THEORETICAL
POPULATION
BIOLOGY
8,
Some Basic Elements
301-313
of Continuous JAMES
Department
(1975)
of Mathematics,
L. Iowa
Received
Selection
Models*
CORNETTE State
April
University,
Ames, Iowa
50010
21, 1975
We present a general framework for the discussion of continuous population genetic models of monoecious diploid populations that incorporate one or more of age structure, mortality selection, mating structure, and fertility of matings. Within this framework, we then develop the specific models recently presented by Charlesworth (1970, Models 1 and 2) and Nagylaki and Crow (1974) and establish conditions under which the Malthusian parameters of these papers are equivalent.
Continuous models in. population genetics have been published recently that are reasonably complex and rich in structure, but in which limitations are imposed, as indicated by the statements: “the occurrence of mating events between members of the two sexes is ignored” (Charlesworth, 1970, Model l), or “does not allow for differences in fecundity,” (Charlesworth, 1970, Model 2), or “neglecting age structure” (Nagylaki and Crow, 1974). Despite the different restrictions imposed, there is a common structure to the models of Charlesworth and the model of Nagylaki and Crow, and the objective of this paper is to delineate a model that will encompass at least these models and provide a framework for determining the significance of the restrictions imposed. We present in Section 1 a general continuous model for a monoecious diploid population that incorporates age structure, differential viabilities, differential fecundities, and specialized mating patterns. We give in Section 2, an interpretation in our general framework of the two models presented by Charlesworth (1970) and in Section 3 we present special conditions under which the more general model of Section 1 becomes that of Nagylaki and Crow (1974). To the extent possible, our notation reflects that of Charlesworth and of Nagylaki and Crow; in particular, we use the notation of ordered genotype and ordered mating pattern as used by Nagylaki and Crow. 1.
A
GENERAL
MODEL
We consider a single locus in a monoecious diploid population with alleles A 1 ,..., A, possible at that locus. For each ordered genotype AiAi , we assume * Journal Paper No. J-8 180 of the Iowa Agriculture Station, Ames, Iowa, Project 1669. Partial support Grant GM 13827.
301 Copyright A\1 rights
Q 1915 by Academic Press, Inc. of reproduction in any form reserved.
and Home by National
Economics Institutes
Experiment of Health,
302
JAMES
L.
CORNETTE
that there is an age density function qii(t, x) with the property that, for any time t and age interval [x, x + b], the number of A& individuals with ages in [x, x + Zr] at time t is
We also assume that there are life tables eij(t, x), such that 8&t, X) is the proportion of A&& individuals born at time t - x that survive to age x (at time t). Often, eij(t, X) is interpreted as a probability, the probability of survival to age x of an individual selected at random among the A,Ai individuals born at time t - X, but in a deterministic model such as we have, it seems clearer and more direct to think of lfj(j(t, X) simply as a proportion. The functions r),(t, x) and Z&t, x) are assume to be nonnegative and to have all necessary continuity and derivative properties and it is assumed that for fixed t, cij(t + X, X) is a monotone decreasing function of x with eii(t, 0) = 1 and lim2+m /Jt + x, x) = 0. With this notation, we next show that &t, 0) is appropriately interpreted as the rate at which newborns of genotype A& are produced at time t. Temporarily, let rii(t) denote this rate and assume that r&t) is continuous. At time t, for any age b, the number of A& individuals in the age interval [0, b] is b &,
4
dx.
Each such individual, of age x in [0, b], must have been born at time t - x and have survived to be of age x at time t. Therefore, b s0
Tij(t, x) dx =
t rij(7) tjj(t, t - T) d7. s t-b
By a mean value theorem for integrals, T’ E [t - b, t] such that
there are numbers
x’ E [0,6]
and
[7&(t, x’)]b = [Tjj(T’) ejj(t, t - T’)]b. We cancel b from both sides and take the limit as b approaches zero and obtain Qj(t, 0) = r,j(t) fij(t, 0) = r,j(t) because ljj(t, 0) = 1. Hence, we have the interpretation of Tjj(t, 0) as the birth rate of A,Aj individuals at time t. Should the birth rate not be continuous, but, for example, piecewise continuous, then we could conclude by a similar analysis that
BASIC
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MODELS
Next, we develop the following general relation between ~~(t, X) and eij(t, x), which is one of the two basic equations that constitute our model:
Consider a time interval [t, t + a] and an age interval [x, x + b]. The number of A,Aj individuals with age in [x, x + b] at time t is zfb
s
~
vii(t,
5)
d5.
The survivors among these individuals at time t + a will be of age [X + a, x + a + b] and the number that survive can be written two ways:
The right-hand side expresses the fact that an individual of age [ + a at time t + a must be among the proportion of those of age .$ at time t that survive to time t + a. Again, by a mean value theorem for integrals, there are numbers t1 and 5, , J + a < f, < x + a + b, x < f2 < x + b, such that
We cancel b from both sides and use a mean value theorem for derivatives of functions of two variables for tij(t + a, [a + a) and we obtain, with t < 7 < t + a, E, < e < 5, + 4
We let b + 0, so that f1 --+ x + a, E, -+ x and we can assume that there is a point (7, m4 E3 PL U
eij(t,
x,
304
JAMES
L.
