A comparison of four systems of group mating for avoiding inbreeding

Une comparaison de quatre systemes d'accouplements par groupe pour eviter la consanguinite. Un regime d'accouplement par groupe de type circulaire est considere comme l'un des systemes les plus efficaces pour eviter la consanguinite. Dans ce systeme, la population est entretenue en groupes separes et les mâles sont transferes de leur groupe au groupe voisin d'une maniere circulaire. D'autres systemes, tels que le systeme de Falconer, le systeme rotatif de HAN et le systeme de Cockerham, ont ete proposes par ailleurs. Ces derniers sont regroupes sous l'appellation de systemes cycliques, puisque le transfert des mâles d'un groupe a l'autre obeit a un rythme cyclique. Dans cette etude, on compare le systeme circulaire aux trois systemes cycliques du point de vue de l'augmentation de la consanguinite au cours des premieres generations et au bout d'un nombre tres eleve de generations. On montre que les systemes cycliques entrainent une consanguinite moindre que le systeme circulaire au cours des premieres generations et tant que le nombre de generations reste faible. Le systeme circulaire donne une consanguinite legerement plus faible a partir de la 100 e generation, stade auquel le coefficient de consanguinite atteint ou depasse 60 %. Si on considere que la consanguinite dans les premieres generations est le facteur determinant de persistance d'une population, on peut conclure que les systemes cycliques sont a recommander de preference a un regime d'accouplement de type circulaire. La consanguinite dans les systemes cycliques s'accroit selon des rythmes oscillatoires d'amplitude variable, mais autour de moyennes tres peu differentes. Parmi les trois systemes cycliques, on peut recommander le systeme de Cockerham pour des nombres pairs de groupes et le systeme rotatif de HAN pour des nombres impairs, qui sont ceux qui montrent les plus faibles amplitudes d'oscillation.

Summary -Circular group mating has been considered one of the most efficient systems for avoiding inbreeding. In this system, a population is conserved in a number of separate groups and males are transferred between neighbouring groups in a circular way. As alternatives, some other mating systems such as Falconer's system, HAN-rotational system and Cockerham's system have been proposed. These systems as a whole are called cyclical systems, since the male transfer between groups changes in a cyclical pattern. In the present study, circular group mating and the three cyclical systems were compared with respect to the progress of inbreeding in early and advanced generations after initiation. It was derived that the cyclical systems gave lower inbreeding coefficients than circular group mating in early and not much advanced generations. Circular group mating gave slightly lower inbreeding after generations more advanced than 100, when the inbreeding coefficient became as high as or higher than 60%. Considering that it is primarily inbreeding in early generations that determines the persistence of a population, it is concluded that cyclical systems have a wider application than circular group mating. Inbreeding in the cyclical systems increased in oscillating patterns, with different amplitudes but with essentially the same trend. Among the three cyclical systems, Cockerham's system for an even number of groups and the HAN-rotational system for an odd number are advisable, since they exhibited the smallest amplitudes of oscillation.

INTRODUCTION
The number of individuals maintainable in most conservation programmes of animals is quite restricted due to financial and facility limitations. In such a situation, inbreeding is expected to seriously harm the viability of populations, and thus the development of strategies for minimizing the advance of inbreeding is one of the most important problems to be solved. Circular group mating, sometimes called a rotational mating plan, is one of the systems proposed for avoiding inbreeding (Yamada, 1980;Maijala et al, 1984;Alderson, 1990aAlderson, , b, 1992. In this mating system, a population is subdivided into a number of groups and males are transferred between neighbouring groups in a circular way. This system has been adopted in several conservation programmes of rare livestock breeds (Alderson, 1990a(Alderson, , b, 1992Bodo, 1990).
The theoretical basis of circular group mating was established by Kimura and Crow (1963). In their theory, the rate of inbreeding in sufficiently advanced generations after initiation was shown to be smaller with circular group mating than with random mating. This ultimate rate of inbreeding, however, is not the only criterion for measuring practical use. Mating systems which reduce the ultimate rate of inbreeding tend to inflate inbreeding in early generations after initiation (Robertson, 1964). If circular group mating causes a rapid increase of inbreeding in early or initial generations, its application should be limited. Besides circular group mating, some other systems of male exchange, such as Poiley's system (Poiley, 1960), Falconer's system (Falconer, 1967), Falconer's maximum avoidance system (Falconer, 1967), the HAN-rotational system (Rapp, 1972) and a series of Cockerham's systems (Cockerham, 1970), have been proposed.
