THE DYNAMICS OF FINITE HAPLOID POPULATIONS WITH … · 2003. 7. 28. · THE DYNAMICS OF FINITE HAPLOID POPULATIONS WITH OVERLAPPING GENERATIONS. I. MOMENTS, FIXATION PROBABILITIES

THE DYNAMICS OF FINITE HAPLOID POPULATIONS WITH OVERLAPPING GENERATIONS. I. MOMENTS, FIXATION

PROBABILITIES AND STATIONARY DISTRIBUTIONS1

TED H. EMIGH

Department of Statistics, Iowa State University, Ames, Iowa 50011 Present address: Department of Statistics, University of Georgia,

Athens, Georgia 30602

Manuscript received November 15, 1977 Revised copy received July 31, 1978

ABSTRACT

Much of the work on finite populations with overlapping generations has been limited to deriving effective population numbers with the tacit assumption that the dynamics of the population will be similar to a population with nonoverlapping generations and the appropriate population number. In this paper, some exact and approximate results will be presented on the behavior of the first two moments of the gene frequencies. The probability of fixation of a neutral gene is found equal to the initial average reproductive value of the gene, and the means and covariances of the stable distribution with mutation in both directions are found by a simple extension of the values found by assuming nonoverlapping generations.

I N populations with overlapping generations, it is hoped to be able to sum- marize the complex mating, selection and mutation structure of the population

to a few easily understood parameters. FELSENSTEIN (1971) calculated one such parameter, effective population number, for a haploid population with overlapping generations. In this and a subsequent paper, we explore the use of his effective number in describing the dynamics of such a population. To do this, we will introduce several other summary parameters, such as average reproductive value, which are useful in describing the population.

In this paper, we will define the model and present some results under the assumption of no selection. In a subsequent paper, the action of selection will be investigated through the use of the diffusion approximation.

T H E MODEL

The model used in this discussion, which was first introduced by FELSENSTEIN (1971) , is based on the assumption of a fixed life table and reproductive structure with independent births.

The model is clearly specified in FELSENSTEIN (1971) , and only a review is given here. The number of haploid individuals in each age group (i = 1 ,2 , . . . , k)

Journal Paper No. 5-9009 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Projeci 1669. This work was supported by Public Health Service grant GM 13827 while the author was at Iowa State University.

Genetics 92 : 323-337 May, 1979.

324 T. H. EMIGH

is constant over time ( t = 0, 1, 2, . . . ) and is equal to N ( i ) . Newborn individuals are independently chosen from parents, with pi equal to the probability that the parent of any particular newborn individual at time t + 1 came from age group i at time t. Individuals in age group i + 1 at time t + 1 are chosen without replacement from individuals in age group i at time t. Although we call age one the newborn age group, it is really the group at which the zygote is formed. We can consider this group as distinct individuals or as merely the zygote. If the latter is chosen, we may consider birth at age group two and thus allow for selection among zygotes.

FELSENSTEIN did not consider mutation or selection, but it is easy to incor- porate this in the framework of his model. The age-specific mutation rates from A, to A, and A, to A, are pi and vi, respectively, for parents in age group i. The age-specific fertility selection coefficient of A , over A, is denoted i i for parents in age group i, and the viability selection coefficient of A, over A, in surviving from age group i-1 to i is denoted si. Let Nl( i , t ) be the number of individuals with allele A, in age group i at time t. Then, for the newborn age group,

(1) independently of survival, where Nl(t) = (Nl(l,t), . . . ,Nl(k, t ) )’, r= ( rl, . . . , ik)’ and

N,(l,t+l) IN,(t)-Binomial [N(I), Pr ( t f l ) ] ,

1 IG (l-pci) (I+?-j)Nl(i,t) +vi [ N ( i ) -N1 1 i=l

P (t+l) = z Pi { N (i) + riNi (i,t)

(l+rdN,(i,t) z N ( i ) + r i N 1 ($1 = v* + 3 pi (1 - pi - v i )

with v+ = z p i v j .