CORNETTE
Now, with a -+ 0, we obtain
from which (1) follows. If one supposes the life tables t’iii(t, X) are determined by external conditions imposed on the population, so that in (I), &(t, X) can be considered as known functions, then, with appropriate boundary conditions (such as specification of Tii(t, 0) for all t), it is a straightforward matter to solve (1) for vii(t, x). If vij(t, 0) for all t is the specified boundary condition, one obtains as the solution to (1) a standard relation in mathematical demography,
dt, x) = r]i&, 0) &(t, x). The life table lij(t, x) could, of course, involve TM(T, f) for all K, e, 7, 4 (a model of competition, for example) so that (1) would be quite a complex system whose solution would be difficult to compute, or for which even existence of solutions may be in question. We now consider the structure of reproduction. To this stage, any interaction between genotypes has been masked by the hypothesized Zli(t, x). For reproduction, however, we assume mating-density functions Yij,,(t, x, y) such that, for each time interval [t, t + a] and for age intervals [x, x + b], [y, y + c], the number of matings between A,A, individuals of age [x, x + 61 and AJA, individuals of age [y, y + c] that occur during [t, t + a] is
To express random mating, for example, one might have Yiiskl(t, x, y) = R(t) ?lij(t, X) qka(t, y), where R(t) is assumed to be known, perhaps in the form of a constant divided by the total population size N(t), or perhaps a periodic function divided by N(t) to reflect seasonal variation in breeding. For all mating schemes, Yii,az(t, x, y) will depend on the age and genotypic structure of the population and will therefore at least implicitly involve r]&t, z) and vkl(t, y) in its expression. We also assume that each such mating produces aii,,,(t, X, y) viable offspring and by assumption of the ordered genotype notation, all such offspring will be A,A, genotype. Therefore, we may write the rate of births of AaAi individuals as (2) where d is the maximum age at death of any individual.
BASIC
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305
From another viewpoint of the process of counting births, we may write the rate at which the A,Aj genotypes contribute offspring as
all of which have genotype of the form AiA- . Equations (1) and (2) constitute a minimal general model incorporating age structure, mortality selection, mating structure, and differential fecundities. In addition to the four primitive functions Tii(t, x), tfj(t, x), Yij,kl(t, x, y), and ~~~,~~(t,x, JJ), we define certain accessory functions as: nij(t) = j” qij(t, x) dx 0
N(t)
= c %(4 ij
Then, +(t) is the number of A,Aj genotypes of all ages alive at time t, and N(t) is the total number of individuals in the population at time t; note that r](t, 0) is the total birth rate of all genotypes at time t. The rate of change of nu(t) with time is important and may be calculated with the use of Eq. (1) as: ‘ij(t)
= $
J”* Tij(t, X) dx 0
Because no individual lives beyond age d, Tij(t, d) = 0. Therefore, liij(t) = 77ij(t, 0) + jod jrlii(t, x) [“‘&
x”;j(;
~~@’ x)‘axl 1 dx.
(3)
The first term of (3) has been interpreted as the birth rate of A,Aj individuals, therefore, the second term may be seen to be the negative of the total death rate of all AiAj individuals. 653/W3-5
306
JAMES
2.
MODELS
1
AND
L. CORNETTE 2 OF CHARLESWORTH
With the notation and general model as established in Section I, we now turn to an interpretation of the two models of Charlesworth (1970). Because of the specific definition of Malthusian parameters that Charlesworth considers, his models are written to compute the rate at which the individuals of A& genotype and age x at time t contribute offspring (zygotes) to the population, a quantity that we may write as
if we attribute all the offspring to the “first”
partner of an ordered mating, or as
if the offspring are attributed equally, half to each partner of a mating. Charlesworth does not use ordered genotype notation, therefore, the expressions (4) or (4’) for A,A, and for AiA, must be combined if i #j. To avoid this duplication, we consider the analogues to his Models 1 and 2, which assume ordered genotype notations, and we use (4) instead of (4’). Charlesworth’s initial equations of his Models 1 and 2, which are the only equations that we consider, apply to both unordered and ordered genotype notation schemes. Model 1 As Charlesworth observes, “the occurrence of mating events between the two sexes is ignored;” this suggests a randomness, which we account for by assuming that
Then, (4) becomes
BASIC
ELEMENTS
OF CONTINUOUS
307
MODELS
where the last line is precisely the notation of Charlesworth’s Model 1. In his notation, B(t - x) is birth rate at time t - x,fii(t - X) is the frequency at birth of the &A, genotype, and mij(t, X) is “the rate at which a member of the ij genotype aged x at time t (and formed at time t - x by fusion of the ith and jth types of gametes)producesgameteswhich enter into zygotes.” We have then provided at least one interpretation of this somewhatbroad and elusive term as %A4 4 = R(t) ; jod?Ikl(4Y> %m(t, x>Y) dY Model
(6)
2
Model 2 explicitly assumesrandom mating and there is introduced a function x(t, X, y) (Charlesworth usesz in place of y) defined by, “Let the proportion of all zygotes produced by matings between individuals of age x to x + dx and individuals of age .a to .a + dx be z-(t, X, z) dx dz.” In the notation of Section 1,
we write this as(with z = y)
Then, the expressionin the Model 2 statement, “the number of zygotes produced as a result of matings between ij individuals aged x to x + dx and kZ individuals aged z to x + dz is B(t) +,
x, 2) B(t - x)&(t
- 2) &(t, x) B(t - z)f,,(t w, x) N(t, 4
- z) I&
z) dt dx dz ,, ,
may be written as
(with y = a), because B(t) = q(t, 0), Tij(t, X) = B(t - z)fij(t - X) l&t, x), etc. as in Model 1. Despite the assumption of random mating, in the initial expression(7) mating structure is random within agegroups, but random mating between age groups is not yet imposed. Age dependenceof matings can occur in Cmn.rsYmn.& x, Y) amn.& x, y>If we sum the expression(7) for all kZ genotypesand agesy, we obtain ~ntn.&
x, Y> dr
I
hiAt> x)lrl(t,
4) dx dt
(8)
as “the total number of offspring contributed by zj individuals of age x to x + dx.” Expression (8), without dt dx, is comparableto (4), which is the basis
308
JAMES
L.
