,

$$\overline {{\rm SD}} ^* = \overline {{\rm SD}} \times {\rm antilog}\left[ {\overline{\rm R}_{{\rm SD}} } \right].$$

(B.20)

APPENDIX C: DERIVATION OF THE COMBINED RESPONSE MODEL

Equations (A.1) and (A.2) in Appendix A describe the mean responses to selection of male and female size ($\overline{\rm R}_{\rm M}$ and$\overline{\rm R}_{\rm F}$, respectively) in terms of the genetic correlation between the sexes, sex-specific heritabilities, sex-specific standard deviations, and sex specific selection intensities. Leutenegger and Cheverud (1982 , 1985) use these equations to model the response of size dimorphism to conditions of pure sexual selection and pure variance dimorphism. Similarly, one can use the change in sex-specific variance, as described in Eqs. (A.19) and (A.20), to model the response of differences in sex-specific variance, i.e., variance dimorphism, to conditions of pure sexual selection and pure variance dimorphism. It can then be shown that these two sets of responses covary in predictable ways depending on the selective forces applied.

In Appendix B, I defined p _XM, p _XF, and $p_{\overline {{\rm SD}} }$ as offspring:parent ratios of arithmetic mean size of log-transformed data, which are shown to be equivalent to ratios of the geometric mean of raw data. Here I define analogous ratios for offspring and parent variances of log-transformed data:

$$p_{s{\rm M}}^2 = \frac{{s_{\rm M}^{*2} }}{{s_{\rm M}^2 }},$$

(C.1)

$$p_{s{\rm F}}^2 = \frac{{s_{\rm F}^{*2} }}{{s_{\rm F}^2 }},$$

(C.2)

and

$$p_{s{\rm SD}}^2 = \frac{{\big(s_{\rm M}^{*2} \big/\big(s_{\rm F}^{*2} \big)\big)}}{{\big(s_{\rm M}^2 \big/\big(s_{\rm F}^2 \big)\big)}} = \frac{{\big(s_{\rm M}^{*2} \big/\big(s_{\rm M}^2 \big)\big)}}{{\big(s_{\rm F}^{*2} \big/\big(s_{\rm F}^2 \big)\big)}} = \frac{{p_{s{\rm M}}^2 }}{{p_{s{\rm F}}^2 }},$$

(C.3)

where the asterisk refers to the offspring generation. (Note: the square roots of these three ratios are equal to the offspring:parent ratios of standard deviation rather than variance.) One may define response variables as log-transformations of the three ratios as follows:

$$R_{s^2 {\rm M}} = \log \big[p_{s{\rm M}}^2 \big] = \log \left[ {\frac{{s_{\rm M}^{*2} }}{{s_{\rm M}^2 }}} \right] = \log \left[ {s_{\rm M}^{*2} } \right] - \log \left[ {s_{\rm M}^2 } \right],$$

(C.4)

$$R_{s^2 {\rm F}} = \log [p_{s{\rm F}}^2 ] = \log \left[ {\frac{{s_{\rm F}^{*2} }}{{s_{\rm F}^2 }}} \right] = \log \left[ {s_{\rm F}^{*2} } \right] - \log \left[ {s_{\rm F}^2 } \right],$$

(C.5)

and

$$R_{s^2 {\rm SD}} = \log \left[ {\frac{{p_{s{\rm M}}^2 }}{{p_{s{\rm F}}^2 }}} \right] = \log \left[ {p_{s{\rm M}}^2 } \right] - \log \left[ {p_{s{\rm F}}^2 } \right] = R_{s^2 {\rm M}} - R_{s^2 {\rm F}} ,$$

(C.6)

where$R_{s^2 {\rm M}}$ is the response of male variance to selection,$R_{s^2 {\rm F}}$ is the response of female variance to selection, and$R_{s^2 {\rm SD}}$ is the response of variance dimorphism to selection. As Eq. (C.6) shows, these three response variables are related to each other in exactly the same way as the three response variables for the mean (Eq. (B.13) in Appendix B).

