A mathematical analysis of the effects of movement on the relatedness between populations

Ann. Hum. Genet., Lo&. (1O6O), 32, 237

Printed in Great Bri tain 237

A mathematical analysis of the effects of movement on the relatedness between populations

BY 1%. W. HIORNS," G. A. HARRISON,t A. J. BOYCEI AND C. F. KUCHEMANNf

* Department of Biomathematics, University of Oxford, f Anthropology Laboratory, Department of Human Anatomy, University of Oxford, $ Department of Biological Sciences,

University of Surrey

The concept of relatedness is fundamental to many problems in genetics and anthropology, and physical anthropologists, in particular, have devoted much effort to attempting to establish the relationships between present and past populations of man. Relatedness between populations has been measured in many different ways (see Sokal & Sneath, 1963 and Yasuda & Morton, 1967, for extensive reviews of taxonomic and genetic aspects respectively) but common to most approaches is the view that degrees of relatedness are to be defined in terms of degrees of genetic similarity. In evolutionary terms, however, genetic similarity can arise from convergence, whilst rates of genetic divergence are far from constant. Under these circumstances, there may be little concordance between relatedness based on genetic similarity and relatedness in the genealogical sense of number of generations since populations had a common ancestry.

The purpose of this paper is to examine this latter concept of relatedness, but instead of viewing the situation of diminishing relatedness under progressive temporal and spatial isolation, the converse of increasing relatedness resulting from genetic exchange between populations will be considered. It is the gene flow between populations of a species which maintains the integrity of that species and obviates the evolutionary forces of diversification.

The type of model situation we propose to deal with is one of a series of populations which are assumed to have had no common ancestry or an ancestry so distant that it can be ignored, but between which there is now movement in the form of mate exchange, and the question of concern is, how many generations will it take under a particular pattern of exogamy for the populations to become homogeneous in their ancestry and how does this pattern of relatedness develop? It will be assumed that each pair of marriage partners produces only two offspring, one of each sex, who themselves reproduce with a standard generation time. Thus from the point of view of common ancestry, the concepts of drift and selection are not relevant. Two populations are considered to be completely related or 'homogeneous ' when the spatial distributions of ancestors of the individuals in the two populations are essentially the same. For convenience, essential similarity is taken arbitrarily as the stage when 95 % of the ancestry of the two populations is shared in common.

In this paper the model is applied to a group of eight neighbouring Oxfordshire parishes, each of which is taken as a population. The data is drawn from the marriage registers kept by the Anglican Churches in each of the parishes, which contain information on the place of residence of the marriage partners just prior to marriage. Perforce, these are taken to represent birth- places, and if both the partners in a marriage were resident in the same parish it is regarded as an endogamous union. Exogamous unions can be considered as of two types: those where the

238 R. W. HIORNS AND OTHERS

exchange is between pairs of parishes in the system, and those where a partner is chosen from another parish not represented in the eight from which data has been obtained. The latter present an important theoretical problem. One knows of the contributions that other parishes have made to those in the system, but there is no direct information of the counter-flow, and one is dealing anyway with an expanding and all-embracing system since there are no distinct boundaries to the system which, in a sense, includes all the populations in the British Isles and beyond.

1851-1966 100 -

h

0 5 10 15 20 25 Time (generations)

Fig. 1. Mean common ancestry calculated from comparative marriage movement with outside world included.

Prior to 1851 100 - 1851-1 966 h

I 1

150 200 Time (generations)

Fig. 2. Mean common ancestry calculated from comparative marriage movement with outside world excluded.

For practical purposes it seems necessary to group together all the exogamous unions outside the system of eight parishes as ‘exogamy with the “outside world”’, and to regard the ‘outside world’ as a ninth population in the system. However, by comparison with the other populations in the system, the outside world is infinitely large and it therefore seems appropriate to assume that the composition of the outside world is unaffected to any detectable extent by movement from the parishes into it. Admittedly, the main ‘outside world’ exchange occurs between the parishes in the system and those which are just outside. Analyses of the distributions of the

Eflects of movement on relatedness between populations 239

marriage distance (Boyce, Kuchemann & Harrison, 1968) have shown that most of the exogamous marriages involving the ‘outside world’ are with partners within a radius of about 30 miles of Otmoor. But because one is dealing with a completely open system the contribution of the ‘Otmoor’ parishes to their immediate neighbours will be constantly diluted and it seems not unreasonable to assume that the contributions of the outside world to each of the parishes will be qualitatively very similar, and not profoundly affected in composition by the counter- flow. The properties therefore which have been ascribed to the ‘outside world’ are that its composition is stable, its contribution to the other populations in the system is qualitatively, though not quantitatively, the same, and that although it is not necessarily homogeneous in composition (in contrast with the assumption made about the founding populations for the other eight populations) no part of this composition is shared with any of the other populations.

