,Journal of J. Math. Biol. (1988) 26:73-92
Mathematical
Biology
9 Springer-Verlag 1988
Separable models for age-structured population genetics Marcel Ovidiu Vlad Bucure~ti, Cfisuta po~talfi 77-49, R.S. Romania
Abstract. This paper is concerned with the applications of nonlinear age-
dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not. Key words: Age-dependence- Separable m o d e l s - Population genetics
1. Introduction
Most deterministic models dealing with the genetics of age-structured populations (Pollak and Kempthorne 1970, 197l; Charlesworth 1980; Pollak and Callanan 1982; Pollak 1984) are based on Leslie approach (1945). One assumes that the time is discrete and that the population may be divided into discrete age classes. Due to the mathematical difficulties the models with continuous time and age structure are less used (or dealt with) (Fisher 1930, Charlesworth 1970, 1980; Demetrius 1983; Demetrius and Ziehe 1984). Recently a new type of continuous age-dependent models were introduced, for which the underlying evolution equations areseparable, affording a detailed analytical treatment. The separability is mathematically nice but there is no reason why it is biologically interesting. However, these models were used by Gurtin and MacCamy (1982), Busenberg and Iannelli (1983a, b; 1985) and by Vlad and Popa (1985) in population dynamics and by Hadeler and Dietz (1984) in the theory of epidemics. The purpose of the present paper is to extend this approach to certain problems of population genetics. 2. Basic notions
We consider the case of a single autosomal locus with m alleles A ] , . . . , Am. We shall use the following state variables:
74
M . O . Vlad
The number of individuals with genotype (. 9 .) having the age between and ,z+ d~((. 9 -)= i, i = 1 , . . . , m for haploid populations and (. 9 . ) = / j , i,j = 1 , . . . , rn for diploid populations): dN(...)(~, t) = n(...)(~z, t) dc~;
(1)
The number of individuals with genotype (...)
N(...)( t) =
n(...)(~z, t) d~;
(2)
The total number of individuals having the age between ~ and ~ + d~:
d N ( ~ , t) = n(a, t) dc~ = ~ n(...)(~, t) c/~;
(3)
(...)
The total number of individuals:
N ( t ) = Y~ N(...)(t)=
n(~, t) d~;
(...)
(4)
The age-genotype density function:
c(...)(a, t) = n(...)(~z, t ) / N ( t ) ,
~
c(...)(~, t) d~ = 1;
(5)
~, c(...)(~,t),
c(a~,t) d ~ = l ;
(6)
(...)
The age density function
c(~,t)=n(~,t)/N(t)=
(..,)
The frequency of the genotype (. 9 .) for the individuals with age 4 :
f (...)(a, t)~- n(...)(a, t)/ n(~z, t)= c(...)(~, t)/ c(~z, t),
(7)
Y~ f(...)(a, t) = 1;
(8)
(...)
The frequency of the genotype (. 9 .) within the whole population
F(...)(t) = g(...)(t)/N(t) =
c(~, t)f(...)(~, t) d~,
~, F(...)(t) = 1;
(...)
(9),(10) The frequency of the gene A~ within the individuals with age ~z:
qi =qi(a, t),
(11)
qg(~, t ) = f ( ~ , t),
(12)
we have
for haploids and
qi(g, t ) : f i ( ~ ,
t) +1 Z fj(,z, t), j~i
(13)
Separable models for population genetics
75
for diploids; The frequency of the gene Ai within the population Qi = Q,(t) =
c(~, t)qi(~, t) d~;
(14)
we have
Qi(t) = F~(t),
(15)
Qi(t) = G i ( t ) + 1 ~ F,~(t), j#i
(16)
for haploids and
for diploids. Our aim is to describe the time evolution of these state variables in terms of the following parameters: The natality and mortality functions corresponding to different genotypes: A(...) = A(...)(~,...),
(17)
/~(...) = ~(...)(~, 9 9 .);
(18)
The mutation rates
fl(Ai--)A~) =/30(~,.- .),
i,j= 1 , . . . , m,
i#j.
(19)
We shall make the following assumptions concerning the functions A(...), ~(...), and fl/j: The natality and mortality functions A(...), and ~(...) can be factorized as A(...) =
(20)
/z(...) =/~...)(~) + 6/z,
(21)
where A~...) and/z~...) depend only on age and on genotype and 6/z may depend on time and on time evolution of population densities n(...)(~, t) but is independent of age and of genotype. A possible choice for 6/z would be the following •/.Z = ~. (91
+ Z
~7(...)1(Xl, tl,
to Y.
(''')1 (''')2
If I;o
t)n(...h(Xl, tl) dx 1 dt,
~(...h(...)2(xl, x2, tl, t2, t)
fO
X n(...)l(Xl, ti)n(...)2(x2, t2) dx 1 dx2
(22)
dti dt2
(Vlad and Popa 1985). The mutation rates fl~j depend only on the subscripts i and j and on age ~: 0
fl/j = fl~(~),
1 ~>fl~
~>0,
i,j= 1 , . . . , m,
i#j.
(23)
The Eqs. (20)-(23) define a broad class of population models for which the underlying evolution equations are separable, in the sense of Busenberg and Iannelli (1985).
