A comparison of four systems of group mating for avoiding inbreeding

HAL Id: hal-00894127

https://hal.archives-ouvertes.fr/hal-00894127

Submitted on 1 Jan 1996

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

A comparison of four systems of group mating for

avoiding inbreeding

T Nomura, K Yonezawa

To cite this version:

Original

article

A

_comparison

of four

_systems

of

_group

mating

for

_{avoiding inbreeding}

T

Nomura,

K Yonezawa

Faculty of Engineering, Kyoto Sangyo University, Kyoto 60.3, Japan

(Received

22

_{February 1995; accepted}

15 November

₁₉₉₅₎

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.

inbreeding avoidance

_/

_inbreeding coefficient

_/

group mating

/

conservation of animal

population

/

effective _population size

Résumé - Une _comparaison de quatre systèmes d’accouplements par groupe pour

éviter la consanguinité. Un

_{régime d’accouplement}

par groupe de type circulaire est

considéré comme l’un des

_systèmes

les

plus efficaces

_pour éviter la

_{consanguinité.}

Dans ce

_système,

la

population

est entretenue en _groupes

_séparés

et les mâles sont

_transférés

de leur groupe au _groupe voisin d’une manière circulaire. D’autres

_systèmes,

tels que

le

système

de

Falconer,

le

système rotatif

de HAN et le

système

de

Cockerham,

ont

été

proposés

par ailleurs. Ces derniers sont

regroupés

sous

_{l’appellation}

de

systèmes

cycliques, puisque

le

_transfert

des mâles d’un _groupe à l’autre obéit à un

_{rythme cyclique.}

Dans cette

_étude,

on _compare le

_système

circulaire aux trois

_{systèmes cycliques}

du point

de vue de

_{l’augmentation}

de la

_{consanguinité}

au cours des

_{premières générations}

et

entraînent une

consanguinité

moindre _que le

_système

circulaire au cours des

_premières

générations

et tant que le nombre de

_{générations}

reste

_faible.

Le

_système

circulaire donne

une

_{consanguinité légèrement plus faible}

à partir de la 100e

_{génération,}

stade

auquel

le

coefficient

de

consanguinité

atteint ou

dépasse

60 %. Si on considère _que la

consanguinité

dans les

premières générations

est le

facteur

déterminant de persistance d’une

population,

on _peut conclure _que les

_{systèmes cycliques}

sont à recommander de

_préférence

à un

régime d’accouplement

de _type circulaire. La

_{consanguinité}

dans les

_{systèmes cycliques}

s’accroît selon des

_rythmes

oscillatoires

_{d’amplitude variable,}

mais autour de moyennes

très peu

différentes.

Parmi les trois

_{systèmes cycliques,}

on _peut recommander le

_système

de Cockerham pour des nombres pairs de groupes et le

_{système rotatif}

de HAN pour des

nombres

impairs,

qui sont ceux

_qui

montrent les

_{plus faibles amplitudes}

d’oscillation.

évitement de la _{consanguinité}

_{/ coefficient}

de _{consanguinité}

_/

accouplement par groupe

/

conservation animale

/ effectif

génétique

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,

_{1990a, 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, b, 1992; Bodo,

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

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,

HAN-rotational,

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

sys-tem,

maximum avoidance

_system

of group

mating

(Falconer, 1967),

is not

inves-tigated,

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

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

t

c =

1, 2, ... ,

TRUNC(log

2

m),

where

TRUNC(log

2

m)

is the

largest

integer

which is smaller than or

equal

to

₁

₀₉₂

m. In this

_series,

the function

d(i, t)

is defined as

In an extreme case of

_t

_c

=

1,

the male transfer follows the _same

_pattern

as in circular group

mating.

The

_system

with the maximum

_length

of

_cycle,

ie,

t

c

=

TRUNC(log

2

m),

is

investigated

in this

_study.

Under this

_system,

_genes of mated individuals have no common ancestral _{groups in}

_t

_c

_preceding

_generations

(Cockerham, 1970).

When m is an

_integral

_power of

_2,

Cockerham’s

_system

is identical to the HAN-rotational

_system.

The

_pattern

with m = 5 is illustrated in

figure

1(e).

For

_{comprehension,}

male transfers from _group

_1,

_ie,

the values of

_{d(1, t)}

in the three

_cyclical

_systems,

are

presented

in table I for m = 4-20.

