Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol168.Issue1.Nov2000:

The e€ective population size of some age-structured

populations

q

Edward Pollak

*

Department of Statistics, Statistical Laboratory, Iowa State University, 111 Snedecor Hall, Ames, IA 50011-1210, USA

Received 28 January 2000; received in revised form 28 July 2000; accepted 31 July 2000

Abstract

It was shown in a previous paper that if generations are discrete, then the e€ective population size of a large population can be derived from the theory of multitype branching processes. It turns out to be proportional to the reciprocal of a term that appears in the denominator of expressions for survival probabilities when there is a supercritical positively regular branching process for which the dominant positive eigenvalue of the ®rst moment matrix is slightly larger than 1. If there is an age-structured pop-ulation with unchanging proportions among sexes and age groups, then the e€ective poppop-ulation size is shown to be also obtainable from the theory of multitype branching processes. The expression for this parameter has the same form as in the corresponding model for discrete generations, multiplied by an appropriate measure of the average length of a generation. Results are obtained for dioecious random mating populations, populations reproducing partly by sel®ng, and populations reproducing partly by full-sib mating. Ó _{2000 Elsevier Science Inc. All rights reserved.}

Keywords:E€ective size; Age-structure; Branching processes

1. Introduction

The subject matter of this paper is the derivation of what Crow [1] called the variance e€ective population number of a population of constant size and unchanging demographic structure. Let there be no selection andqand q1 be the frequencies of an alleleAin generations 0 and 1. Then

q

Journal Paper No. J-18135 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa, Project No. 3201, and supported by Hatch Act and State of Iowa Funds.

*_{Tel.: +1-515 294 7765; fax: +1-515 294 4040.}

E-mail address:pllk@iastate.edu (E. Pollak).

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

if _Dqˆq1ÿq, this measure, Ne, of the e€ective population size satis®es the approximate equation

Var…_Dq† q…1ÿq†

2Ne :

The equality is exact only in special cases, but holds approximately if Ne is large. References to further work on this subject by Crow and his colleagues, as well as other workers, are given by Pollak [2].

The approach that will be used to calculate Ne was developed for populations with discrete generations by Pollak [2]. It will be shown to be applicable also to age-structured populations. The method is to take some well-known results from the theory of multitype branching processes and adapt them to obtain an approximation of the properties of a stochastic process that governs the change in numbers of an initially rare allele in a large ®nite population.

In Section 2, consequences of the standard population genetic theory will be derived for large populations with a rare allele A. In Section 3, the branching process approach that leads to an approximation toNewith discrete generations will be developed. This will be shown in Section 4 to be generalizable to age-structured populations by making use of discrete generation results, supplemented by an appropriate measure of the mean length of a generation. In Sections 5±7, the theory will be applied to obtain expressions for Ne for random mating dioecious populations, monoecious populations reproducing partly by sel®ng, and dioecious populations with permanent couples that reproduce partly by full-sib mating.

2. Standard theory

Let us ®rst suppose that at time 0 the frequency of an alleleAin a large diploid population is equal to q. Then, if Ne is the e€ective population size, the variance in the frequency of A in generation t is

Var…qt† q…1ÿq† 1

ÿ 1

ÿ 1

2Ne t

:

Thus, if q andt=…2Ne† are small,

Var…qt† q

t

2Ne: …1†

Now, let K be the number of copies of a gene among fertilized eggs. Then, if Nm and Nf are, respectively, the numbers of males and females in a dioecious population,Kis equal to 2…Nm‡Nf†

if there is an autosomal locus and toNm‡2Nfthere is a sex-linked locus. If there is a monoecious population of sizeN, thenK ˆ2N. Thus, ifqˆz=K, then it follows from (1) that the variance of the number of copies of Ain generation t is

Var…zt† K2

z K

t 2Neˆ

Kzt

3. The derivation ofNe ± discrete generations

It has been assumed in the derivation of (1) and (2) that the alleleAis rare between generations 0 andt. It is thus reasonable to assume, as an approximation, that units having at least one copy ofAin their genetic makeup reproduce independently.

