Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue2.Aug2000:

Mathematical Biosciences 166 (2000) 1±21
www.elsevier.com/locate/mbs

The e€ective number of a population that varies cyclically in
size. I. Discrete generations
Yufeng Wang a,1, Edward Pollak b,*
a

b

Department of Zoology and Genetics, Iowa State University, Ames, IA 50011-1210, USA
Statistical Lab, Department of Statistics, 111 Snedecor Hall, Iowa State University, Ames, IA 50011-1210, USA
Received 28 September 1999; received in revised form 9 May 2000; accepted 16 May 2000

Abstract
We consider a dioecious population having numbers of males and females that vary over time in cycles of
length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the e€ective population number (or size), Ne , is approximately equal to the harmonic
mean of the e€ective population sizes during any given cycle. This result holds whether the locus under
consideration is autosomal or sex-linked and whether inbreeding e€ective population numbers or variance
e€ective population numbers are involved in the calculation of Ne . If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding e€ective population number, NeI , di€ers from the variance e€ective population number, NeV . The
mutation e€ective population number turns out to be the same as the number derived using calculations

involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue e€ective population number is the same as NeV . Ó 2000 Published by Elsevier Science Inc. All
rights reserved.
Keywords: E€ective population size; Cyclic variation

1. Introduction
Random genetic drift is an important in¯uence on the genetic variability of a ®nite population
and a numerical measure of its in¯uence is the e€ective population size. Wright [1±3] presented the
®rst example for a dioecious population and a general expression for monoecious populations that

*

1

Corresponding author. Tel.: +1-515 294 7765; fax: +1-515 294 4040.
E-mail addresses: ywang@iastate.edu (Y. Wang), pllk@iastate.edu (E. Pollak).
Tel.: +1-515 294 9053.

0025-5564/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 2 7 - 4

2

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

do not change in size. Later, Crow [4] showed, for a monoecious population, that the e€ective
population size measuring the increase in the probability that two copies of a gene are identical by
descent di€ers from the e€ective population size involved in the variance of allele frequency
changes, except if the size of the population remains constant. Other early work on this subject is
discussed in the text by Crow and Kimura [5].
Recently, Caballero [6] and Nagylaki [7] derived general expressions for the e€ective population
sizes of random mating dioecious populations, whether the locus under consideration is autosomal or sex-linked. For an autosomal locus, the results are consistent with those obtained by
Crow and Denniston [8]. Caballero [6] and Nagylaki [7] assume, however, that the numbers of
males and females do not change between generations. However, wild populations can ¯uctuate
considerably in size from year to year. A simple way to describe such populations, which may not
be too unrealistic, is to assume that these changes occur in repeated cycles. Wright [2,3] showed
that, for monoecious populations in which individuals have approximate Poisson distributions of
numbers of successful gametes in o€spring, the e€ective population size is approximately equal to
the harmonic mean of the populations sizes in a cycle, provided the length of a cycle is short in
comparison to those sizes. Pollak [9] has shown that this result also holds generally for o€spring
distributions with ®nite variances. However, to the best our knowledge, no such results have yet

been presented for dioecious populations. In the next two sections we shall derive e€ective population sizes for autosomal and sex-linked loci, by generalizing the reasoning used by Caballero [6]
and Nagylaki [7], whereby recurrence equations are obtained for probabilities of identity by
descent. Next, there will be alternative derivations for autosomal and sex-linked loci, which rely
on obtaining the variance of the change in the frequency of an allele in a generation. Here the
reasoning will be a discrete generation version of the reasoning used by Hill [10] and Pollak [11],
respectively, for an autosomal and a sex-linked locus. This alternative approach leads to the same
expressions as before for e€ective population sizes if the population is followed through an entire
cycle, although the individual terms in the resulting harmonic mean are not quite the same as the
corresponding terms that arise from the identity-by-descent approach. Mutation e€ective population numbers are also derived, as are eigenvalue e€ective population numbers in one special case. 2
2. Autosomal loci
We consider an autosomal locus in a population whose size undergoes repeated cycles of length
k. Let us suppose that at times t and t ‡ 1 the population is respectively in generations i and i ‡ 1
of a cycle. Then, at time t ‡ 1,
Fi‡1;t‡1 ˆ the inbreeding coefficient of a random individual
and
fvw;i‡1;t‡1 ˆ the coefficient of coancestry of a random pair of separate individuals of sexes
v and w;

2

Journal paper No. J-18628 of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA, Project
No. 3201, and supported by Hatch Act and State of Iowa Funds.

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

3

where v and w can be replaced by either of the symbols m and f, denoting, respectively, male and
female. Let
…i†
ˆ P ‰two copies of a gene in random separate offspring of sexes
Pu;vw

v and w were from the same parent of sex u generation i of a cycleŠ:
Then
Fi‡1;t‡1 ˆ fmf ;i;t

…1†

and

fvw;i‡1;t‡1



1
…i† 1
…i†
…i† 1
…i†
P
‰1 ‡ Fi;t Š ‡ ‰1 ÿ Pm;vw Šfmm;i;t ‡ Pf ;vw ‰1 ‡ Fi;t Š ‡ ‰1 ÿ Pf ;vw Šfff ;i;t ‡ 2fmf ;i;t ; …2†
ˆ
4 m;vw 2
2

because half of the copies of a gene in an individual come from a male and half come from a
female. These equations can be simpli®ed if we set
Ui;t ˆ 1 ÿ Fi;t ;
dvw;it ˆ 1 ÿ fvw;i;t :
Then (1) and (2) reduce, respectively, to

Ui‡1;t‡1 ˆ dmf ;i;t

…3†

and
h
i
o
i
1 nh …i†
…i†
…i†
…i†
Pm;vw ‡ Pf ;vw Ui;t ‡ 2‰1 ÿ Pm;vw
Šdmm;i;t ‡ 2 1 ÿ Pf ;vw dff ;i;t ‡ 4dmf ;i;t :
8
Eqs. (3) and (4) can be rewritten in matrix notation as
dvw;i‡1;t‡1 ˆ

…4†

di‡1;t‡1 ˆ Ai di;t ;

…5†

where
i 8
6 P …i† ‡ P …i†
…i†
4
2 1 ÿ Pm;mm
6 m;mm
f ;mm
16
h
i
Ai ˆ 6
…i†
…i†
…i†

4
86
6 Pm;mf ‡ Pf ;mf 2 1 ÿ Pm;mf
4
h
i
…i†
…i†
…i†
Pm;ff ‡ Pf ;ff 2 1 ÿ Pm;ff
4
2
3
2
0
0 0 1 0
…i†
…i†
6
6 0 1 1 1 7 1 6 Pm;mm ‡ Pf ;mm

6
4
2
47
ˆ6
7 ‡ 6 …i†
…i†
4 0 14 12 14 5 8 6
4 Pm;mf ‡ Pf ;mf
…i†
…i†
0 14 12 14
P
‡P
2

0

0h

m;ff

ˆ A0 ‡ Di :