i For survival,

N,(i+l, t f l ) 1 N,(i,t) -Noncentral hypergeometric ( N ( i ) , N l ( i , t ) , N (i+l), si+l), (2)

i=l, . . . ,&I, and N,(i+l, t f l ) = N(iA-1) - Nl(i+l, tSI), (3) independently o€ the other age groups and reproduction. The noncentral hypergeometric distribution is defined such that the probability of drawing gene A, if there are ( a ) A , genes and ( b ) A, genes left before that draw is a (1 + si+,)/ [a(l+si+,) + b ] , instcad of a/[&b] for the central hypergeometric distributions (see WALLENIUS 1963, or JOHNSON and KOTZ 1969). FELSENSTEIN (1971 ; see also EMIGH 1976) has shown that, for this type of population, the effective population number is, ignoring selection,

where qi = 2 pj =probability of reproducing at or after age i, 1 3 =N(i)/N (1) = probability of surviving to age i, and T = ipi, the generatiox time. The

jzi

qi = 1 *

HAPLOID OVERLAPPING GENERATIONS I. 325

quantity qi is related to FISHER’S (1930) reproductive value as U+ = qi/li. Letting

li+’ and d+ = 1 - wg, the effective population number can be written as

N ( 1 ) T 1 + 3 1 4 wi dj u2i+l

wi =li Ne =

z

M O M E N T S O F THE G E N E FREQUENCIES

Let X (i , t) = N I (i,t) / N (i,t) , the frequency of gene A, in age group i at time t , and define X ( t ) = [ X ( l , t ) , . . . , X ( k , t ) ] ’ . At the next time interval, t+l, the gene frequencies will change, governed by the probability distribution ( 1 ) and (2) . The means and covariances are important characteristics of this change. The means

-J

Consider the newborn age class. E [ X ( l , t + l ) I N , ( t ) ] = P . ( t + l )

If {pi}, {vi} and {r i } are 0 - , then (6) can be written as (;e )

E [ X ( l , t + l ) I N , ( t ) ] = v* + Z pg( l -p i -vg)X( i , t )

+I: piriX(i , t ) [ l - X ( i , t ) ] + o ( 4 ) . (7)

(8)

(9)

Ne If there is no selection, then (6) can be written as

and with no selection or mutation, E [ X ( l , t + l ) I N , ( t ) ] = v * + s p i ( l - p i - ~ i ) X ( i , t ) ,

E [ X ( l , t + l ) I N i ( t ) ] =? p i X ( i , t ) .

Consider age group i + 1. If is 0 - and N ( i + l ) is large, then the

mean of the noncentral hypergeometic distribution given in equation (2) is approximately.

(2 E [ X ( i + l , t + l ) I N I ( i , t ) ]

= ( I + s i + J X ( i , t ) - s++lXz(i , t ) + 0 (&) (10) This approximation is valid if either (1) N ( i + l ) is small in comparison to N ( i ) , such as with some trees; or (2 ) N ( i + l ) is close to N ( i ) , such as with humans. As N (i) becomes very large, the noncentral hypergeometric distribution approxi- mates the binomial. Caution must be exercised when a moderate drop in population number occurs during one year, although with very small si+,, even this is not serious. If no selection is present, the mean is exactly

E [ X ( i + l , t + l ) 1 N , ( i , t ) ] = X ( i , t ) . ( 1 1 ) If there is no selection present in the population, ri = si = 0, all i, the means

E [ X ( t + l ) I X ( t ) ] =E* + PX(t) ; (12) can be written in matrix form as

T. H. EMIGH 326

where

and

P =

- v* = ( V * , o,o, . . . 7 O)’,

‘pl(l-Pl-vl) p?(l-pP-vP) . . . pk-l(l-pk-I-vL1) pk(1-pk-vk) ‘

1 0 . . . 0 0 0 1 . . . 0 0

0 0 . . . 0 0 0 0 ... 1 0

This same type of matrix (without mutation) is described by HILL (1972), and is a special case of the type discussed by LESLIE (1945). Hence, if Mt = E [X(t) ] , then a recurrence relation can be written for the means when there is no selection as