Let us then consider a T-type branching process with the ®rst moment matrix M and ®nite second moments. Let M be irreducible and aperiodic. The following consequences follow, as discussed for example by Harris [3]. First, M has a simple dominant eigenvalue q. In all the examples to be discussed in this paper,qˆ1. Second, there correspond toqunique left and right eigenvectorsp0 and v that have only positive elements and satisfy the equations

p0M ˆp0; Mvˆv;

p01ˆX T

iˆ1

pi ˆ1;

p0vˆX T

iˆ1

piviˆ1:

…3†

Now lete0_ibe equal to a row vector that has 1 in theith position and zeros elsewhere. This means that there is a single ancestor of type i and none of any of the other types. The vector of ®rst moments of the Ttypes at time t is then e0_iMt. Finally, it can be shown that

e0_iMtˆe0_ivp0‡Mt₂ˆvip0‡e0_iMt₂; …4†

where the sum of the absolute values of the elements ofMt₂is of the same order of magnitude asat

for some a, where 0<a<1.

In what follows, the types in the branching process consist of individuals having one copy ofA in their genotypes if there is complete random mating. In this case, type refers to sex. If, however, there is partial inbreeding at least one type could have more than the copy ofA. Under random mating or a mixture of random mating and sel®ng, the reproducing units are individuals. But if there is partial full-sib mating, full-sibs mated to each other cannot give rise to independently developing lineages. However, separate couples with at least one copy ofAamong the genotypes of the mates can be considered to reproduce approximately independently. Thus, if there is partial full-sib mating, then the units are taken to be couples.

As (2) and the branching process approximation are both based on the assumption that A remains rare for a considerable time, it is reasonable to suppose that branching process theory may be useful in derivation ofNe. This will now be shown to be the case. Let us suppose ®rst that each type has only one copy of A, as is the case when there is random mating. Let Yij be the

number of o€spring of typejof a single individual of typeiand one of the types with a single copy ofAbe labeled by iˆ1. Then if the population is observed at timest1 andt1‡t2, where botht1 andt2are large, the variance of the number of copies ofAat timet1‡t2, given a single ancestor of type 1 is, by (2) and (4)

E‰Var…zt1‡t2jzt1†Š

Kv1t2 2Ne t2v1

XT

iˆ1 piVar

XT

jˆ1 Yijvj

!

Hence,

K 2Ne

XT

iˆ1 piVar

XT

jˆ1 Yijvj

!

: …5†

Now, let us consider a population in which there is partial inbreeding. Then each unit of typeiin a population hasnicopies ofAin its genetic makeup,n1ˆ1, and it turns out thatvi ˆniv1. Then, in

the long run, the expected number of descendants of a single ancestor of type 1 isv1, whereas the expected number of copies of Aamong the descendants is

v1X

T

iˆ1 nipiˆ

X

T

iˆ1

vipiˆ1:

Thus iftis large it follows from (2) and (4) that the variance of the number of copies ofAat time t‡1 is

Var…zt‡1† K

2Ne…t‡1† tv1 XT

iˆ1 piVar

XT

jˆ1 Yijnj

!

ˆ t

v1 XT

iˆ1 piVar

XT

jˆ1 Yijvj

!

:

Therefore,

Kv1 2Ne

XT

iˆ1 piVar

XT

jˆ1 Yijvj

!

: …6†

4. The derivation of Ne ± overlapping generations

When there is an age-structured population the branching process modeling can be done in two di€erent ways. One way is to de®ne the types of all possible genotype±age combinations. A second approach is to consider only o€spring of the youngest age group (age group 0) produced throughout the lifetime of a parent. The types are then reduced to all genotypes of age group 0 and the branching process models the development over time of a population with discrete generations, having the e€ective population sizeNed. It will now be shown thatNed, supplemented by a measure of the generation interval, will allow us to derive an expression forNe, the e€ective population size of the age-structured population.

number of copies ofAin the long run is approximatelyKz=…2Ne†. Therefore, if the mean length of a generation isL, then the rate of increase per generation, divided by Kz, is

L 2Ne

ˆ 1

2Ned

and

NeˆLNed: …7†

Now, it is generally the case for age-structured populations thatTis large. Since it is necessary to calculate the eigenvectors p0 and v for such populations as a means toward the derivation of an expression forL, we need assurance that there does not exist an eigenvalueq>1. Let us suppose that such an eigenvalue exists, but that we can also ®nd vectorsp0 andvthat satisfy (3) and have only positive elements. Let the right eigenvector with positive elements that corresponds toqbey. Then, we would have

p0Myˆp0yˆqp0y;

which implies that p0_{yˆ0. This contradicts the assumption that y has only positive elements.}

Therefore,qˆ1 if vectors vand p0, which satisfy (3), can be found.