These systems as a whole are called cyclical systems in the sense that the pattern of male exchange between groups changes cyclically (Rochambeau and Chevalet, 1982). Assuming a simple model of each group consisting of only one male and one female, Rapp (1972) computed the progress of inbreeding in the initial ten generations for the first four of the systems mentioned above, leading to the conclusion that the HAN-rotational system gave the smallest rate of inbreeding.
The four cyclical systems were compared also by Eggenberger (1973) with respect to genetic differentiation among groups in various combinations of population size and number of groups. He showed that, while there are only small differences among the four systems in the initial 20 generations, Poiley's system ultimately caused a larger genetic differentiation among groups. Matheron and Chevalet (1977) studied inbreeding in a simulated population maintained with a system called the third degree cyclical system of Cockerham. They found that by this system the inbreeding coefficient in the first ten generations was lowered by 3% compared to a random mating system. Rochambeau and Chevalet (1985) investigated some particular types of cyclical system, some of which belong to the HAN-rotational system, and showed that the cyclical system for some numbers of groups causes lower inbreeding than circular group mating in the initial 20 years. A comparison of various types of cyclical systems with circular group mating is important for choosing an optimal mating system but remains to be investigated more systematically. In this paper, three cyclical systems (ie, Falconer's, HANrotational, and Cockerham's), in which the rule of male transfer is explicitly defined, will be compared with circular group mating taking account of the progress of inbreeding in both early and advanced generations. Based on this, optimal mating systems for animal conservation will be discussed.

MODEL AND METHOD
A population which is composed of m groups with N m males and N f females in each is considered. The total numbers of males and females in this populations are then N M = mN m and N F = mN f respectively. Mating within groups is assumed to be random with random distribution of progeny sizes of male and female parents, so that the effective size (N e ) of a group is 4N m N f /(N m + N f ) . Circular group mating and three cyclical systems, ie, Falconer's system, the HAN-rotational system, and Cockerham's system, are investigated. Another system, maximum avoidance system of group mating (Falconer, 1967), is not investigated, because in this system males are exchanged among groups in the same way as in Wright's system of maximum avoidance of inbreeding (Wright, 1921). Application of this system is limited to cases where the number of groups equals an integral power of 2, and the inbreeding coefficient under this mating system increases at the same rate as in Cockerham's system. Poiley's system, although it was proposed earlier than the other cyclical systems, is not investigated either. In Poiley's idea, the rule of male transfer was not consistently defined; different rules seem to be used with different numbers of groups. Also, as pointed out by Rapp (1972), the inbreeding coefficient under this system converges to different values in different groups, meaning that the progress of inbreeding cannot be formulated by a single recurrence equation.
In circular group mating, all males in a group are transferred to a neighbouring group every generation. Figure 1(a) shows a case where the population is composed of four groups. In the three cyclical systems, all males in group i (= 1, 2, ... , m) in generation t (= 1, 2, ... , t c ) are transferred to group d(i, t), a function defined below. This male transfer pattern is repeated with a cycle of t c generations. In Falconer's system, t c and d(i, t) are given as and Figure 1 (b) describes the pattern in one cycle of male transfer for m = 4.
In the HAN-rotational system, the male transfer system is different according to whether the number of groups (m) is even or odd. With an even value of m, t c and d(i, t) are given as and where CEIL(log 2 m) is the smallest integer greater than or equal to 1 092 m. When the number of groups is odd, t! is obtained as the smallest integer which makes (2 t c -1)/ m equal to an integer, eg, t c = 4 for m = 5. The function d(i, t) for odd values of m is given as where MOD(2 t -1 , m) is the remainder of 2!!! divided by m. Figure l(c) and (d) illustrate one cycle of the male transfer, for m = 4 and 5 respectively.