Mt+l=z* + P Mt . (13)

Variances; covariances and higher moments Now, consider the variance-covariance matrix of the gene frequencies. If the

selection coefficients are small [s,,.r; are O(l/Ne)], then the variance will be approximately the same [O(l/N,z)] as if there was no selection. Let

The (i,j) t h element of V ( t+ 1 ) is

where ML,t+l is the ith element of Mt+l. By first conditioning on X(t), and then taking the expected value over X( t ) , we obtain,

V(t+I) = E[ (X(t+l) - Mt+i) (X(t+l) - Mt+i)’] . V,j (t+I ) = E[ (X (i,t+I ) -Mi,t+i) (X (i.t+l) -hi’j,t +I)] e

Vzj(t+I) == E,,,, {Cov[X(i,t+l), X(j,t+l) j x(t>l} . + Cov,(,, {E[X(i.t+l) I x( t> l ,E [X( i ,~+ l ) I x( t> ]> ,

where E g ( , ) is the expected value over X(t) and CovXctt is the covariance over

Define X ’ ( l ~ + l ) = v * +,E p2(1-pL,-v,) X ( i , t ) , and X’(i,t+l) =X(i-l,t), X(t>.

z

‘i#l. Then,

Hence, E[X(i,t+l) I X(t)] =X’(i,t>.

ViJ(t+l) = E , ( , ) {Cov[X(i,t+l), X(i,t+l) [ X(t ) l } + COV,,,, Cx’(i,t>, x’(i,t>I -

Consider the newborn age group (i = i = 1 ) , X‘( 1 ,t) (I,-Xf( 1 ,t) )

N(1) Var [X(l,t+l) 1 X ( t ) ] =-

The expected value of this quantity is


Since v* + + pi( I-pI-vi) Mj,t, this can be rewritten as

The variance of X’( 1,t) is easily calculated as ~.PiPdl-w-vi) (l-Pj-vj)v$j 0).

Combining (1 6) ai;d ( 17) into (14) ,

For i = 1, j > 1, since reproduction is independent of survival, cov (X(1, t+l), X(j, t+l) I X ( t ) } = 0,

V,j(t+l) = $. pi(l--pi--vi)Vi,f-I(t). (19)

cov [X(i, t+l), X(j, t+l) I X(t)] = 0,

vi, (t+l ) = vi-1, j-1 ( t ) .

so

If i#j, i, j > 1

so (20)

Finally, if i=j#l, N (i- 1) -N (i)

[N (i-I ) -11 N (i) Var [X(i,t+l) I X(t)] = X(i-1,t) [l-X(i-IJ)] ’ so

1 -N(i-1) I -N(i-I) Using elementary properties of the binomial and hypergeometric distributions,

it is easy to show that the third and fourth moments are 0(l/Ne2) (see EMIGH 1976), although the skewness and kurtosis may not be negligible. In the next sections, these moments will be used to obtain elementary properties of the population.

PROBABILITY O F FIXATION O F A N E U T R A L G E N E

Assume that there is no mutation or selection in the population. Since the population has finite size either A, or A, will eventually become fixed in the population if there exists a j such that pi, pj+l > 0. The condition that pj, > 0 assures that the population will not be cyclic. Populations where this condition may not hold are periodic cicadas (Magicicada sp.), with different populations emerging each year. It is then possible to have some populations fixed with gene A, and other populations fixed with gene A,. The above condition assures that there will be proper mixing across years.

For neutral genes, equation (12) becomes E[X(t+l) 1 X(t)l = P X ( t ) , (22)

328

where - - P I P2 ... pk-1 Pk 1 0 ... 0 0 0 1 ... 0 0

P =

0 0 ... 0 0 0 0 ... 1 U

L -

T. H. EMIGH

.4 1 - _- qix(i,o) . T ”

Since there is no mutation, if the population ever becomes entirely A,, or entirely A?. it will stay that way. Hence, the states X ( t ) = 0.0 , . . . , 0)’ and X ( t ) = ( 1 , 1 , . . . , 1)’ are absorbing states. It is not difficult to see that: all states com- municate with st least one of the two absorbing states. Therefore, the population eventually will become entirely A , or entirely AP, so long as it is not cyclic. Hence,

lim E[X(i,t) I X ( O ) ] = &‘[Fixation of A,] - 1 t -r m

3. &‘[LOSS of A,] . O = &‘[Fixation of A , ] .