Thus for all r
di‡r;t‡r ˆ ‰Ai‡rÿ1 Ai‡rÿ2


Ai Šdi;t


f ;ff

0h

i

3

…i†
2 1 ÿ Pf ;mm 7
7
h
i7

7
…i†
2 1 ÿ Pf ;mf 7
7
h
i 5
…i†
2 1 ÿ Pf ;ff

0

0

0

…i†
ÿ2Pm;mm

0

…i†
ÿ2Pf ;mm 7
7
7
…i† 7
ÿ2Pf ;mf 5

…i†

0

…i†

0

ÿ2Pm;mf
ÿ2Pm;ff

3

…i†

ÿ2Pf ;ff

4

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21
…i†

We shall show later that if, at each stage i in the cycle, the numbers Nm…i† and Nf of males and
…i†
in the matrix Di are small. It can then be proved by
females are large, all the elements Pu;vw
induction that
Ai‡rÿ1 Ai‡rÿ2


Ai ˆ …A0 ‡ Di‡rÿ1 †…A0 ‡ Di‡rÿ2 †


…A0 ‡ Di †  Ar0 ‡

rÿ1
X
Di‡j Aj0 ; r P 1;
Arÿ1ÿj
0
jˆ0

…i†

…i†

if r  max‰Nm…i† ; Nf Š. Thus, if k  max‰Nm…i† ; Nf Š,
"
#
kÿ1
X
di‡k;t‡k  Ak0 ‡
A0kÿ1ÿj Di‡j Aj0 di;t ˆ ‰M ‡ Di Šdi;t ;

…6†

jˆ0

where M ˆ Ak0 and the elements of
Di ˆ

kÿ1
X

Di‡j Aj0
Akÿ1ÿj
0

jˆ0

are all small.
Now
2
0
60
6
Aro ˆ 6
40

1
4
1
4
1
4
1
4

1
2
1
2
1
2
1
2

13
4
17
47
17
5
4
1
4

0
2 3
1
6 1 7 1 1 1 
6 7
ˆ6 7 0
;
415
4 2 4

r P 2;

1

and it can also be shown that the eigenvalues of Ar0 are k1 ˆ 1; k2 ˆ k3 ˆ k4 ˆ 0. The left and right
eigenvectors p0 and v that correspond to k1 ˆ 1; and satisfy the normalization conditions
p0 v ˆ p0 1 ˆ 1 are


1 1 1
0
p ˆ 0
;
4 2 4
and
0

v ˆ 1 ˆ ‰1 1 1 1Š :
Since all the elements of Di are small, the dominant eigenvalue of M ‡ Di is
q ˆ 1 ‡ d;
where d is small. By a standard result from perturbation theory, as discussed, for example, by
Franklin [12, Section 6.12],

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

5

d  p0 Di v
2 3
1
!6 7

 X
kÿ1
617
1 1 1
7
Dj 6
ˆ 0
617
4 2 4
5
4
jˆ0

1
h
i 30
…j†
…j†
…j†
…j†
…j†
kÿ1
…j†
2 3
P
‡
P
‡
2…P
‡
P
†
‡
P
‡
P
f ;mm
m;mf
f ;mf
m;ff
f ;ff
m;mm
jˆ0
7 1
76 7
i
Pkÿ1 h …j†
7 617
…j†
…j†
‡ 2Pm;mf ‡ Pm;ff
ÿ2 jˆ0 Pm;mm
76 7
76 7
7 415
7
0
h
i
5
Pkÿ1 …j†
…j†
…j†
1
ÿ2 jˆ0 Pf ;mm ‡ 2Pf ;mf ‡ 2Pf ;ff

2P

6
6
1 6
6
ˆ
6
32 6
6
4
ˆ ÿ

kÿ1 nh
i h
io
1 X
…j†
…j†
…j†
…j†
…j†
…j†
‡ 2Pm;mf ‡ 2Pm;ff ‡ Pf ;mm ‡ 2Pf ;mf ‡ Pf ;ff :
Pm;mm
32 jˆ0

…7†

Thus, after many cycles, the probabilities of non-identity of pairs of copies of a gene shrink at a
steady rate that is approximately equal to 1 ‡ d per cycle, where d, the right-hand side of (7) is not
dependent on i. This rate may then be set equal to ‰1 ÿ 1=…2Ne †Šk , so that
kÿ1 nh
io
i h
1
1 X
…j†
…j†
…j†
…j†
…j†
…j†

Pm;mm
‡ 2Pm;mf ‡ Pm;ff ‡ Pf ;mm ‡ 2Pf ;mf ‡ Pf ;ff :
Ne 16k jˆ0

…8†

…j†
to be the number of successful gametes contributed by a parent of sex u in
We now de®ne Guv
generation j of a cycle to an o€spring of sex v. Since we are considering neutral alleles and there
are Nu…j† parents of sex u and Nv…j‡1† o€spring of sex v,

 …j†  Nv…j‡1†
ˆ …j† :
E Guv
Nu

Since there is a random mating, any pair of gametes in o€spring of sexes v and w coming from
parents of sex u is just as probable as any other pair, even if both gametes come from a single
individual. Thus,
…j†
ˆ
Pu;vv

ˆ

ˆ

Nu…j†
ÿ 1†

 …j† ÿ …j†


Guv ÿ 1
E Guv

…j‡1†

ÿ 1†

n
 …j† o
ÿ …j†
 ÿ …j†
2
ÿ E Guv
‡ E Guv
Var Guv



Nu…j†

…j‡1†
…j‡1†
Nv …Nv

Nu…j†
…j‡1†

Nv

…Nv

1
…j‡1†

Nv

ÿ1


ÿ …j†
Nv…j‡1†
Var Guv ‡ …j† ÿ 1 ;
…j‡1†
Nv
Nu

…9†

6

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

and
Nu…j†

…j†

Pu;mf ˆ
ˆ

…j‡1†

Nm

h
i
…j† …j†
G
E
G
um uf
…j‡1†

Nf

Nu…j†



1
…j†
…j†
;
G
Cov
G
‡ …j† :
uf
um
…j‡1† …j‡1†
Nm Nf
Nu

…10†

…j†

We assume that Nm…j† and Nf are all large so that the elements of Di are indeed small for all i. If,
therefore, (9) and (10) are substituted in (8), we obtain
8
!
"
>
kÿ1


ÿ …j†

1
1 X< Nm…j†
Nm…j‡1†
…j†
…j†
Cov
G

;
G

2 Var Gmm ‡ 2
mf
mm
…j‡1†
Ne
16k jˆ0 >
Nf
: Nm…j‡1†
3
!2
 …j‡1† 2
 …j‡1† 


…j‡1†
Nm
Nm
Nm
…j†
5
‡
ÿ2
Var Gmf ‡ 4
…j‡1†
…j†
…j†
Nf
Nm
Nm
"
!
…j†
…j‡1†




Nf
N
f
…j†
…j†
…j†
‡
Cov
G
;
G
2 Var Gff ‡ 2
ff
fm
…j‡1†
…j‡1†
Nm
Nf
!2
!2
!3 9
…j‡1†
…j‡1†
…j‡1†
=


