Journal of J. Math. Biology 5, 281-291, (1978)

(~) by Springer-Verlag1978

A Multi-Locus Continuous-Time Selection Model* Michael Moody Department of Biophysics and Theoretical Biology, The University of Chicago, 920 East 58th Street, Chicago, Illinois 60637, USA Summary. A continuous time selection model is formulated for a diploid monoecious population with multiple alleles at each of an arbitrary number of loci, incorporating differential fertility and mortality as well as arbitrary mating and age structure. The model is simplified in the case of age-independence and for the case of a stable age distribution. The age-independent model is examined in detail for the special case of multiple alleles at each of two loci. This model is analyzed under the assumptions of random mating and additive fertilities, with close attention given to the behavior of the system with respect to HardyWeinberg proportions and linkage equilibrium.

I. Introduction

Population genetic models in continuous time with overlapping generations probably provide a more realistic approximation for the evolution of certain biological systems than do their discrete analogs. Until recently, considerations of such models were generally restricted to the single-locus case and involved either explicit or implicit simplifications neglecting such factors as mating and age structure, differential fertility and mortality. See, for example, Haldane [5], Fisher [4], Norton [11], and Kimura [6, 7]. Charlesworth [2], for the case of multiple loci with multiple alleles, generalized the treatment somewhat by examining two models, one of which considered differential fertility among genotypes but neglected mating structure while the other did the reverse~ He also suggested a more complete interpretation of the Malthusian parameter than had previously been given. Nagylaki and Crow [8] considered for a single locus with multiple alleles and for two alleles at each of two loci models which allow for arbitrary mating structure, differential fertility and mortality, but neglect age structure. Cornette [3] generalized these models for the single-locus case, presenting a model that incorporates arbitrary mating and age structure as well as differential fertility and mortality; he recovered the models of Nagylaki and Crow [8] and of Charlesworth [2] as special cases.

*

M.M. was supported by a U.S. Public Health Service training grant (Grant No. GM780). 0303-6812/78/0005/0281/$02.20

282

M. M o o d y

In this paper we will show how the single-locus approach of Cornette [3] and Nagylaki [10] can be extended to the case of an arbitrary number of alleles at each of an arbitrary number of loci in a diploid monoecious population. The ageindependent model of Nagylaki [10] for two multiallelic loci will be derived for the two cases of age-independence and a stable age distribution. Particular attention will be given to the behavior of this system with respect to Hardy-Weinberg proportions. It should be noted that biological constraints on the total population size will in general limit the applicability of the model when the number of loci under consideration becomes large. Assuming at least two alleles per locus, n loci will generate at least 3n possible genotypes. Hence for n sufficiently large, the number of possible genotypes will far exceed the size of normal biological populations, in which case we would not expect the evolution of all genotypic numbers to be described by a deterministic function. Rather, we would expect that individual genotypes may appear in the population in a discrete, stochastic manner and disappear likewise. II. General Formulation Consider an arbitrary number of loci in a diploid monoecious population with an arbitrary number of alleles at each locus. Denoting the ith gametic type by Gt, we follow the convention of Nagylaki [10] and denote the ordered genotype formed from gametes Gl and Gj by GIGj. For each ordered GIGj genotype we assume the existence of an age density function, vu(t, x), such that the number of GIGj individuals at time t with ages in the interval Ix1, x2] is given by

ff

~ vii(t, x) ax.

(1)

1

Assuming that vu(t, x) is for each i, j and t integrable with respect to x, we obtain the total number of GtGj individuals at time t, nij(t), as

nu(t) =

f0~

vt~(t, x) dx.

(2)

We observe that under these conditions, vii(t, x)--~O as x--~ oo, necessarily. However, in any real population there exists some maximum age X such that v~j(t, x) = 0 for x > 1" and for every i, j. Since the v~ vanish for x > X, taking the upper limit as infinity in (2) introduces no complications. From (2) we obtain the total population number at time t, N(t), as

N(t) = ~ ni,(t).