Therefore, P[Fixation I X(O)] = x ( 0 )

While the model used here assumes a constant number of individuals in the age groups, we need only the relationship (22) and that the gene will always go toward fixation or loss. In particular, the values for N ( i ) do not appear. In a related model, EMIGH (1976) has shown that (27) holds for the case with N ( 1 ) fixed and the p’s fixed. Survival is accomplished through the binomial distribution rather than the hypergeometric distribution.

It is possible to allow N ( l ) to vary, either stochastically or deterministically, so long as N ( i , t ) < N * , for some N*. This guarantees that the gene will be either lost or fixed in the population. If the population were allowed to pow fast enough, it could be possible to have both genes present in the population for every t.

If the p’s also are considered random variables independent of t and X ( t ) , then the p’s in ( 5 ) are replaced by their expected values and

s(0) =- z E ( q i ) X( i ,O) . E ( T )

The joint distribution of the p’s does not matter, but the condition on the p’s then becomes a condition on the p’s in probability.

It is obvious that, as FISHER (1930) surmised, the reproductive value of a population is an important quantity in describing the population.

STABLE DISTRIBUTION W I T H MUTATION A N D N O SELECTION

Consider the population to have mutation but no selection. With mutation in both directions (i.e., from A, to A, as well as from A, to A,) ~ one gene cannot be fixed in the population. The Markov chain describing the population can be seen to be an irreducible positive recurrent Markov chain, since, with mu tation, any state of the population is possible and the population always will return to any given state after a sufficient time. Therefore, the population will reach a stable distribution and

Hence, from (12), M , + , = M , = M .

M I X * + P M , or, so long as ,p* f Y * > 0, then (I-P)-I exists, and

M = (I - P)-’x*. Since E* is a column with only one non-zero element, this being in the first

polsition, M is the first column of (I - P)-, multiplied by?*. Designate the first column of (I - P)-, by F, with the elements of F denoted by f i , i = 1, . . . , k, and the itn row of (I - P) by Gi. Then, since (I - P) (I - P)-I = I,

and G , F = l ,

G i F = O , i = 2 , . . ., k. (29)

330 T. H. EMIGH

For i#l, Gi = (0, . . . , - 1 , 1 , . . . , 0), with -1 in the the ith position, with the other elements zero.

position and 1 in

Therefore, for i f l , GiF == a f i -1 + f i = 0,

or

The its term of GI is -pi(l - pi--vi) if i f 1 and l-pi(l--pi-vi) if i = 1. Therefore,

f i =f, for all i. (30)

1 = GiF = - B pi (1 - pi - v i ) fi fi

= -f 3 pi (1 - pr - Vi) + f = f ( P * + v*>,

(31)

where p* = z pi pi and v* = Hence,

or

pi v i . 1.

f + = 1/(p* + v*), for all i,

M = M j , (32) V*

where M = p* f v '

That is, for every i,

and j is a column of ones.

(33) V*

lim E[X(i,t)] = t+ m ,p* + v*

Notice that this is independent of the initial gene frequencies.

and Mt+l =Mt =M, whereM = M j, with M = v * / ( p * + v * ) , as above.