The real situation obviously is an open one and involves a considerable outside world effect, but, as has been indicated, consideration of this effect necessitates the making of a number of assumptions about the properties of the outside world. By artificially closing the system and ignoring the exchange with the outside world, it is possible to examine separately just the effects of the marriage movements between the eight parishes. In some of the ensuing analyses this has been done by excluding marriages which have taken place with the outside world and distributing this component of exogamy proportionately to the endogamy and inter-parish exogamy.

The parish records extend back in time for at least 250 years, and for some parishes for much longer periods-up to 400 years. Analysis of the marriage data for each of the parishes (Boyce et al. 1968) reveals that the mean marriage distance was more or less constant at around 6-8 miles until about 1850. But after this time there was a considerable rise in marriage distance due mainly no doubt to the advent of mechanized transport. It therefore seemed appropriate to examine separately the effects of marriage movement on relatedness before and after this date of 1860, using over-all figures of exchanges between the populations in the two periods as the measures of the effective exchange rates.

MATHEMATICAL MODELS

The marriage exchanges between every pair of populations are depicted by a square stochastic matrix M of order N , the number of populations, with elements m,, representing the probability that a marriage settling into population i comprises one partner from population j; the other partner is here assumed to originate in population i . The elements mi, will then be the endogamy rates and mij for i =t= j will be exogamy rates. It will be convenient to define effective exc:change rates, pi , to represent the proportion of individuals in population i who, prior to their marriages, belonged to population j. These exchange rates comprise a square stochastic matrix P of order N .

Exchange rates between any two communities in opposite directions may be the same and in such cases the system will be symmetric, in accordance with the symmetric nature of the matrices M and P. Assuming constant population sizes in time but different sizes for the communities, however, the systems will not in general be symmetric.

Relatedness as mentioned is measured in kinship terms, in particular, in the distribution of ancestors in the system. We shall therefore refer to ancestor frequencies which, rn already

16 Hum. Gen. 32, 3


mentioned, for any particular population would mean the relative frequency or probability at any given time of an individual with an ancestor from another population being present in that population.

U

Fig. 3. Geographical pattern of developing relationship between the parishes calculated from comparative marriage movement before 1860.

The ancestor frequencies in a system may be conveniently represented by a vector a(,,, of order N , using the bracketed suffix to denote time as n generations after some arbitrary com- mencing time, an element at(,) being the frequency of ancestors from a given population in the ith population. Using the above definitions of endogamy rates and assumptions concerning the manner of exchange we may now establish the ancestor frequencies in the system after one further generation in terms of a(n). After the marriages of the nth generation, which we assume to take place instantaneously at the end of that generation, of the marriages settling into the ith population, all will contain at least one partner from that population and some proportion mid will have two. The proportion of marriages in population i having one partner from population j is mgj and the proportion of individuals in population i originating in j is therefore $mgi. The ancestor frequencies are related thus :

N

I 1 ai(n+l) = at(,) + 4 C. ~ j a j ( n ) , (1)

Effects of movement on relatedness between populations 241

where the unbracketed suffix on the ancestor frequencies refers to the population with this frequency in the generation indicated. It follows from the definitions of the matrices M and P that this relation is equivalent to the matrix equations

(2)

From the ancestor frequencies for each population may be constructed a matrix A , each column of which contains the ancestor frequencies from a population in all populations. The frequencies in successive generations are then determined by the relation

a(,+,) = M + M ) a c , , = Pa(,,*

4,) = P4,4 (3)

4,) = P"40,. (4)

or, in terms of the initial ancestor frequencies by,

In order to describe the progress of populations in a system which behaves in this way, it will be convenient to define a measure of ancestral relationship between a pair of populations. Mathematically, the simplest computable measure of this kind, which expresses the proportion of their ancestry which two populations, i and j, have in common, would seem to be