(3)

iS

We also posit the existence of life table functions, ljj(t, x), where for t >t x, ljj(t, x) is the probability that an individual of genotype GiGj born at time t - x survives to age x and for t < x, the probability that a GiG~ individual aged x - t at time t = 0 survives to age x. We have immediately that ll~(t, O) = lu(O, x) = 1 for each

A Multi-Locus Continuous-Time Selection Model

283

t, x > 0 and for each i, j. Since the analysis of Cornette [3], Section I, and of Nagylaki [10], Eqs. (4.120) to (4.121), is independent of the classification criteria for individuals subscripted by i and j, the results derived for the l~j and ~7,jof Cornette and the l~j and v,j of Nagylaki in the single locus case apply here with the notational changes i ~ i, j -+ j. In particular, the age structure of the population is determined by the life table, birth rate and initial age structure via the relationships

vis(t, x) = ~vi,(t - x, 0)lu(t, x), t t> x, (4) /yu(0, x - t)ln(t, x), t < x, where vu(t, 0) is interpreted as the birth rate of G~Gj individuals at time t. The l~j(t, x) also satisfy the differential equation alis(t, x) + altt(t, x) = _ dis(t, x)lu(t ' x), ~x ~t

(5)

where the d~s(t, x) are functions such that for each t, x >>.O, du(t, x ) A x is the probability that a GtG 1 individual aged x at time t dies between ages x and x + Ax. The solution to (5) is found to be (Nagylaki [10])

f lis(t, x) = ~exp /

d,~(~+t-x, Dd~], t>x, (6)

exp [ --f ~x-t dij($ +

t - - x, ~)d~],

t < x,

which implies that the age structure is determined by the birth rates, age-specific death probabilities and initial age distributions. The analogous equation to (5) with vii(t, x) in place of 1u(t, x) also applies (Nagylaki [10], Cornette [3]),

~vis(t,8_..___x~X)+ 3vl~(t,8t x) = -dis(t, x)vo(t, x).

(7)

An equation and its solution analogous to Eq. (7) for an age but not genetically structured population has been given by yon Foerster [13], and Trucco [12]. Differentiating (2) with respect to t and employing (7), we obtain the analog of Eq. (3) of Cornette [3] and Eq. (4.121) of Nagylaki [10]

t~j(t ) = v,j(t, O) -

dis(t, x)vij(t, x) dx,

(8)

where the dot indicates differentiation with respect to time. Here we have used the previously mentioned fact that vu(t, oo) = O. To analyze the birth term, vu(t, 0), in (8), we must consider the mating structure of the population. Following Cornette [3], we posit the existence of mating density functions, Yk~,mn(t, x, y), such that the number of ordered matings in the time interval [tl, t2] between GkGt individuals in the age interval [xl, x2] and GmGu individuals in the age interval [yl, Y2] is

y2 Yk,,m(t, x, y) dt dx dy. 1

(9)

284

M. Moody

Let aki,~(t, x, y) be the average number of progeny from the mating of GkGl individuals aged x at time t and GmGn individuals aged y at time t. Let Ri,k~(t, x) be the frequency of gamete GI in the gametic output of GkGI individuals aged x at time t. In general, R~,k~(t,x) is independent of t and x and will henceforth be written as Rl.~a (Bodmer [1], Nagylaki [10]). Obviously, ~l Rl,kl = 1 for each k, l. With this device, the birth term vu(t, 0) of Eq. (8) reduces to

v~(t, O) ~ ~ Ri,k,Rj,~ kllan

////

Ykl,m(t, x, y)a~a,~,~(t, x, y) dx dy.

(10)

From (8) and (10) our basic equation now reads

Rl,vaRj,,~

tiu(/)= ~ k~

--

R

///

Y~a,mn(t, x, y)a~a,m(t, x, y) dx dy

0

//

djj(t, x)v~(t, x) dx.

(11)

Without further assumptions, (11) does not in general represent a closed system of differential equations for the n~(t). In order to obtain such a closed system, we impose restrictions upon the mating structure, fertilities, and mortalities which render Eq. (11) independent of age structure. As in Nagylaki [10], we accomplish this with the assumptions that fertilities and mortalities are age-independent

ak,..~,.(t, X, y) = a~a,m(t),

dki(t, x) = dk,(t),

(12)

and that genotype and age be functionally independent in the mating structure in the particular sense that

Y~a.m(t, x, y) = ?k,,~(t)vk~(t, X)vm(t, y),

(13)

where Yk~.m(t) is some non-negative function of t. As in the simpler case, we cannot assume that Yva,m(t, x, y) = Y~a,m(t) as a function of t alone, since it must be true that Ykl,~,~(t,X, y) = 0 whenever either of the age densities vk~(t, x) or v~(t, y) vanish at x or y. Denoting the total number of matings at time t by M(t) and the proportion of these matings which are between GkG~ and G~Gn individuals by Xkl,~(t), we find from (13) that