VI, ( t f 1 ) = ( 1 - l / N ( 1 ) ) pipj ( 1 -pEL(-v2 ) ( 1 -pj -vj ) vi j ( t ) + - * M(l-M)

To find the variance of the stable distribution, note that V(t+l) = V ( t ) = V

Replace Mt with M in equations (18) and (21). Then, at equilib~um,

23 N(1) and (34)

1 1

1 - N(i-1) 1 - N(i-I)

Now, consider equation (35). Replacing Vi-l,~-l(t) by its value in terms of Vi-z,i-z (t-I ) , and simplifying,

1 - 1 1

Vii(t+l) = N(i), VikZ,iC2(t+1) + 1 M(l-M) . N(i) N(i-2) I--

I 1- I

N (i-2) 1- N (i-2) By repeating this process,

1 1 1


This is easily seen from noting that in sampling theory, a two-stage subsampling scheme is equivalent to a one-stage sampling scheme. Also, for i < j ,

from (20). While equations (34), (35) and (37) can be solved directly at equilibrium,

we will reduce the number of equations from k2 to k by considering the covariances Vlj, j=1, . . . , k. First, consider equation (18),

Vij(t+l) = Vl,j-i+l(t-i+2), (37)

Vl,(t+l) = (1 --) 1 z pip j ( l -prv i ) (l-pj-vj)vij(t) N ( l )

+- M(1-M) N(1)

For simplicity, assume that p k = 0. This will not exclude any type of age structure; we need only add another age group that does not participate in reproduction. For example, the case of nonoverlapping generations would have p1 = 1 and pz = 0. Then

+- I M (l-M) N(1)

Using equation (19) and letting Vk,j ( t ) = 0, we obtain

M(1-M) (38) 1 V1,(t+l)=(l --) pi(l-~EL(-Y~)vl,i+l(2+1) + - N(1) N(1)

Since the population is in equilibrium, Vij(t+l) = Vij(t) = . . . = Vij. Therefore,

332 T. H. EMIGH

1 1

It is obvious that only Vl = (Vll, VI,, . . . , Vlk)’ is needed in finding the variances and covariances of the stable distribution. Using (4.0) and (41), it is possible to obtain the remaining elements of V.

Equations (39) and (42) can be written in matrix form as Vi = A Vi + B,

where

B = M(1-M:

and

1

0 N ( 1 )

1 1

1 1 N ( k - l ) - m

Pk-l(l -pk-l-vk-l) 1

If I - A is nonsingular, then the solution is V I = (I,-A)-’B .

APPROXIMATE COVARIANCES FOR T H E STABLE DISTRIBUTION

While it is possible to sol\-c (44) exactly for particular choices of N ( l ) , p , . pi

and v t , approxiiiiale analytic solutions for populations in terms of the N’s, p’s, p’s, V’S, etc. are possible.

As a first approximation, assume that V,j = V,.,,, for all i, i’, i, y. This approximation is a ieasomble first step, considering that this is essentially the assumption needed to obtain the effective number. Recalling that I: q1 = T , consider the following weighted average of the V1,’s.

1

H A P L O I D OVERLAPPING G E N E R A T I O N S I. 333

I h

2

2 7r

I

k . . . 0 0 . . . 0

n a m

I

I . . . .

n

a .-I

.C

n 2 I:

7 4 W

. . .

L ?- a A

r( ;r

I o f

d W

&

.. .

n 4

I 'N

i I I - 1

334 T. H. EMIGH

(6) 1

V(tS1) =- I: qiqj Vij(t+l). T2 13

Let From (18), (19), (20), and (21), the right hand side of (45) is

refer to the summation over i and j not equal and greater than one. i , J > l

1 T' 1>1 -{q;vll(t+l) +q1 .. QjVli(t+I) +q1 . E qivil(t+l)

+ .?+ QiQI Vij(t+l) + .z q2. Vii(t+l)) *J>1 2>1 L.

M(1-M)

+ q1 j;l Q, & pi(l-Pi-Vi)Vi,j-l(t) + 41 ,x qi z pj(1-Pj-vj)Vi-l,j(t) %>I 3

1 1 - M(1-M)} . N ( i ) N(i-1)

1 + &4; 1 -

N(i-1) Define q k f l = 0 and recall that ql = 1 and qj = ,z pi, so qi+l + pi = qi.