When there is no common ancestry rir takes the value zero and when the ancestor frequencies correspond exactly it is unity, being the sum of these frequencies for either of the two populations. As an example of this measure in application, suppose that there are three populations X , Y and 2 with ancestor frequencies shown in the matrix

X Y z X 0.7 0'2 0' I Y 0.4 0.6 0

z 0' I 0 0.9

so that 70 yo of the ancestors of the present X population derived from the founder population of X, 20 % from that of Y and 10 % from that of 2. The ancestral relationship between X and Y is then

r,, = 0.4+ 0.2 + 0 = 0.6,

so that 60 yo of the ancestry of X has ancestry corresponding to it in Y and vice versa. The other relationships are

rxz = 0.2 and rpz = 0.1.

Ancestral relationship between two populations may thus be said to be accomplished if some prescribed proportion of the ancestry of the populations is common to them both. This proportion in practice may be some such value aa r = 0.95.

For several populations, perhaps a system of N populations, the ancestral relationship is obtained by slightly extending the above definition (5 ) . It will be the proportion of the ancestry of each population which is common to them all. Mathematically, this may be defined aa

N rT = 2 min (as).

8=1 i (6)

I 6-2


Following this definition, a fundamental result appears, namely that the total ancestral relationship between the populations of a system is not greater than the least amount of relationship between any pair of populations. This may be seen immediately, for let

ruv = min (Ti,), & I

N N N

8=1 s a= 1 24% v 8= 1 rT = x min (ai,) = 11 min (uh,u,,uv8) < min (uw,ue8) = ruv.

In general, the total relationship will be less than the least 'paired relationship' and these will be equal when two particular populations are slightly related whilst the remainder are more closely related, i.e. when for each 8 , f in at, = min (a&) takes one of only two values, u. and v

as defined above. For the above example this is the case. i

rT = min (0.7,0.4, O.l)+min (0-2, 0.6, O)+min (0.1, 0, 0.9) = 0.1

and rue = min (0.6, 0.2, 0.1) = 0.1.

The average or mean relationship between pairs of populations will be useful in describing the behaviour of a system. This quantity is not dependent on outlying populations being related to the same degree as central ones and may be more useful as an indicator of developing relationship at the centre of a system.

For the deterministic model outlined above, some analysis follows of the behaviour of simplified symmetric systems with constant and changing exchange rates. This analysis provides results for finite and infinite systems in terms of the ancestral relationship measure.

SYMMETRIC SYSTEM WITH EQUAL ENDOGAMY RATES

In a symmetric system with equal endogamy rates y for all populations, exogamy rates also being equal to p suppose, the exchange matrix P has elements

but provides a means of investigating the distance measure. Its simplicity allows its determi- nant, a well-known one used in the theory of experimental design, to be written down as

I PI = [ Y + P ( N - l ) l ( Y - P ) N - ' . (8) The distance function may be derived from a general form of Pn which is first determined by

means of the spectral resolution of P. This matrix has eigenvalues 1 and 1 - N p with multipli- cities 1 and N - 1 respectively. Hence by an established technique (see e.g. Bartlett, 1956), the nth power of P may be expressed in terms of these eigenvalues and the adjoint adj #(A) , where 4 = A I - P and A is an eigenvalue; this technique applies to give

where = (A-&JN-' and +2(A) = (A-A,). Since P is a stochastic matrix with equal off diagonal elements so is P" and it may be fully determined once a diagonal element PE is known.

P ( A - l + p ) ( A - l+Np)N-Z A- 1

( A - 1 + p ) ( A - 1 + p; =

1 - N - -+(l-Np)" J

(9)

and

Effects of movement on relatedness between populations 1 1 1

N - 1 N N pz = - ( 1 - P z ) = --- ( ~ - N P ) ~ .

243

(10)

Prom these expressions it is clear that, because N p < 1 ,

1 lim PE = lim P& = - -00 -03 N

and that PFd and P$ are respectively monotonically decreasing and increaaing functions of n. Also we may observe that for n = 0, P& = 1 and P!, = 0 corresponding to the earlier assumptions.

