Supportive Breeding and Inbreeding Effective Number:
Reply to Nomura
NILS RYMAN,* PER ERIK JORDE,** AND LINDA LAIKRE
Division of Population Genetics, Stockholm University, S-106 91 Stockholm, Sweden
Introduction
The discrepancy between Nomura’s (1998) formula for
inbreeding effective number (NeI ) and our original one
(Ryman & Laikre 1991) is due to different definitions of
NeI. Therefore, contrary to what is implied by Nomura,
neither approach is necessarily erroneous. Nevertheless,
we consider ours most appropriate for practical management applications because it bears a direct relationship to
the actual degree of inbreeding (and thereby to potential
inbreeding depression) which the inbreeding effective
number is typically thought to do. We also consider Nomura’s concept of the “total effect” of supportive breeding to be misleading because it is not clear how such a
quantity should be defined. The issue may be regarded as
conceptually intricate, and, because of the direct implications for management, we provide a detailed explanation
of the differences between the two approaches.
We introduced the concept of supportive breeding (Ryman & Laikre 1991) to address practical genetic problems
encountered when weak natural populations are managed through the release of individuals bred in captivity
(Fig. 1). Previously, it was largely considered genetically
harmless to stock a population with individuals from the
same gene pool, but our results showed that supportive
breeding may result in a significant increase of inbreeding
in the total (wild 1 captive) population which would not
occur had it been left alone. We provided a formula for
assessing the inbreeding effective population size (NeI )
of a nonselfing population subject to supportive breeding, and in a subsequent paper (Ryman et al. 1995) we
presented an expression for variance effective number
(NeV) and compared the immediate effect of supportive
breeding on inbreeding and genetic drift. A primary intention of these studies was to provide an analytical
* email nils.ryman@popgen.su.se
** Current address: Division of Zoology, Department of Biology, University of Oslo, P.O. Box 1050 Blindern, N-0316 Oslo, Norway.
Paper submitted August 24, 1998; revised manuscript accepted December 28, 1998.
background for supplying management recommendations on how to avoid a sudden excessive increase of inbreeding and drift in particular generations.
Nomura (1998) proposes an alternate formula for inbreeding effective number which is identical to our expression for NeV, a result indicating that the impact of
supportive breeding is the same on both effective numbers. He suggests that our comparisons of the effect of
supportive breeding on NeI and NeV may be misleading
because our formula for NeI does not account for the “total effect” of supportive breeding, whereas the one for
NeV does. Clearly, Nomura’s derivations demonstrate that,
using his definition of NeI , the overall effect of supportive
breeding can be considered the same on NeI and NeV . This
result is expected because Nomura’s definition disregards
inbreeding that is due to the nonrandom mating caused
by population subdivision in generation t 1 1. Under random mating, inbreeding and drift are both due to the
same process of random sampling of genes during reproduction, and when the appropriate generations are considered the corresponding effective sizes should coincide.
Reduction of heterozygosity and inbreeding then follows
directly from the change of gene frequency caused by genetic drift. Inbreeding and drift may become manifest in
different generations, and NeI may differ from NeV when
population size is changing. Nevertheless, in the long run
the two quantities are expected to be the same on average when mating is random.
Definitions of Inbreeding Effective Size
Nomura’s definition of NeI differs from ours in that it
does not relate directly to the change of inbreeding that
actually occurs from one particular generation to the
next. When deriving our formula we applied the common definition of NeI : the inbreeding effective size of a
population is defined as the size of an ideal population
exhibiting the same rate of inbreeding (F ) as the actual
one (Crow & Kimura 1970). Letting nF denote the
673
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Volume 13, No. 3, June 1999
674
Supportive Breeding and Inbreeding Effective Number
Ryman et al.
Figure 1. Schematic representation of a supportive
breeding situation. Before breeding, the total population in generation t 1 1 is divided into a captive and
a wild group of size Nc,t 1 1 and Nw,t 1 1 that reproduce
in captivity and in the wild, respectively. The Nc,t 1 2
and Nw,t 1 2 offspring are mixed before breeding and
mate at random in generation t 1 2. Nt 1 1 5 Nc,t 1 1 1
Nw,t 1 1 and Nt 1 2 5 Nc,t 1 2 1 Nw,t 1 2 . Modified from
Ryman et al. (1995).
change of average inbreeding between two consecutive
generations (t 2 1 to t),
Ft – Ft – 1
∆F t = ---------------------,
1 – Ft – 1
(1)
and temporarily dropping the t subscript the inbreeding
effective number is defined as
1
N eI = ----------.
2∆F
(2)
In contrast, Nomura uses a definition based on the
change of average coancestry (nu), such that
1
N eI ,θ = ----------,
2∆θ
(3)
the u subscript added to mark the distinction from the
NeI defined in equation 2.
Citing Caballero and Hill (1992), Nomura states that in
nonrandom mating populations “the rate of inbreeding
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Volume 13, No. 3, June 1999
should be computed with the average coancestry among
individuals instead of the actual inbreeding coefficient.”