M(t)X~a,~(t) =

f///

Ykl,~(t)vkl(t, X)v~(t, y) dx dy

= Fkl,~(t)n~a(t)nm(t), which shows that Xk~,m(t) is indeed independent of age structure. Note also that the X~a,m.(t) are normalized = klmll

since

-ol

1,

A Multi-Locus Continuous-Time Selection Model

285

Employing (12) and (13) in (11) yields the basic equation of the age-independent model tit)(t) = M(t) ~ X)a,ma(t)am,ma(t)Rl,klRl, ~ - dl~(t)ntj(t). (14) klmn

This is the analog of Eq. (4.126) of Nagylaki [10] for the two-locus case. Implicit in (14) is the possibility that the time dependence may involve genotypic numbers and any other relevant quantities. It is also true in the more general setting that Xkl,mn(t ) = Xmaa(t), aal,mn(t)= a~aa(t), and that nu(0)= nil(0) implies ni)(t) = nil(t) for all t > 0. As in Nagylaki [10], Eq. (14) may be deduced from the assumptions that the population has attained a stable age distribution and that the age and time dependence of the mating frequencies are multiplicative. To this end, suppose that for each i, j there exist equilibrium age density functions, An(x), with the property that

vt)(t, x) = ntj(t ))q)(x). With this definition, we interpret for any xl and x2, x2 > xl, the expression ~= au(x ) dx, as the proportion of GiGj individuals with ages in the interval [x,, x2]. Similarly, we suppose that there exists functions S~a,m=(x,y), independent of t, such that the integral

represents the proportion of matings between GkG, individuals with ages in the interval [xl, x2] and GmG~ individuals with ages in the interval [yl, Y2], relative to the total number of matings between GkGt and GroG. individuals. Assuming that Ym,mn(t, x, y) = M(t)X~a,m(t)Sk~,~(x, y), we obtain from (11) the result (14) with the aid of the identities

J0Jo

ku(x ) dx = 1,

Sk~.,n(x, y) dE dy = 1,

and the identifications

f; ara,m(t) =

fofo"

ara,m~(t, x, y)Sra,mn(x, y) dx dy.

In this case, di)(t) and ak~,mn(t) are just the mean mortality and fertility, respectively. Defining the fertility relative to the number of individuals rather than the number of matings, by A,,=n(t) = M(t)[N(t)]- ~ara,n,.(t),

(15)

286

M. Moody

we obtain from (14) ft u = N ~

Xk,,,j~,,,mR,,~aRj,m, - auni j,

(16)

klmll

where we have suppressed explicit dependence on t for notational convenience. Writing Pu for the ordered genotypic frequency of G~Gj individuals where, evidently, Pu = n~jN-1, we define the mean mortality and fertility by = ~ Pudu,

f=

(17)

(18) klmn

where in (18) we have used the fact that the Xk~,m~are normalized. Summing Eq. (16) over i and j and employing (17) and (18) along with the previously mentioned identity Ya R~,k~ = 1 yields the result = (f-

d)N.

(19)

Defining the mean fitness, ~, by ~ = f - d we obtain from (19) 2~ = raN.

(20)

Hence, with the proper identifications, Eq. (9) of Nagylaki and Crow [8] in the single locus case is completely general. Differentiating Pu = N-ln~ yields the equation for the genotypic frequencies Pij = k ~ , Xk~.mfm,m.Rl.k,Rj,m - (dn + m)Pu.

(21)

We define dt as the average death rate of individuals carrying gamete G~ with pia, =

e.a,j,

(22)

J

where p~ = ~j Pu is the frequency of gamete Gl. We also define bkl as the average rate at which GkG~ individuals give birth by Prabkt = ~, Xm,mf~a,mn.

(23)

Summing Eq. (21) over j and employing (22) and (23) we obtain the equation for the gametic frequencies P, = ~, Pk~bk,R,,k, -- (di + m)p~.

(24)

kl

Equations (14), (21) and (24) appear to be the most general possible, without age dependence, governing the evolution of the relevant parameters. However, without more explicit information about the linkage map and recombination structure of the system, the reduction in terms of gametic fitnesses as effected by Nagylaki and Crow [8] in the single-locus case cannot be accomplished here.