For any two populations there are ( N - 2 ) identical pairs of ancestor frequencies in this system so that the relationship between any pair of populations in the system is

r = ( N - 2)Kj+2 min (nil P$) = NP& = 1 - ( ~ - N P ) ~ . ( 1 1 )

The simplification is brought about by a practical constraint on the symmetric system. The effective endogamy rate is y > + and it follows that N p < 1 for N 2 2 but more strictly N p < p + 3 and p < 4 for N > 2. Furthermore the relationship between any pair of populations is equal to the total ancestral relationship of the system because of this constraint. In Fig. 4 this function is illustrated for various endogamy rates and numbers of populations in a system. The time of homogeneity is taken to be the time from having no common ancestry to having a 95% total common ancestry.

0.5 0.6 0.7 0.8 0.9 1.0 Effective endogamy rate

Fig. 4. Times to homogeneity for a symmetric system under various constant endogamy rates and with different numbers of population units.

For very large systems, where p is infinitesimal and N infinite, N p may be replaced by some finite p, so that

limr = 1-pn. (12 ) N-03

Here p is approximately the effective endogamy rate. At homogeneity, for some prescribed value of r , 1 - a suppose,

n = a b p ,

e.g. a = 0.05, r = 0.95 after 28 generations for p = 0.9.

(13)


NON-CONSTANT EXCHANGE RATES

Here we may consider the endogamy rates in the simplified symmetric system above to vary with time. In particular, in real systems the endogamy rates will decrease with time. Taking the simplest function of time, a linear decrease, and assuming only a small decrease c in the endogamy rate in each generation we may define the matrix gn corresponding to P n in the constant exchange case.

n

i = O 9" = n Pi = Po(Po+cA)(Po+2cA) ...( Po+cnA) = P$-l[Po+$cn(n+l)A], (14)

where -1 (i = j) A = {A,,} and A,, =

From the results (9) and (10) of the previous section it follows that

Furthermore, as in the previous section r may be expressed directly as

r = N 9 n . $3 = 1 - (1 -Np)n+ 4 cn (n+ 1) [ N / ( N - l)] (1 -Np)'+l. (16) Once more the time to homogeneity has been evaluated by solving iteratively for n the

equation for r = 0.95. The rate of decrease c, of the endogamy rate per generation, is assumed to be 0.001. The results are illustrated in Pig. 4.

For large systems, again replacing 1 - N p by p in (16)

lim r = 1 -pn-lh- 8 cn(n+ l)] (17)

and in this limiting case when the initial endogamy is near unity, putting ,u = 1 - IS with IS extremely small, it can be shown that

(18)

so that in the limit as IS + 0 r is composed of the first term only, which is due to the declining endogamy. From this expression we may determine the limiting time to homogeneity by solving equation (18) for n given some criterion for homogeneity T . Doing this gives for all but small integer solutions n,

When r = 0.95 and c = 0.001 this provides the solution n = 44 generations. If the decrease in endogamy is larger at c = 0.01, the solution is fourteen generations. These values of c seem to be realistic for some situations and these times which result from assuming complete endogamy initially are interesting if only because they are upper bounds to the times for lesser amounts of endogamy and smaller systems. This feature is illustrated in Fig. 5.

h'+m

r = + cn (n + 1) + nIS[ 1 - &(nz - I)],

n = ,i(2r/c). (19)

RESULTS

The data obtained from the parish registers on the numbers of endogamous, exogamous marriages between the parishes, and exogamous marriages with the outside world are presented for the two time-periods in Tables 1 and 2. The inter-parish exogamy is presented as PROM one parish TO another, but as already mentioned, the contribution of the parishes TO the outside

Effects of movement on relatedness between, populations 245

world is unknown. It is, however, of some interest to note that the movement of hereditary consequence in all cases is probably mainly in the opposite direction to that shown in the tables, since it appears to be customary for a woman to be married in her own parish and then sub- sequently to take up residence in that of her husband. This nevertheless is not invariable; men do marry in their own parish women of other parishes and also take up residence in the parish of their wives, as can be shown by linking up subsequent baptisms to these marriages. It

- VI

C 0 .- 2 80- aJ C aJ M v

x 60 0)

u .-

100 r I

- N = w

Effective endogamy rate

Fig. 5. Times to homogeneity for a symmetric system with diminishing endogamy rates of 0.001 per generation for each of the initial endogamy rates.