76
M.O. Vlad
3. Haploid populations
For haploid populations the constitutive equations (20)-(23) lead to the following evolution equations:
(Ot + cg~)n~(a,, t) + (/x~ n~(O, t)=~j
+ 8/x) n,(z~, t) = O,
vj,(e)A~
(24)
t) d~,
(25)
with the initial conditions n,(~, to) = n~
(26)
In (25) the functions ~,jz(e) are given by
z,j,(e) = fljo(~),
j # i,
1 I> t,ji(~) i> O,
(27)
1 >i ~,z(e) ~> O.
Vjs(e) = 1 - 5", fl~
i#j
(28)
Obviously uj,(~) = 1.
(29)
i
Using as state variables the total population size N ( t ) and the age-genotype density function c~(e, t) the Eqs. (24)-(26) can be rewritten as:
( Ot + O.)ci(e, t) + Ci(~, t)tz~
)
foo (h~(e)-/x~
+ c~(e, t ) ~
e~(0, t)=2 J
t) d~=O,
(30)
fo~ ~,Ae)~Y(~)cAe,t) d~.
(31)
and
dtN(t) = N ( t ) ~
(A~
t) d ~ - N ( t )
#~
81~.
(32)
with the initial conditions
c,(e, to) = c~
= n~
n~
de,
(33)
~ N(to) = N O= We see that the evolution of the total population size reduced to a system of linear suggested by Vlad and Popa
n~(a0 de.
(34)
equations for ci (30)-(31) and (33) are independent N(t). Although nonlinear, these equations can be integral equations. Indeed, making use of a method (1985) the solutions ci(a, t) can be expressed as:
ci(e, t)
h( t - to- ,z)Zi( t - e)p,(e) + h(e - t - to)C~ - t+ to)
fO tOZi(t 9
- ~ ) p i ( e ) d~ +~i
I~ t-,o
c~
- t+ to)
p~(e) Pi(a, - t + to)
pi(e) d~ pi(a, - t+ to)
'
(35)
Separable models for population genetics
77
where
), and Z~(t) are the solutions of the linear integral equations:
Z~(t) = ? IO ' ~to g~(~z')Zj(t-~') d ~ ' + ~ ( t ) ,
(37)
with g~(
,)=
o , ~:~(~, )a:(~ )p:(~ , ),
(38)
fco gq(~ t )cj(~ 0 r - t + t o ) d~' J/,(t) = ~ ,-to pj( d - t + to)
(39)
and h(t) is the usual Heaviside function. The detailed computation is tedious and is left to the reader (see also Vlad and Popa 1985, Appendix 1). Combining (6), (7), (10), (12) and (14)-(15) we can also express the functions qi(~, t ) = f ( ~ , t) and Qi(t)= F~(t) in terms of Zi(t). We get: qi(z~, t)=f(z~, t) P,(~) p~(~ - t+ to) pi(z~) t-I- to) Pi (~ - - t + to) (40)
h ( t - to- ~)Z~( t - g)pi(~) + h (z~ - t+ to)C~ - t+ to) h( t - t o - ~ ) Y~ Zi( t - ~ )p~(z~) + h(~z - t-k- to) Z i
c O ( z~ - -
i
Qi(t) = F~(t) =
Io -'~
:
2 "
fo"
Z i ( t - ~ ) p i ( ~ ) d~+
ci(~-t+
I t--to ~ ~
z,(t-~)pi(~) d~+E "
I
to)
p,(zQ P i ( ~ - t+ to) pi(a,)
c~ - t+ to)
t -
to
Pi ( ~
d~
(41)
d~
-- t + to)
The Eqs. (37) are amenable to solution through Laplace transform. Introducing the notations Z(t) = IIz,(t)ll,
Z*(y) = Z(to+y),
(42)
~r162 =,//(to+y),
(43) (44)
g ( ~ ) = [Ig,j(~)II,
~ ( t ) = II~t,(t)[[, the Eqs. (37) lead to:
5fZ*(y)=f:e-~YZ*(y)dy=(I-f:e-S~g(c~)d~)-lffCl(s),
(45)
with Jr(s) =
fo i;
o
, ,,) pc ~: (( ~a'/--yy) ) e -sy g~A~
d,~'
.
(46)
78
M.O. Vlad
It turns out that the behavior of the secular equation:
Z~(t) and c~(~, t) depends on the solutions of
detl6o.-I;e-~go.(z~)d~l=O.
(47)
Assuming the validity of the following conditions:
gij(z~) > O,
(a) (b)
(C)
i,j = 1,..., m,
g , ( a ) d ~ > 1.
;o ogij(~) e-qad~
(48)
for some i.
for some q > 0 ;
(49)
i , j = l , . . . , m,
(50)
Bellman and Cooke (1963) proved that there is a positive vector X and a positive number So such that
[f;e-'o~
d~]X=X
(51)
and So is the root of (47) with largest real part. So is a simple root. Making use of these results it follows that, if the conditions (48)-(50) are fulfilled, then the age-genotype density function and the genotype frequencies f(z~, t) and Fg(t) evolve towards time persistent forms. From (35), (40), (41) and (45) we come to:
Zi(t) -~ Y~' e'o('-'o) for t >>to,
(52)
c,(a, t)~cT'(~)
as t~oo,
(53)
f(~, t)~fT'(~) F,(t)-V;'
as t ~ ,
(54)
as t~oo,
(54')
where Y~' = lim (s - So)
I-
g(e) e s~ d~z
s - > So
a~C(s
,
(55)
i
c~'(~) = b~'p,(a) e -so~,
(56)
f~'(~) = b~ p~(~ )/z b~s,Pi(~).