The

_inbreeding

coefficient and the

_inbreeding

effective

_population

size for circular

group

mating

are

_computed

_by

the method of Kimura and Crow

_(1963).

With

appropriate

modifications as described in the

_Appendix,

this method can be

applied

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 2n - 1 common ancestral _{groups in} the

n — 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 gen-erations under the four group

mating

systems

with numbers of groups

size

_N

_M

+

N

F

was added as a

_check,

_being

_computed

by Wright’s

formula

(Wright,

1931),

where

_E

₍

_RM

₎

_N

is the effective

_population

size,

(N

M

4N

F/

N

M

+

_N

_F

_).

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

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

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

DISCUSSION

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.

In the numerical

_computations

_presented

in

_figure

2 and tables III and

_IV,

the best

_system

_{of group}

_mating

_changes

with duration

_{(generations)}

of

popu-lation maintenance. The

_cyclical

_systems

would be recommended for short- or

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

ap-pears

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

inbreed-ing

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

practi-cal

_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

avail-able,

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

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.

REFERENCES

Alderson L

_(1990a)

The work of the rare breeds survival trust. In: Genetic Conservation

of Domestic

Livestock

_(L

Alderson,

ed),

CAB

_{International, Wallingford,}

32-44 Alderson L

_(1990b)

The relevance of

_genetic

improvement programmes within a

_policy

for

_genetic

conservation. In: Genetic Conservation

_of

Domestic Livestock

_(L

_Alderson,

ed),

CAB

_{International, Wallingford,}

206-220

Alderson L

₍₁₉₉₂₎

A system to maximize the maintenance of

_{genetic variability}

in small

populations.

In: Genetic Conservation

_{of Domestic}

Livestock

_(L

_Alderson,

I

_Bodo,

_eds),

Vol

_2,

CAB

_{International,}

_Wallingford,

18-29

Beilharz RG

₍₁₉₈₂₎

The effect of

_inbreeding

on

_reproduction

in mice. Anim Prod

_{34, 49-54}

Bodo I

₍₁₉₉₀₎

The maintenance of

_Hungarian

breeds of farm animals threatened

_by

extinction. In: Genetic Conservation

_of

Domestic Livestock

_(L

Alderson,

ed),

CAB

International,

Wallingford,

73-84

Cockerham CC

₍₁₉₇₀₎

Avoidance and rate of

_inbreeding.

In: Mathematical

_Topics

in

Population

Genetics

_(K

_Kojima,

_ed),

_Springer,

New

_York,

104-127

Eggenberger

Von E

₍₁₉₇₃₎

_{Modellpopulationen}

zur

_Beurteilung

von

_{Rotationssystmen}

in

Falconer DS

₍₁₉₆₇₎

Genetic aspects of

_breeding

methods. In: The UFAW Handbook on

the Care and

Management of Laboratory

Animals

_(W

Lane-Petter,

ed),

3rd

edition,

Livinstone, Edinburgh,

72-96

Farid

_A,

Mararechian

_M,

Stobeck C

₍₁₉₈₇₎

_Inbreeding

under a

cyclical mating

_system.

Theor

Appl

Genet

_73,

506-515

Kimura

M,

Crow JF

₍₁₉₆₃₎

On the maximum avoidance of

_inbreeding.

Genet Res Camb

4, 399-415

Lande

R, Barrowclough

GF

₍₁₉₈₇₎

Effective

population

size,

genetic variation,

and their

use in

_population

_management. In: Viable

_{Populations for}

Conservation

_(ME

_Soul6,

ed),

Cambridge University Press, Cambridge,

87-123

Maijala K,

Cherekaev

AV,

Devillard

JM,

Reklewski

Z, Rognoni G,

Simon

DL,

Steane D

(1984)

Conservation of animal

_genetic

resources in

_Europe.

Final report of the

_European

Association of Animal Production

_{Working Party.}

Livest Prod Sci

11,

3-22

Matheron

_G,

Chevalet C

₍₁₉₇₇₎

_Design

of a control line of rabbits:

_Expected

evolution of

individual

_inbreeding

coefficients

_according

to various

mating

schemes. Ann Genet Sel

Anim 9, 1-13

Poiley

SM

₍₁₉₆₀₎

A

systematic

method of breeder rotation for non-inbred

laboratory

animal colonies. Proc Anim Care Panel

10,

159-166

Rapp

G

₍₁₉₇₂₎

_{HAN-rotation,}

a new _system for

_{rigorous outbreeding.}

Z Versuchstierk

_14,

133-142

Robertson A

₍₁₉₆₄₎

The effect of non-random

_mating

within inbred lines on the rate of

inbreeding.