Table 1. Marriage numbers within and between the various parishes prior to 1850

From

BHS CFM 0 M WS WN AAB B OW T A r v

To

BHS CFM

303 I 0 6 297 8 8 I

3 20 132 I 3 I 2 I 82 3 2 I 2 260 6 2 4 3

3 5 4 5 3 I 2

4 6 104 3 I 0 3 109 2 I 54 2 7 32 2 4 78 98 5 89

4 115 65 4 432 3 168

427 445 216 I39 349 207

190 624

BHS, Beckley, Horton and Studley; CFM, Charlton, Fencott and Murcott; 0, Oddington; M, Merton; w s , Weston; WN, Wendlebury; AAB, Ambrosden, Arncott and Blackthorn; B, Boarstall; OW, ‘Outside World’. See Figs. 1 and 2 for geographical arrangement of these parishes.

Table 2. Marriage n7dmbers within and between the various parishes from, 1851 to 1966

From A

I 3

BHS CFM 0 M WS WN AAB B ow T

To

[62 I 3 I94 360 5 I22 8 4 3 4 2 I 26 274

20 16 I I I 2 I 44 86 I 6 37 3 13 81 141 I I * I05 2 I I33 243 I I 2 2 59 78 I43 I 7 I 0 I 6 196 2 172 395 4 5 4 34 I59 109

R. W. HIORNS AND OTHERS

therefore seems most appropriate to take the marriage register data at its face value and to count an entry of an exogamous marriage in a parish register as a contribution to that parish. One might have reasonably expected the number of mates exchanged between two parishes to be symmetrical, irrespective of comparative population sizes, and there is no indication that this is not so in the present data.

It is necessary to point out here that the different parish records begin a t different times, and while for some parishes data has been obtained from 1601 onwards, for others there is no information prior to the early seventeen hundreds. This means that the data on marriage movement prior to 1850 (Table 1) is not based on equivalent time periods for the different parishes and therefore the number of entries for the parishes is not a reflexion of their population sizes. It also follows that in this time period one cannot test for symmetry of movement. It might be argued that some standard time period should have been used, but this would have meant discarding information and limiting the analysis to the time period encompassed by the latest register (Merton, 1739). Problems would also arise from ‘gaps’ which occur at some time or other in most of the early registers. From the point of view of the analysis, the inequality of time periods and these gaps arenot of primary concern since the analysis is based on comparative endogamy and exogamy rates. In calculating these rates adjustment has been made for the fact that in an exogamous union only one of the partners moves, the other remaining in his native parish, and thus the relationship between two parishes is only established through one individual.

Table 3. Generations to homogeneity

Prior to 1850 Including ‘outside world’ Excluding ‘outside world ’

BHS BHS 19 CFM IOO CFM

20 16 16 M 112 60 60 M 23 22 22 21 WS 142 133 128 127 WS 21 20 19 20 23 WN 115 68 66 38 123 WN

20 20 20 21 23 15 19 B 96 85 82 88 128 89 99 B 23 23 23 23 25 16 22 18 OW

I9 I5 0 104 44 0

20 19 19 19 23 19 AAB 107 78 85 77 I37 85 AAB

BHS 11 CFM 11 6 0 10 10 9 M

‘I0 I1 I 1 9 I 0 I1 I 0 I 0

I2 I 1 I1 I 0

9 9 1 0 8 I 0 I 2 I 1 I 0

1851-1966 inclusive BHS 177 CFM 181 16 0 188 73 50 M

ws 207 146 140 130 WS

12 12 AAB 192 93 73 31 124 80 AAB 10 9 I I B 162 75 95 118 146 136 1 3 0 B I 0 I 0 I 2 I 0 ow

9 w N 195 109 99 86 105 WN

The effects, as determined from the mathematical model, of these comparative endogamies and exogamies in producing homogeneity are presented in Table 3 in terms of the number of generations required to bring any pair of parishes to a state where 95% of their ancestry is common. The pattern by which homogeneity is produced is also represented in Figs. 1 and 2 where, M a measure of over-all relatedness, the proportion of common ancestors between all possible pairs of parishes are averaged after different numbers of generations.