(57)
F ~ ' - b" --
i
(58)
e-'o~pi(~) d~. dO
and b~'= c~'(0) = Y~'
Y~'
Pi(~) e "o~ d~,
i = 1. . . . , rn,
(59)
are the per capita birth rates of the different genotypes i, i = 1 , . . . , m. We observe that the determination of c~'(~),f~'(,z) and FT' reduces to the finding out of So and b~', i -- 1 , . . . , m. The determination of b~t from (55) and (59) is cumbersome.
Separable models for population genetics
79
It is more convenient to derive a system of linear algebraic equations for bT'. To do this we substitute (52), (55) and (59) into (37) and make to-->-co. We get:
(;o
~t ~it --
)
gi/(z~)e -'o~ dz~ b;'t =0,
i = 1 , . . . , m.
(60)
As so is the solution of (47), the linear equations (60) are not linearly independent. The normalization condition ~t ~o cSt(e) d~z = 1 provides the additional relationship necessary for the determination of b~':
b~'
e-'o~p~(e) d~ = 1.
(61)
If the functions ci(~, t), f ( ~ , t) and F~(t) reach their "stable" forms c~'(,z), fT'(~) and FT', they remain time persistent. Indeed, we can prove that:
c,(~, tic ~ c}'(~),j = 1,.. 9 m ) = cTt(e),
(62)
f ( e , t[c~ c~'(z~),j = 1 , . . . , m) = f s ' ( e ) ,
(63)
~(t[c ~ = c}'(e),j = 1 , . . . , m) = F s'.
(64)
(see Appendix 1). Our results concerning the behavior of age-genotype density functions and of genic and genotypic frequencies are consistent with the linear theory of Pollak and Kempthorne (1970, 1971). However, the time evolution of the total population size may be different. Indeed, Pollak approach leads, in the limit t-> co, to an exponential law of growth. Within the framework of our theory, as ~/x may depend on N (see (22)), the differential equation (32) may be nonlinear, and thus the occurrence of time persistent age density functions does not necessarily lead to an exponential law of growth.
4. Diploid populations: selfing Within this section we consider the case of self-fertilization. For simplicity, we assume that the selection and mutation are absent: of i,j,
(65)
+ 6/z = independent of i,j,
(66)
A~ = A~ /z~ =/z~
~~
= O,
i,j = 1 , . . . , m,
i ~j.
(67)
The evolution equations take the following form:
(Ot-[-Cg,~)l'lij(zT~,t ) + ( I z ~ nij(O, t)=71 n,,(O, t ) =
;o
;o
a o ( e ) ,n , , ( e , t ),
t)=O,
Ao( ,)nij(d,t)dd, d~ ' +z~ E j~-i
;o~
h~
i,j= 1,..., m,
i,j=l, .. ., m, i r ', t) d e ,
(68) (69)
i = 1 , . . . , m, (70)
80
M.O. Vlad
with the initial conditions:
n~(a,, to)= n ioj ( e ) .
(71)
(68)-(71) allow us to derive a closed system of equations for the total age density. Summing in (68)-(71) over ij and making use of (3) we get:
(O,+O~)n(a,, t)+ (/z~ n(O, t ) =
6/z)n(~, t ) = 0 ,
Ioo A~
(72)
t) de,
(73)
with the initial condition
n(e, to) = n~
= E n.(a,)+~}~ o 1 E no-(a,). o i
(74)
i jr
The equations (72)-(74) were studied by Vlad and Popa (1985). The age profile can be expressed as: P(a) t+ tO)p(a,
h( t - to- ~ ) Z ( t - a,)p(e) + h ( a - t+ to)C~
t + to) ,
Z ( t -a,)p(a,) da,+
t-,o
c~
- t+ to)
P(~) da, p(a, - t+ to)
(75) with (76) ~
c~
-- c(~, to),
(77)
and Z ( t ) obeys a linear integral equation similar with (37):
Z ( t ) = IO t6 g ( a , ' ) Z ( t - d ) d ~ ' + ~ ( t ) ,
(78)
g ( J ) = )t (a,,)p(d),
(79)
. . . - . ~- to) da,' ~((t) = [~ t ~g ( J ) . c o.(a, J,-to p( a, - t+ to)
(80)
where
and the asymptotic behavior of Z ( t ) and c(e, t) is described by:
Z ( t ) ~ - Y S t e "~176 , yst= =o
t >>to,
g(x------~) c ~ e -rr~ dx dy =y p ( x - y) c(a,, t ) ~ cSt(~) as t-->~,
(81)
zg(z) e -z'~ dz,
(81')
~=o (82)
with cS'(e) = p ( ~ ) e -r~
p(~) e -r~ de,
(83)
Separable models for population genetics
81
and r0 is the unique real root of the Lotka equation ong(a) e -sada` = 1.
(84)
(Lotka 1939). It is convenient to use as state variables the genotypic frequencies fj(a`, t). Combining (6). (7) and (68)-(75) we get a system of integrodifferential equations for these functions: (Or + Oo)fj(a`, t) = 0, fj(O,t)= 89 ff
r
f,(o,
(85)
.or ( ~ z , t ) ~J ~ ( a `, , t ) d d , .. ~a`,,~c(-'~-~.t) ~ '
i#j.