A series of Cockerham's system is defined depending on the length of cycle NUMERICAL COMPUTATIONS In contrast with the progress of inbreeding in circular group mating, where males are transferred from the same neighbouring groups each generation, the inbreeding in a cyclical system is expected to increase in an oscillating pattern, since the relatedness of any two mated groups differs with generations. The oscillation pattern has been confirmed by Beilharz (1982) for a mouse population maintained by Falconer's system with each group being composed of one pair of parents. The oscillation was studied theoretically by Farid et al (1987). To evaluate the advantage of the cyclical systems, not only the average rate (trend) of inbreeding per generation but also the amplitude of oscillation should be taken into account. Numerical computations revealed that, in all of the cyclical systems, the rates of inbreeding in generations within a cycle of male transfer (Af t' i t' = 1,2,..., t c ) converge to steady values of their own after the third or fourth cycle. The steady values in generations in the fifth cycle were obtained for the cases of 4 to 21 groups of size N m = 2 and N f = 4, and are presented in table II. The rates of inbreeding with different sizes of N m and N f (data not presented) showed the same oscillation pattern though with different amplitudes. In Falconer's system, a large increase of inbreeding takes place in the first generation regardless of the number of groups (m). When m is odd (table II(b)), this large increase occurs also in the (m + 1)/2-th generation. The large positive values of AF t' are followed by large negative values, indicating that the inbreeding coefficient increases in a cyclically oscillating pattern of rise and fall. AF t' takes a large positive value in generations when males are transferred back to the groups from which their fathers came. In such generations, two individuals paired may have a common grandparent, so that the progenies produced are more highly inbred than those in the previous generation. This high inbreeding, however, is reduced immediately in the next generation, since females are mated with males from less related groups.
In HAN-rotational and Cockerham's systems, the pattern of AF t , varies with the number of groups. When m is an integral power of 2, the two systems show no oscillation in AF t , and give the same rate of inbreeding each generation. With other even numbers of m, the two systems show different patterns of AF t' ? in the HAN-rotational system, AF t , starts with a negative value and ends with a positive value (table II(a)), whereas in Cockerham's system, AF t' takes positive values in all cases but that of m = 14, indicating that the inbreeding coefficient increases steadily without oscillation. When m is odd, the HAN-rotational system always gives a constant A!'. In Cockerham's system, AF t , is constant only when m = 5, 9 and 17. In general, in Cockerham's system with m = 2' + l, (n = 1, 2, 3, ...), groups mated in any generation share 2 n -1 common ancestral groups in the n &mdash; 1 th previous generation, so that the relatedness among the groups mated, and therefore AF t ,, stays constant in all generations. Figure 2 illustrates the increase of inbreeding coefficients in the initial 30 generations under the four group mating systems with numbers of groups varying from 4 to 15. Inbreeding in an undivided randomly-mating population of a size N M + N F was added as a check, being computed by Wright's formula (Wright, 1931), where N E ( RM ) is the effective population size, 4N M N F/ (N M + N F ).
When the number of groups m is 4 or 5, there are only small differences between the four systems for any total population size. In circular group mating an increase in m leads to increased inbreeding. When m is larger than 6, the inbreeding coefficient in circular group mating surpasses that in random mating. The inflation of inbreeding is ascribed to an increased occurrence of single cousin mating. In circular group mating, females in a group are mated with males coming from the same group in all generations, and so single cousin mating occurs with a probability of Since N!, and N F are both > m, this probability is an increasing function of m.
For the same reason, the oscillation amplitude of inbreeding under cyclical systems is enhanced with an increase in the number of groups. However, except in some generations when the inbreeding increases sharply, the inbreeding coefficients are smaller than those with random mating. This superiority of cyclical systems over random mating increases with increasing number of groups, though the difference is fairly trivial when the total population size is large. Table III presents the effective population size for random and four group mating systems. (Average effective size N E , defined in the Appendix, is presented for cyclical systems). These effective population sizes are asymptotic ones and determine the progress of inbreeding after sufficiently many generations. The effective population sizes in the four group mating systems are always greater than that of random mating. In all group mating systems an increase in m results in an increased effective population size. With a given number of groups and total population size, the effective population size in circular group mating is larger than those in the three cyclical systems, meaning that the inbreeding under circular group mating will eventually become smaller, although it is larger in early generations (cf fig 2), than under the cyclical systems.
The critical generations in which the inbreeding coefficients in circular and cyclical mating reverse in rank, together with the inbreeding value per se in these generations, are presented in table IV. The critical generations depend largely on the total population size; the reversal occurs at later generations with larger populations sizes. The values of inbreeding in this turning-point generation, however, do not differ much from each other, ie, the inbreeding coefficient lies within the range of 60-73% over all of the group numbers and total population sizes calculated.