Although such an approach may be appropriate in some
instances, we have difficulty seeing that it represents a
universal truth. When interest is focused on the actual
change of inbreeding (nF ) or on inbreeding depression,
it seems more natural to use the traditional definition of
NeI (equation 2). It is interesting to note in this context
that Wright (1922) in his original treatment of inbreeding
considered it important that the inbreeding coefficient relate directly to contemporary inbreeding depression.
In a subdivided population, inbreeding increases for
two different reasons, because of the finite size of the total population (i.e., drift) and because breeding is restricted to within subpopulations (i.e., nonrandom mating). In the absence of immigration, the inbreeding due
to finite population size is “permanent,” whereas the inbreeding caused by subdivision may be considered “temporary” because it disappears if subdivision is removed
and random mating established.
In some instances it may be helpful to distinguish between permanent and temporary inbreeding. For such
situations Caballero and Hill (1992) have shown that the
inbreeding effective size defined on the basis of coancestry (NeI,u, equation 3) bears a direct relationship to the
inbreeding due to the finite size of the total population.
Thus, although NeI is connected with the change of total
inbreeding, NeI,u relates only to the portion explained by
restricted population size, ignoring the part due to nonrandom mating.
To illustrate the difference between NeI and NeI,u numerically, we have simulated the accumulation of inbreeding for the Pacific salmon supportive breeding situation used as an example by Nomura (1998), keeping
the same designation of generations as in his paper (Table 1). In generation t 1 1, a population of 50 breeders
is divided into a captive and a wild group of Nc,t 1 1 5 30
and Nw,t 1 1 5 20 breeders, and these two groups produce Nc,t 1 2 5 5000 captive and Nw,t 1 2 5 50 wild progeny, respectively. In t 1 2 the Nt 1 2 5 5050 breeders
mate at random and produce a total of Nt 1 3 5 50 progeny, and this population size of 50 panmictic breeders is
maintained in subsequent generations. Our primary interest refers to nonselfing species (such as the Pacific
salmon), but for comparisons with exactly the situation
modeled by Nomura we also conducted simulations permitting a random amount of selfing.
When selfing is excluded there is the usual delay of
one generation before inbreeding is manifest, because
individuals cannot be homozygous for alleles that are
identical by descent from generation t 1 1 until generation t 1 3 (Table 1, Fig. 1). The change of total inbreeding experienced in t 1 3 (nFt 1 3) corresponds exactly
to that predicted by our original formula for NeI (Ryman
& Laikre 1991), such that nFt 1 3 5 1/(2NeI, t 1 1), as it
should (equation 2). Similarly, the inbreeding coefficient
Ryman et al.
Supportive Breeding and Inbreeding Effective Number
675
Table 1. Results of pseudo-random number computer simulations of the accumulation of inbreeding (F) in selfing and nonselfing
populations subject to supportive breeding.a
No selfing
Generation
t11
t12
t13
t14
Nc
Nw
F
30
20
5000
50
5050d
50
50
0
DF
Selfing
NeI 5 1/(2DF)
30.6c
0
0
0.01634
0.01644
0.01634
0.00010
5134.3
F
DF
0
NeI 5 1/(2DF)
29.9
0.01675
0.01675
0.01644
0.02627
20.00032
0.01000
21572.8
50
Nomura’s
NeIb
50
76.9
50
a
Cf. Fig. 1 and Nomura (1998). All individuals are uniquely heterozygous in generation t 1 1 ( Ft11 5 0), and subsequent generations were created through random sampling of the number of gametes necessary to create the stipulated number of diploid offspring. Simulated F values
represent the mean of one million replicates, and deviations between those means and the ones calculated exactly are noticeable in the sixth
decimal place only. The NeI estimates for particular generations are based on the D F observed one and two generations later in the case of selfing and nonselfing, respectively.
b
From Table 1 in Nomura (1998); the value designed NeI, t11→t12 is assigned to generation t 1 1, etc.
c
The formula of Ryman and Laikre (1991) yields 30.60.
d
Captive and wild progeny mix and mate at random in t 1 2. In subsequent generations all matings take place in the wild and there is no further release of captive progeny.
in t 1 4 is determined both by the number of breeders
in t 1 2 and by the inbreeding in t 1 3 through
2N t + 2 – 1
1
F t + 4 = ---------------- +  ------------------------- F t + 3
2N t + 2  2N t + 2 
(4)
(Crow & Kimura 1970; equation 3.11.3). Thus, in accordance with expectation for the early generations, the increase of F from t 1 3 to t 1 4 corresponds approximately, but not exactly, to the number of breeders in t 1 2
[1/(2nFt 1 4) 5 5134.3 rather than 5050; Table 1].
In the case where a random amount of selfing occurs,
the split in t 1 1 has the effect that substantial inbreeding due to supportive breeding is manifest already in t 1
2. In some situations, such as the present one, there is
even a reduction of inbreeding from t 1 2 to t 1 3. The
pattern for accumulation of inbreeding is quite different
when selfing is allowed or disallowed in the early generations following an event of supportive breeding.