(86)
(a`', t) da`'.
(87)
,, c(a` ,___2)~. , t) da`' f f .o~ A t~ ~ c(O,' tt) J , t ~ ,
=
+88 2
ao(,)
j~i
The solutions of (85)-(87) can be expressed in terms of n o ( t ) =fj(O, t ) Z ( t).
(88)
Integrating (85) and making use of (75) and (86)-(88) we get: fj(a`, t) = h( t - to-a`) nij( Z ( t -ta- `a)` ) ~ - h ( ~ - t + t ~ 1 7 6 1 7 6
(89)
where j~j(g) =fj(a`, to),
(90)
and ngj are the solutions of the integral equations: t -- to
fO I/o
n o ( t ) =1
g ( g ) n o ( t - a ` ) da`
"
f o~ c~ t + to) o +89 t_tog(a`)-f-~a`~i+to)fij(a`-t+to)
n~i(t) =
g ( a ` ) n i ~ ( t - ~ ) da`+88 ~
I/o
d~,
i#j,
(91)
g ( a ` ) n u ( t - a ` ) da`
j~i
+
f o~
C~ -- t + to) g(a`)?(-~Zi-~-_f--o)fi(a`-t+to) da`
-- to
+~ J~ •
~ c~ - t + to) ~ , -to g(~) .p ( a. ` _. t +. t o .) J O. ( ~
ff
t+to) da`.
(92)
To investigate the asymptotic behavior offj(a`, t) we introduce the Laplace transforms (~*(s) =
fo
e-Synq(to+y) dy,
y = t - to.
(93)
82
M. 0 Vlad
(91)-(92) lead to:
(
~*(s)= 89 1- 89
IO~ e-S'~g(~)da, I--lfoX3;yO-syz\CO(ffc-y) "~ e g~,a,)-~_~jo~a~-y)da,
dy, (94)
O~(s) = ( 1 - I o
e-S~g(a,)da,) -110~ IyX~e-Syg(a,)~-'~-~Jii,a,-y)da, c~ - Y) -"~ "'~
+ 88189 x
;o l;
dy
-1
co(o, y. fo(~ _y) d~ dy. p(~-y) j~i
e-,Yg(a0_
Therefore, the time evolution of f~0(t); i,j = 1,..., two transcendental equations of Lotka type:
(95)
m depends on the roots of
fog(a) e-S~ d~ = 2,
(96)
fog(~z) e-S~ d~z = 1.
(97)
Taking into account the results available in literature (Lotka 1939; Keyfitz 1968) it turns out that: (1) Each of Eqs. (96) and (97) can have a single real root, respectively r~ and ro,
g(a0 e -r;~ d~ = 2,
g(a0
0
e -r~ d~ = 1.
(98)
0
These roots fulfill the conditions if Ro ~ 2 ifRo~l
then r;-~ 0, thenro-~0,
No =
fo g(x) dx,
(99) (100)
where Ro is the net reproduction rate of the population. (2) The complex roots must come in conjugate pairs
s~t = u;+jwl,
s• = ut+jwt,
j =~-1,
(101), (102)
and the real part of any complex root must be less than the corresponding real root ul < r~, (3) As
ul < ro, Vl.
(103)
d/ds ~o e-~g(~) da <0, the roots r~ and ro fulfill the inequality r~< ro.
(104)
From (89) and (94)-(104) we get the following expressions for the asymptotic behavior of ~ j ( t ) and fj(~, t):
no(t ) -~ e to(t-to)fo~ ~ e-r~ v / V y e - % Y g ( y ) dy, /30
e,'-".y,C~ t >>to,
--y) d~ dy iysj
(105)
Separable models for population genetics
12ii(t) =
83
e-r~163
e '~176
/foYe-r~
-
dy,
f j ( o , t)-+0
--
do, dy
q ~ ) = qi( ~, to),
t>>to,
i#j,
as t-+m,
(106) (107)
f , ( o , t), q,(o, t) -+f~[ = q~' as t-+ oo,
(108)
where
fi"[=qSit= fOlly x~ e r~ /f?
C0(~Z"--Y)q~(o-y)dody o
fy ~e-r~176 c~ -y) (109)
= independent of o. 5. Diploid populations: selfing with mutation Removing the restriction (67), the Eqs. (69)-(70) become:
fo
nkk(O, t) = ~i ~ Uik(O)h 2 o(zQnii(o, t) do +1~/ •
j~i
fo o V~k(O)(Vik(O)+ vjk(~z))h~
t) do,
(110)
o
nkt(O, t)----~i
Vig(o)vi;(o)h~
+1~ 2
j~i
t) do
fo o V~k(O)(V~t(~z)+vj;(o))A~
t) do.
(111)
Summing in (110)-(111) over kl we recover (73), so that the relationships (72)-(85) and (88)-(89) remain valid. After some calculus, we get the following system of linear integral equations in 12~(t):
~kt(t) =~i frO-- to gktd~(o)~i~(t--o) do --to
frO gkt,O(o)~ij(t--~) do+ ~kl(t),
+ ~ j~>i
(112)
gk,,.(o) = ~ k ( o ) . , , ( o ) g ( o ) .
(113)
gkl,O(o) = 89
(114)
where
+ Ujt(o))g(o),
84
M.O. Vlad f o~ c~ t + to) o t-to V,k(~)V~t(~)g(~)~--~-- ~ t - ~ / , ( ~ - - t + to) d~
~k,(t) : ~
+1~
j~>i
t--t o
c~ o x p ( ~ _ t+ to) f O ( ~ - t+ to) da,.