CORNETTE
for Model 1. It may be seen that (4) and (8) b ecome identical under the assumptions that
and that the common expression for the two is
3. THE MODEL OF NAGYLAKI AND CROW We now consider the model of Nagylaki and Crow in which they “disregard agestructure” and we give conditions under which the model of Eqs. (I) and (2) will specializeto their model. The basic equation for the model of Nagylaki and Crow is their equation
liij = M 2 Xi,,,jZi,,,j 161
- dijNPij
,
(9)
where nii = q*(t) and N = N(t) are such aswe have defined, Pii = nij/N, di, is the probability per unit time of the death of the genotype AiAj , M is the number of matings per unit time (generation), and Xik,lj is the fraction of those matings that is between genotypes AiA, and A,Aj . Except for the lack of age dependence,&Zj is the sameas aik,u(t, x, y) as in Section 1. Also, it may be seen that except for the lack of age dependence, MXik,u is Yik,li(t, x, y) of Section 1. Nagylaki and Crow usethe ordered genotype and mating notation and assumethat initially nij = nji and observe that, if Xik,u = Xlj,ik and aik,lj = ulj,ik , then for all time, nij(t) = nji(t). Perhaps it is too obvious to state, but one also needs equalities such as Xik,Cj = Xki,,j , dij = dji , etc. The question now is: Is there a conceptual population that does have age structure and is modeled by Eqs. (1) and (2) and from which the more simple model of Eq. (9) is obtained ?We begin by assumingan age-structuredpopulation for which, within the mating structure, there is functional independencebetween genotypes and agesand alsowithin the fecundities, there is functional independence between genotypes and ages.That is, we assumethat there are functions W, Y> and Q(x, Y>, wh ich are functions of agesx andy, and Tij,kl(t) and Sij,kl(t), which are functions of time (perhaps constant), such that Yij,kl(t9 x~Y) = p(x, Y> Fiiij,7cl(t)
(10)
aij,!d(t,
(11)
x, Y) = !2Cx7 Y) G,kdt)*
BASIC
ELEMENTS
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CONTINUOUS
MODELS
309
Next, we consider eij(t, x). When age structure is neglected, during a time interval [t, t + At], the proportion of A,A, individuals of age xi that die will be the same as the proportion of A,& individuals of age x2 that die; we let that proportion be &(T) At, t < r < t + At, where &(t) is a continuous function of time appropriately interpreted as the death rate at time t of the &Ii individuals. If d&t) = dij is constant in time, we have the familiar example of exponential decay and the life table is
More generally, we may write
ejj(t, x) - ejjp + At, x + At) = djj(T)At, 2jj(t, x, and with At --f 0, obtain [&tjj(t,
X)/at
+
With the boundary condition ejj(
atjj(t,
t,
X)/aX]/l~j(t*
X)
=
--d,(t).
(12)
0) = 1, (12) has the unique solution
tjj(t, x) = exp I-
S,:, &(T) dt/ .
(13)
With d,(t) continuous for all time, Ljj(t, x) > 0 for all t and x and we must take as the upper bound of the age at death of an individual d = co. Consider now the conditions [( 10, (1 l), and (13)] imposed on the model of Section 1 represented by Eqs. (2) and (3); Eq. (3) was derived from (1). We use the fact that (13) is the solution to (12) and have
cij(t) =Tij(t, 0)+ImTij(t, x){d,j(t)> dx 0
Except for the constant ]F ]r P(x, y) Q(x, y) dx dy, which can be eliminated by appropriate scaling, Eqs. (14) are those of Nagylaki and Crow, Eqs. (9). Therefore, if we accept the conditions that age and genotype be functionally independent in the reproductive structure and allow for infinite maximum age at death, d = co, it appears that the age structure of the population may be neglected. These are fairly obvious requirements and it wouId seem that they
310
JAMES
L.
CORNETTE
would be acceptable requirements for some problems in population genetics. Observe that although infinite maximum age at death, d = co, is required, allowance may be made for a finite maximum age, a, , at which reproduction is possible by selecting appropriate P(x, y) or Q(x, y) (=0 for x or y > a,,). This affects only the constant s: sr P(x, y) Q(x, y) dx dy. Note also that although we assumed above that dij(t) was continuous, one may wish to allow, for example, periodic infinite discontinuities in dij(t) to model the effect of periodic disaster among certain genotypes. The author first thought that it would be necessary to require d,,(t) to be constant in time, in order that the general model of Section 1 specialize to that of Nagylaki and Crow. Professor Nagylaki very kindly corrected this error and provided analysis leading to the more general expression of Eq. (13). He also observed that Eq. (15) below may be derived on the same basis as (12) with &t, x) instead of dij(t), and that with the boundary condition &(t, 0) = 1, (15) has as its solution &ij(t, x) = exp I-- ‘”
4.
t-z
dij(7, T - (t - x)) dT/.