Nf
Nf
Nf
…j†
5 :
‡
…11†
‡
4
Var
G
ÿ
2
fm
…j‡1†
…j†
…j†
;
Nm
N
N
f

f

3. Sex-linked loci
…i†

…i†
The notation remains the same as in the previous section, but now Pm;mm
ˆ Pm;mf ˆ 0 because
males get their copies of a gene only from their mothers. Females get half their copies of a gene
from parents of each sex. Therefore,

Fi‡1;t‡1 ˆ fmf ;i;t ;

…12†

fmm;i‡1;t‡1 ˆ

…i†
Pf ;mm ‰12…1

fmf ;i‡1;t‡1 ˆ

…i†
1
fPf ;mf ‰12…1
2

fff ;i‡1;t‡1 ˆ

…i†
1
fPm;ff
4

‡ Fi;t †Š ‡ ‰1 ÿ

…i†
Pf ;mm Šfff ;it ;

‡ Fi;t †Š ‡ ‰1 ÿ

‡ ‰1 ÿ

…i†
Pm;ff Šfmm;i;t

…i†
Pf ;mf Šfff ;i;t

‡

…i†
Pf ;ff ‰12…1

…13†
‡ fmf ;i;t g;
‡ Fi;t †Š ‡ ‰1 ÿ

…14†
…i†
Pf ;ff Šfff ;i;t

‡ 2fmf ;i;t g:

…15†

As before, the recurrence equations can be simpli®ed if we set
Ui;t ˆ 1 ÿ Fi;t ;
dvw;i;t ˆ 1 ÿ fvw;it :
Eqs. (12)±(15) are then replaced by
di‡1;t‡1 ˆ Ai di;t ;

…16†

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

7

where
Ui;t‡1

2

3

7
6d
6 mm;i‡1;t‡1 7
di‡1;t‡1 ˆ 6
7;
4 dmf ;i‡1;t‡1 5
dff ;i‡1;t‡1
0

2

6 4P …i†
f ;mm
16
Ai ˆ 6
…i†
6
8 4 2Pf ;mf

0

8

0

0

0

…i†
8‰1 ÿ Pf ;mm Š 7
7
7
…i†
4‰1 ÿ Pf ;mf Š 7
5

0

0
60
6
ˆ6
40
2

0

4

…i†

…i†
Pm;ff Š

…i†
Pf ;ff

3

4 2‰1 ÿ Pf ;ff Š
2‰1 ÿ
3
2
3
0
0
0 0
0 1 0
…i†
6 4P …i†
0
0 ÿ8Pf ;mm 7
f ;mm
7
0 0 17
7 16
7
6
‡ 6 …i†
…i† 7
1
17
0 ÿ4Pf ;mf 5
0 2 2 5 8 4 2Pf ;mf 0

1
4

1
2

1
4

…i†

Pf ;ff

…i†

ÿ2Pm;ff

…i†

0 ÿ2Pf ;ff

ˆ A0 ‡ Di :

It follows from (16) that
"
#
kÿ1
X
j
kÿ1ÿj
k
di‡k;t‡k  A0 ‡
Di‡j A0 di;t ˆ ‰M ‡ Di Šdi;t ;
A0

…17†

jˆ0

because, as in the previous section, it will be shown that all of the elements of Di are small. The
characteristic equation corresponding to A0 is


 ÿk
0
1 


1
1
 ˆ k…k ÿ 1†‰k2 ‡ 1k ÿ 1Š ˆ 0
ÿk
jA0 ÿ kIj ˆ …ÿk† 0
2
2

4
8

 1
1
1
ÿ
k
4
2
4

and has the roots k1 ˆ 1; k2 ˆ ÿ12; k3 ˆ 14; k4 ˆ 0. The eigenvectors of Ar0 associated with any power
r of Ar and the dominant eigenvalue 1 are, then,


1 4 4
0
p ˆ 0
9 9 9
and
v ˆ 1 ˆ ‰1 1 1 1Š0 :

It follows from perturbation theory that if all the elements of Di are small, the dominant eigenvalue of M ‡ Di is q ˆ 1 ‡ d, where because of periodicity
!
iÿ1
kÿ1
kÿ1
kÿ1
X
X
X
X
d  p0 Di v ˆ p0 Di‡j v ˆ p0
Dj ‡
Dj v ˆ p0 Dj v:
jˆ0

jˆi

jˆ0

jˆ0

8

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

This is the same for all i. Thus
2


 kÿ1 6
6
1 4 4 X
6
d  p Di v ˆ 0
9 9 9 jˆ0 6
4
0

0
1 …j†
ÿ 2 Pf ;mm

3

…j†

ÿ 14 Pf ;mf
…j†

…j†

ÿ 18 ‰Pf ;ff ‡ 2Pm;ff Š

7
7
7
7
5

kÿ1
1 X
k
…j†
…j†
…j†
…j†
ˆÿ
:
‰P
‡ 2Pf ;mf ‡ Pf ;ff ‡ 2Pm;ff Š  ÿ
18 jˆ0 f ;mm
2Ne

…18†

As (9) and (10) still hold for a sex-linked locus, all the elements of Di are small. Their substitution
in (18) leads to
8
!
"
>
…j†
…j‡1†
kÿ1
Nf
1
1 X< Nf
…j†
…j†
…j†

Cov…Gff ; Gfm †

2 Var…Gff † ‡ 2
…j‡1†
Ne
9k jˆ0 >
Nm
: N …j‡1†
f
!2
!2
!#
…j‡1†
…j‡1†
…j‡1†
Nf
Nf
Nf
…j†
‡
Var…Gfm † ‡ 4
ÿ3
…j‡1†
…j†
…j†
Nm
Nf
Nf
39
2
!2
 …j‡1† 2  …j‡1†  >
=
…j†
…j‡1†
N
Nm
Nm
Nm
…j†
5
‡  m  2 42
‡
2
ÿ
Var…G
†
:
…19†
mf
…j‡1†
…j†
…j†
>
…j‡1†
N
N
N
;
m
m
f
Nm
4. Earlier results that are related to (11) and (19)
…j†
…j†
Let us assume that Pu;vw
ˆ Pu;vw , Nu…j† ˆ Nu and Guv
ˆ Guv for all j. There is then no cycle and k is
replaced by 1 in (11) and (19). Expressions (11) and (19) then, respectively, reduce to
(
"
#



2
1
1
1
Nm
Nm

Var…Gmm † ‡ 2
Var…Gmf † ‡ 2
Cov…Gmm ; Gmf † ‡
Nf
Nf
Ne 16 Nm
"
#)



2
1
Nf
Nf
‡
Var…Gff † ‡ 2
Var…Gfm † ‡ 2
…20†
Cov…Gff ; Gfm † ‡
Nf
Nm
Nm

and
#
"



2
1
Nf
Nf
Cov…Gff ; Gfm † ‡
Var…Gfm † ‡ 1
Var…Gff † ‡ 2
Nf
Nm
Nm
" 
#)
2
1
Nm
Var…Gmf † ‡ 1 :
2
‡
Nf
Nm

1
1

Ne 9

(

…21†

Expressions (20) and (21) are respectively the discrete generation versions of results derived by
Hill [10,13] and Pollak [11,14].