(115)
Through a convenient labelling the Eqs. (112) may be reduced to (37). Introducing the notations
(ij)->u,
u=l,...,m+m(m-1)/2=m(m+l)/2,
~kl(t)-->~~u(t),
gkl.ij(~)'->guv(~),
(116)
J[/~m(t)-->J~u(t),
(117)
the relationships (112) lead to (37) with the difference that Zj(t) are replaced by
~kl(t) = Ftu(t). Therefore, the functions flkl(t) and f~j(~, t) depend on the roots of the characteristic equation (47). Imposing to gkl, i j ( ~ ) -= guv(~) a set of restrictions similar with (48)-(50), it follows that there is a positive vector X and a positive root So with the largest real part obeying (47) and (51). Moreover, the root So is identical with the real root ro of (84) (see Appendix 2). It turns out that the asymptotic behavior off~j(~, t) and Fij(t) is given by
F o ( t ) ' f u =st F ~ , ,,
fij(gb, t),
as t->oo,
(118)
where the genotypic frequencies f ~ = F~-t= independent of a,
(119)
are the solutions of the linear equations f kstl = ~ f i i ~t ~0 ~ 1.'ik(~)Vil(a,)g(z~ ) i
e -~~ d~
+88 ~. f~] [oo vik(~z)(vu(a.)+vj,(a.))g(*Q e -r~ da., 9
j#i
(120)
dO ~ r + L V" st Jii 2 s ~'~ f/j = 1. i i j#i
(121)
6. Diploid populations: mating For simplicity, in this section we consider a monoecious population obeying the conditions (65), (66) and (67). In this case the "death equations" (68) remain valid. The "birth equations" relating the birth rates B~(t) = no.(O, t) to the population densities depend on the mating mechanism. In particular, considering that the matings between the different age classes are forbidden, we have (I):
fo
ne(0, t) = ( 2 - ,5~)
qi(a,, t)qj(a~, t)A~
t) d~,
(122)
whereas for random mating: (II):
nij(O, t) = ( 2 - ~u)Qi(t)Qj(t) [oo A~ do
,z, t) d~z.
(123)
Separable models for population genetics
85
It is easy to check that the Eqs. (68)-(122) or (68)-(123) are consistent with the population model defined by (72)-(73) and thus we can use the integration method developed in Sect. 4. The main results are the following: (a) The birth rates are given by (122). The genic and genotypic frequencies are equal to: qi(z~, t) =
h(t - t o - a ) -U i ( t - e )
Z(t-e) +h(e-t+to)q~
fj(e, t)= h(t-to-e)(2-8ij)
(124)
[f/ ,o-~A(x) Oi(;(t-e- ~ - x ) x)Z(t-e) a/t-e-x)
p(x) dx
hO(x)qO(x_t+e+to)qO(x_t+e + to) c O ( x -t+e+to)
+ f oo
z(t-e)
,-,o o
p(x)
p(x- t+e+ to)
de]+h(e_t+to)fO(e_t+to),
(125)
where the functions ~,(t) =
Z(t)qi(O, t),
(126)
are the solutions of the integral equations f~(t) =
fo'-'~ Oi(t-e)g(a,) de+ f~t -- to g(e)CT-~ ,,-to)C~176 p , e -U-yTzq~
de. (127)
The asymptotic behavior of the functions ~i(t),fj(e,
t) and F0(t) is the following:
g(x)--qO(x--y) f fly ~176 p(x-y)
e
/ f o Z g ( z ) e-zr~
t>>to,
~'~i(t) ~-~er~176
qi(e,
t), Qi(t)~ q~t= Q~t
fj(e, t), Fij(t) ~ f ij~'=F~S'
yr~
(128)
as t~oo,
(129)
a s t --->oo,
(130)
where
q~t= Q~.,=fo ~ jy [~ p(~-_y) g(x) c~176
e -yr~dx dy
/fo~ foo g(x____))cO(x_y ) e -'to dxdy=independent 9]0 ,Jy p(x-y)
of,z,
(131)
and fgst= F~ut= (2 (see Appendix 3).
8o)qiStqjSt
=
independent of e.
(132)
M. O. Vlad
86 (b) The birth rates are given by (123). We have:
qi(z~,t)=h(t-to-~) fj(e,t)=h(t-to-~)
-_
t-h(~-t+to)q~
. D.i(t-~)~j(t-~) --~t--~)
(133)
~-h(~-t+t~176176
(134)
and the functions fL(t) are the solutions of the integral equations -- to
l),(t) = b(t)
frO f
+b(t)
l ) , ( t - ~ ) p ( a ) d~
~
c~ t + to) o p(z~)-p-~_--f+~o) q~(~z-t+to)d~ ,
(135)
t-- to
where b(t) is the per capita birth rate:
b(t)=Z(t)
/(fo -,op(~z)Z(t-z~) da~+
Ho
p(z~) c~176 p(e-t+to)
}
d~z . (136)
We shall consider two particular cases: (bl) 0
qO= QO = independent of ~,
0
fij =f0(~).