Calculations of the inbreeding coefficients after this critical generation (data not presented) showed that the superiority of circular mating over cyclical matings is rather trivial, ie, less than 2% in all of the cases studied (4-35 groups and 300-2 000 generations). Kimura and Crow (1963) studied three types of circular mating, ie, circular individual, circular pair, and circular group mating, and concluded that the effective population size of the three circular mating systems is larger than that of random mating. Robertson (1964) obtained a more generalized conclusion, saying that not only the circular mating systems as studied by Kimura and Crow (1963), but any other types of population subdivision, like the cyclical systems discussed in this paper, enlarge the effective population size. Superiority of group matings over random mating is rather small, however, unless the total population size is small. It is shown in figure 2 that when N E is as large as 250, a population may be maintained either in one location as one population or in separate groups (by cyclical mating). In many conservation programmes, the number of animals maintainable in one location is severely restricted due to the limited facilities. In such cases the population must be maintained in a number of small groups located in different stations such as zoological gardens and natural parks. As Yamada (1980) suggested, group mating should be used in this situation, since it is not only effective in suppressing the progress of inbreeding but also much easier to practise than random mating of the whole population (Rochambeau and Chevalet, 1990). Maintenance in different locations has the additional merit of reducing the risk of accidental loss of the population.

DISCUSSION
In the numerical computations presented in figure 2 and tables III and IV, the best system of group mating changes with duration (generations) of population maintenance. The cyclical systems would be recommended for shortor intermediate-term maintenance, ie, a few hundreds or fewer generations as far as a population with an effective size of about 100 is concerned. The advantage of the cyclical systems is more prominent when the population is partitioned into more groups. For long-term conservation, on the other hand, circular group mating appears superior, though only slightly, to the cyclical systems (tables III and IV).
Application of circular group mating is rather limited, however. The superiority of circular group mating is attained only after some generations where the inbreeding coefficient has already exceeded the critical level of inbreeding (50-60%) for survival of animal populations (Soul6, 1980). When, as is the case in many practical projects, the population is initiated from relatively highly related animals, its survival will be determined by the progress of inbreeding in early generations. To avoid extinction due to inbreeding depression in early generations, cyclical systems should be preferred. When pedigree information of the starting animals is available, the initial groups may be constructed with the minimum relatedness among groups, leading to a further reduction in the rate of inbreeding in initial generations (Rochambeau and Chevalet, 1982).
Inbreeding proceeds in any population of a limited size. This does not necessarily mean that the population will eventually become extinct by inbreeding depression. Inbreeding, if it proceeds gradually, may not cause serious inbreeding depression, because deleterious alleles can be eliminated gradually through selection. The slower the progress of inbreeding, the greater the opportunity for the deleterious alleles to be eliminated (Lande and Barowclough, 1987). In this respect also, cyclical group mating is superior to circular mating. It is concluded that cyclical systems have much wider application than circular group mating. Of the three cyclical systems examined, the one which exhibits the least oscillating pattern of inbreeding should be chosen. A sharp increase in inbreeding in one generation may cause a serious inbreeding depression, as evidenced in the experiment of Beilharz (1982) using a mouse population. As seen in table II, the best system differs with the number of groups (m): with an odd number of groups the HAN-rotational system is recommended, while Cockerham's system is advisable for an even number.
The number of groups is a key factor determining the result of a conservation programme (Rochambeau and Chevalet, 1982). As indicated in figure 2, a large number of groups is more favourable. The number of groups, however, cannot be large when the total number of animals is limited; each group must be sufficiently large to ensure its persistence. Optimal allocation of the animals must be chosen, taking account not only of the genetic factor as discussed in this paper but also demographic parameters like reproductive and survival rates.
Only inbreeding has been investigated in the present study. When conserving animals as genetic resources, the maintenance of genetic variability must also be taken into account. Theoretically, the largest genetic variability will be maintained when the population is split into a number of completely isolated groups, but on the other hand a high rate of inbreeding will be caused within the groups (Kimura and Crow, 1963;Robertson, 1964). To compromise between these two conflicting effects, not all but some appropriate proportion of males or females may be exchanged among groups. The mating system proposed by Alderson (1990b) is one possible system to cope with this problem. In his system, a proportion of females in each group are transferred to a neighbouring group in the same pattern as in the circular mating, the remaining females being mated to males of the same group. The effectiveness of systems with partial transfer of males or females is another problem to be addressed, and will be thoroughly discussed elsewhere.