ON SOME MODELS OF FERTILITY SELECTION

(1)

Copyright 0 1983 by the Genetics Society of America

O N SOME MODELS OF FERTILITY SELECTION

MARCUS W. FELDMAN,* FREDDY B. CHRISTIANSEN' and URI LIBERMAN*

*Dp@rtinPnt of Biological Sciences, Starqord University, Stanford, Calfurnia 94305; +Department of Ecology and Genetics, University of Aarhus, DK-8000 Aarhus C., Denmark; and *Department of

Statistics, Tel Aviv University, Tel Aviv, Israel

Manuscript received November 29, 1982 Revised copy accepted July 30, 1983

ABSTRACT

Additive, multiplicative and symmetric models of fertility controlled by one diallelic gene are studied. For the completely symmetric fertility system a complete equilibrium and local stability analysis is possible. Contrary to previous conjectures, asymmetric equilibria can be stable. Conditions are derived under which a multiplicative model can be regarded as equivalent to a symmetric fertility system.

HE purpose of this article is to reexamine the models of fertility selection

T

at one genetic locus that are in the population genetic literature. In

particular, we address how these models are related to viability selection and how they may be reduced to a multiplicative form first studied in detail by

BODMER (1 965).

T h e fertility of a given mating will have its generally accepted meaning, namely, the relative proportion among the newly produced offspring attrib- utable to that mating. Fecundity will be used to refer to the proportions of offspring among female parents, and virility will be used to refer to those among the male parents. Previous theoretical studies of this form of selection

are those of PENROSE (1949), BODMER (1965) and HADELER and LIBERMAN

(1 975). Some special examples of these models can be placed within the mod-

eling framework of sexual selection developed by O'DONALD (1977, 1978) and

KARLIN (1978).

The brtsic fertilzty model: sex symmetry

We shall consider autosomal genes and an array of gametes {yf), which may

include as many loci as desired, and we shall assume equal recombination in

males and females. T h e male and female genotypes are then { y 7 / y j ] which have

frequencies {g,]) as zygotes. In general the fertility of a mating between a male

y J y j and female yJyl is F y , h , , for instance. But every genotype, yI/ys, produced

by this mating is produced in the same frequency by the mating between a

female y7/yJ and a male yh/y/, which has fertility F M , ~ . Thus, among the zygotes the frequency g,q of y,/rs is the same as if the reciprocal matings had the same fertility, namely, (Fll,k/

+

F k l , l j ) / 2 . This property, which holds independently of

(2)

the number of autosomal loci studied, permits us to assume that the fertilities of reciprocal matings are equal. Therefore, we call it sex symmetry.

The additive fertility model

PENROSE (1949) studied one locus with two alleles and proposed that the parameter for the fertility of a mating pair should be the sum of parameters for the fecundity and virility of the participants. Sex symmetry allows the

assumption that the virility and fecundity of a given genotype are the same,

without loss of generality. PENROSE showed that there is an equilibrium of this

model with the genotypes in Hardy-Weinberg proportions with the gene frequency given by the gene frequency equilibrium of the one-locus two-allele viability model with viabilities equal to the virilities (equal to the fecundities). This result is a corollary of the fact, first noted by BODMER (1965), that the additive fertility model is formally equivalent to a model in which viability selection acts in one sex only, the other sex being selectively neutral.

This result can be generalized to an arbitrary number of alleles at an arbitrary number of loci. T h e mating y l / y J X y k / y / has fertilityJ;,

+fk/,

and this is

the only selection on the system. Now consider a similar model in which the

genotype y J y I has the viability U , , in the females and unit viability in males. In

this viability model the frequency of the mating r 7 / y J X y k / y / is v,g,gkl/77, where it =

E,

zliigV. Thus, the proportions among zygotes produced in this model are the same as in a fecundity model with FtJ,kl = v V r which is again equivalent to the model with F,.k/ = ( U ,

+

uk/)/2. Thus, by assigning the relative viabilities

z',, = 2J, to the females of the viability model, we establish the equivalence between the one sex viability model and the additive fecundity model.

Now consider one locus with alleles A I , .

.

, A,, (y, = A7). Then, in the next generation we have

and

with

#J

Following BODMER (1965) the systems (1) and (2) are equivalent to a viability

selection regimen specified by viabi1itiesJJ of one sex only, the other sex being

selectively neutral. Then, at equilibrium the gene frequencies are the same in

(3)

ONE-LOCUS FERTILITY MODELS 1005

the same selection on both sexes (see also

Roux

1977). T h e genotype fre-

quencies are the appropriate products of the gene frequencies. This result can

be generalized to the case in which the male and female additive contributions

are not the same. Here, the analogous one-locus viabilities are the averages of the male and females additive genotype contributions to fertility. CANNINGS (1969a) showed that, for the model with viability selection on one sex only, this equilibrium is locally stable. With two alleles he showed (1969b) that it is globally stable.