5. Dioecious random mating populations

Let us assume ®rst that generations are discrete. Consider a large random mating population with, initially, a single copy ofAin a fertilized egg, whereas all other copies of the gene among fertilized eggs consist of other alleles, collectively denoted byA. Let types 1 and 2 be, respectively, males and females that have one copy ofAin their genotypes. Then, ifNmandNfare, respectively, the numbers of males and females in each generation, the ®rst moment matrix of the branching process is

M ˆ

1 2b

c

2

Nf

Nm

1 2

Nm Nf

1 2

" #

; …8†

wherebˆcˆ1 if the locus under consideration is autosomal andbˆ0; cˆ2 if it is sex-linked. It can then be shown after some algebra that (5) and (8) imply that

K 2Ne

Nm‡cNf

…1‡c†2 Nm

1 N2

m

Var…Y11†

‡ 2

NmNfCov…Y11;Y12† ‡ 1 N2

f

Var…Y12†

‡cNf 1

N2 m

Var…Y21†

‡ 2

NmNfCov…Y21;Y22† ‡ 1 N2

f

Var…Y22†

: …9†

Now, letkij denote the total number of successful gametes, contributing to o€spring of sexjthat

emanate from a parent of sexi. Let us suppose ®rst that the locus is autosomal or that it is sex-linked, but that the parent of sexiis a female. Then, the conditional distribution ofYij, givenkij, is

Var…Yij† ˆE‰Var…Yijjkij†Š ‡Var‰E…Yijjkij†Š ˆ

1

4‰E…kij† ‡Var…kij†Š; …10†

whereE…kij† ˆ2E…Yij†. In addition, Mendelian segregation takes place independently in gametes

contributing to male and female o€spring, so that

Cov…Yi1;Yi2† ˆE‰Cov…Yi1;Yi2†j…ki1;ki2†Š ‡Cov‰E…Yi1jki1†;E…Yi2jki2†Š ˆ

if the locus is autosomal, and

1

if the locus is sex-linked.

If the population is age-structured, then we assume that it is observed at times 0;1;2;. . . Let there be age groups 0;1;. . .;K1among males and age groups 0;1;. . .;K2among females, where an individual will be said to be in age groupaat timetif it is betweenaanda‡1 units of age at that time. Hence, there areK1‡K2‡2 age±sex classes (ai) in the population, whereihas values 1 and 2, respectively, for males and females. These will be ordered by listing younger individuals before older individuals within a sex and males before females. As before, we only consider individuals with one copy of a rare alleleAin their genotypes. If the population is large, then such individuals are almost certain to mate with individuals that do not haveA.

The branching process model that will be used is the one that was introduced by Pollak [4] and is a generalization of a one-sex model due to Goodman [5]. First, let

PaiˆP‰individual of sexisurvives to age groupa‡1jindividual has survived to age groupaŠ

`0iˆ1

`aiˆP0i;P1i;. . .;Paÿ1;i; a>0;

so that`ai is the probability that an individual of sexisurvives to at least age group a. Next, we

observe that a parent of age±sex class (ai) at time t can produce o€spring in two mutually ex-clusive ways. One is to live to age groupa‡1 and have o€spring and the second is to fail to live to age group a‡1 but, nevertheless, have o€spring. Let the associated probability generating functions of the two conditional o€spring distributions be f…ai†S…x01;x02† and f…ai†D…x01;x02†. In

t‡1 is considered to produce a single o€spring of age±sex class (a‡1;i) at timet‡1. Therefore, the unconditional generating function of the distribution of o€spring of an individual of type (ai) is

fai…x† ˆ …1ÿPai†f…ai†D…x01;x02† ‡Paixa‡1;if…ai†S…x01;x02†: …14†

By di€erentiating (14) with respect tox0j and then setting all of thexs equal to 1, we obtain

b…ai†j ˆ …1ÿPai†bD…ai†j‡PaibS…ai†j;

whereb…ai†j,bD…ai†j, andbS…ai†jare, respectively, the mean number of o€spring of sexjand age group

0 produced by a parent of type (ai) at timetand the corresponding conditional means, given that the parent dies or lives to reach age groupa‡1. In addition, (14) implies that

o_fai…x†

o_xa‡1;j

xˆ1

ˆdijPai;

wheredij is Kronecker's delta. The matrix of ®rst moments is thus of the form

M ˆ M11 M12

M21 M22

; …15†

where

Mijˆ

b…0i†j dijP0i 0 . . . 0

b…1i†j 0 dijP1i . . . 0

.. .