DISCUSSION
We believe the model presented in Section 1 consists of elements fundamental to continuous models in population genetics. The model of Charlesworth provides for age structure and fecundities of matings and the model of Nagylaki and Crow provides for fecundities of matings and specialized mating structure. We now have a model that incorporates all three concepts and that will, therefore, more accurately approximate real biological populations. The model also is more complex mathematically and its solution will require quite simple choices of eij(t, x), Yi&t, X, y) and acj,,r(t, X, y). Of course, the models of Charlesworth and of Nagylaki and Crow already represent certain choices of these functions, but we would wish to explore choices that maintain the three biological characteristics of the model. We have shown that the elements presented can be used to interpret parameters in other presentations, such as Charlesworth’s m,?(t, X) as described by Eq. (6). Charlesworth mentions briefly the consequences relative to his Malthusian parameters of m,(t, X) being constant and suggests that this is very unlikely to occur in nature. Eq. (6) provides a basis for examining conditions under which mij(t, x) might be constant. The differential equations of the model also can be used to provide interpretations of parameters in other models. Again, in Charlesworth’s discussion of his Malthusian parameters, he introduces pii(t, X) as the death rate of A&l, individuals aged x at time t. In Eq. (3) of our model, the second term has been interpreted as the negative of the total death rate of all &A, individuals. An
BASIC
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311
MODELS
inspection of that second term suggests that Charlesworth’s defined by /Q(t, x) rzz -
&t,
x) should be (15)
It may be seen, then, that pij(t, 2) is not a new parameter to his system since he has previously introduced life tables ezj(t, x). Paired with his discussion of m,?(t, X) as a constant is a consideration that pCLij(t,X) be constant; it may be seen from (15) that a sufficient condition for p&t, x) to be constant is that f&t, x) = exp(-difx), dii constant in time. It alsois possibleto usethe structure presentedhere to compareparametersas they appear in separatemodels. The Malthusian parametersof genotypes AiAf and of alleles Ai have been variously defined and when one encounters these parametersin two separatemodels, he may ask whether the two are essentially the same or have an essential difference. The Malthusian parameter is the underlying concept of the papers by Charlesworth and by Nagylaki and Crow and we consider the possiblesimilarities of this parameter asit appearsin those two papers. We consider specifically the genotypic Malthusian parameter olij of the genotype AiAj . The comparison must be made in the context of the special conditions peculiar to either of the two models. Thus, to make the comparisonsuggested,we must assumethe random mating of Charlesworth and must suppressagestructure, at least partly, asin Nagylaki and Crow. Therefore, we impose, from Sections2 and 3, Yij,kl(t2
and
x, Y) = R(t)7idt,
x)rlkl(t, Y),
~G(CX) = exp f - If:, dij(T) dT/.
(5) (13)
Charlesworth definesolij by olij =
Ji B(t - X).fii(f - X) cif(t, x)(Q(t, J; B(t - x)f&
X) - ~ij(t, x)} dx
- x) t&(t, x) dx
All this notation has been previously described.We may use B(t - X)fij(t - x> zij(tr x) = 7ijCt, x)7 Eq. (6) to describe ~(t, and (13), and obtain
x), Eq. (15) to describe pLij(t,x), imposeconditions (5)
312
JAMES
L.
CORNETTE
Nagylaki and Crow do not define aii explicitly, but it is clear from their discussion of the population Malthusian parameter and the Malthusian parameter of an allele that, in their development, aij
=
b,,
-
d..
23
3
where dij is the death rate of the A,A, genotype as used above and bij is “the number of offspring of a single AiAj individual per generation” and is defined by (using only the notation of theirs as given in Section 3)
Now, n,b, is the number of offspring and this was written in Section 1 as
nijbij
=
F
sff
J:
Yij,kdt,
of all A,A, individuals
XP
Y)
%,kdt,
Y)
X,
4
per generation
dx-
Therefore, we may write, using conditions (5) and (13),
Oljj
=
Ckl
St
St
Yii.kl(t,
2,
Y)
%.kdt,
X9
Y> dY
dx
_
ndt)
d,,(t) 23
(16’) =
R(t)
ckl
.I,” C
Tdt,
Xj
Y) ‘?kdt, ndt>
X,
Y> %,kdt,
2,
Y> dY
dx
- dij(t).
Equations (16) and (16’) are identical, and thus, we have found a common ground (random mating (5) and the life table (13)) on which the Malthusian parametersof the two papers are identical. Perhapsthe most severelimitation of our model is the lack of explicit distinction between the sexes. Examples of such structure appear for discrete time modelsin Pollak and Kempthorne (1970, Part II) and, for continuous models,in a section of the paper by Nagylaki and Crow. It is apparent from these examples that sexual distinction will rather severely increasethe complexity of the model.
ACKNOWLEDGMENTS The author gratefully acknowledges the stimulation of Professors Oscar Kempthorne and Edward Polk&, who provided direct suggestions leading to the model in Section 1 and very helpful discussions and review of the work. He is also grateful to Professor Thomas Nagylaki for providing the correction as noted at the end of Section 3.
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REFERENCES CHARLESWORTH, B. 1970. Selection in populations with overlapping generations. I. The use of Malthusian parameters in population genetics, Theor. Pop. Biol. 1, 352-370. NAGYLAKI, T. AND CROW, J. F. 1974. Continuous selective models, Theor. Pop. Bid. 5, 257-283. POLLAK, E. AND KEMPTHORNE, 0. 1971. Malthusian parameters in genetic populations. II. Random mating populations in infinite habitats, Theor. Pop. Biol. 2, 357-390.