Symmetric fertility model

HADELER and LIBERMAN (1975) analyzed a symmetric fertility model de-

scribed by Table l . For this symmetric model HADELER and LIBERMAN showed

that there are five possible polymorphic equilibria. Three of these are sym-

metric, that is, AlAl and AzAZ occur equally frequently, and two are asymmet-

ric. It is possible that all five polymorphic equilibria exist, although the maximum number of stable points remains an open question. Under extreme conditions on the parameters it is possible that none of the fixation or polymorphic equilibria be locally stable. Under these circumstances the fundamental theo- rem of natural selection clearly would not be expected to hold, and, indeed,

it does not (POLLAK 1978).

If a = y, a complete analysis of the model is possible. Let

4,

6,

z5

be the

equilibrium frequencies of A l A l , A1A2, AzAz, respectively. Then, there is a

single symmetric equilibrium with Zi = Zj = t/4 and

3

=

Yz.

T w o asymmetric equilibria given by

2

+

z5

= (6

-

p)/(a

+

b

-

2p),

3

= ( a

-

P)/(a

+

6

-

2P),

(4)

where the values of

Zi

are the roots of

exist if

as shown by HADELER and LIBERMAN (1975). Now under the assumption a =

7, a complete stability analysis is possible. There are six parametric inequalities

(4)

TABLE 1

Syinmetnr oiie-locu.s fcrtilitzrs

Males

Note that Roux’s conjecture (1977) that the asymmetric equilibria cannot

be stable is incorrect. We might conjecture that, in the more general case

a # y, two is the maximum number of stable polymorphic equilibria, and, if

fixations are included, the maximum is three stable equilibria.

The multipliccitive model

BODMER (1 965) introduced a model in which the matings between 7,/yI

males and yk/y/ females had relative fertility F& = m , , f k l , where m,, is the

virility of yJy, males and

fk,

is the fecundity of yh/y( females. Sex symmetry allows us to assume without loss of generality that Ft,,k/ = Fkl,rl = (my&/

+

m k l J ; I ) /

2.

For the case of two alleles at one locus with virilities m11 = 1

-

P I I , m12 =

1, mZ2 = 1

-

pZ2 and fecunditiesfil = 1

-

4 1 1 , f i p

= l , f 2 2 = 1

-

4 2 2 for A I A I , A1A2 and A2A2, respectively, BODMER showed that if ~ 2 2 , 4 2 2

>

0 and ,alI, 411

<

0, fixation of both sexes in A l A l is globally stable, whereas, if the

inequalities are reversed, A2A2 is globally stable. More generally A l A l is locally

stable if

Pull

+

411

<

P11411,

(7)

(8) and A2A2 is locally stable if

P22

+

4 2 2

<

P 2 2 4 2 2 .

BODMER showed that the multiplicative fertility assumption produced genotype recursions identical with those for a model in which the selection is through viabilities acting differently on the two sexes. That model has been

analyzed in some cases by OWEN (1953), MANDEL (1971) and KARLIN (1972).

BODMER paid special attention to the case in which the selection on males would produce the same equilibrium as that on females. This is equivalent to the assumption that P I I / P Z ~ = k = 4 1 1 / 4 2 2 . In addition to the equilibrium E, at

which the frequency of A I is k / ( l

+

k ) in both sexes ( i e . , the equilibrium of a system in which selection is the same on both sexes) there may exist two other polymorphic equilibria E,’ and E;. When E, is stable, E,’ and E; do not exist, and neither fixation can be stable, in contrast to the symmetric case (i) described earlier. It is possible, however, for the fixations and E, to be unstable, in which case E: and E; exist and are stable, as in case (iv) for the symmetric model. When the selection is overdominant in one sex and underdominant in

the other one, fixation and one of E: and E; can be stable. When the selection

(5)

ONE-LOCUS FERTILITY MODELS 1007

fixations may be unstable and the only polymorphic equilibrium point is one

of E: and E; which is then stable. This is by no means a complete discussion

of the multiplicative model and is intended primarily to point out some features that distinguish it from the symmetric case. We now address ways in which the two models can be regarded as equivalent.

Equivalence of the symmetric model to the multiplicative model

In the multiplicative model for two alleles the virilities of AIAI, A l A 2 and A2A2 are m l l , 1, mZ2 and the corresponding fecundities a r e f i l , 1 , h 2 , which by

sex symmetry in fertility produce Table 2.