The three variance ratios of Eqs. (C.1)–(C.3) may be restated by substitution of Eqs. (A.19) and (A.20):

$$p_{s{\rm M}}^2 = \frac{{s_{\rm M}^2 + \left(\frac{1}{4}h_{\rm M}^4\right) ds_{\rm M}^2 + \left( \frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm M}^2 }}{{s_{\rm F}^2 }}\right) ds_{\rm F}^2 }}{{s_{\rm M}^2 }},$$

(C.7)

$$p_{s{\rm F}}^2 = \frac{{s_{\rm F}^2 + \left( \frac{1}{4}h_{\rm F}^4\right) ds_{\rm F}^2 + \left( \frac{1}{4}h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm F}^2 }}{{s_{\rm M}^2 }}\right) ds_{\rm M}^2 }}{{s_{\rm F}^2 }},$$

(C.8)

and

$$p_{s{\rm M}}^2 = \frac{{s_{\rm M}^2 + \left(\frac{1}{4} h_{\rm M}^4 \right)ds_{\rm M}^2 + \left( \frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm M}^2 }}{{s_{\rm F}^2 }}\right) ds_{\rm F}^2 }}{{s_{\rm F}^2 + \left( \frac{1}{4} h_{\rm F}^4\right) ds_{\rm F}^2 + \left( \frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm F}^2 }}{{s_{\rm M}^2 }}\right) ds_{\rm M}^2 }} \times \frac{{s_{\rm F}^2 }}{{s_{\rm M}^2 }}.$$

(C.9)

One can also express Eq. (C.9) as

$$p_{s{\rm M}}^2 = \frac{{s_{\rm M}^2 s_{\rm F}^2 + \left(\frac{1}{4} h_{\rm M}^4\right) s_{\rm F}^2 ds_{\rm M}^2 + \left(\frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm M}^2 }}{{s_{\rm F}^2 }}\right) s_{\rm F}^2 ds_{\rm F}^2}}{{s_{\rm M}^2 s_{\rm F}^2 + \left(\frac{1}{4} h_{\rm F}^4\right) s_{\rm M}^2 ds_{\rm F}^2 + \left(\frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm F}^2 }}{{s_{\rm M}^2 }}\right) s_{\rm M}^2 ds_{\rm M}^2 }}.$$

(C.10)

Substituting Eq. (C.10) into Eq. (C.4) yields

$$R_{s^2 {\rm SD}} = \log \left[ {\frac{{s_{\rm M}^2 s_{\rm F}^2 + \left( \frac{1}{4} h_{\rm M}^4\right) s_{\rm F}^2 ds_{\rm M}^2 + \left( \frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm M}^2 }}{{s_{\rm F}^2 }}\right) s_{\rm F}^2 ds_{\rm F}^2 }}{{s_{\rm M}^2 s_{\rm F}^2 + \left( \frac{1}{4} h_{\rm F}^4\right) s_{\rm M}^2 ds_{\rm F}^2 + \left( \frac{1}{4} h_{\rm M}^2 h_{\rm F}^2 r_{\rm a}^2 \frac{{s_{\rm F}^2 }}{{s_{\rm M}^2 }}\right) s_{\rm M}^2 ds_{\rm M}^2 }}} \right],$$

(C.11)

a description of the response of relative variability ratio in the offspring generation to selection, expressed in terms of parameters drawn exclusively from the parental generation. One can obtain a similar description of the response of mean sexual size dimorphism by substituting Eqs. (A.1) and (A.2) into (B.13) (this equation appears in Leutenegger and Cheverud, 1982, 1985):

$$\overline{\rm R}_{{\rm SD}} = \left( {\frac{1}{2}} \right)\left[ {\left( {h_{\rm M}^2 s_{\rm M} i_{\rm M} - h_{\rm F}^2 s_{\rm F} i_{\rm F} } \right) + h_{\rm M} h_{\rm F} r_{\rm a} \left( {s_{\rm M} i_{\rm F} - s_{\rm F} i_{\rm M} } \right)} \right].$$

(C.12)

Equations (C.11) and (C.12) jointly comprise the combined response model. Given the input parameters for these two equations, all of which are drawn only from the parental population, one can calculate the response of male:female ratios of sex-specific mean size and sex-specific variance for one or more generations. Constraining the input parameters to conditions consistent with various selective forces, e.g., sexual selection with selection for increasing size in both sexes, variance dimorphism with selection for decreasing size in females and no selection on males, etc., generates the set of possible responses of$R_{s^2 {\rm SD}}$ and$\overline {R_{{\rm SD}} }$ for those forces, which in turn allows for the identification of categories of outcomes that must correspond to particular selective forces.