The profound effect of the exogamy between the various Otmoor parishes and the ‘outside’ world is clearly seen in Table 3 and by comparison with Figs. 1 and 2, from which it is evident that, whether one considers the pre- or the post-1850 periods, the time taken for pairs of parishes, or the whole assemblage, to become homogeneous is strikingly shorter when the outside world is included in the analysis than when it is excluded. There is nevertheless also a marked difference (when the outside world is included) in the two periods, and the mating patterns in the post- 1850 period lead to homogeneity much more rapidly, and with less variation in the time taken for pairs of parishes to become homogeneous, than in the early period. This is accounted for mainly by the over-all increase in exogamy and the concomitant decline in endogamy. It follows from the assumption that the outside world is stable, that at over-all homogeneity the Otmoor region is indistinguishable from the outside world, and one can regard the situation as a ‘flush system’ in which the whole Otmoor ancestry is ultimately distributed to an undetectable extent in the outside world and replaced by people with only an outside world ancestry. Nevertheless, although it is the contributions of the outside world which are mainly responsible for bringing pairs of parishes together, it is evident that the last population in the system with which the others become related tends to be the outside world. This, of course, also follows from assuming that the outside world is stable, since whilst there is any heterogeneity in the parish system at all, this must act to prevent any of the parishes coming into homogeneity with the outside world. The geographical pattern of developing relatedness is shown diagrammatically in Fig. 3, for the pre-1850 period.

Once the outside-world effect is ignored, the striking difference between the two periods in the generations to homogeneity disappears, and indeed in many instances it takes a longer time for two parishes to become completely related under the post-1850 exchange pattern than under the pre-1850 one. This situation arises from the fact that the increased total exogamy in the later period is due to increased exogamy with the outside world, no doubt arising from greater ease of movement with the advent of mechanized transport and improved road conditions. Ignoring this exchange actually leads in many cases to an apparent increase in the endogamy rate and hence a longer time to homogeneity.

However, the general pattern of developing homogeneity is not dissimilar in the two periods and has a broad concordance with the situations in which the outside world effects are included. In particular, Charlton and Oddington are the first two parishes to become related, whilst Weston and Beckley tend to be relatively late in becoming homogeneous with the other parishes. The ultimate equilibrium situation is of course different from that when the outside world is considered, and is determined by the over-all comparative absolute contributions that the different parishes make to one another. In other words, assuming other factors to be equal, it is related to population size; but as has been indicated, the estimates of population size from marriage numbers in the pre-1850 period are severely distorted by the inadequacies of the parish registers.

It is therefore meaningless to indicate the equilibrium frequenoy established on the pre-1850 exchange pattern, but the final composition determined for the post-1850 period is

BHS CFM 0 M WS WN AAB B 0.498 0.125 0.011 0.032 0.067 0.098 0.126 0.041


DISCUSSION

The prime purpose of this paper was to examine in general terms how relatedness between populations can be expressed, analysed and determined from knowledge of movement patterns, but the issues arose from consideration of a particular situation concerning a group of neigh- bowing Oxfordshire Parishes, and the analysis of this situation has highlighted a number of interesting problems.

In many ways the analysis presented of the Oxfordshire case is unreal. Assumptions have had to be made about the ‘nature of the “outside world”’, or the effects of the ‘outside world’ have been ignored. Only the effects resulting from marriage movement have been considerd. Artificial conditions have been imposed on family size and structure. But most important of all, in determining the ‘generations to homogeneity’ it has been assumed that at the ‘starting point’ all the populations in the system were unrelated; whjch is an almost impossible situation. Nevertheless, it seems to us that this analysis does afford some real insight into the concept of relatedness between populations and how this may be measured from demographic data,

The model corresponds in some ways to the genetic one in which each of a series of founding populations has its own unique allele at a 100 % frequency but between which a system of gene flow through intermarriage is established. However, calculating rates at which genetic horno- geneity is attained from knowledge of exogamy alone presents the difficult problem of esti- mating the cumulative effect of genetic drift which, even with the assumptions made about family structure in the present analysis, would still operate within the genetic model. By considering deterministically the ancestor situation this difficulty is circumvented.

The analysis clearly indicates the importance of exogamy as a faotor determining the composition of the Oxfordshire populations, even before the advent of mechanized transport, and the assumption of a starting-point of no relatedness between the parishes means that all the estimates of ‘generations to convergence’ are maximal. One might therefore expect no genetic diversity between the various parishes or between them and the surrounding areas, but we have no direct information on this. It certainly seems unlikely in the face of the very strong homo- genizing effect of the exogamy that within such 8 restricted and apparently environmentally constant area disruptive selection would produce diversity, or that drift would operate to any diversifying effect even though the populations are not large. However, the analysis has been based on the assumption of random movement and there is some evidence from the frequency of surnames in the parish registers that each of the parishes tends to have a nucleus of ‘old families’ in whioh the males at least are not mobile. So far as marriage movement is concerned, with the customary habit of married couples residing in the husband’s parish this is not un- expected, but it is possible that some families are more mobile than others, and this might permit some retention of initial diversity, especially if the old families tended to be the endogamous ones. It would certainly seem true that most of the relatedness between populations arising &om marriage movement is established through females.