For the multiplicative matrix of Table

2

to fit a normalized sex symmetric

fertility matrix ( F f J , ~ , ] with F 1 2 , 1 2 = 1 we must have

F11.11

= f i l m l l , F22,22 = f 2 v "

F11.12 = ( f i l

+

m11)/2, F 22 , 12 = ( $ 2 2

+

m22)/2.

Thus,

fi

and ml1 must be the roots of the quadratic equation

y2

-

2 F i i . i ~ ~

+

Fii.11 = 0 ,

.- 2F322.122

+

F22,22 = 0.

(9)

(10)

(1 l a )

whereash2 and m22 are the roots of

T h e two possible configurations forfil and m l l are

f??

= F11,12

+

(F:1,12

-

F 1 1 ~ 1 ) ~ / ~ ,

f':1

= F I I , ~ Z

-

(F?1,12

-

F11,11)~",

f ( * ) 22

-

F z z , i z

+

( F h . 1 2

-

F 2 2 , ~ 2 ) ~ / ~ 1

my? = F11,12

-

( F ? ~ , I z

-

F I I , ~ ~ ) " ~

mi? = F11,12

+

(F:1,12

-

F,1,11)"* ( I l b )

m(') 22

-

F 22,12

-

( ~ 2 2 2 , 1 2

-

F22,22)'/' ( 1 % ) and

T h e two possible configurations for f 2 2 and mZ2 are

and

fF2)

= F 2 2 , 1 2

-

( F & , ~ P

-

F z ~ , ~ z ) ~ / ' , m(') 22

-

F P P , I P

+ ( F & J , ~ z

-

F ~ z , z z ) ~ / ~ . _(12b) T h e four gossible roducts f y ? m $ )

+

f(:2my;, fyim$$

+

f(:3m\'?:), f\';m&

+

fYim\'/,

f\'/m$2)

+

f P 2 m l l (2) (27 take the two possible values

2(F11,12F22,12

+

[ ( F : i , i z

-

Fii,ii)(F222,12

-

F z z , ~ z ) ] " ~ )

2(Fll,l2F22,12 - [(F:i,lP

-

F l l , l 1 ) ( ~ ~ 2 , 1 *

-

F22,*2)11/21

( 1 3 4

(1 3b) and

because of the sex symmetry. If one of these is equal to 2F11,22, then the multiplicative model fits the given Fli,hl configuration.

It is clear from (11) and (12) that, for a successful fit by the multiplicative model, we require of the original model that

(6)

TABLE 2

Multiplicntive hrtilities

Males

Returning now to the symmetric model of Table 1 , normalized by assuming

6 = 1, the conditions of (14) require that

p2

L a. Under the assumption that

6'

2 cy the choice

entails

j i

1 m 2 2

+

f22inl = 2a. These values substituted into the multiplicative

model Table

2

produce the symmetric model Table 1.

We can now examine the six cases of the symmetric model, with 6 = 1, to

determine how they relate to the possibilities for the multiplicative case. Note first that cases (i) and (ii) of the symmetric model both have a!

>

p2, so

no

appropriate multiplicative model can exist. Cases (iii) and (iv) both have

p2

>

cy and so allow a multiplicative representation. Both cases correspond tofil =

f;22

>

1. T h e relation m I 1 = m22 = /3

-

(p2

-

a)'/*

<

1 holds if, and only if,

p

>

(1

+

a ) / 2 , which is obviously the case in both (iii) and (iv). Thus, these cases correspond to overdominance in one sex and underdominance in the other.

Cases (v) and (vi) both contain models that have equivalent multiplicative

formulations and models that have no such representation. In case (v) the

condition

p'

>

a! is satisfied for ,6 large enough, in which case f i l = f 2 2

>

1 .

In addition the relation m i l = 11222

>

1 holds if, and only if, ( 1

+

a ) / 2

>

p.

Thus, case (v) contains representations in which both sexes are underdominant

in fertility and others in which one sex is underdominant and one is overdominant. In case (vi) the condition

p2

>

a is again fulfilled only for @ large enough and inll = 11222

<

1 . T h e relationhi = f22

<

1 holds if, and only if, (1

+

a)/2

>

p,

so this case contains representations in which both sexes are overdominant in fertility, and others in which one sex is overdominant and the other underdominant.