.. .

.. .

.. .

b…Kiÿ1;i†j 0 0 . . . dijPKiÿ1;j b…Kii†j 0 0 . . . 0 2

6 6 6 6 6 4

3

7 7 7 7 7 5

:

If the locus under study is sex-linked, the ®rst column ofM11has elements that are all equal to 0. It was shown by Pollak [4] that a sucient condition forM to have a simple positive dominant eigenvalueqis to have two successive age groups in which progeny of both sexes can be produced. It will be assumed that this condition holds. If there exist vectors

p0ˆ …p01;p11;. . .;pK11 jp02;p12;. . .;pK22†

and

vˆ …v01;v11;. . .;vK11 jv02;v12;. . .;vK22† 0

satisfying (3), it follows from (15) that

pai ˆpaÿ1;iPaÿ1;iˆ`aip0i; a>0; …16†

p0iˆp01

XK1

aˆ0

`aib…a1†i‡p02

XK2

aˆ0

`a2b…a2†i; …17†

foriˆ1;2 and

The expressionPKj

aˆ0`ajb…aj†iis equal to the mean number of o€spring of age group 0 and sexithat

are produced throughout the lifetime of an individual of sex j. Then if at all times there are, respectively,nm andnf males and females of age group 0 and Ais a neutral allele

XK1 Thus, it follows from (17) that

p01ˆ1

Also, if we set aˆ0 in (18), then these equations imply that

v01ˆ1

Eqs. (3), (16) and (18) together imply that

1ˆX

is equal to the mean, among o€spring of sexj, of ages of parents of sexiwhen the o€spring are in age group 0. Thus, by (19)±(21),

Thus,

p01v01ˆ 4

2…L11‡L12‡L21‡L22†ˆ

1

2L …23†

when there is an autosomal locus and

p01v01ˆ1

when the locus is sex-linked, where L, the mean age of a parent when an o€spring is born, is de®ned to be the generation interval.

The e€ective population size Ned, which is associated with age group 0 progeny produced throughout the lifetime of parents, is given by (12) for an autosomal locus and (13) for a sex-linked locus ifNmand Nf are, respectively, replaced bynmandnf. Thus, by (7), (12) and (23), we obtain

for an autosomal locus. Likewise, for a sex-linked locus, (7), (13) and (24) lead to

1

Eqs. (25) and (26), respectively, were derived by Hill [6,7] and Pollak [4,8] in other ways.

6. Monoecious population reproducing partly by sel®ng

We ®rst assume that generations are discrete. Consider a large monoecious population of sizeN in each generation, in which there is self-fertilization with probabilityband the union of gametes from random separate individuals with probability 1ÿb. If there is reproduction by sel®ng, then an individual contributes both a male and a female gamete to a fertilization, whereas if there is random mating, then the average numbers of male and female gametes it contributes to separate o€spring are each equal to one. Thus, each individual has, on the average,bo€spring from self-fertilization and 2…1ÿb† from random mating. Let types 1 and 2 be de®ned to be, respectively, the genotypesAAand AA. We assume that initially all individuals except one have the genotype

AA. Thus, the array of o€spring of a randomly mated AA parent is approximately

and all o€spring are of genotype AA if an AA parent reproduces by sel®ng. Hence, the ®rst moment matrix of the associated branching process is

M ˆ 1ÿb=2 b=4

2…1ÿb† b

: …27†

Eq. (27) implies that

X 2

iˆ1 piVar

X 2

jˆ1 Yijvj

!

