357
Ann. Hum. Genet. Lond. (1978), 41, 357
Printed in Great Britaan
Asymptotic behaviour of the mean of a continuous phenotypic diffusion process with overlapping generations BY S. R. WILSON Department of Statistics, I.A.S., The Australian National University, Box 4, P.O., Canberra, A.C.T., 2600, Austra,lia INTRODUCTION
As many of the selection forces such as infant mortality and fertility differentials are being reduced in human populat,ions, it is necessary that geneticists develop a broader understanding of the effects of various sociological processes on the gene pool, and especially those effects concerned with mating systems. It is well known that mating is not appreciably influenced by most polymorphic characters, such as blood groups. However, there is strong assortative mating for a number of continuous characters such as height and intelligence. These characteristics are believed to have, a t least in part, 8 multifactorial genetic basis. I n this article we find explicitly the asymptotic equation for the phenotypic equilibrium of a population which reproduces both randomly and assortatively for a continuous characteristic. Following the method given by Rossi (1 976) we find the asymptotic behaviour of the process as a function of the pmameters of the assortation distribution as well as the initial composition of the population. This method uses a similar form of the integral reproduction equations for the continuous Characteristic model with overlapping generations as given by Rossi ( 1 976) for simple Mendelian inheritance. I n the equilibrium position we consider the random mating case and the two major positive assortative mating models for a continuous characteristic to have been considered so far, namely the model proposed by Fisher (1918) and that proposed by Wilson (1973a, b ) . The advantages and disadvantages of both these models are discussed and neither model is found to be particularly satisfactory in light of present day observation.
GENERAL MODEL AND ASYMPTOTIC RESULTS
The phenotypic selection with assortative mating process has been studied extensively for a single locus with 2 alleles by de Finetti (1926, 1927), de Finetti &, Rossi (1974) and Rossi (1975, 1976). I n this section a similar model is proposed for the phenotypic selection with assortative mating process for a continuous phenotypic characteristic. Following Rossi (1976) we write the general integral reproduction equation for such a process in the following form:
E ( x ;t ) d x =
jm1-1P ( x ;y , z ) ! . ~ S n A E ( yt;- r ) E ( z ;t - s ) N ( y , -w
3
z ; t , r , s)drdsdydzdx, (1)
1
where E ( x , t )dx is the expected density of children born at time t , with phenotypic value in the range ( x , x+dx), P ( x ; y , z)’dx is the probability of a child with phenotypic value in the range ( x , x + d x ) from parents having phenotypic values y and z , [j,k] represents the female fertility period, [ l , m] represents the male fertility period, and N(y, z ; t , r , s) is a weighted selection 24
H G E 41
S. R. WILSON
358
(i.0. survival and fertility) and mating function of individuals with phenotypic values y and aged r and s respectively at time t. We shall assume that the assortative mating function is independent of time t and of ages r, s and denote such a function by A(y, z ) , being the relative frequency of matings between individuals having phenotypic values y and z. Assortation is supposed to act during the fertility period. Discussion of the form of A(y, z ) which is taken to be fixed will be given later. Now writing N(Y, z ; t , r, s) = A(Y, z )B(r ,s , t ) then B(r,s, t ) can be completely calculated on the basis of demographic tables. I n fact w0 will assume B to be of the form
B(r, s, t )
= n(Z- n) r ( r )A(s - r ) @(r)/E(t - s),
where r(r)is the survival function for an individual to age r, and is taken to be independent of sex and phenotype; A(s - r ) is the distribution function of the difference between ages of husband, s, and wife, r, @ ( r ) is the female fertility function, that is the probability that a woman aged r has a child during the year, and E(t - s) is the expected density of the number of children born at time t - s. In the following we shall assume both the functions A and B to be known. Also we shall mainly be concerned with estimating the function f (x; t ) dx which represents the proportion of individuals having phenotypic values in the range (x, x + dx) among all individuals alive at time t . In this article we will study the integral equation (1) from an algebraic point of view to obtain general results concerning the asymptotic behaviour of the process. Now we can rewrite ( 1 ) in the following form :
= s_s-mmP(z;y,
j- E ( y ; t - r ) h ( z ; t - s ) D ( r , s)drdsdydzdx, (2) j-mm E ( z ; d z , and D(r, s) B(r, E(t - s) which is
z)A(y, z ) /
k
3
where h(z;t ) = E ( z ;t ) / E ( t )where , E(t)=
m
1
t)
=
s, t )
independent oft. So h(z;t ) dz represents the proportion of individuals having phenotypic value in the range ( z ; z+dx) among all children born at time t. Now consider the rate of increase of the function
E(t) = given by the logarithmic derivative, which we assume to be a continuous function, so lim -= &(a;t ) , E ( t )At
A t 4
where a represents the parameters of the function A(y, 2). Writing E(to) = E,, we have
Asymptotic behaviour of a continuous phenotypic digusion process
359
If we assume that the limit of this process is a stationary process then l i m b ( a ; t ) = €(a), l&(a)l