POPULATION
BIOLOGY
8,
Some Basic Elements
301-313
of Continuous JAMES
Department
(1975)
of Mathematics,
L. Iowa
Received
Selection
Models*
CORNETTE State
April
University,
Ames, Iowa
50010
21, 1975
We present a general framework for the discussion of continuous population genetic models of monoecious diploid populations that incorporate one or more of age structure, mortality selection, mating structure, and fertility of matings. Within this framework, we then develop the specific models recently presented by Charlesworth (1970, Models 1 and 2) and Nagylaki and Crow (1974) and establish conditions under which the Malthusian parameters of these papers are equivalent.
Continuous models in. population genetics have been published recently that are reasonably complex and rich in structure, but in which limitations are imposed, as indicated by the statements: “the occurrence of mating events between members of the two sexes is ignored” (Charlesworth, 1970, Model l), or “does not allow for differences in fecundity,” (Charlesworth, 1970, Model 2), or “neglecting age structure” (Nagylaki and Crow, 1974). Despite the different restrictions imposed, there is a common structure to the models of Charlesworth and the model of Nagylaki and Crow, and the objective of this paper is to delineate a model that will encompass at least these models and provide a framework for determining the significance of the restrictions imposed. We present in Section 1 a general continuous model for a monoecious diploid population that incorporates age structure, differential viabilities, differential fecundities, and specialized mating patterns. We give in Section 2, an interpretation in our general framework of the two models presented by Charlesworth (1970) and in Section 3 we present special conditions under which the more general model of Section 1 becomes that of Nagylaki and Crow (1974). To the extent possible, our notation reflects that of Charlesworth and of Nagylaki and Crow; in particular, we use the notation of ordered genotype and ordered mating pattern as used by Nagylaki and Crow. 1.
A
GENERAL
MODEL
We consider a single locus in a monoecious diploid population with alleles A 1 ,..., A, possible at that locus. For each ordered genotype AiAi , we assume * Journal Paper No. J-8 180 of the Iowa Agriculture Station, Ames, Iowa, Project 1669. Partial support Grant GM 13827.
301 Copyright A\1 rights
Q 1915 by Academic Press, Inc. of reproduction in any form reserved.
and Home by National
Economics Institutes
Experiment of Health,
302
JAMES
L.
CORNETTE
that there is an age density function qii(t, x) with the property that, for any time t and age interval [x, x + b], the number of A& individuals with ages in [x, x + Zr] at time t is
We also assume that there are life tables eij(t, x), such that 8&t, X) is the proportion of A&& individuals born at time t - x that survive to age x (at time t). Often, eij(t, X) is interpreted as a probability, the probability of survival to age x of an individual selected at random among the A,Ai individuals born at time t - X, but in a deterministic model such as we have, it seems clearer and more direct to think of lfj(j(t, X) simply as a proportion. The functions r),(t, x) and Z&t, x) are assume to be nonnegative and to have all necessary continuity and derivative properties and it is assumed that for fixed t, cij(t + X, X) is a monotone decreasing function of x with eii(t, 0) = 1 and lim2+m /Jt + x, x) = 0. With this notation, we next show that &t, 0) is appropriately interpreted as the rate at which newborns of genotype A& are produced at time t. Temporarily, let rii(t) denote this rate and assume that r&t) is continuous. At time t, for any age b, the number of A& individuals in the age interval [0, b] is b &,
4
dx.
Each such individual, of age x in [0, b], must have been born at time t - x and have survived to be of age x at time t. Therefore, b s0
Tij(t, x) dx =
t rij(7) tjj(t, t - T) d7. s t-b
By a mean value theorem for integrals, T’ E [t - b, t] such that
there are numbers
x’ E [0,6]
and
[7&(t, x’)]b = [Tjj(T’) ejj(t, t - T’)]b. We cancel b from both sides and take the limit as b approaches zero and obtain Qj(t, 0) = r,j(t) fij(t, 0) = r,j(t) because ljj(t, 0) = 1. Hence, we have the interpretation of Tjj(t, 0) as the birth rate of A,Aj individuals at time t. Should the birth rate not be continuous, but, for example, piecewise continuous, then we could conclude by a similar analysis that
BASIC
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303
MODELS
Next, we develop the following general relation between ~~(t, X) and eij(t, x), which is one of the two basic equations that constitute our model:
Consider a time interval [t, t + a] and an age interval [x, x + b]. The number of A,Aj individuals with age in [x, x + b] at time t is zfb
s
~
vii(t,
5)
d5.