ˆ 4ÿ3b

…4ÿ2b†2‰4…1ÿb†Var…Y11‡2Y12† ‡bVar…Y21‡2Y22†Š: …28†

Now Yi1‡2Yi2 is equal to ki, the number of copies of allele A that an individual of type i

con-tributes to an o€spring. If iˆ2, then this is equal to k, the total number of successful gametes produced by an individual. If iˆ1, then the conditional distribution of k1, given k, is binomial withk trials and a probability 1=2 of success. Since E…k† ˆ2,

Var…Y11‡2Y12† ˆVar k 2

‡E k

4

ˆ1

4‰Var…k† ‡2Š:

As Kˆ2N, it follows from (6) and (28) that

v1

Ne

v1 N

4…1ÿb†

4ÿ2b 1

4…Var…k†

‡2† ‡ b

4ÿ2bVar…k†

ˆ v1

2…2ÿb†N‰Var…k† ‡2…1ÿb†Š: …29†

Haldane [9] showed that if there is an in®nite population with respective probabilitiesband 1ÿb of reproduction by sel®ng and by random mating, the inbreeding coecient F of a random in-dividual is equal to b=…2ÿb† in the long run. Thus, 1‡F ˆ2=…2ÿb† and 1ÿF ˆ2…1ÿb†=

…2ÿb†, so that (29) can be recast as 1

Ne

1

4N‰…1‡F†Var…k† ‡2…1ÿF†Š: …30†

Eq. (30) was derived, using the approach in this paper, by Pollak [2] and earlier, in other ways, by Caballero and Hill [10] and Pollak and Sabran [11]. IfF ˆ0, then (30) reduces to an expression for Ne obtained by Haldane [12], which is almost the same, when N is large, as one derived by Wright [13].

If the population is age-structured, then let us suppose that it has age groups 0;1;. . .;K, where Kis the last group in which it is possible to reproduce. LetPa be the probability that an individual

survives from age group a to age groupa‡1, regardless of its genotype, and b…ai†j ˆE‰number of genotype joffspring at time t‡1

of a parent of genotypei and agea at time tŠ:

The symboli (orj) is set equal, respectively, to 1 and 2 for genotypesAAandAA. There are then 2…K‡1† age±genotype classes, where types 0;1;. . .;K refer to age groups among individuals of genotypeAAand typesK‡1;. . .;2K‡2 to age groups among individuals of genotypeAA. The ®rst moment matrix of the associated branching process thus has the form

M ˆ M11 M12

M21 M22

where

Pollak [14] showed that sucient conditions for M to have a simple dominant positive ei-genvalueqare that both genotypes are fertile at the same ages and that there are two successive fertile age groups. We will assume henceforth that these conditions hold. In addition, qˆ1 if vectors

p0ˆ …p01;p11;. . .;pK1jp02;p12;. . .;pK2†

and

vˆ …v01;v11;. . .;vK1 jv02;v12;. . .;vK2†0

that satisfy (3), and have only positive elements, can be found.

The notation that has been employed in writing the elements of (31) and the vectors p0 _{and v}

indicate that, from a formal algebraic point of view, Eqs. (16)±(18) hold if we replaceK1andK2by Kand Pai by Pa for iˆ1;2. Thus, if `a ˆ

produced throughout the lifetime of an individual of genotypej. Thus,

PK

It then follows from (33) that

where

In addition, the special case of (34) for which aˆ0 implies that

Md

Now, the special case of (22) that applies in this section is

1ˆX

is equal to the mean, among o€spring of genotype j and age group 0, of the ages of parents of genotypei, (35)±(38) imply that

1ˆv01

Ne 4LN0

…1‡F†Var…k† ‡2…1ÿF†; …41†

whereN0 is the number of individuals of age group 0. IfbˆF ˆ0, then (41) reduces to an ex-pression obtained by Hill [6,7].

7. Populations reproducing partly by full-sib mating

We assume that there are N=2 permanent couples in the population at any time, of which a fractionbare expected to be full-sibs and a fraction 1ÿbto be random male±female pairs. If an allele is initially rare, then it is reasonable to assume, as an approximation, that separate couples reproduce independently. If there is an autosomal locus the mating types with at least one copy of Ain the genotypes of the mates are (i)AAAA, (ii) AAAA, (iii) AAAA, (iv) AAAAand (v) AAAA. If a locus is sex-linked, then the mating types are (i) AAAY , (ii) AAAY, (iii) AAAY , (iv)AAAY, and (v) AAAY.