Most important, however, Nomura’s NeI,u of 76.9 that is
based on coancestry does not correspond to the change
of total inbreeding (nF) between any two generations, regardless of selfing or nonselfing (Table 1). Rather, 76.9
corresponds to the change of coancestry (nu ) from t 1 1
to t 1 2 where ut 1 1 5 1/(2Nt 1 1) and ut 1 2 5 Ft 1 3. It
should be noted that using these definitions ut 1 1 is different from the actual level of inbreeding in t 1 2 (Ft 1 2 5
0.01675): the inbreeding due to nonrandom mating in t 1 1
is disregarded.
“Total Effect” of Supportive Breeding
We consider the issue of defining a total effect of supportive breeding potentially less straightforward than indicated by Nomura. An obvious problem encountered
when trying to compare “total” effects on drift and actual inbreeding (F) is that they become manifest in dif-
ferent generations. For a nonselfing organism the effect
on drift is displayed in generation t 1 2 (Fig. 1), whereas
an effect on inbreeding does not occur until t 1 3.
Our formula for NeI (Ryman & Laikre 1991) relates to
the increase of inbreeding that is manifest in t 1 3, and
this change (nFt 1 3 ) depends only on the number of
wild and captive parents in t 1 1 and their relative contribution to t 1 2. Nomura argues that the number of individuals in t 1 2 (Nt 1 2 ) is also an effect of supportive
breeding that should be accounted for when evaluating
the total effect. We have not “neglected” this aspect, but
it is not clear to us why the number in t 1 2 is the only
one that should be considered. The goal of a supportive
breeding operation typically is to increase the population size over several generations, and we see no reason
why the total effect should account for only two generations as a general rule.
The example chosen by Nomura (Table 1) depicts only
one of a series of possible scenarios: a population that increases dramatically in t 1 2 and then crashes to its original size in t 1 3. In this particular case it may seem natural
to consider the larger size in t 1 2 and the correspondingly smaller nFt 1 4. In other situations, however, there
may be no crash or the crash may occur much later. It
would then seem equally natural to regard the larger size
persisting over multiple generations as an effect of successful supportive breeding. The rate of inbreeding would
then be reduced over a range of subsequent generations,
and it is not obvious how to define a generation in which
the effect of supportive breeding ends.
In other situations supportive breeding may be at least
partially successful yet not result in an increased number
of breeders in t 1 2. This would occur, for example, in a
salmonid population in which the area available for
spawning can accommodate only a restricted number of
redds. Releasing large numbers of hatchery juveniles
might then result in a disproportionately large fraction
of captive progeny in the pre-spawning population, and
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Supportive Breeding and Inbreeding Effective Number
this over-representation may persist among the breeders. In such a case supportive breeding might be considered successful because of the larger abundance of fish,
although the absolute number of breeders remains the
same as in previous generations.
Of course, a manager who is interested in the average
generational change of inbreeding from t 1 2 to t 1 4
may compute that quantity (and the average of nFt 1 3
and nFt 1 4). Here, nFt 1 3 follows from the formula of
Ryman and Laikre (1991), and nFt 1 4 is given by equation 4. For the simulation in Table 1 the average nF is
(0.01634 1 0.00010)/2 5 0.00822, corresponding to an
average NeI of 60.83; this approximate number can also
be obtained through the harmonic mean of the corresponding NeI s (the harmonic mean of 30.6 and 5134.3 is
60.84; Crow & Kimura 1970).
We see a potential danger, however, in dealing with
an NeI that represents an average over multiple generations rather than with the separate ones reflecting the
change from one generation to another. First, in followup studies of a supportive breeding project it may be imperative to have accurate information about the actual
inbreeding of each particular generation. For the operation pictured in Table 1, for example, the inbreeding
“strikes” in t 1 3, whereas the change from t 1 3 to t 1
4 is negligible. Thus, if the actual values of nF are
known, the observation of, for example, a high mortality
in t 1 3 may correctly alert a manager that the population may be notably susceptible to inbreeding depression. In contrast, the possibility of inbreeding depression may go unnoticed if inbreeding is erroneously
believed to increase uniformly over both generations.
Further, supportive breeding projects should be planned
to avoid an excessive increase of inbreeding in any particular generation. For example, in line with general recommendations for conservation genetics (Frankel & Soulé
1981), we suggested that operations resulting in an NeI
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Ryman et al.
smaller than 50 should be modified to provide less dramatic increases in inbreeding (Ryman et al. 1995). In this
respect the Pacific salmon scenario (Table 1) must be
considered a failure in that the inbreeding being expressed in t 1 3 corresponds to an NeI of only about 30.
By using our NeI formula when planning the operation,
such a large increase of inbreeding could have been
avoided by modifying the number of wild and captive
breeders in t 1 1 and their proportionate contribution to
t 1 2. In contrast, Nomura’s equation does not provide
this opportunity.
Acknowledgments
This note was prepared with support from the Swedish
Natural Science Research Council, and part of the work
was conducted within the scope of the Swedish research program on Sustainable Coastal Zone Management, SUCOZOMA, funded by the Foundation for Strategic Environmental Research, MISTRA.
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