Replacing each Vij ( t ) by the approximated value V and simplifying, equation (45) can be written as

a 3

1 1 T v = 7 { [T2 + (l-p*-v*)2-1 - __ N(1) ('-p*-'*)'

(46) 1 1 -2(T--l)(p*+v*) -- qi+1 2 ( - - - 9 ] v

N( l ) li+l li

+- 1 M(1-M) + - - - Z q i + 1 2 1 (---) 1 M(1-M)}. N(1) N(1) li+l Li

From (4) , recalling that qi = T, the generation time, 1 1 - T - 1

1 xqi+,Z(---)-- ___ . N(l ) li+l li Ne N(1)

(47)

Hence, solving for V in (46))

(48) M(1-M)

3 [(l - ----) (1 - ( l - p * - ~ * ) ~ ) + 2(T-1) (JL*+v*)] + 1 T N(1)

By ignoring terms of order ,L*~, v * ~ and (p*+v*)/N(l), sinceN(1) is large, this can be approximated by

1 V =

M(1'- M) 1 + 2Ne(/.L*+v*) V = (49 1

This is the value found by WRIGHT (1931 ) for populations with nonoverlapping generations and is valid for large numbers in each of the age classes.


The approximation (48) appears to be quite good. This approximation can be improved somewhat by using (38) through (42) and iterating. The improved estimates are

(50 )

1 1 vij = (1-p*-v*) v+ pli-jl ( l - p ~ ~ - j [ - V ~ & j ~ ) (N(li-il) -4 N(1) . [M(l-M) - (I-@*-'*) VI, i#i, (51)

where V is given in (48). A comparison of the exact and approximate covariances of Vlj, i=1, . . . , k is

presented in Table 1. The relative fertilities follow a similar pattern to that observed in females in the United States (EMIGH and POLLAK 1979) although the life table is different. The approximations are extremely good (within order @* + Y*).

SUMMARY

The moments of the gene frequencies in a haploid model with overlapping generation are found through recurrence formulas that are easy to evaluate. In a population without mutation and selection, the unconditional mean reproduc-

tive value of the gene in the population, E ( - x q i X ( i , t ) ) , is unchanged with

time. This leads to the result that the probability of fixation of a neutral gene is equal to the relative initial reproductive value of the gene in the population

1 T i

With mutation present, but without selection, we are able to derive an exact mean and approximate covariances of the gene frequencies in the resulting stable distribution. The mean is exactly that found for populations with nonoverlapping generations, with p* = Z p i p i and Y* = Fpivi, average mutation rates, replacing p and V. The correlations are seen to tend to unity, although not exactly (within order p* + Y*) . The common approximate variance is the same as for a population with nonoverlapping generations with p*, V* and N e replacing p, Y and N , respectively.

It seems reasonable, therefore, to believe that haploid populations with overlapping generations can be sufficiently described through the reproductive value, with the proper modifications to the parameters, in the same way as a population with nonoverlapping generations. This is investigated in a subsequent paper.

There has been some initial work done on diploid populations by HILL (1972), JOHNSON (1977a,b) and EMIGH and POLLAK (1979). It is expected that diploid populations will give results similar to this paper.

Portions of this paper appeared in my thesis (EMIGH 1976) under the direction of OSCAR KEMPTHORNE; I greatly appreciate his guidance. Helpful comments and suggestions by EDWARD POLLAK on the thesis and this paper also are appreciated. Comments on an earlier draft of this paper by JOSEPH FELSENSTEIN and an anonymous reviewer are appreciated.

P Z

336 T. H. EMIGH

TABLE 1

Comparison of true and approximated covariances for a population with overlapping generations

4 1 1.0 2 1.0 3 0.9 4 0.9 5 0.8 6 0.7 7 0.6 8 0.5 9 0.5

10 0.5 N(1) M V Ne . I”* /y*

f c Vli

pi pi Exact Approx.

0 0 0 1 3 3 2 1 1 0

- - 0.249137 - - 0.249088 - - 0.249084

10-5 l W 0.249088 10-5 10-5 0.249101

10-5 10-5 0.249082

10-5 10-5 0 . ~ 1 0 1 10-5 10-5 o . 2 4 . ~ ~ ~ 4 10-5 10-5 0.249086 - - 0.24Q089

U) 0.5000 0.2490993

91.16 10-5/10-5

0.249140 0.24-9094 0.249094 0.249094 0.249095 0.249098 0.249100 0.249099 0.249097 0.249097

ci pi

- - - - - - 10-5 10-5 10-5 10-5 10-5 10-5 10-5 10-5 10-5 10-5 10-5 10-5 - -

100 0.5000 0.2455310

1 0-5/10’5 455.78

Vli Exact Approx.