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

9

Another special case of (11) is where there are independent Poisson distributions of male and
female o€spring of an individual in each generation. Then
…j†
†ˆ
Var…Guv

Nv…j‡1†
…j†

Nu

and
…j†

…j†
; Guf † ˆ 0:
Cov…Gum

Hence (11) reduces to
"
#
kÿ1
1
1 X
1
1
4
2
1
1
4
2
‡
‡
‡
ÿ

ÿ
‡
‡
Ne 16k jˆ0 Nm…j‡1† Nf…j‡1† Nm…j† Nm…j‡1† Nf…j‡1† Nm…j‡1† Nf…j† Nf…j‡1†
"
#
kÿ1
1 X
1
1
ˆ
‡
;
4k jˆ0 Nm…j† Nf…j†
so that Ne is the harmonic mean of the k e€ective sizes in a cycle, each of which is of the form
derived by Wright [1].
Analogously, a special case of (19) is one for which there are independent Poisson distributions
of male and female o€spring of a female and the number of daughters produced by a male has a
Poisson distribution. Then
#
#
"
"
kÿ1
kÿ1
1
1 X
1
1
4
3
2
2
1
1 X
4
2
ˆ
;
‡
‡
‡
ÿ
ÿ
‡
‡

Ne 9k jˆ0 Nf…j‡1† Nm…j‡1† Nf…j† Nf…j‡1† Nf…j‡1† Nm…j† Nm…j‡1†
9k jˆ0 Nf…j† Nm…j†
so that Ne is the harmonic mean of the k e€ective population sizes in a cycle, each of which is of
the form derived by Wright [15].
5. Another characterization of Ne
In this section, we will express the results given by (11) and (19) in terms of variance e€ective
numbers within a cycle. The reasoning will be based on that used by Hill [10]. Thus, we take
account of three sources of variability when gametes are transmitted from one generation to the
next: between numbers of gametes produced by di€erent individuals, between genotypes of individuals, and among alleles carried by gametes when parents are heterozygous.
We consider a population in which parents and o€spring are, respectively, in phases j and j ‡ 1
…j†
be the number of successful gametes contributed by the rth parent of sex u to
of a cycle. Let Guvr
o€spring of sex v, and
Xur…j† ˆ the frequency of allelle B1 in the rth parent of sex u;
…j†

duvr` ˆ the difference between Xur…j† and the frequency of B1 in the
`th gamete contributed to an offspring of sex v by the rth parent of sex u:
So if the population has alleles B1 and B2 , Xur…i† will respectively be equal to 1, 1/2 and 0 in indi…j†
viduals of genotypes B1 B1 , B1 B2 and B2 B2 . In addition duvr` will be 0 if the parent of sex u is a

10

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

homozygote or a male with an X-linked locus and 1=2 with equal probabilities if it is a heterozygote.
If there is an autosomal locus the male and female o€spring, respectively, originate from 2Nm…j‡1†
…j‡1†
successful gametes. These two sets of gametes are drawn from a population whose
and 2Nf
frequency of B1 is p. The frequency p0 of B1 among the o€spring is the unweighted average of the
frequencies among males and females. Thus
8 2
2
39
3
…j†
…j†
…j†
…j†
Nf
Gfmr
>
>
Gmmr
Nm
<
=
X
X
X
X
1
6 …j† …j†
…j† 5
…j† 7
…j†
…j†
0
4
d
‡
X
X
G
G
d
p ˆ
‡
‡
4
5
fmr fr
mmr`
fmr`
mmr mr
…j‡1†
>
4Nm >
: rˆ1
;
`ˆ1
rˆ1
`ˆ1
9
8 2
3
3
2
…j†
…j†
…j†
…j†
Nf
Gffr
Gmfr
>
>
N
=
<
m
X
1
6 …j† …j† X …j† 7 X6 …j† …j† X …j† 7
…22†
dffr` 5 :
dmfr` 5 ‡
‡ …j‡1†
4Gffr Xfr ‡
4Gmfr Xmr ‡
>
>
4Nf
;
: rˆ1
rˆ1
`ˆ1
`ˆ1
…j†

When Nm…j† and Nf are large, the covariance between Guvr and Gu0 v0 r0 is approximately 0 if r 6ˆ r0 , and
…j†
E…Guvr
†ˆ

Nv…j‡1†
…j†

Nu

…23†

:

…j†
As we are considering neutral alleles the distribution of G…j†
uvr is the same for all r, so that Guvr and
…j†
…j†
…j†
…j†
Xur…j† are independent, as are Guvr
and duvr` . Also, if Nm…j† and Nf are large, the variance of both Xmr
…j†
and Xfr is

Var…Xur…j† † ˆ 12 p…1 ÿ p†;

…24†

because the frequencies of B1 B1 ; B1 B2 ; and B2 B2 in both sexes are approximately p2 ; 2p…1 ÿ p† and
…1 ÿ p†2 . In addition
…j†

…j†

Var…duvr` † ˆ E……duvr` †2 † ˆ 14 P …B1 B2 † ˆ 12 p…1 ÿ p†:

…25†

After some algebra, it follows from Eqs. (22)±(25) that
2

Var…p0 j p† ˆ E‰…p08
ÿ p† j pŠ
2
!
>
<
…j†
…j‡1†
p…1 ÿ p†
Nm
N
…j†
…j†
…j†
m
; Gmf †
Cov…Gmm


2 4Var…Gmm † ‡ 2
…j‡1†
>
2
Nf
: 16 Nm…j‡1†
3
2
!2
…j†
…j‡1†
…j‡1†
N
Nm
N
f
…j†
…j†
‡
Var…Gmf † ‡ 2 m …j† 5 ‡ 
2 4Var…Gff †
…j‡1†
…j‡1†
Nf
Nm
16 Nf
39
!
!2
…j‡1†
…j‡1†
…j‡1† >
=
Nf
Nf
Nf
…j†
…j†
…j†
5
†
‡
;
G
†
‡
2
Var…G
Cov…G
‡2
fm
ff
fm
…j‡1†
…j‡1†
…j†
>
Nm
Nm
Nf
;
ˆ

…j†

p…1 ÿ p†
…j†

2NeV

;

…26†

where NeV is the variance e€ective population number when parents are in generation j of a cycle.