To summarize, these multiplicative models exhibit the two types of dynamics

known from simple selection modeis with equal viabilities in the sexes, i.e., (iii),

(v) and (vi). In addition, we have properties of case (iv) with two interior stable

equilibria. T h e nonmultiplicative symmetric models also exhibit the dynamics

of the standard viability selection model, i . e . , (ii), (v), (vi), as well as the properties of case (i) in which both fixations and a polymorphism may be stable

simultaneously. Thus, both the multiplicative and nonmultiplicative versions of

(7)

ONE-LOCUS FERTILITY MODELS 1009

Con cl ud

ing

remarks

Most population genetic theory of selection incorporates only the differences among the genotypes of zygotes in survival to maturity, the zygotic selection

component (BUNGAARD and CHRISTIANSEN 1972; CHRISTIANSEN and FRYDEN-

BERG 1973). These viability differences are properties of individuals and can properly be interpreted in terms of individual fitness. Fertility differences, however, are properties of pairs of individuals, and it is not proper, in general, to describe this selection component in terms of individual fitness differences (KEMPTHORNE and POLLAK 1970; Roux 1977; CHRISTIANSEN 1983).

We have seen earlier that, in some cases, selection mediated by differential

fertility can be formally equivalent to viability selection, in which cases the selection can be described in terms of individual fitnesses. This must be regarded merely as an aid in understanding the evolutionary significance of the interaction between indidduals during the breeding process. It is certainly not of any help in describing observed fecundity differences in natural or experimental populations. In particular, the comparison between those fertility models interpretable in terms of individual fitness effects, and the simple symmetric case that cannot be so interpreted, underscores that the simplest inter- actions between individuals in the process of selection can produce evolutionary conclusions not expected from standard individual fitness models.

Selection by differential fertility usually occurs after the time at which most natural or experimental populations are observed: it is postobservational selec-

tion in the sense of PROUT (1965). In evaluation of individual fitnesses by

comparison of genotypic proportions between generations, it is important to remember that postobservational selection can introduce substantial biases. In the simplest cases these biases can lead to the erroneous inference that frequency-dependent selection, with an advantage to rare genotypes, is occurring

(PROUT 1965; BUNGAARD and CHRISTIANSEN 1972; CHRISTIANSEN, BUNGAARD

and BARKER 1977). These biases will certainly be present if the fertility selection is in fact multiplicative, and there is no reason to believe that they will be absent if the true fertility selection is of a more general type.

GM 10452.

This research was supported in part by National lnstitutes of Health grants GM28016 and

LITERATURE CITED

BODMER, W. F., 1965

BUNGAARD, J. and F. B. CHRISTIANSEN, 1972

CANNINCS, C., 1969a

Differential fertility in population genetics models. Genetics 51: 41 1-424. Dynamics of polymorphisms. I. Selection compo-

The study of multiallelic genetic systems by matrix methods. Genet. Res.

Unisexual selection at an autosomal locus. Genetics 62: 225-229. nents in an experimental population of Drosophila melanogaster. Genetics 71: 439-460.

1 4 167-183.

CANNINGS, C., 1969b

CHRISTIANSEN, F. B . , 1983

CHRISTIANSEN, F. B., J. BUNCAARD and J. S. F. BARKER, 1977

The definition and measurement of fitness. In: Evolutionary Ecology.

On the structure of fitness estimates

B.E.S. Synposiutn 23, Edited by B. SHORROCKS. Blackwell, Oxford. In press.

(8)

CHRISTIANSEN, F. B. and 0. FRYDENBERG, 1973 Selection component analysis of natural populations using population samples including mother-offspring combinations. Theor. Pop. Biol. 4: 425-445.

Selection models with fertility differences. J. Math. Biol.

Some mathematical models of population genetics. Am. Math. Monthly 79:

Comparisons of positive assortative mating and sexual selection models. Theor.

Concepts of fitness in Mendelian populations. Genetics

Owen’s model of a genetic system with differential viability between HADELER, K. P. and U. LIBERMAN, 1975

2: 19-32.

KARLIN, S., 1972 699-739.

KARLIN, S., 1978

Pop. Biol. 14: 281-312.

KEMPTHORNE, 0. and E. POLLAK, 1970 64: 125-145.

MANDEL, S. P. H . , 1971

O’DONALD, P., 1977

O’DONALD, P., 1978 sexes. Heredity 26: 49-63.

Theoretical aspects of sexual selection. Theor. Pop. Biol. 12: 298-334.

A general model of mating behavior with natural selection and female

A genetical system admitting of two distinct stable equilibria under natural

T h e meaning of fitness in human populations. Ann Eugen. (Lond.) 1 4

With selection for fecundity the mean fitness does not necessarily increase.

T h e estimation of fitness from genotypic frequencies. Evolution 19: 546-551.

Fecundity differences between mating pairs for a single autosomal locus, sex

Corresponding editor: W. J. EWENS preference. Heredity 40: 427-438.

selection. Heredity 7: 97-102. OWEN, A. R. G . , 1953

PENROSE, L. S . , 1949 301-304.

POLLAK, E., 1978 Genetics 9 0 383-389. PROUT, T., 1965

Roux, C. Z., 1977