If generations are discrete and the locus under consideration is autosomal it was shown by Caballero and Hill [10] that

1

Ne

1

4N‰…1‡3F†Var…k† ‡2…1ÿF†Š; …42†

wherekis the number of o€spring of a couple andF ˆb=…4ÿ3b†, the inbreeding coecient in the long run of an in®nite population with partial full-sib mating. This expression for F was derived by Ghai [15]. IfF ˆ0, then (42) reduces to the special case of (12) for which NmˆNf ˆ

N=2 andk11‡k12ˆk21‡k22ˆk. If the locus is sex-linked, then

1

Ne

2

9N‰2…1ÿF† ‡ …1‡F†Var…kf† ‡2Var…km† ‡4FCov…kf;km†Š; …43†

whereFis the same as for an autosomal locus andkf andkmare, respectively, equal to the total numbers of gametes withXchromosomes contributed by females to their o€spring and by males to their daughters. Eq. (43) is a special case of an expression for partially inbred populations derived by Wang [16]. It was also derived independently for this case by Pollak [2]. IfF ˆ0, then (43) reduces to the special case of (13) that applies when NmˆNf ˆN=2.

For both autosomal and sex-linked loci, the ®rst moment matrix of the approximating branching process has ®ve rows and ®ve columns. When the population is age-structured, the elements in these matrices will each be replaced by matrices in the following manner. We assume that the ages of members of a couple are the same and that they mate for life. Thus, a couple has an age equal to that of the mates that belong to it. Let

PaˆP‰couple of age a lives to age a‡1jcouple has survived to age group aŠ

and

B…ai†jˆE‰number of typej offspring male±female pairs

where aˆ0;1;. . .;K and i and j can assume values between 1 and 5. Hence, the ®rst moment matrix is of the form

M ˆ

and dij is Kronecker's delta.

It will be shown in Appendix A that M has a simple dominant positive eigenvalue q if all genotypes are fertile at the same age groups and there are two successive fertile age groups. This eigenvalue is equal to 1 if there exist vectors

p0ˆ ‰p01;. . .;pK1 jp02;. . .;pK2 j jp05;. . .;pK5Š

and

vˆ ‰v01;. . .;vK1 jv02;. . .;vK2 j jv05;. . .;vK5Š0

that satisfy (3). If this is the case, then

paiˆPaÿ1paÿ1;i ˆ

kˆ0`kB…ki†j is equal to the mean number mij of o€spring male±female pairs of type j

produced in the lifetime of a parental couple of type i. This is also the elementmij in theith row

andjth column ofMd, the ®rst moment matrix when there are discrete generations. Thus, by (46),

p0i ˆCpi, where pi is theith element ofp0when generations are discrete. It thus follows from (3)

1ˆX

generations are discrete. It was shown by Pollak [2] thatvi ˆniv1, whereniis equal to the number

of copies of Ain genotypes of couples of typei. Now

Lijˆ

PK

kˆ0…k‡1†`kB…ki†j

PK

kˆ0`kB…ki†j

is the mean of the ages of parent couples of typeiamong o€spring male±female pairs of age group 0 and type j. Thus,

the long-term mean, among copies ofAin o€spring male±female pairs of age group 0, of ages of parent couples, and

8. Discussion

In this paper, expressions have been derived forNe when there are dioecious populations that mate randomly or undergo full-sib mating with probability b, and also when there are mono-ecious populations that reproduce partially be sel®ng. Populations with both discrete and over-lapping generations were considered.

If there is a dioecious population reproducing partly by full-sib mating, thenNeis a decreasing function ofFwhen Var…k†>2=3 if the locus under consideration is autosomal. If the locus is sex-linked, then the condition for Ne to be a decreasing function of F is to have Var…kf† ‡ 4 Cov…kf;km†>2. These conditions are satis®ed if, in particular, there are Poisson distributions of numbers of male and female progeny of a mating, for then Var…k† ˆVar…kf† ˆ2 and there is a positive covariance betweenkf andkm. Caballero and Hill [10] have shown in this case that when there is an autosomal locus and discrete generations NeˆN=…1‡F†. They also found this ex-pression to hold if there is a monoecious population, partial sel®ng and a Poisson distribution of family size. If there is partial full-sib mating and a sex-linked locus Var…km† ˆ1, Var…kf† ˆ2, and, since both members of a couple have the same number of progeny of either sex and mothers produce daughters with probability 1/2,

Cov…kf;km† ˆE‰Cov…k22;kmjkf†Š ‡Cov‰E…k22jkf†;E…km†Š ˆ1

2Var…km† ‡1ˆ3=2:

Thus, (43) reduces to

1

Ne

2

9N…6‡6F† ˆ 4

3N…1‡F† ˆ 1‡F

NeR

;

whereNeR is the e€ective size of a population withN=2 couples whenF ˆ0.