A Multi-Locus Continuous-Time Selection Model

287

n l . Special Cases

A. One Locus In the case of one locus, denote alleles by G~ and ordered genotypes by GzGj. Inserting the Mendelian formula, R~aa = 89 + 3~z), where 3o is the Kronecker delta function, into (16) recovers the single-locus equation of Nagylaki and Crow

[7] it~j = N ~ Xn,r~Sf~,m~ -- d,inij. lm

B. Two Loci Consider two loci, A and B, with alleles A~ at A and Bj at B and assume ageindependence. Denote the number of ordered (A~B 5 [ AkB~) genotypes at time t by nt(ij, kl). The fraction of matings at time t which are between individuals of genotype (AmB~ I AqB~) and genotype (Am, B~, I Aq, B~,) will be written Xt(mn, qr; m'n', q'r'), and the fertility of each such mating, relative to individuals, is written f~(mn, qr; m'n', q'r'). The death rate of (A~Bj I A~B~) individuals at time t is similarly denoted dt(ij, kl), and the frequency of gamete A~Bj in the gametic output of (A~B~IA~B~) individuals will be written as R(ij; ran, qr). With this notation (16) reads

ht(i.]',kl) = N

Xt(mn,qr,m n ,q r )ft(" ;.)R(ij;mn,qr)R(kl;m'n',q'r') rr'

r~

(25)

- d,(ij, kt)n,(i], ~t),

where for notational simplicity the arguments of~(. ; 9) are understood to be those of the Xt function multiplying it immediately on the left. This is just Eq. (8.75) of Nagylaki [10]. In the two-locus case, the recombination functions have been given by Nagylaki [10] as

R(ij; mn, qr) = 89

- c)3j~ + c3jr) + 21-8~q((1 - c)3jr + cSj~),

(26)

where c is the recombination probability between loci A and B. Substituting (26) into (25) and observing that the summands are symmetric under the transformations (m r), (m' ~ q', n' ~ r') we obtain the two-locus, multiple-allele analog of Eq. (56) of Nagylaki and Crow [7]

U-~i~(ij, kl) = c 2 ~

X(in, qj; m'l, kr')f(. ; .)

nflpff['r"

+ (1 - c) 2 ~

X(ij, qr;m'n',kl)f(.;

.) + c(1 - c)

ffr~m'n"

• [ ~ X(ij, qr; m'l, kr')f(.; .) kffr,m'T ~ +

X(in, qj; m'n', kl)f(. ;.)] - d(ij, kl)P(ij, kl). d

(27)

288

M. Moody

Expanding (27) and collecting terms we obtain

N-~(ij, kl) = ~

X(U, qr; m'n', kl)f(.; .)

qr,m'~,"

~ ~,

X(ij, q~;m'l, kr')f(., .1

qr

~, X(ij, qr;m'n',kOZ(.; .) m'lV

Xqn, qj;m'n',kl)f(.; .1 - ~X(U, qr,.... mn,klg(.,

+ ~, m'1~" L n , q

'l)

q1"

+c2{~qr [m~,~,X(ij'qr;m'n"kl)f(';')- ~X(ijq' r;ml'kr')f(;')],r, + ~ [m~'r,X(in'qj;m'l'kr')f('; ")-- ~m',v X(in, qj;m'n',kl)f(.; . ) ] ) -- d(ij, kl)P(ij, kl).

(28)

Denoting the sum of all but the first and last terms on the right-hand side of (28) by S, we obtain from (28), (17) and (18) the equation for the genotypic frequencies,

P(ij, kl) = ~

X(ij, qr; m'n', kl)f(.; .) + S

- (d(ij, kl) + m)P(ij, kl).

(29)

Summing (29) over k and 1 and employing (22), we obtain for the gametic frequencies

[~(ij) =

~

X(ij, qr; m'n', q'r')f(. ; .)

q r , rn'n'q'r"

c ~,

x(in, qj; m'n', kOf(" ; ")

t'lVn "

X(ij, qr;m'n',kl)f(. ; ")J~ - (d(ij) + ~)p(ij),

(30)

where

p(ij) = ~ P(ij, kl),

p(ij) d(ij) = ~ P(ij, kl) d(ij, kl). kl

kl

Defining b(ij, kl) as the rate at which (AiBj [ AkBz) individuals give birth we have

P(ij, kl)b(ij, kl) =

~

X((], kl; m'n', q'r')f('; ")

(31)

m'n'q'r"

If b(ij) is the average birth rate of individuals carrying gamete A,Bj, then

p(ij)b(ij) = ~ P(/j, kl)b(ij, kl). Hence we have immediately =~p(U)b(U) =f.