(137)
In this case the Eqs. (135) have the solutions
(138)
~i(t) = Z ( t ) q ~ and thus
qi(z~, t) = Qi(t)= qO= QO = constant,
f/j(~, t), Fij(t) o ( 2 - 6 o ) q i q j 0
0
as t o ~ ;
(139) (140)
(b2) qO = qO(~),
cO(zQ = c~t(c~).
(141)
In this case the asymptotic behavior of q~, f j and F~ is given by
qi(~,t),Qi(t)-~q~it=Q~it
as t~o0, St
st
fj(~, t), Fij(t)~(2-60)qi qj
(142)
as t o ~ ,
(143)
where
q~, = Q~, =
fof/ qO(~_y) e-ro~ p(~z) d~ dy
~p(z~) e -ro~ d~.
(144)
(see Appendix 4). Although the two mating mechanisms lead to Hardy Weinberg distribution (see (132) and (143)), the values of equilibrium genic frequencies may be different (see (131) and (144)). If qO = QO = independent of ~, the two mating mechanisms lead to identical results even if c~ r cSt(z~). In both cases we have:
fij(a, t ) = h ( t - t o - a . ) ( Z - 6 i j ) q ~ 1 7 6 1 7 6
(145)
Separable models for.population genetics
87
Fu(t)
W op ( e ) Z ( t - g ) de+ ;o,Op ( e ) Z ( t - a ) de+
( 2 - 6o)q~ y
,_,~ p(g) c~176 ----~ p (e f~ t~ f ij(e~ _ t + to) dz~ p(e) c~176 d~ t-,o p(e-t+to) (146)
Moreover, if f o = F o = independent of ~,
(147)
Fu0
(148)
(1 - ~O)(QO)2+ 0~OQO,
=
F ~ = 2(1 - ~O)QOQO,
(149)
where ~-o is an inbreeding coefficient (Li 1955), then (146) may be written as F . ( t ) = (1 - ff(t))(Q~
+ o%(t)Q~
(150)
Fo(t ) = 2(1 - o~(t))Q~ ~
(151)
where the inbreeding coefficient @(t) is given by ~(t)
:
f
~-o
IO t -- to p(e)Z(t
In particular, for c~
,~(t) = ~o
oo c~ t~
p(e) p ( e - t + to) de
t
- e ) d~+ I t c~ c~ - t+ to) -to
(152)
p(e)
p (e - t+
to)
d~
= c " ( ~ ) we have
e-'o~p(e) d~ t-t o
/fo
e-romp(e) d~ = ~-o
c't(e) de.
(153)
t-t o
7. Diploid populations: mating with mutation If mutations are present, the birth equations (122)-(123) should be replaced by (I'): nq(O, t) = ( 2 - 8~) ~ ~
9, j1
(II'):
fo
vi,i(~)vfj(~)qi,(e, t)qy(e, t)A~
t) d~,
(154)
no(O, t) = ( 2 - 6/j)
f a~
vi,i(z~ )c(e, t)qi,(~z, t)v;j(z~ )c(e , t)qj,(~ , t) de' de"
t) d,~.
(155)
By applying the integration method used in the preceding sections, we obtain: (a) The matings between two different age classes are forbidden:
qi(~,t)=h(t-to-~Z)~(tt~)) + h ( ~ - t + t o ) q ~
(156)
M. O. Vlad
88 and fit(e, t) = h( t - to-z~)(2 - 6/j)
{fo -'~
x
, , , , f ~ i ,- (Zt(-~~t-:x-)~~)))j , ( t - z ~ - x ) g(x) 2 2 vi,itx)vj,j(x)
dx
i'j'
+
g(x) Z • vi'i(x)vj'j(x)q~ x - t+ t o + ~ ) @ ( x - t+ to+ ~). t--to--~
c~
i' j '
- t+ to+~Z) d x } + h(~ - t+ to)f~
- t+ to),
(157)
where the functions f~(t) fulfill the integral equations
ft,(t) =} IO--tog.,(~)a,,(t -~,) d~+~,(t).
(158)
g,,(~) = vi,,(~)g(a),
(159)
with
~,(t) = ~ '
f
oo
g(~)
t-to
cO(~z_t+ to) p(~-t+to)
vv,(~z)q~
- t + to) d~.
(160)
(see Appendix 5). (b) Random mating. The evolution equations (133)-(134) remain valid, with the difference that the functions fli(t) are the solutions of the integral equations:
fOt- togiv(~, t)~)v(t-~z) d~z+ Jgi(t),
f~(t) = ~
i'
(161)
with (162)
g.,(a, t) = b(t)p(a.) vi,~(a~), ~ i ( t ) = b(t) Y. v""
i oo t_to
c~ ' - t + to) v~,~(d)p(~') p( ~z' q~ t + to)
t+ to) d~'.
(163)
(see Appendix 6). Making use of (158), (161) and assuming that the functions g,, fulfill the conditions (48)-(50) we can prove that the genic and genotypic frequencies evolve towards time persistent forms. We get: (a) The Eqs. (154) are valid qi(~, t), Q i ( t ) ~ q ~ t fj(~,t),Fo(t)~(2-6ij)qi,
st
i' j"
qj,st
io
as t~oo,
e -ro~ g ( ~ ) v i , i ( ~ ) v f j ( ~ ) & z
(164) as t~oo, (165)
where the frequencies q~t are the solutions of the linear equations: q~t= 2 q~t
uvf(a,)g(a,) e -'o'~ da,,
(166)
Separable models for population genetics
89
and q~'= 1.