11

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21
…j†

Since the changes in allele frequencies in di€erent generations are independent, Nm…j† and Nf are
large for all j and k  min…Nm…j† ; Nm…j† †, the variance
of the total change in allele frequency
P
…j†
throughout a cycle is approximately equal to kjˆ1 p…1 ÿ p†=…2NeV †. Hence, we can de®ne the
e€ective population size to be
k
1
1X
1

…j†
NeV
k jˆ1 NeV
8
2
!
kÿ1 >
< N …j†
…j‡1†
1 X
N
…j†
…j†
…j†
m
m

; Gmf †
Cov…Gmm

2 4Var…Gmm † ‡ 2
…j‡1†
16k jˆ0 >
…j‡1†
N
: Nm
f
3
!2
Nm…j‡1†
2
…j†
‡
Var…Gmf †5 ‡ …j‡1†
…j‡1†
Nm
Nf
2
!
…j†
…j‡1†
Nf
Nf
…j†
…j†
…j†
Cov…Gff ; Gfm †
‡
2 4Var…Gff † ‡ 2
…j‡1†
…j‡1†
Nm
Nf
9
3
!2
>
…j‡1†
Nf
2 =
…j† 5
‡
Var…G
†
‡
:
…27†
fm
…j‡1†
…j‡1† >
Nm
Nf
;
Note that the individual expressions being summed in (11) di€er, with respect to terms not involving variances and covariances, from corresponding terms in (27). However, the sum of such
expressions in (11) is
8
9
2
" 
!2
!3 >
#
>



…j†
…j‡1†
…j‡1†
2
kÿ1
<
=
X
Nf
Nf
Nf
Nm…j†
Nm…j‡1†
Nm…j‡1†
4
5
4
4
ÿ
2
‡
ÿ
2

2

2
…j†
…j†
…j†
…j†
>
>
…j‡1†
…j‡1†
Nm
Nm
Nf
Nf
;
jˆ0 : Nm
Nf
"
"
#
#
kÿ1
kÿ1
X
X
4
2
4
2
2
2
ÿ …j‡1† ‡ …j† ÿ …j‡1† ˆ
‡ …j‡1† ;
ˆ
…j†
…j‡1†
Nm
Nf
Nf
Nf
jˆ0 Nm
jˆ0 Nm

so that (11) and (27) give identical expressions for the e€ective population number.
…j‡1†
female o€spring
Now let us suppose that there is a sex-linked locus. In this case the Nf
…j‡1†
originate from 2Nf
successful gametes, but the Nm…j‡1† males are derived from only Nm…j‡1† gametes, all of which were contributed by their mothers. Another new feature is that the frequency p0
among the o€spring is a weighted average, with weights 1/3 and 2/3 for males and females. Thus
3
2
…j†
…j†
Gfmr
Nf
1 X6 …j† …j† X …j† 7
dfmr` 5
p0 ˆ
4Gfmr Xfr ‡
…j‡1†
3Nm
`ˆ1
rˆ1
8
39
2
…j†
…j†
…j†
G
N
>
N
ffr
f
=
m
X6 …j† …j† X …j† 7>
2
1
3 2Nf…j‡1† >
;
: rˆ1
`ˆ1
rˆ1

12

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

Eqs. (24) and (25) still hold if u ˆ f , but now
…j†
Var…Xmr
† ˆ p…1 ÿ p†;

…29†

because the frequencies of B1 Y and B2 Y are p and 1 ÿ p.
…j†
Eq. (23) also remains valid except for the fact that E…Gmmr
† ˆ 0. It therefore follows from
Eqs. (23)±(25) and (29) that
Var…p0 jp† ˆ E‰…p0 ÿ p†2 jpŠ
8
2
!
>
…j†
…j‡1†
Nf
N
p…1 ÿ p† <
f
…j†
…j†
…j†

Cov…Gff ; Gfm †

2 4Var…Gff † ‡ 2
…j‡1†
>
2
…j‡1†
N
:9 N
m
f
3
39
2
!2
!2
>
…j‡1†
…j‡1†
Nf
Nf
Nm…j†
Nm…j‡1†
Nm…j‡1† 5=
…j†
…j†
5
4
‡
Var…Gfm † ‡ …j†
Var…Gmf † ‡ …j†
‡ 
2 2
…j‡1†
…j‡1†
>
…j‡1†
Nm
Nf
Nf
Nm
;
9 Nm
ˆ

p…1 ÿ p†
:
2NeV

…30†

Hence
8
2
!
…j†
…j‡1†
kÿ1 >
<
X
Nf
Nf
1
1
…j†
…j†
…j†
Cov…Gff ; Gfm † ‡


2 4Var…Gff † ‡ 2
…j‡1†
>
NeV 9k jˆ0 :
…j‡1†
Nm
Nf
9
2
3
!2
>
1
Nm…j† 4 Nm…j‡1†
1 =
…j† 5
‡ …j‡1† ‡ 
Var…Gmf † ‡ …j‡1† :
2 2
…j‡1†
…j‡1†
N
Nf
Nm >
;
f
Nm

…j‡1†
Nf
…j‡1†
Nm

!2

3

…j†
Var…Gfm †5

…31†

The terms in (19) that do not involve variances and covariances add to
#
#
"
"
kÿ1
kÿ1
X
X
4
3
2
1
1
1
ÿ …j‡1† ‡ …j† ÿ …j‡1† ˆ
‡ …j‡1†
…j†
…j‡1†
Nf
Nm
Nm
Nm
jˆ0 Nf
jˆ0 Nf
so that, while individual summands in (19) and (31) di€er, Ne , given by (19), and NeV , given by
(31), are the same.

6. Inbreeding and variance e€ective population numbers
…j†
are small and set duv;0;0 ˆ 1 for all u and v, Eqs. (4)
If we assume that all the probabilities Pu;vw
imply that

dmm;i‡1;t‡1  dff ;i‡1;t‡1  dmf ;i‡1;t‡1  14‰dmm;it ‡ 2dmf ;i;t ‡ dff ;i;t Š
for all t. Thus, if we rede®ne the time scale so that t ˆ 0 and t ˆ 2 respectively indicate times at
which grandparents and grandchildren live, then

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21



i
1 h …0†
1 h …0†
…0†
…0†
U2;2 ˆ dmf ;11  Pm;mf ‡ Pf ;mf U0;0 ‡ 1 ÿ Pm;mf ‡ Pf ;mf
8
4

i
i
1 h …0†
1 h …0†
…0†
…0†
ˆ Pm;mf ‡ Pf ;mf U0;0 ‡ 1 ÿ Pm;mf ‡ Pf ;mf U1;1 :
8
4

i

13

dmf ;00
…32†

This equation is a generalization of the usual recurrence equation connecting three successive
values of the panmictic index when there are independent Poisson distributions of numbers of
…0†
male and female o€spring of an individual. It thus makes sense to replace 14 Nm…0† ‡ 14 Nf in that
special case by
i
1
1 h …0†
…0†
ˆ Pm;mf ‡ Pf ;mf ;
…33†
NeI 4
which is the probability that both gametes uniting to produce an individual in generation 2 came
from the same grandparent in generation 0. If we simplify our notation, by setting

and


 Nv…1†
ˆ luv
E G…0†
uv ˆ
…1†
Nu


…0†
…0†
Cov Gum
; Guf ˆ rum;uf ;

it follows, when (10) is substituted in (33), that
#
"
1
1 rmm;mf ‡ lmm lmf rfm;ff ‡ lfm lff
;
‡
ˆ
…0†
…0†
NeI 4
Nm lmm lmf
Nf lfm lff