The survivors among these individuals at time t + a will be of age [X + a, x + a + b] and the number that survive can be written two ways:
The right-hand side expresses the fact that an individual of age [ + a at time t + a must be among the proportion of those of age .$ at time t that survive to time t + a. Again, by a mean value theorem for integrals, there are numbers t1 and 5, , J + a < f, < x + a + b, x < f2 < x + b, such that
We cancel b from both sides and use a mean value theorem for derivatives of functions of two variables for tij(t + a, [a + a) and we obtain, with t < 7 < t + a, E, < e < 5, + 4
We let b + 0, so that f1 --+ x + a, E, -+ x and we can assume that there is a point (7, m4 E3 PL U
eij(t,
x,
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Now, with a -+ 0, we obtain
from which (1) follows. If one supposes the life tables t’iii(t, X) are determined by external conditions imposed on the population, so that in (I), &(t, X) can be considered as known functions, then, with appropriate boundary conditions (such as specification of Tii(t, 0) for all t), it is a straightforward matter to solve (1) for vii(t, x). If vij(t, 0) for all t is the specified boundary condition, one obtains as the solution to (1) a standard relation in mathematical demography,
dt, x) = r]i&, 0) &(t, x). The life table lij(t, x) could, of course, involve TM(T, f) for all K, e, 7, 4 (a model of competition, for example) so that (1) would be quite a complex system whose solution would be difficult to compute, or for which even existence of solutions may be in question. We now consider the structure of reproduction. To this stage, any interaction between genotypes has been masked by the hypothesized Zli(t, x). For reproduction, however, we assume mating-density functions Yij,,(t, x, y) such that, for each time interval [t, t + a] and for age intervals [x, x + b], [y, y + c], the number of matings between A,A, individuals of age [x, x + 61 and AJA, individuals of age [y, y + c] that occur during [t, t + a] is
To express random mating, for example, one might have Yiiskl(t, x, y) = R(t) ?lij(t, X) qka(t, y), where R(t) is assumed to be known, perhaps in the form of a constant divided by the total population size N(t), or perhaps a periodic function divided by N(t) to reflect seasonal variation in breeding. For all mating schemes, Yii,az(t, x, y) will depend on the age and genotypic structure of the population and will therefore at least implicitly involve r]&t, z) and vkl(t, y) in its expression. We also assume that each such mating produces aii,,,(t, X, y) viable offspring and by assumption of the ordered genotype notation, all such offspring will be A,A, genotype. Therefore, we may write the rate of births of AaAi individuals as (2) where d is the maximum age at death of any individual.
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From another viewpoint of the process of counting births, we may write the rate at which the A,Aj genotypes contribute offspring as
all of which have genotype of the form AiA- . Equations (1) and (2) constitute a minimal general model incorporating age structure, mortality selection, mating structure, and differential fecundities. In addition to the four primitive functions Tii(t, x), tfj(t, x), Yij,kl(t, x, y), and ~~~,~~(t,x, JJ), we define certain accessory functions as: nij(t) = j” qij(t, x) dx 0
N(t)
= c %(4 ij
Then, +(t) is the number of A,Aj genotypes of all ages alive at time t, and N(t) is the total number of individuals in the population at time t; note that r](t, 0) is the total birth rate of all genotypes at time t. The rate of change of nu(t) with time is important and may be calculated with the use of Eq. (1) as: ‘ij(t)
= $
J”* Tij(t, X) dx 0
Because no individual lives beyond age d, Tij(t, d) = 0. Therefore, liij(t) = 77ij(t, 0) + jod jrlii(t, x) [“‘&
x”;j(;
~~@’ x)‘axl 1 dx.
(3)
The first term of (3) has been interpreted as the birth rate of A,Aj individuals, therefore, the second term may be seen to be the negative of the total death rate of all AiAj individuals. 653/W3-5
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2.
MODELS
1
AND
L. CORNETTE 2 OF CHARLESWORTH
With the notation and general model as established in Section I, we now turn to an interpretation of the two models of Charlesworth (1970). Because of the specific definition of Malthusian parameters that Charlesworth considers, his models are written to compute the rate at which the individuals of A& genotype and age x at time t contribute offspring (zygotes) to the population, a quantity that we may write as
if we attribute all the offspring to the “first”
partner of an ordered mating, or as
if the offspring are attributed equally, half to each partner of a mating. Charlesworth does not use ordered genotype notation, therefore, the expressions (4) or (4’) for A,A, and for AiA, must be combined if i #j. To avoid this duplication, we consider the analogues to his Models 1 and 2, which assume ordered genotype notations, and we use (4) instead of (4’). Charlesworth’s initial equations of his Models 1 and 2, which are the only equations that we consider, apply to both unordered and ordered genotype notation schemes. Model 1 As Charlesworth observes, “the occurrence of mating events between the two sexes is ignored;” this suggests a randomness, which we account for by assuming that
Then, (4) becomes
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where the last line is precisely the notation of Charlesworth’s Model 1. In his notation, B(t - x) is birth rate at time t - x,fii(t - X) is the frequency at birth of the &A, genotype, and mij(t, X) is “the rate at which a member of the ij genotype aged x at time t (and formed at time t - x by fusion of the ith and jth types of gametes)producesgameteswhich enter into zygotes.” We have then provided at least one interpretation of this somewhatbroad and elusive term as %A4 4 = R(t) ; jod?Ikl(4Y> %m(t, x>Y) dY Model
(6)
2
