:~iii':!Special feature ii:~:i:ii:ii:i:i::~:~ i:i:::;:i::::i:i:i~i:i:i~;i~i:i:!;: i::::;~i::::;:;ii::;:;:i:i~:;~:::!;~i::::;i~:+:!~r
Extinction risk and life history evolution
Res. Popul. Ecol. 4 0 ( 3 ) , 1 9 9 8 , p p . 2 7 9 - 2 8 6 . ~% 1998 by the Society of P o p u l a t i o n E c o l o g y
18th Symposium of the Society of Population Ecology
Theoretical Aspects of Extinction by Inbreeding Depression Yoshinari
TANAKA 1)
Institute of Environmental Science and Technology, Yokohama National University, Tokiwadai 79-7, Hodogaya-ku, Yokohama 240-0067, Japan
Populational extinction due to inbreeding depression is analyzed with simple population genetic and population ecological models. Two alternative genetic mechanisms of inbreeding depression, i.e. recessive deleterious genes and overdominant genes, are assumed in separate analyses in order to examine their relative importance. With both mechanisms the population size and the coefficient of inbreeding are maintained at stable equilibria if there is no non-genetic demographic disturbance or stress. With a certain amount of demographic disturbance the population declines rapidly due to interaction between the decrease of population size and the increase of inbreeding coefficient. Such rapid extinction occurs with both genetic mechanisms. However, in the case of overdominant genes extinction happens only if the equilibrium population size is small and the selection coefficient is large such that segregation load is large. In nature, extinction due to overdominant genes is considered to be much less likely than extinction due to recessive deleterious genes. Abstract.
Key words: deleterious mutation, extinction, genetic load, inbreeding depression, overdominance.
Introduction Inbreeding depression may be an important source of extinction of small populations (Frankel and Soule 1981; Gilpin and Soule 1986). If there is a positive correlation between mean fitness and mean heterozygosity, populations declining due to anthropogenic disturbances (e.g. destruction of habitats and over-hunting) may go extinct by an interaction between population size and inbreeding depression (Gilpin and Soule 1986). A previous theoretical study has suggested that such rapid extinction by inbreeding depression can happen only under demographic disturbances with moderately large initial (equilibrium) population size (Tanaka 1997). Demographic disturbances that cause perturbation of populations from equilibria are essential because the population size and the inbreeding coefficient are locally stable at equilibria. Two genetic hypotheses are proposed for the mechanism of inbreeding depression, i.e. recessive deleterious mutations and overdominant genes. The overdominant genes achieve higher fitness when heterozygous but do not have any deleterious genic effect. A problem of the recessive deleterious gene hypothesis is that the deleterious mutations may not be maintained in a sufficient amount so as to 1) E-mail: tanaka@kan, ynu.ac.jp
cause inbreeding depression since they are eliminated by purging selection. The overdominant genes are free from purging selection because they cause no reduction in genic fitness. However, the overdominant genes cause large segregation load if the number of loci is large. Although recessive deleterious genes may explain the majority of inbreeding depression, there is some evidence supporting the importance of overdominant genes (Mitton 1993). A parallel analysis employed by Tanaka (1997) for the case of recessive deleterious mutation applies to overdominant genes in this study. A previous numerical study of extinction by recessive deleterious mutations demonstrated that "the extinction vortex" (Gilpin and Soule 1986) by inbreeding depression can occur only when populations are reduced below carrying capacity due to demographic stress and equilibrium population size is moderately large (Tanaka 1997). The present study extends the numerical method into overdominant genes, and further analyzes local stability of population size and inbreeding coefficient.