…34†

as found by Crow and Denniston [8].
In the particular use in which rum;uf ˆ 0, (34) reduces to
1
1
1
‡ …0† ;
ˆ
…0†
NeI 4Nm
4Nf
which would hold, for example, if each individual had independent Poisson distributions of
numbers of male and female o€spring. But then (27) implies, for k ˆ 1, that
"
#
1
1 1
1
‡
:
ˆ
NeV 4 Nm…1† Nf…1†
In general, (34) seems to be di€erent from the special case of (11) that results when k is set equal
to 1. But the two approaches that are used to calculate NeI can be reconciled as follows. For
simplicity, we set
ÿ


ˆ r2uv
Var G…0†
uv
and



…0†
…0†
r2u ˆ Var Gum
‡ Guf :

14

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21
…0†

…0†
Next, we note that, given Gum
‡ Guf ˆ Gu ˆ g; G…0†
um has a binomial distribution with mean
…1†

…1†

…1†

gNm…1† =…Nm…1† ‡ Nf † and variance gNm…1† Nf =…Nm…1† ‡ Nf †2 . Hence
…1†

r2um

r2uf

ˆ E‰Var…Gum †jGu Š ‡ Var‰E…Gum jGu †Š ˆ h

…1†

Nm…1† Nf
…1†

…1†

Nm ‡ Nf

"
#2
lum luf
lum
‡
r2u ;
ˆ
lum ‡ luf
lum ‡ luf
"
#2
lum luf
luf
r2u ;
‡
ˆ
lum ‡ luf
lum ‡ luf

Nm…1† ‡ Nf
i2

…0†
Nu

‡

"

Nm…1†
…1†
Nm

‡

…1†
Nf

#2

r2u

and
rum;uf ˆ E‰Cov…Gum ; Guf † j Gu Š ‡ Cov‰E…Gum j Gu †; E…Guf j Gu †Š
…1†
…1†
…1†
Nm…1† Nf
Nm…1† ‡ Nf
Nm…1† Nf
lum luf
lum luf
2
ˆ ÿh
‡
‡
r2 :
i2
h
i2 ru ˆ ÿ
2 u
…0†
l
‡
l
…1†
…1†
…1†
…1†
‰l
‡
l
Š
Nu
um
uf
um
uf
Nm ‡ Nf
Nm ‡ Nf

Thus

# "
#2 "
#
2 "
l
l
l
‡
l
l
l
um
uf
um
uf
um
uf
ˆ
r2uf ÿ
r2um ÿ
lum ‡ luf
luf
lum ‡ luf
#
"
2
‰lum ‡ luf Š
lum luf
ˆ
:
r2um;uf ‡
lum ‡ luf
lum luf

lum ‡ luf
r2u ˆ
lum


…35†

It, therefore, follows from Eqs. (34) and (35) that
(
"
"
#
#)
1
1
1
1
1
1
1
1
‡ …0†
ˆ
r2 ‡ 1 ÿ
r2 ‡ 1 ÿ
2 f
NeI 4 Nm…0† …lmm ‡ lmf †2 m
lmm ‡ lmf
lfm ‡ lff
…lfm ‡ lff †
Nf
(
"
#
r2mf
lmf
1
1 r2mm
rmm;mf
lmm
2
‡2
ˆ
ÿ
‡4
‡
ÿ
ÿ
16 Nm…0† l2mm
lmm lmf l2mf lmm …lmm ‡ lmf † lmf …lmm ‡ lmf † lmm ‡ lmf
‡

1
…0†

Nf

"

r2fm
r2ff
lff
lfm
rfm;ff
2
‡
2
ÿ
‡
ÿ
ÿ
‡4
2
2
lfm
lfm lff lff lfm …lfm ‡ lff † lff …lfm ‡ lff † lfm ‡ lff

#)

:

Hence
(

"

#
r2mf
r2mm
rmm;mf
4
2
‡2
‡ 2
‡ …0† ÿ …1†
…0† l2
lmm lmf lmf
Nm
Nm
Nm
mm
)
"
#
r2ff
1 r2fm
rfm;ff
4
2
‡
‡ …0†
‡
2
‡ …0† ÿ …1† ;
2
2
lfm lff lff
lfm
Nf
Nf
Nf

1
1
ˆ
NeI 16

1

…36†

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

15

because
1
…0†

Nu

"

luf
lum
2
‡
‡
lum …lum ‡ luf † luf …lum ‡ luf † lum ‡ luf

ˆ

1
…0†
Nu

"

luf ‡ lum
lum luf

#

ˆ

1
…0†
Nu

"

1
1
‡
lum luf

#

ˆ

#

1
…1†
Nm

‡

1
…1†

Nf

:

Expression (36) is consistent with (11).
If a locus is sex-linked and we set k ˆ 1 and j ˆ 0
"
!

#
1
1
1
1
1
1
1
ÿ

4
ÿ …1†
:
ÿ …1† ‡ 2
…0†
…0†
NeI NeV 9
Nf
Nf
Nm
Nm
Suppose, in particular, that there are independent Poisson distributions of male and female o€spring of a female, and the number of daughters produced by a male has a Poisson distribution.
Hence, by the discussion in Section 4,
#
"
1
1 4
2
‡

NeI 9 Nf…0† Nm…0†
and
#
"
1
1 4
2

:
‡
NeV 9 Nf…1† Nm…1†

7. Mutation e€ective population numbers
…i†
Let Pu;vw
have the same meaning as in Sections 2 and 3. But we now assume that the in®nite
alleles model holds, so that genes mutate at a rate u, in such a way that each mutant is to an
entirely novel allelic type. Suppose that a population is in generation i ‡ 1 of a cycle at time t ‡ 1.
Then, at this time, Gi‡1;t‡1 and gvw;i‡1;t‡1 respectively denote the probabilities that two copies of a
gene in one individual, and in two random separate individuals of sexes v and w, are identical in
state. Then, if the locus under consideration is autosomal, Eqs. (1) and (2) are replaced by
Gi‡1;t‡1 ˆ …1 ÿ u†2 gmf ;it ;
…37†

and
gvw;i‡1;t‡1 ˆ

i
…1 ÿ u†2 nh …i†
…i†
…i†
Šgmm;i;t
Pm;vw ‡ Pf ;vw ‰1 ‡ Gi;t Š ‡ 2‰1 ÿ Pm;vw
8
o
…i†

‡ 4gmf ;i;t ‡ 2‰1 ÿ Pf ;vw Šgff ;i;t :

…38†

If we now set Hi‡1;t‡1 ˆ 1 ÿ Gi‡1;t‡1 and hvw;i‡1;t‡1 ˆ 1 ÿ gvw;i‡1;t‡1 , (37) and (38) imply that
Hi‡1;t‡1 ˆ 1 ÿ …1 ÿ u†2 ‡ …1 ÿ u†2 hmf ;i;t ;

…39†

16

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

and
2
i
i
h
…1 ÿ u† nh …i†
…i†
…i†
hmm;i;t ‡ 4hmf ;i;t
Pm;vw ‡ Pf ;vw Hi;t ‡ 2 1 ÿ Pm;vw
8
h
o
i
…i†
‡ 2 1 ÿ Pf ;vw hff ;i;t :

hvw;i‡1;t‡1 ˆ 1 ÿ …1 ÿ u†2 ‡

…40†

Then if hi;t ˆ ‰Hi;t ; hmm;i;t ; hmf ;i;t ; hff ;i;t Š0 , Eqs. (39) and (40) can be written in matrix notation as
hi‡1;t‡1 ˆ ‰1 ÿ …1 ÿ u†2 Š1 ‡ …1 ÿ u†2 Ai hi;t ;