Ann. Hum. Genet. Lond. (1978), 41, 357
Printed in Great Britaan
Asymptotic behaviour of the mean of a continuous phenotypic diffusion process with overlapping generations BY S. R. WILSON Department of Statistics, I.A.S., The Australian National University, Box 4, P.O., Canberra, A.C.T., 2600, Austra,lia INTRODUCTION
As many of the selection forces such as infant mortality and fertility differentials are being reduced in human populat,ions, it is necessary that geneticists develop a broader understanding of the effects of various sociological processes on the gene pool, and especially those effects concerned with mating systems. It is well known that mating is not appreciably influenced by most polymorphic characters, such as blood groups. However, there is strong assortative mating for a number of continuous characters such as height and intelligence. These characteristics are believed to have, a t least in part, 8 multifactorial genetic basis. I n this article we find explicitly the asymptotic equation for the phenotypic equilibrium of a population which reproduces both randomly and assortatively for a continuous characteristic. Following the method given by Rossi (1 976) we find the asymptotic behaviour of the process as a function of the pmameters of the assortation distribution as well as the initial composition of the population. This method uses a similar form of the integral reproduction equations for the continuous Characteristic model with overlapping generations as given by Rossi ( 1 976) for simple Mendelian inheritance. I n the equilibrium position we consider the random mating case and the two major positive assortative mating models for a continuous characteristic to have been considered so far, namely the model proposed by Fisher (1918) and that proposed by Wilson (1973a, b ) . The advantages and disadvantages of both these models are discussed and neither model is found to be particularly satisfactory in light of present day observation.
GENERAL MODEL AND ASYMPTOTIC RESULTS
The phenotypic selection with assortative mating process has been studied extensively for a single locus with 2 alleles by de Finetti (1926, 1927), de Finetti &, Rossi (1974) and Rossi (1975, 1976). I n this section a similar model is proposed for the phenotypic selection with assortative mating process for a continuous phenotypic characteristic. Following Rossi (1976) we write the general integral reproduction equation for such a process in the following form:
E ( x ;t ) d x =
jm1-1P ( x ;y , z ) ! . ~ S n A E ( yt;- r ) E ( z ;t - s ) N ( y , -w
3
z ; t , r , s)drdsdydzdx, (1)
1
where E ( x , t )dx is the expected density of children born at time t , with phenotypic value in the range ( x , x+dx), P ( x ; y , z)’dx is the probability of a child with phenotypic value in the range ( x , x + d x ) from parents having phenotypic values y and z , [j,k] represents the female fertility period, [ l , m] represents the male fertility period, and N(y, z ; t , r , s) is a weighted selection 24
H G E 41
S. R. WILSON
358
(i.0. survival and fertility) and mating function of individuals with phenotypic values y and aged r and s respectively at time t. We shall assume that the assortative mating function is independent of time t and of ages r, s and denote such a function by A(y, z ) , being the relative frequency of matings between individuals having phenotypic values y and z. Assortation is supposed to act during the fertility period. Discussion of the form of A(y, z ) which is taken to be fixed will be given later. Now writing N(Y, z ; t , r, s) = A(Y, z )B(r ,s , t ) then B(r,s, t ) can be completely calculated on the basis of demographic tables. I n fact w0 will assume B to be of the form
B(r, s, t )
= n(Z- n) r ( r )A(s - r ) @(r)/E(t - s),
where r(r)is the survival function for an individual to age r, and is taken to be independent of sex and phenotype; A(s - r ) is the distribution function of the difference between ages of husband, s, and wife, r, @ ( r ) is the female fertility function, that is the probability that a woman aged r has a child during the year, and E(t - s) is the expected density of the number of children born at time t - s. In the following we shall assume both the functions A and B to be known. Also we shall mainly be concerned with estimating the function f (x; t ) dx which represents the proportion of individuals having phenotypic values in the range (x, x + dx) among all individuals alive at time t . In this article we will study the integral equation (1) from an algebraic point of view to obtain general results concerning the asymptotic behaviour of the process. Now we can rewrite ( 1 ) in the following form :
= s_s-mmP(z;y,
j- E ( y ; t - r ) h ( z ; t - s ) D ( r , s)drdsdydzdx, (2) j-mm E ( z ; d z , and D(r, s) B(r, E(t - s) which is
z)A(y, z ) /
k
3
where h(z;t ) = E ( z ;t ) / E ( t )where , E(t)=
m
1
t)
=
s, t )
independent oft. So h(z;t ) dz represents the proportion of individuals having phenotypic value in the range ( z ; z+dx) among all children born at time t. Now consider the rate of increase of the function
E(t) = given by the logarithmic derivative, which we assume to be a continuous function, so lim -= &(a;t ) , E ( t )At
A t 4
where a represents the parameters of the function A(y, 2). Writing E(to) = E,, we have
Asymptotic behaviour of a continuous phenotypic digusion process
359
If we assume that the limit of this process is a stationary process then l i m b ( a ; t ) = €(a), l&(a)l