Analysis We assumed two genetic systems as causes of inbreeding depression, i.e. recessive deleterious and overdominant
280
TANAKA
genes. For both cases we assumed diallelic autosomal loci. In the case of recessive genes it is assumed that three genotypes, AA, Aa, and aa, have mean fitnesses as 1, 1, and 1 - s, respectively. For simplicity, we assumed complete dominance for the wild-type allele. Incomplete dominance of the wild-type alleles must reduce inbreeding depression and make an extinction vortex by inbreeding depression more unlikely to happen. As for the overdominant genes, the fitnesses of the three genotypes AA, Aa, and aa are 1 - s, 1, and 1 - s. For simplicity we assumed that both homozygotes have the same fitness, and then equilibrium gene frequency of either gene is 0.5. Small deviations from the equal fitness of homozygotes may not bring about large differences in liability to inbreeding depression. In both cases, we assume n independent and identical loci so that linkage disequilibrium is neglected. For simplicity we assumed no synergistic epistasis among loci. The number of loci in the case of recessive genes is assumed to be 5,000. The per-locus mutation rate is assumed to be p -- 4 x 10 s for all analyses. The empirical evidence supporting these values are briefly reviewed in a previous paper (Tanaka 1997; c.f. Simmons and Crow 1977). The assumed number of loci and per-locus mutation rate comprise a total mutation rate of deleterious genes, U -- 0.4, that is compatible with experimental estimates in various organisms. With deleterious mutations or homozygous overdominant genes the mean fitness of a population decreases from its maximum value 1 by L. This value is called the genetic load in population genetics (Crow and Kimura 1970; Nei 1987). In order to predict extinction due to inbreeding depression the genetic load must be translated into demographic parameters. Here we assume that the intrinsic growth rate r and the carrying capacity K decrease linearly with the genetic load: r -- rmax (1 - L) and K = Kmax (1 - L) where rm~x and Kmax are maximum values of the intrinsic growth rate and the carrying capacity. If deleterious genetic effects influence reproductive output regardless of population density, K must decrease as well as r as the genetic load increases. A population is assumed to be at demographic and genetic equilibria. Demographic disturbances due to anthropogenic factors (e.g. destruction of habitats and hunting) cause perturbation of a population from the equilibrium. If the perturbation of population size diminishes, the population reaches a new equilibrium population size and will not become extinct. But if the perturbation is amplified over generations, the population will quickly go extinct by a process called the extinction vortex (see later sections). At equilibrium the inbreeding coefficient and the gene frequency are kept unchanged by a balance between mutation, inbreeding and selection. The genetic load is
also at an equilibrium. The equilibrium population size is equal to the equilibrium carrying capacity. Thus N = Kma~ ( 1 - / , ) . Hereafter, the tildes represent equilibrium values. The equilibrium genetic load is calculated from an arbitrarily determined equilibrium population size (see the next section). Hence the equilibrium population size is the primary assumption. The maximum carrying capacity Kmax is set as Kmax = N/(1 - L ) . If the equilibrium genetic load is large, Kmaxis set much larger than .N. This part of the assumptions is made to prevent the equilibrium genetic load from confounding with inbreeding depression resulting from demographic disturbances. The same manner of parameterization was taken for the maximum intrinsic rate of natural increase rmaxand the equilibrium intrinsic rate ?. We disregard the effect of the equilibrium level of inbreeding and genetic load on the population size and the intrinsic rate of natural increase. We then focus on the effect of inbreeding depression newly caused by demographic disturbances to induce an extinction vortex. The per-locus genetic load 1 is defined as l = 1 - W~ Wm~x, where Wmax is the maximum fitness ( = 1 in this study). The mean fitness is W = 1 - s q 2 - F s q ( 1 - q ) for - -
S
the case of recessive genes, and W = 1 - - ~ - (1 + F ) for the case of overdominant genes, where q is the gene frequency of the recessive deleterious gene and F is the inbreeding coefficient. The total genetic load per individual is L =
nl. Demographic and genetic equilibrium Without demographic disturbance and environmental stochasticity the population is kept at a demographic and genetic equilibrium, in which the population size, the gene frequency and the inbreeding coefficient do not change. To calculate the equilibrium values we need to know the dynamics of these variables. The change of gene frequency by selection is derived from Wright's formula, h~q =
q ( 1 - - q ) ~W For the 2W Oq " 9W case of recessive genes the partial derivative is Oq -s{2q(1-F)+F}. Hence selection is influenced by F and continues to act until q = 0 (purging selection). On the other hand, selection does not operate for the case OW of overdominant genes, ~q - 0, as long as the gene frequency is 0.5. In the case of recessive genes the gene frequency changes by mutation. The rate of change is equivalent to the perlocus per-gamete mutation rate Amq = I2. On the other hand, frequency of the overdominant genes does not change by mutation, i.e. Amq = 0, because mutations are likely to be reversible and the gene frequency is inter-
EXTINCTION BY INBREEDING
mediate. At equilibrium due to mutation and selection (Asq § Amq = 0), the gene frequency must meet a quadratic equation: (F--1)~2--(F/2)~l+l#s ~- 0 for the case of recessive deleterious genes. The gene frequency of overdominant genes is 0.5. Evolution of the inbreeding coefficient is mostly determined by inbreeding and is slightly modified by selection and mutation. The per-generation change in F is expressed by the following recursion equation: Ft+l =
t-~--~+
-
for the case of recessive genes (Tanaka 1997), and
Ft+l = { ~ N § ( 1 - ~-N-N)Ft}(1-2/J)(1-- l s ) for the case of overdominant genes. At equilibrium, the coefficient becomes F = t4N(/L + 1 0 s ) + l t 1 for the recessive genes and ff = {4N(tl + s / 4 ) §
1
for the
overdominant genes. Unfortunately, it is not possible to find a simple solution of O for the case of recessive genes. Numerical calculation using Asq + Amq = 0 evaluated O.