…41†
0

where Ai is the same matrix as in Eq. (5) and 1 ˆ ‰1 1 1 1Š . We assume that u is very small. Hence,
it follows from (6) and (41) and the fact that the sum of the elements of each row of any power of
A0 is 1, that
hi‡k;t‡k ˆ ‰I ‡ …1 ÿ u†2 Ai‡kÿ1 ‡


‡ …1 ÿ u†2…kÿ1† Ai‡kÿ1


Ai‡1 Š‰1 ÿ …1 ÿ u†2 Š1
2k

‡ …1 ÿ u† Ai‡kÿ1


Ai hi;t  2ku1 ‡ …1 ÿ 2ku†…M ‡ Di †hi;t :
…i†

Thus, if ck  max‰Nm…i† ; Nf ; uÿ1 Š,
hi‡ck;t‡ck





4Ne u ‡ 1
 2cku1 ‡ …1 ÿ 2cku†q 1p hi;t  2cku1 ‡ 1 ÿ ck
2Ne
c

0


1p0 hi;t ;

…42†

where 1 and p0 are the eigenvectors associated with the dominant eigenvalue 1 of
Ar0 ; q ˆ ‰1 ÿ 1=…2Ne †Šk , and Ne is given by (8) and (11). The stationary value of hi;t is hi , where, by
(42),
 


 
4Ne u ‡ 1 0
0
hi  ck 2u ÿ
p hi ‡ p hi 1:
2Ne
Hence, all the elements of hi are equal to Hi and, because p0 1 ˆ 1,
4Ne u
Hi ˆ H 
; i ˆ 1; 2; . . . ; k:
4Ne u ‡ 1

…43†

The deviations from equilibrium are ei;t ˆ hi;t ÿ hi and, by (42),



4Ne u ‡ 1
1p0 ei;t ;
ei‡ck;t‡ck  1 ÿ ck
2Ne
which shows that the equilibrium given by (43) is stable.
If the locus under consideration is sex-linked, it can be shown that
hi‡1;t‡1 ˆ ‰1 ÿ …1 ÿ u†2 Š1 ‡ …1 ÿ u†2 Ai hi;t ;

…44†

where Ai is now the same matrix as in (18). It then follows from the same sequence of steps as
those leading from (41) to (43) that
4Ne u
1;
…45†
h i ! Hi 1 
4Ne u ‡ 1
where Ne is given by (19). Therefore, whether the locus is autosomal or sex-linked,
Gi;t ! Gi 

1
;
4Ne u ‡ 1

t ! 1:

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

17

Thus, Ne , the mutation e€ective population number, as de®ned by Ewens [16], is the same as the
inbreeding e€ective population number.
8. Eigenvalue e€ective population numbers
…i†
In what follows, let Guvr
have the same meaning as in Section 5. Suppose that this random
variable has a binomial distribution with Nv…j‡1† trials and a probability 1=Nu…j† of success. In ad…j†
…j†
dition, we assume that Gumr
and Gufr are independently distributed if there is an autosomal locus
or when there is a sex-linked locus and u ˆ f . Then


Nv…j‡1†
1
…j†
Var…Guvr † ˆ …j† 1 ÿ …j† :
Nu
Nu
If a locus is autosomal (27) reduces to
"
#
kÿ1
1
1 X
1
1
‡

;
…46†
NeV 4k jˆ0 Nm…j‡1† Nf…j‡1†
…j‡1†

if Nm…j‡1† and Nf

are large for all j ‡ 1. If a locus is sex-linked G…j†
mmr  0 and we assume that

…j‡1† 
Nf
1
…j†
Var…Gmfr † ˆ …j† 1 ÿ …j† ;
Nm
Nm
Eq. (31) implies that
"
#
kÿ1
1
1 X
4
2
‡
;
…47†

NeV 9k jˆ0 Nf…j‡1† Nf…j‡1†

if the numbers of males and females are always large.
Under the assumptions we have made about population sizes the covariance between the
outputs of successful gametes from two separate individuals of the same sex is approximately 0. It
is then also consistent with the special cases in the foregoing paragraph to consider the distribution of copies of B1 passed on to adults of the next generation by parents of sex u to be binomial. Let Yu …t ‡ 1† be the number of copies B1 among fertilized eggs that lead to adults of sex v
in generation t ‡ 1, and Xuv …t† be the number of copies of this allele passed on to these fertilized
eggs by parents of sex u. Then, if there is an autosomal locus
Yv …t ‡ 1† ˆ Xmv …t† ‡ Xfv …t†;

…48†

where Xmv …t† and Xfv …t† have independent binomial distributions. If Nu and Nu0 respectively denote
the number of adults of sex u in generation t and t ‡ 1


0 Yu …t†
:
…49†
Xuv …t†  Bin Nv ;
2Nu
It follows from (48) and (49) that



Yv …t ‡ 1† 
Ym …t† Yf …t†
Yu …t†; u ˆ m; f ˆ
E
‡
;

0
2Nv
2Nm
2Nf

…50†

18

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

#
2 
Yv …t ‡ 1† 
E
Yu …t†; u ˆ m; f

2Nv0
"
(

2


2 
2 #)
1
Yf …t†
Ym …t†
Yf …t†
0
0
0 Ym …t†
‡
Nv
‡ Nv …Nv ÿ 1†
‡
ˆ
2Nv0
2Nm
2Nf
2Nm
2Nf
"

‡

1 Ym …t† Yf …t†
;
2 2Nm 2Nf

…51†


#
Ym …t ‡ 1† Yf …t ‡ 1† 
E
Yu …t†; u ˆ m; f
2Nm0
2Nf0 
(
2


 
2 )
1
Ym …t†
Ym …t†
Yf …t†
Yf …t†
‡2
:
‡
ˆ
4
2Nm
2Nm
2Nf
2Nf
"

…52†

If, therefore, we average the right-hand sides of Eqs. (50)±(52) over the joint distributions of Ym …t†
and Yf …t† and set mv …t† ˆ E‰Yv …t†=…2Nv …t††Š and muv …t† ˆ E‰Yu …t†Yv …t†=…4Nu Nv †Š; we obtain the system of equations
mt‡1 ˆ T…t†mt ;

…53†

where
mt‡1 ˆ ‰mm …t ‡ 1†; mf …t ‡ 1†; mmm …t ‡ 1†; mmf …t ‡ 1†; mff …t ‡ 1†Š0 ;
mt ˆ ‰mm …t†; mf …t†; mmm …t†; mmf …t†; mff …t†Š0 ;
and
2

1
6 21
6 2
6
6 1
6 4Nm0

T…t† ˆ 6
6
6 0
4

1
4Nf0

1
2
1
2
1
4Nm0

0
1
4Nf0

1
4Nm0

0

0

0

0

…Nm0 ÿ 1†

1
4
1
0
…N
f
4Nf0

ÿ 1†

1
2
1
2
1
2

0

3

7
7
7 
7
T11 …t†
1
0
…N
ÿ
1†
7ˆ
0
m
4Nm
7
T21 …t†
7
1
7
4
5
1
…Nf0 ÿ 1†
4N 0
0

0
T22 …t†



:

f

The characteristic equation of T…t† can be shown to be
(
"
!#
1 1
1
1
2
2
‡
ÿ
jT…t† ÿ kIj ˆ k …k ÿ 1† k ÿ k 1 ÿ
4 Nm0 Nf0
8

1
1
‡
Nm0 Nf0

!)