An extreme situation in whichNeincreases withFis that in which there is no variability in the number of o€spring. It can then be shown thatNe2N0L=…1ÿF†if the population is monoecious and there is partial sel®ng. If there is partial full-sib mating, then (42) and (43) imply that NeNe0R=…1ÿF†, where Ne0R is the e€ective size of a population with F ˆ0. Ne0R is equal to 2N

when the locus is autosomal and to 9N=4 when it is sex-linked.

In all the cases discussed in this paper, the expressions for Ne when the population is age-structured have N0Lin place of N, which is what appears in the corresponding expressions that apply when generations are discrete. In addition,L is always the mean, among copies of a par-ticular allele in individuals of the youngest age group, of ages of parental units, whether they are individuals or monogamous couples.

The approach used in this paper to derive Ne for age-structured populations has, in common with the reasoning used by Hill [6,7], the need to calculate L. For this purpose Hill needed to derive a matrix whose elements are probabilities that genes in age±sex classes at timetcome from genes in age±sex classes at time tÿ1. In this paper,Lis computed from a matrix that expresses how genes are transferred from age±type classes at time tto age±type classes at timet‡1. Thus, one way or another, it is necessary to do computations with matrices that are of comparable complexity to those considered in this paper.

have more complicated forms, which could make the analysis unwieldy. It is not clear to this author, at present, how to obtain general results for partially inbreeding age-structured popula-tions.

Hill [6,7] did not need to invoke the theory of multitype branching processes in his derivations. However, an advantage of using branching process theory is that approximate expressions can be obtained for survival probabilities if the simple positive dominant eigenvalueq is slightly larger than 1. It was shown by Pollak [2] that in this case approximate probabilities of survival of alleleA of a gene in the long run are proportional toqÿ1 times the e€ective population size and inversely proportional to the number of copies of the gene among fertilized eggs.

Appendix A

Letnai…t†be the expected number of couples of typeithat are in age groupaat timet. Then, if

n0_t ˆ ‰n01…t†;. . .;nK1…t† j jn05…t†;. . .;nK5…t†Š;

n0_t ˆn0_tÿ1M;

whereM is given by (44). Hence,

n0i…t† ˆ

X5

jˆ1 XK

aˆ0

naj…tÿ1†B…aj†i ˆ

X5

jˆ1 XK

aˆ0

n0j…tÿ1ÿa†`aB…aj†i:

It will be assumed that all types are fertile at the same ages.

First, consider a special case in whichB…aj†iˆBai for each j, provided B…aj†i 6ˆ0. Then

n0i…t† ˆ

X 5

jˆ1 XK

aˆ0

n0j…tÿ1ÿa†`amiBai;

wheremi is the number ofB…aj†i's not equal to zero. Then

n0…t† ˆX

5

iˆ1

n0i…t† ˆ

X5

jˆ1 XK

aˆ0

n0j…tÿ1ÿa†`a

X5

iˆ1

miBaiˆ

XK

aˆ0

n0…tÿ1ÿa†`a

X5

iˆ1

miBai: …A:1†

The auxiliary equation corresponding to Eq. (A.1) can be written in the form

1ˆX K

aˆ0

kÿ…1‡a† `a

X 5

iˆ1 miBai

" #

: …A:2†

This is an equation of a type that appears in the theory of population growth that was ®rst discussed in detail by Leslie [17].

As the right-hand side of (A.2) is a decreasing function of k when k>0, it is clear that this equation has a single positive rootkˆq_{. Leslie [17] stated that a sucient condition for this root}

If we assume that there are two successive age groups in which all types of couples are fertile, n0…t†, and hence numbers of all genotype±age classes, are asymptotically equal to multiples of…q†

t

astincreases. If, however, matrix algebra had been used, then the same result would hold. Hence, q _{is also the dominant positive simple eigenvalue of the matrixM corresponding to this special}

case.

Now note that M is a matrix having only nonnegative elements and that all of its positive elements are in the same positions as the positive elements ofM. Since there is sometsuch that

…M†t has only positive elements it then follows thatMt also has only positive elements for this value of t. Thus M, like M, is a positively regular matrix and hence has a simple dominant positive eigenvalue q.

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