(167)
i
c~
(b) The Eqs. (155) are valid. In this case, for simplicity, we supppose that = dt(z~). We have:
qi(~, t), Qi(t)~q~'. as t~oo, fj(,z, t),Fo(t)~(2-6ij)qS'q} ' as t~oc,
(169) (170)
where q~' are the solutions of the linear equations
fo
q~t=~ q~,,
d'(a,)vi,i(,z) da,,
(171)
q~.'= 1.
(172)
i
We see that the presence of mutations lead to time persistent genic and genotypic frequencies which are independent of initial conditions. In the case (I'), due to mutations, the genotypic frequencies are not in Hardy-Weinberg equilibrium. However, even in this case the valuesfj can be viewed as a superposition of equilibrium frequencies which obey the Hardy-Weinberg law. Indeed, from (154) follows that the equilibrium genic frequencies q~t(birth)for a group of new borns from ancestors having the age a ' are equal to:
qs'~bi~th)(z~')= 2 q~;(~') VV,(~').
(173)
i
On the other side,
w(~') d~'= e-~'r~
da/,
w(~') d a / = 1,
(174)
is the density function of ancestors' ages at childbearing (Keyfitz 1968; Vlad and Popa 1985), so that (165) can be rewritten as
fj(~,
t),
Fo.(t ) ~
;oow ( ~ ' ) ( 2 - t~ij)q~t(birth)(~t/)q;t(birth)(~') d~' =((2--3ij)q~t~,
q.e.d.
(175)
8. Conclusions
Within this paper we constructed several continuum age-dependent models for haploid and monoecious diploid populations. The basic features of the considered models are the following: (a) in spite of their nonlinearity they are amenable to analytical treatments; (b) irrespective of the time evolution of the population size all the models lead to asymptotically time persistent genic and genotypic frequencies. Although we considered the case of a single locus, our approach may be applied to more complex systems such as multi locus systems or sexual populations.
M. O. Vlad
90
Appendix 1 If cj-~ c]t(,~),
j = 1.....
(A1)
m,
(37)-(39) and (60) yield to I ~ to
e-~(t-to)J/t~(t) dt = ~ b]'
J =
IO~fy ~
& j ( ~ ) e -(s-ro)y-ro~ d,~ d y
b~' - ~
S--ro
e - ' ~ g i j ( ~ ) dry.
j S--rodo
(A2)
Combining (45) and (A2) gives ~ Z * ~ ( y ) = b ~ ' / ( s - ro) ,
Z i ( t ) = b~it e 'o('-'o).
(A3)
Inserting (A3) into (35) and (40)-(41) after some calculus we come to (62)-(64).
Appendix 2 Provided that the conditions (48)-(50) are fulfilled, from (112) and (116)-(117) it follows that the asymptotic behavior of f ~ k l ( t ) = f~u(t) is given by l)kt(t)~e
so(t-'o)
as t - ~ .
(A4)
Summing in (88) over i and j and taking into account of (8) and (A4) we get: Z(t) =~ ~ii(t)+ 89
i
~, I~ij(t ) - eSo('-'0)as t-+oo. i j~i
(A5)
By comparing (81) with (A5) we come to (A6)
s o = r o.
Appendix 3 Integrating (85) and taking into account of (13) gives fj(~,
t) = h ( t - t o - a)f/j(0, t - ~ z ) + h(~z - t + t o ) f ~ ( a - t + to) ,
q i ( ~ , t) = h ( t - t o - ~) qi(0, t - ~ ) + h ( a - t + t o ) q ~
- t + to) ,
(A7) (A8)
Combining now (7), (75), (122), (A7) and (A8) we have: Z(t)fiij(O, t) = (2 - 6ij)
Io t~A~
9 Z(t-~)p(~) 9
t-a)qj(O,
t-~z)
d~ + ( 2 - 6,j)
q?(~ - t + to)q~
- t+
to)
t- to
9 A~176
- t + to)p(~)/p(~
- t + to) ] d~,
(A9)
wl~refrom summing over j and making use of (13) and (126) yield to the integral equations (127). On the other side, inserting (126) and (A9) into (A7) and (A8) gives (133) and (134). The integral equations (127) are similar with (78). Operating in (78) the substitutions Z ( t)--> ~ ( t),
c~
- t + to) -+ c~
- t + to)qO(~ _ t + to) ,
(A10) (A11)
(78) yields to 027). Therefore, using (A10) and ( A l l ) , (81) leads to (128). Combining (124)-(125) and (128) we come to (129)-(130).
Separable models for population genetics Appendix
91
4
From (7), (13), (14), (73) and (123) we have: fj(0, t) = (2 - t~,j)O~(t)Qj(t),
(A12)
q~(O, t) = Qi(t) =
(A13)
c~(~,, t)q~(~, t) d~.
Inserting into (A13) the equations (75), (126), (t36) and (A8) yields the integral equations (135). Combining (126), (A7), (A8), (A12) and (A13) we come to (133)-(134). If c~ = cS~(~) we have
b(t) = b st=
F
]_1
e-ro~p(~) da~
(A14)
d~.
(A15)
and the integral equations (135) become:
O,(t) = fo -'o ~,(t-~)~7-st d~ p(e) o
+
I
~ q~
)
~
c~ p ( ~ ) ~ ~
t-t o
The secular equation attached to (A15) is the following:
fo e-~p(,z) d,z /Io oe-~o~p(a) d~ = 1.