Model 2 explicitly assumesrandom mating and there is introduced a function x(t, X, y) (Charlesworth usesz in place of y) defined by, “Let the proportion of all zygotes produced by matings between individuals of age x to x + dx and individuals of age .a to .a + dx be z-(t, X, z) dx dz.” In the notation of Section 1,
we write this as(with z = y)
Then, the expressionin the Model 2 statement, “the number of zygotes produced as a result of matings between ij individuals aged x to x + dx and kZ individuals aged z to x + dz is B(t) +,
x, 2) B(t - x)&(t
- 2) &(t, x) B(t - z)f,,(t w, x) N(t, 4
- z) I&
z) dt dx dz ,, ,
may be written as
(with y = a), because B(t) = q(t, 0), Tij(t, X) = B(t - z)fij(t - X) l&t, x), etc. as in Model 1. Despite the assumption of random mating, in the initial expression(7) mating structure is random within agegroups, but random mating between age groups is not yet imposed. Age dependenceof matings can occur in Cmn.rsYmn.& x, Y) amn.& x, y>If we sum the expression(7) for all kZ genotypesand agesy, we obtain ~ntn.&
x, Y> dr
I
hiAt> x)lrl(t,
4) dx dt
(8)
as “the total number of offspring contributed by zj individuals of age x to x + dx.” Expression (8), without dt dx, is comparableto (4), which is the basis
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for Model 1. It may be seen that (4) and (8) b ecome identical under the assumptions that
and that the common expression for the two is
3. THE MODEL OF NAGYLAKI AND CROW We now consider the model of Nagylaki and Crow in which they “disregard agestructure” and we give conditions under which the model of Eqs. (I) and (2) will specializeto their model. The basic equation for the model of Nagylaki and Crow is their equation
liij = M 2 Xi,,,jZi,,,j 161
- dijNPij
,
(9)
where nii = q*(t) and N = N(t) are such aswe have defined, Pii = nij/N, di, is the probability per unit time of the death of the genotype AiAj , M is the number of matings per unit time (generation), and Xik,lj is the fraction of those matings that is between genotypes AiA, and A,Aj . Except for the lack of age dependence,&Zj is the sameas aik,u(t, x, y) as in Section 1. Also, it may be seen that except for the lack of age dependence, MXik,u is Yik,li(t, x, y) of Section 1. Nagylaki and Crow usethe ordered genotype and mating notation and assumethat initially nij = nji and observe that, if Xik,u = Xlj,ik and aik,lj = ulj,ik , then for all time, nij(t) = nji(t). Perhaps it is too obvious to state, but one also needs equalities such as Xik,Cj = Xki,,j , dij = dji , etc. The question now is: Is there a conceptual population that does have age structure and is modeled by Eqs. (1) and (2) and from which the more simple model of Eq. (9) is obtained ?We begin by assumingan age-structuredpopulation for which, within the mating structure, there is functional independencebetween genotypes and agesand alsowithin the fecundities, there is functional independence between genotypes and ages.That is, we assumethat there are functions W, Y> and Q(x, Y>, wh ich are functions of agesx andy, and Tij,kl(t) and Sij,kl(t), which are functions of time (perhaps constant), such that Yij,kl(t9 x~Y) = p(x, Y> Fiiij,7cl(t)
(10)
aij,!d(t,
(11)
x, Y) = !2Cx7 Y) G,kdt)*
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309
Next, we consider eij(t, x). When age structure is neglected, during a time interval [t, t + At], the proportion of A,A, individuals of age xi that die will be the same as the proportion of A,& individuals of age x2 that die; we let that proportion be &(T) At, t < r < t + At, where &(t) is a continuous function of time appropriately interpreted as the death rate at time t of the &Ii individuals. If d&t) = dij is constant in time, we have the familiar example of exponential decay and the life table is
More generally, we may write
ejj(t, x) - ejjp + At, x + At) = djj(T)At, 2jj(t, x, and with At --f 0, obtain [&tjj(t,
X)/at
+
With the boundary condition ejj(
atjj(t,
t,
X)/aX]/l~j(t*
X)
=
--d,(t).
(12)
0) = 1, (12) has the unique solution
tjj(t, x) = exp I-
S,:, &(T) dt/ .
(13)
With d,(t) continuous for all time, Ljj(t, x) > 0 for all t and x and we must take as the upper bound of the age at death of an individual d = co. Consider now the conditions [( 10, (1 l), and (13)] imposed on the model of Section 1 represented by Eqs. (2) and (3); Eq. (3) was derived from (1). We use the fact that (13) is the solution to (12) and have
cij(t) =Tij(t, 0)+ImTij(t, x){d,j(t)> dx 0
Except for the constant ]F ]r P(x, y) Q(x, y) dx dy, which can be eliminated by appropriate scaling, Eqs. (14) are those of Nagylaki and Crow, Eqs. (9). Therefore, if we accept the conditions that age and genotype be functionally independent in the reproductive structure and allow for infinite maximum age at death, d = co, it appears that the age structure of the population may be neglected. These are fairly obvious requirements and it wouId seem that they
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would be acceptable requirements for some problems in population genetics. Observe that although infinite maximum age at death, d = co, is required, allowance may be made for a finite maximum age, a, , at which reproduction is possible by selecting appropriate P(x, y) or Q(x, y) (=0 for x or y > a,,). This affects only the constant s: sr P(x, y) Q(x, y) dx dy. Note also that although we assumed above that dij(t) was continuous, one may wish to allow, for example, periodic infinite discontinuities in dij(t) to model the effect of periodic disaster among certain genotypes. The author first thought that it would be necessary to require d,,(t) to be constant in time, in order that the general model of Section 1 specialize to that of Nagylaki and Crow. Professor Nagylaki very kindly corrected this error and provided analysis leading to the more general expression of Eq. (13). He also observed that Eq. (15) below may be derived on the same basis as (12) with &t, x) instead of dij(t), and that with the boundary condition &(t, 0) = 1, (15) has as its solution &ij(t, x) = exp I-- ‘”
4.
t-z
dij(7, T - (t - x)) dT/.