(32)

A Multi-Locus Continuous-Time Selection Model

289

Defining the fitness of the gamete A~Bj and the mean fitness by m(ij) - b(ij) -- d(ij),

m =- b - d,

we find from (30) with the aid of (31) and (32) that [~(ij) = p(ij)(m(ij) - m) - cD(ij),

(33)

where D(ij) = ~ [P(ij, kl)b(ij, kl) - P(ik, lj)b(ik, lj)],

(34)

kl

are the linkage disequilibria. With the allelic frequencies at loci A and B are given by p(i) = ~ p(ij),

q(j) = ~, p(ij),

we define the allelic fitness for allele A~ at locus A, m(i), by p(i)m(i) = E p(ij)m(ij); (36) J an analogous definition applies to locus B. Evidently, re(i) is just the conditional expected fitness given that an individual carries allele A~. Summing Eq. (33) over j yields the usual equation (8.85 in Nagylaki [10]) for the allelic frequencies p(i) = p(i)(m(i) - m).

(37)

The comments after Eq. (15) of Nagylaki and Crow [7] are applicable in this more general setting. Specifically, the use of Eq. (33) requires characterization of the genotypic frequencies in terms of the gametic frequencies consistent with (29) and the fertilities b(ij, kl) must be calculated from (31). We proceed by considering the case of random mating and additive fertilities, i.e., X(mn, qr; m'n', q'r') = P(mn, qr)P(m'n', q'r'),

(38)

f(mn, qr; rn'n', q'r') = fl(mn, qr) + fl(m' n', q'r').

(39)

The assumption in (39) is the natural analog for continuous time (Nagylaki and Crow [8]) of Bodmer's [1] multiplicative fertilities in a discrete model. As the simplifications introduced in (38) and (39) eliminate those variables requiring eight indices in their designation, we will now employ subscripts in order to facilitate future manipulations requiring the inclusion of the time argument explicitly. For example, PtOJ, kl)-->P~j,kz(t). Employing (38) and (39) in (31), we conclude that

and from (32) by summing over i and j we have b = 2/~ where ~j~cl

Hence we may write f ( m n , qr; m'n', q'r') = br~,qr + bm,n,,a,r, = b + (br,,~.qr - b) + (bm,n,.q,r,

-- b).

(40)

290

M. Moody

As in the single-locus case, (40) makes clear the biological meaning of the special form (39): the fertility of each mating is taken to be the sum of the average birth rate of the population and the deviations from this mean of each member of the mating pair. From (29), with the aid of (38) and (40), we obtain the equation for the genotypic frequencies P~,~, = p~jp~(b~j + bkz - b)

+ c{p,r

- b,j) -

+ c=(DijB~, -

Dk,] + p~,[D,j(g -- bk,) -- D,j]}

D ~ , , ~ j - / 5 / 3 , f l 3 ~ , ) - (d,j,kz + m)Ptj,k,,

(41)

where we have defined D,, = ~ (P~,.k, - P,~,,,) = p,j -

~ P,~.,,,

which are the linkage disequilibria in the absence of selection. As in the simpler case of Nagylaki and Crow [8], equation (41) reveals that nonvanishing linkage disequilibria will in general not permit Hardy-Weinberg proportions to hold in either the presence or absence of genotypic frequency equilibrium (/~.kz = 0). To illustrate this point explicitly, we exhibit Nagylaki's [10] solution for the neutral case (Eqs. (8.101) and (8.102)), /3,r

= (1 - c)-1[(/3,j(0) - /30(0))e -st + (/3~

/3~

= (1 - c)-1[(/3,~(0) - /3~

e -~* + (/3~

p , j ( t ) = p,j(0) - (1 - c)-l[c(/3,j(0) - b~

+ (~~ Q,j.~l(t) = Q,j,k,(O) e -~t + bc z e -st

- c/3,j(0))e-St*l, - c/3~,(0))e-Sa],

(43)

- e -s*)

- c/3,j(0))(1 - e-~t)],

f2

(42)

e~'~,j(r);~k,('r) dr,

(44) (45)

where the/30 are the linkage disequilibria assuming Hardy-Weinberg proportions, i.e., assuming that P~j.k, = P~jpkz, in which case boj = p~j _ p~qj. The Q,j,~ are measures of the departure from Hardy-Weinberg proportions, Q,y.~ = P,J.e, p~p~.