Genet Res Camb 5, 164-167

de Rochambeau

H,

Chevalet C

₍₁₉₈₂₎

Some aspects of the

genetic

management of small breeds. In: Proc 2nd World

_Cong

Genet

_Appl

Livest

Prod, Madrid, VII,

282-287 de Rochambeau

_H,

Chevalet C

₍₁₉₈₅₎

Minimisation des coefficients de

_{consanguinit6}

moyens dans les

petites

populations

d’animaux

domestiques.

Genet Sel Evol

_17,

459-480 de Rochambeau

H,

Chevalet C

₍₁₉₉₀₎

Genetic

principles

of conservation. In: Proc

_4th

World

_Cong

Genet

_Appl

Livest

_{Prod, Edinburgh, XIV,}

434-442

Soul6 ME

₍₁₉₈₀₎

Thresholds for survival:

_maintaining

fitness and

_{evolutionary potential.}

In: Conservation

_Biology.

An

_{Evolutionary-Ecological Perspective}

_(ME

_Soul6,

BA

Wilcox,

eds),

Sinauer

_{Associates, Sunderland,}

151-169

Wright

S

₍₁₉₂₁₎

_Systems

of

_mating.

Genetics

6, 111-178

Wright

S

₍₁₉₃₁₎

Evolution in Mendelian

populations.

Genetics

_16,

97-159

Yamada Y

₍₁₉₈₀₎

The

_importance

of

_mating

_systems in the conservation of animal

_genetic

resources. FAO Animal Production and Health

_{Paper 24,}

268-278

APPENDIX

Following

Kimura and Crow

_(1963),

we define the

_following

_quantities:

F

t

= the

inbreeding

coefficient _at

_generation

_t,

J

t

(0)

= the

probability

that _two

homologous

genes of two

_newly

born individuals

within a _group are identical

_{by descent, subscript t showing}

_generation,

J

t

(k)

= the

probability

that _two

homologous

genes of two

newly

born individuals in two different groups of distance k are identical

by descent, k being

in the range

1 to n =

m/2

when _m is

a recurrence

_relation,

holds for circular _group

_mating,

where G is the transition matrix. With the

_largest

eigen

value

_(A)

of the matrix

_G,

the

_inbreeding

effective

population

size

_(N

_E

₎

is

computed by

For a

_cyclical

_system

of _group

_mating,

the recurrence relation varies with

generations,

but the same set of relations is

_repeated

with a

_cycle

of

t

c

generations.

Thus,

we need

_t

_c

_matrices,

G

1

,

_{, ... ,}

₂

_G

_G

_te

_,

With these

_matrices,

the recurrence relation is

_generally

written as

where

_{subscript t’ (= 1, 2, ... , t!)}

shows a

_generation

a

_generation

within the

cycle,

formulated

as t’

=

MOD(t -

1, t

c

)

_{+ 1.} The matrices

_G

_t

_,

_are obtained _{in a}

way

similar to Kimura and Crow

_(1963).

For

_example,

the matrices for Cockerham’s

system

with m = 6 _are formulated _as

Since the rate of

_inbreeding

is determined

_{only through}

the effective size of the

group

e

N

=

4N

n

,N

f

/(N

n

,

₊

) ,

N

f

the _sex ratio has _no effect _on the _rate of

inbreeding

so

_long

as

_N

_e

is constant.

In some cases as shown in table

II,

the rate of

_inbreeding

per

generation

changes

from

_generation

to

_{generation. However,}

the rate of

_inbreeding

_per

_cycle,

ie,

(F

t+t

, -

F

t

)

/(1 - F

t

) ,

approaches

an

_asymptotic

value after

_sufficiently

_many

cycles. Defining

a matrix

_G

_c

as

the

_asymptotic

value

_(AF

_C

₎

is

_obtained,

with

A

c

showing

the

_largest

_eigen

value of

Introducing

OF

and

AF

t’

as the _average rate of

_inbreeding

per

generation

and

the rate of

_inbreeding

at

_{generation t’}

within a

cycle, respectively,

a relation

is formulated. From

equations

[2]

and

_!3!,

OF

is obtained as