DISCUSSION
We believe the model presented in Section 1 consists of elements fundamental to continuous models in population genetics. The model of Charlesworth provides for age structure and fecundities of matings and the model of Nagylaki and Crow provides for fecundities of matings and specialized mating structure. We now have a model that incorporates all three concepts and that will, therefore, more accurately approximate real biological populations. The model also is more complex mathematically and its solution will require quite simple choices of eij(t, x), Yi&t, X, y) and acj,,r(t, X, y). Of course, the models of Charlesworth and of Nagylaki and Crow already represent certain choices of these functions, but we would wish to explore choices that maintain the three biological characteristics of the model. We have shown that the elements presented can be used to interpret parameters in other presentations, such as Charlesworth’s m,?(t, X) as described by Eq. (6). Charlesworth mentions briefly the consequences relative to his Malthusian parameters of m,(t, X) being constant and suggests that this is very unlikely to occur in nature. Eq. (6) provides a basis for examining conditions under which mij(t, x) might be constant. The differential equations of the model also can be used to provide interpretations of parameters in other models. Again, in Charlesworth’s discussion of his Malthusian parameters, he introduces pii(t, X) as the death rate of A&l, individuals aged x at time t. In Eq. (3) of our model, the second term has been interpreted as the negative of the total death rate of all &A, individuals. An
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MODELS
inspection of that second term suggests that Charlesworth’s defined by /Q(t, x) rzz -
&t,
x) should be (15)
It may be seen, then, that pij(t, 2) is not a new parameter to his system since he has previously introduced life tables ezj(t, x). Paired with his discussion of m,?(t, X) as a constant is a consideration that pCLij(t,X) be constant; it may be seen from (15) that a sufficient condition for p&t, x) to be constant is that f&t, x) = exp(-difx), dii constant in time. It alsois possibleto usethe structure presentedhere to compareparametersas they appear in separatemodels. The Malthusian parametersof genotypes AiAf and of alleles Ai have been variously defined and when one encounters these parametersin two separatemodels, he may ask whether the two are essentially the same or have an essential difference. The Malthusian parameter is the underlying concept of the papers by Charlesworth and by Nagylaki and Crow and we consider the possiblesimilarities of this parameter asit appearsin those two papers. We consider specifically the genotypic Malthusian parameter olij of the genotype AiAj . The comparison must be made in the context of the special conditions peculiar to either of the two models. Thus, to make the comparisonsuggested,we must assumethe random mating of Charlesworth and must suppressagestructure, at least partly, asin Nagylaki and Crow. Therefore, we impose, from Sections2 and 3, Yij,kl(t2
and
x, Y) = R(t)7idt,
x)rlkl(t, Y),
~G(CX) = exp f - If:, dij(T) dT/.
(5) (13)
Charlesworth definesolij by olij =
Ji B(t - X).fii(f - X) cif(t, x)(Q(t, J; B(t - x)f&
X) - ~ij(t, x)} dx
- x) t&(t, x) dx
All this notation has been previously described.We may use B(t - X)fij(t - x> zij(tr x) = 7ijCt, x)7 Eq. (6) to describe ~(t, and (13), and obtain
x), Eq. (15) to describe pLij(t,x), imposeconditions (5)
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Nagylaki and Crow do not define aii explicitly, but it is clear from their discussion of the population Malthusian parameter and the Malthusian parameter of an allele that, in their development, aij
=
b,,
-
d..
23
3
where dij is the death rate of the A,A, genotype as used above and bij is “the number of offspring of a single AiAj individual per generation” and is defined by (using only the notation of theirs as given in Section 3)
Now, n,b, is the number of offspring and this was written in Section 1 as
nijbij
=
F
sff
J:
Yij,kdt,
of all A,A, individuals
XP
Y)
%,kdt,
Y)
X,
4
per generation
dx-
Therefore, we may write, using conditions (5) and (13),
Oljj
=
Ckl
St
St
Yii.kl(t,
2,
Y)
%.kdt,
X9
Y> dY
dx
_
ndt)
d,,(t) 23
(16’) =
R(t)
ckl
.I,” C
Tdt,
Xj
Y) ‘?kdt, ndt>
X,
Y> %,kdt,
2,
Y> dY
dx
- dij(t).
Equations (16) and (16’) are identical, and thus, we have found a common ground (random mating (5) and the life table (13)) on which the Malthusian parametersof the two papers are identical. Perhapsthe most severelimitation of our model is the lack of explicit distinction between the sexes. Examples of such structure appear for discrete time modelsin Pollak and Kempthorne (1970, Part II) and, for continuous models,in a section of the paper by Nagylaki and Crow. It is apparent from these examples that sexual distinction will rather severely increasethe complexity of the model.
ACKNOWLEDGMENTS The author gratefully acknowledges the stimulation of Professors Oscar Kempthorne and Edward Polk&, who provided direct suggestions leading to the model in Section 1 and very helpful discussions and review of the work. He is also grateful to Professor Thomas Nagylaki for providing the correction as noted at the end of Section 3.
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REFERENCES CHARLESWORTH, B. 1970. Selection in populations with overlapping generations. I. The use of Malthusian parameters in population genetics, Theor. Pop. Biol. 1, 352-370. NAGYLAKI, T. AND CROW, J. F. 1974. Continuous selective models, Theor. Pop. Bid. 5, 257-283. POLLAK, E. AND KEMPTHORNE, 0. 1971. Malthusian parameters in genetic populations. II. Random mating populations in infinite habitats, Theor. Pop. Biol. 2, 357-390.