Although all the ‘times to convergence’ are maximal estimates, the patterns in which relatedness develop are the same as those determined here, whatever the initial levels of relatedness, as long as these levels were the same for all the parishes in the system. It is therefore of some interest that these patterns are essentially similar over $he two historic periods considered and that they are not solely a function of geographic distance. It has been shown else-


where that local patterns of movement have not been profoundly affected by the advent of mechanized transport, and that such features as topography and road systems affect the direction of marriage movement (Boyce et al. 1968).

Whilst zero levels of initial relatedness are usually not found, the situation is not completely artificial. Thus when two or more groups which previously have been long geographically isolated from one another come to take up residence in the same area, the situation must be very like the one analysed here and the model is clearly applicable to the formation of hybrid populations. The converse of this extreme situation is when a single population becomes divided into a series of isolated subgroups and in the genealogical sense there is diminishing relationship between these subgroups with the passage of generations. In the absence of any gene flow between them this amounts to a reduction by one quarter with every generation. Typically, of course, both processes of diminishing relatedness with isolation and increasing relatedness through gene flow are occurring simultaneously, and defined in these terms levels of relatedness depend upon the comparative magnitude of the endogamy and exogamy. However, in this paper relatedness has been defined in terms of the distribution of ancestors and this negates any necessity to consider the diminishing relationship with increasing generation span.

It may be noted in conclusion that the mathematical models present some wider issues when some simplifying assumptions can be made. Thus, for instance, Fig. 4 shows the time to homogeneity at any particular exchange endogamy rate and with differing numbers of populations in the system, when it is assumed that the exchange rates are symmetric, and the exogamy component for each population is distributed evenly with respect to all the other populations. The infinite system indicates that it would require only about thirty generations for the whole world to become homogeneous, even if 80 % of all marriages were endogamous (i.e. effective endogamy 90 yo), but the exogamy component was evenly distributed the world over, and for many of the unit populations this again would be a maximal time. The evidence from the Oxfordshire study indicates a declining endogamy with historic time and this no doubt is a fairly world-wide phenomenon. Figure 6 represents in general terms the effect of such a decline on the times to homogeneity making the same simplifying assumptions about symmetry and even endogamy as mentioned beforehand. Here the effective endogamy rate has been considered for varying initial rates of effective endogamy to decrease per generation by 0.1 yo.

Again, it is impressive to see how little time is taken to produce homogeneity, even when a very large number of population units are considered in the system. Thus, for instance, in an infinite system starting with an effective endogamy of 90 % it requires only twenty-five generations to come to homogeneity.

SUMMARY

A study has been made of the effects of migratory movement between a series of neighbouring parishes in the Otmoor region of Oxfordshire. Using data obtained from the marriage registers on the comparative endogamy and exogamy, and the distribution of the exogamous unions between the different parishes and the ‘outside world’, a deterministic model has been produced of the over-all effects of gene flow on the relationships between the different populations. Relationship for this purpose is taken to mean the extent to which ancestry is shared and the populations are considered to be identical with regard to their ancestors, when some set proportion of their ancestors are common to both populations.


As a starting-point each of the populations is considered as having its own distinctive ancestry. Relationship is measured by means of a matrix constructed from the marriage exchange rates. It is shown that although the Otmoor region is reputedly an isolated one, the exogamy rates with the ‘outside’ world are so high that after approximately twenty generations the different populations are all likely to be indistinguishable from the surrounding areas. On the other hand, if one ignores the exchange with the outside world, the marriage exchange patterns within the area take much longer to produce over-all homogeneity.

The rate of the convergence of the different populations upon each other is shown to be related to their comparative endogamy and spatial distribution, but the ultimate equilibrium of ancestor distribution is determined, through the over-all exogamy, by comparative population size.

Grateful acknowledgement is made to Professor M. S. Bartlett for his comments upon a draft of this paper and to the Nuffield Foundation for a grant which made this study possible.

REBERENOES

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A mathematical analysis of the effects of movement on the relatedness between populations