(A16)
It is easy to check that the root with the largest real part of (A16) is s o = r o.
(A17)
It follows that the asymptotic behavior of l)i(t) is given by
fo~f~q~ ~i(t)-~b~e ~o(~-to)
e-~o~p(~)d~dy ,
o~
as t--)co.
(A18)
e -~o~ d~
From (133), (134) and (A18) we get (142)-(144). Appendix
5
Using (7), (75), (154), (A7) and (A8) we get:
z(t)f,~(o, t) = ( 2 - ~,j)
frot- t~ x~
Y~Z ~',,,(,~)~'j,j(~) i' j'
9 q,,(O, t-~)q;(O, t - e ) Z ( t - , z )
de+(2- 61j)
A~ t-t o
9 Y E ~,,,(,~) vj,j(~)q?,(-i'j'
9 q~
- t + to) [ c ~
t+
to)
- t+ to)/p(~
- t+
to)] c/~,
(A19)
wherefrom summing o v e r j and using (13) and (126) we get the integral equations (158). Similarly, substituting (126) and (A19) into (A7) and (AS) yields (156) and (157). The equations (158) have the same form as (37). Assuming the validity of conditions (48)-(50) and making use of the methods developed in Sect. 3 and in Appendix 2, we get the asymptotic expressions f ~ ( t ) ~ Y~' e s0(t-'o)
as t->oo,
(A20)
where y~t, i = 1. . . . . m are determined by a set of equations similar with (55). From (156)-(157) and (A20) we come to (164)-(167).
92
M.O. Vlad
Appendix 6 Using (7), (13), (14), (73) and (155) we obtain f/j(0, t)= (2-8ij )
( So
~'ci(Z~')c(d, t)qi,(z~' , t) d,z' qi(O, t)= .~
fo
/( Io E
/ \j'
/
~,yj(d')c(d', t)qj,(a/', t) dd' ,
~'i,i(z~')c(z~', t)qi,(d, t) d~'.
(A21) (A22)
Substituting the Eqs. (75), (126) and (A8) into (A22) yields the integral equations (161). Making use of (126), (AT), (A8), (A21) and (A22) we get (133)-(134). Supposing the validity of conditions (48)-(50) and applying the methods presented in Sect. 3 and in Appendix 1 we can prove the validity of (169)-(172).
Acknowledgements. This paper has been supported by CIMC Bucure~ti. The author wishes to thank the referees for helpful suggestions.
References 1. Bellman, R., Cooke K. L.: Differential-difference equations, pp. 216-256. New York: Academic Press 1963 2. Busenberg, S., Iannelli, M: A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlinear Analysis T.M.A. 7, 501-529 (1983a) 3. Busenberg, S., Iannelli, M.: A degenerate nonlinear diffusion problem in age-structured population dynamics. Nonlinear Analysis T.M.A. 7, 1411-1429 (1983b) 4. Busenberg, S., Iannelli, M.: Separable models in age-dependent population dynamics. J. Math. Biol. 22, 145-173 (1985) 5. Charlesworth, B.: Selection in populations with overlapping generations. I. The use of Malthusian parameters in population genetics. Theor. Popul. Biol. 1, 352-370 (1970) 6. Charlesworth, B.: Evolution in age-structured populations. Cambridge: Cambridge University Press 1980 7. Demetrius, L.: Statistical mechanics and population biology. J. Stat. Phys. 30, 709-753 (1983) 8. Demetrius, L., Ziehe, M.: The measurement of Darwinian fitness in human populations. Proc. R. Soc. London B222, 33-50 (1984) 9. Fisher, R. A.: The genetical theory of natural selection. Oxford: Clarendon Press 1930 10. Gurtin, M. E., MacCamy, R. C.: Product solutions and asymptotic behavior for age-dependent, dispersing populations. Math. Biosci. 62, 157-167 (1982) 11. Hadeler, K. P., Dietz, K.: Population dynamics of killing parasites which reproduce in the host. J. Math. Biol. 21, 45-65 (1984) 12. Keyfitz, N.: Introduction to the mathematics of population. Reading: Addison-Wesley 1968 13. Leslie, P. H.: On the use of matrices in certain population mathematics, Biometrika 33, 183-212 (1945) 14. Li, C. C.: Population genetics. Chicago: University of Chicago Press 1955 15. Lotka, A. J.: Th6orie analytique des associations biologiques, vol. II, Paris: Hermann 1939 16. Pollak, E. Kempthorne, O.: Malthusian parameters in genetic populations. I. Haploid and selfing models. Theor. Popul. Biol. 1, 315-345 (1970). 17. Pollak, E., Kempthorne, O.: Malthusian parameters in genetic populations. Part II. Random mating populations in infinite habitats. Theor. Popul. Biol. 2, 357-390 (1971) 18. Pollak, E., Callanan, T.: Convergence of two-locus gamete frequencies in random-mating agestructured populations, Math. Biosci. 62, 179-199 (1982) 19. Pollak, E.: Gamete frequencies at two sex-linked loci in random mating, Math. Biosci. 70, 217-235 (1984) 20. Vlad, M. O., Popa, V. T.: A new nonlinear model of age-dependent population growth, Math. Biosci. 76, 161-184 (1985) Received November 17, 1986/Revised August 15, 1987