ˆ 0:

This equation has a single root equal to 1 and two others equal to 0. Among the remaining roots
the one of largest absolute value is
8
2
!
!2 31=2 9
0
0
0
0
=
<
Nm ‡ Nf
Nm ‡ Nf
1
1
1
1
5
‡
:
‡ 41 ‡
k…t ‡ 1† ˆ
1ÿ
1ÿ
;
4Nm0 Nf0
4Nm0 Nf0
2:
8 Nm0 Nf0

If the numbers of males and females do not change between generations this expression reduces to
1 ÿ 1=…2NeE †, where NeE is the eigenvalue e€ective population number obtained by Ewens [17,18].

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

19

If we now follow the population over a whole cycle, (53) leads to
mt‡k ˆ T…t ‡ k ÿ 1†T…t ‡ k ÿ 2†


T…t†mt :
Thus, if there are s cycles
(
"
#s )
k
Y
k…j†
m0 ;
msk  A ‡ B

s ! 1;

…54†

jˆ1

where A and B are matrices that depend on the phase of the cycle at time 0. Note that, by (46),
"
#
k
k
Y
1X
1
1
k
‡ …j†  1 ÿ
k…j†  1 ÿ
…j†
8 jˆ1 Nm
2NeV
Nf
jˆ1

so that at the level of approximations used in this paper, the recurrence equations on the ®rst and
second moments lead to the variance e€ective population number.
We claim that, at least for the model discussed in this section, NeV is the same as NeE , the eigenvalue e€ective size of Ewens [17,18], which is the largest non-unit eigenvalue of the matrix of
transition probabilities of numbers of copies of B, in males and females when the same phase is
considered in two successive cycles. If second moments are traced through s cycles, and s is large,
they each satisfy
"
# 
ks
1 Ym …0† Yf …0†
1
‡ …0† ‡ 1 ÿ
muv …sk j AC†;
…55†
muv …sk† 
2 Nm…0†
2NeE
Nf

where muv …sk j AC† is the expected value of the uvth moment, given the asymptotic conditional
distribution of allele frequency pairs in males and females when B1 has been neither ®xed nor lost.
Since (54) and (55) are of the same form, NeE is at least approximately equal to NeV .
If the locus under consideration is sex-linked (48) still holds if v ˆ f , but


0 Ym …t†
Xmf …t†  Bin Nf ;
Nm
and


0 Yf …t†
Ym …t ‡ 1† ˆ Xfm …t†  Bin Nm ;
:
2Nf
Again, we assume that any pair of the random variables Xuv …t† are independent. It can then be
shown that
mt‡1 ˆ T…t†mt ;

where mm …t† ˆ Ym …t†=Nm ; mm …t ‡ 1† ˆ Ym …t ‡ 1†=Nm0 and
3
2
0
1
0
0
0
7
6 1
1
0
0
0
7
6 2
2
7
6
1
1
6 0
0
0
1 ÿ Nm0 7
Nm0
T…t† ˆ 6
7:
7
6
1
1
7
6 0
0
0
2
2
5
4
1
1
1
1
1
1
1
…1 ÿ N 0 † 2 4 …1 ÿ N 0 †
4N 0
4N 0
4
f

f

f

f

20

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

The characteristic equation corresponding to T …t† is
!
!

(
1
1
3
1
3
1
1
1
3
2
2
ÿ
ÿ
‡k
k ÿ kÿ
ÿk ‡k
ÿ
‡
2
2
4 4Nf0
8 4Nm0 8Nf0 4Nm0 Nf0
!) 

1
1
1
1
1
1
2
1ÿ 0 ÿ 0 ‡ 0 0
ˆ k ÿ kÿ
ÿ
f …k† ˆ 0:
8
Nm Nf Nm Nf
2
2
The ®rst factor has roots 1 and ÿ12, whereas
!
1
1
‡
;
f …1†  ÿ
8Nm0 4Nf0

 

1
1
1
1
1
f ÿ
f …0† < 0; f
ˆ ;
ˆ
‡ 0;
0
2
8
2
8Nf 4Nm
and
9
1 1
;
f 0 …1† ˆ ÿ ‡ O
8
Nm0 Nf0

!

:

Hence, by using Newton's method and the intermediate value theorem, we ®nd that the largest
root of f …0† ˆ 0 is approximately
!
1
2
1ÿ
‡
9Nm0 9Nf0
and the other two roots are between 0 and 12 in magnitude, with one being negative and the other
positive. Thus, if we trace the population through s cycles we ®nd, as for an autosomal locus, that
!
k
1
1 X
2
4
1
‡ …j† 

:
…j†
NeE 9k yˆ1 Nm
NeV
Nf
We have not yet been able to generalize the foregoing reasoning to arbitrary o€spring distributions, but conjecture that then also NeE  NeV . Another reason to suspect that this is the case is
that Eq. (31), with k ˆ 1, was ®rst derived by Pollak [19] by a computation of the approximate
rate at which the probabilities in the asymptotic conditional distribution decrease. This method
also led to Eq. (27) when k ˆ 1. For either an autosomal or a sex-linked locus the branching
process approximation used by Pollak [19] involved looking forward in time from parents to
o€spring, so that a variance e€ective population number was obtained.

9. Discussion
In this paper expressions have been derived for the e€ective population number with cyclic
variation in numbers of males and females, for either an autosomal or a sex-linked locus. In either
case, it is approximately equal to the harmonic mean of e€ective population numbers in a cycle,

Y. Wang, E. Pollak / Mathematical Biosciences 166 (2000) 1±21

21

and is the same regardless of whether the harmonic mean is calculated from inbreeding e€ective
numbers or variance e€ective numbers. We obtain the same result by calculating the mutation
e€ective size, and, as far as we can tell, the eigenvalue e€ective size. Of course, as shown in Section
6, the inbreeding e€ective number, NeI , and the variance e€ective number, NeV , are not the same if
only two successive generations are considered and the population changes in the size. As far as
we know, such expressions have not previously been given when there is a sex-linked locus.
In our calculations, we have not restricted ourselves to assuming Poisson distributions of o€spring. Thus complete account has been taken of both ¯uctuations in population size and variability in family sizes. Frankham [20] reviewed 192 published estimates from 102 species of the
ratio of Ne to the actual population size, N, and concluded that the ®rst and second most important variables explaining variation among these estimates were, respectively, ¯uctuation in
population size and variance in family size. These variables both act to reduce Ne =N to the low
values found by him in wildlife populations.

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