As observed by Nagylaki [10], unless both the exact and Hardy-Weinberg (b~) linkage disequilibria are initially 0, linkage disequilibrium will build up, i.e., /3~(t) ~ 0 for t > 0. From (45) he has noted that even if Q,~,~,(O) = 0, linkage disequilibrium will generate departures from Hardy-Weinberg proportions. From (42) and (43) we observe that both the exact and Hardy-Weinberg disequilibria approach 0 exponentially, and from (45) we infer that Hardy-Weinberg holds at equilibrium. From (44) we deduce the equilibrium gametic frequencies to be p~(o~) = p~(O) -

D~

= pt(O)q~(O).

This result indicates that the mating process has no effect on the equilibrium value of the gametic frequencies.

A Multi-Locus Continuous-Time Selection Model

291

N a g y l a k i [10] has analyzed (29) in the case o f slow selection, i.e., under the assumptions t h a t

f(ij, kl; i7', k T ) = b + O(s),

d(ij, kl) = d + O(s),

where b and d are constants a n d s is the intensity o f selection. H e deduced that after a s h o r t t i m e tl ~ - ( b c ) -1 ln s, / 3 ~ j ( t ) = 0(s), / 3 ~ 0(s) for t /> ti. F u r t h e r m o r e , for t >i h, Q~j,k~(t) = 0(s) a n d P~j,~(t) = 0(s). W i t h the a s s u m p t i o n that the explicit t i m e - d e p e n d e n c e o f the fertilities and mortalities is o f 0(s2), he showed t h a t for t I> t2 ~ 2tl, ~ j ( t ) = 0(s2), /3~ -- 0(s 2) and Q~s,~z(t) = O(s~). W i t h the a d d i t i o n a l c o n d i t i o n that the time derivative o f the fitnesses are o f 0(se), he p r o v e d that the rate o f change o f the m e a n fitness is a p p r o x i m a t e l y equal to the genic (additive genetic) variance. Thus the F u n d a m e n t a l T h e o r e m o f N a t u r a l Selection (Fisher [4]; K i m u r a [6], [7]; N a g y l a k i [9]) is a p p r o x i m a t e l y true for weak selection.

Acknowledgment. I am very grateful to Professor Thomas Nagylaki who suggested this problem and made many helpfu~ comments on the manuscript.

References 1. Bodmer, W. F. : Differential fertility in population genetics models. Genetics 51, 411-424 (1965) 2. Charlesworth, B. : Selection in populations with overlapping generations. I. The use of Malthusian parameters in population genetics. Theor. Pop. Biol. 1, 352-370 (1970) 3. Cornette, J. : Some basic elements of continuous selective models. Theor. Pop. Biol. 8, 301-313 (1975) 4. Fisher, R. A. : The genetical theory of natural selection, Oxford: Clarendon Press 1930 5. Hatdane, J. B. S. : A mathematical theory of natural and artificial selection, IV. Proc. Camb. Phil. Soc. 23, 607-615 (1927) 6. Kimura, M. : On the change of population fitness by natural selection. Heredity 12, 145-167 (1958) 7. Kimura, M. : Attainment of quasilinkage equilibrium when gene frequencies are changing by natural selection. Genetics 52, 875-890 (1965) 8. Nagylaki, T., Crow, J. F. : Continuous selective models. Theor. Pop. Biol. 5, 257-283 (1974) 9. Nagylaki, T. : Selection in one- and two-locus systems. Genetics 83, 583-600 (1976) 10. Nagylaki, T.: Selection in one- and two-locus systems, Berlin, Heidelberg, New York: Springer 1977 11. Norton, H. T. J.: Natural selection and Mendelian variation. Proc. Lond. Mat. Soc. 28, 1-45 (1928) 12. Trucco, E. : Mathematical models for cellular systems. The von Foerster equation. Part I. Bull. Math. Biophys. 27, 285-304 (1965) 13. yon Foerster, H.: Some remarks on changing populations. In: The kinetics of cellular proliferation, F. Stohlman, Jr. (ed), p. 187. New York: Grune and Stratton 1959

Received April 18, 1977," Revised August 22, 1977