0.245568 0.246571 0.246520 0.245526 0.245516 0.245526 0.245514 0.245526 0.245520 0.246527 0.245533 0.246530 0.246533 0.245531 0.245525 0.246531 0.245518 0.246529 0.245521 0.245529

VI, P i y+ Exact Approx. pi

Vxi Exact Approx.

1 - - 0.249719 0.249720 - - 2 - - 0.249703 0.249705 3 - - 0.249702 0.249705 4 10-5 10-5 0.249701 0.24!3705 5 10-6 10-6 0.249703 0.24!2705 6 10-6 10-6 0.249707 0.249706 7 10-6 0.249707 0.24Q707 8 10-5 10-5 0.249705 0.249707 9 10-5 10-5 0.249702 0.240706

10 - - 0.249703 0.249706 N(1) 20 M 0.5000 V 0.2497066 N e 91.16 C*/V * 3.25x 1 0-6/3.25 x 10-6

- - - - 10-5 e x 10-5 10-5 2x10-5 10-5 2 x 10-5 10-5 2x10-5 10-5 2 x 10-5 10-5 2x10-5 - -

U)

0.3333 0.2210235

91.16 10-5/2X10-5

0.221074 0.221077 0.221009 0.2.21017 0.221003 0.22101 7 0.221000 0.221017 0.221009 0.221 01 8 0.221026 0.221022 0.221026 0.221024 0.221016 0.221023 0.221006 0.221021 0.221009 0.221021

LITER4TURE CITED

EMIGH, T. H., 1976

EMIGH, T. H. and E. POLLAK, 1979

FELSENSTEIN, J., 1971

The effects of finite population size on genetic populations with overlapping generations. Ph.D. Thesis, Iowa State University.

Fixation probabilities and effective population numbers in diploid populations with overlapping generations. Theoret. Popul. Bid. 15: (in press).

Inbreeding and variance effective numbers in populations with overlapping generations. Genetics 68: 581-597.

Oxford. FISHER, R. A., 1930 The Genetical Theory of Natural Selection. Oxford University Press,

HAPLOID OVERLAPPING GENERATIONS I. 337 HILL, W. G., 1972 Effective size of populations with overlapping generations. Theoret. Popul.

Biol. 3: 278-289. JOHNSON, D. L., 1977a Inbreeding in populations with overlapping generations. Genetics 87 :

581491. -, 1977b Variance-covariance structure of group means with overlapping generations. pp. 851-858. In: Proceedings of the International Conference on Quantitative Genetics. Edited by E. POLLACK, 0. KFLVIPTKORNE and T. B. BAILEY, JR. Iowa State Uni- versity Press, Ames, Iowa.

JOHNSON, N. L. and S. KOTZ, 19691 Distributions in statistics: Discrete Distributions. Houghton Mifflin Co., Boston.

LESLIE, P. H., 1945 On the use of matrices in certain population mathematics. Biometrika 33: 183-212.

POLLAK, E. and 0. KEMPTHORNE, 1970 Malthusian parameters in genetic populations I. Haploid and selfing models. Theoret. Popul. Biol. 1: 315-345.

WALLENIUS, K. T., 1963 Biased sampling: The noncentral hypergeometric probability distribution. Ph.D. Thesis, Stanford University.

WRIGHT, S., 1931 Evolution in Mendelian populations. Genetics 16: 97-159. Corresponding editor: D. L. HARTL

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THE DYNAMICS OF FINITE HAPLOID POPULATIONS WITH … · 2003. 7. 28. · THE DYNAMICS OF FINITE HAPLOID POPULATIONS WITH OVERLAPPING GENERATIONS. I. MOMENTS, FIXATION PROBABILITIES