Local stability at equilibria The standard protocol of local stability analysis indicates that the demographic and genetic equilibria are locally stable for either case regardless of the equilibrium population size, the mutation rate, the selection coefficient and the number of loci (see Appendix A). This result suggests that inbreeding depression does not alone induce extinction of populations under stable environments.
that changes in r and K by genetic factors are not taken account. Actually as the population size is reduced from the original value, inbreeding is enhanced by the decreasing population size. If the gene frequency change is much slower than changes in the inbreeding coefficient, inbreeding depression increases and the population growth rate decreases. The decreased growth rate further reduces the population size as does demographic disturbance. Depending upon parameter values of the equilibrium population size and the mutation, the interaction between the decreasing population size and the increasing inbreeding coefficient leads to rapid extinction (extinction vortex by inbreeding depression). We assumed a sufficiently small rate of demographic disturbance, du = 0.15, and an equilibrium intrinsic growth rate larger than dN, ~ = 0.2, so that demographic disturbance cannot alone cause extinction, i.e. 6 < 1. If the genetic effects are disregarded, populations subject to disturbances must reach new stable equilibria in all cases. Predicted rapid extinctions are attributed to inbreeding depression. In the following sections we employed two approaches, numerical simulations and local stability analysis, to examine the conditions for populational extinction. The numerical simulation evaluates population size, gene frequency, inbreeding coefficient, and genetic load at every generation after the onset of the demographic disturbance. The local stability analysis will be explained in a later section. The dynamics of the three parameters are described by the following joint recursion equations:
Ntq,+,\ [
qt-sq,(1-qt){2qt(l-Ft)+Ft}/(2~)
\
Ft+l]- [ [1/(2Nt) + {1-1/(2Nt)}Ft](1- 2/*)(1 -qts) [ +2] /Ut+,[1 +rmax(1-L,){a-Nt+,/(Kmax(1-Lt))}(l-dN)]] where Wt = 1--sqt2--Ftsqt(1--qt) and Lf = rts{qt2(1--Ft)
Demographic disturbances
+qtFt} for the case of recessive deleterious genes. Equa-
The extinction vortex due to inbreeding depression may be triggered by demographic disturbances. In this study the demographic disturbance is supposed to be caused by artificial factors such as hunting and destruction of habitats. With demographic disturbance the population size tends to reach a new equilibrium which is smaller than the original equilibrium size. We assume that the disturbance is a proportional reduction, du, in population size every generation. The population growth is then
281
Nt+~ = {Nt +
rNt(1- -~-)} (1--dN), and the new equilibrium is R*=
tions for the overdominant genes are parallel to the above. The reason why the description of population dynamics is one generation ahead from the other two parameters is that the genetic effects on demographic parameters through survival and reproduction occur in the next (offspring) generation. Initial values of the gene frequency, the inbreeding coefficient and the population size are assumed to be equivalent to the stable equilibrium values without demographic disturbance. Iterative calculations are repeated until populations become extinct or reach new equilibria (see the next section).
(1--g)N if genetic deterioration is not taken account. Here 3 means g --
dN
r(1--dN) and .N is the original
equilibrium population size equivalent to /s The new equilibrium population size N* is a hypothetical value in
Purging selection In the case of recessive deleterious genes selection purges deleterious genes after the population decreases due to
282
TANAKA
demographic disturbances. If the purging selection is so effective as to overcome the fitness decrements due to the synergistic interaction between the genetic load and decreases o f the population size (extinction vortex), populations escape f r o m extinction by inbreeding depression (Fig. 1). However, once the vortex occurs, purging selection is only effective in very small populations (N < 20 in our runs), in which the recessive deleterious genes are exposed to selection due to the absence of heterozygosity. Our analysis is not effective for gene frequency changes in such very small populations because we disregard stochastic changes in gene frequency. Some deleterious genes are likely to be fixed by drift in small populations. Other sources of extinction (demographic or environmental stochasticity and erosion of adaptive genetic variability) may also be important in such very small populations. As a criteria of extinction we employed N < 20 in this study. Our analysis may overestimate the effect of purging selection by neglecting stochastic gene frequency changes.
Purging selection may be negligible if the extinction vortex dramatically reduces the population size within 100 generations since the gene frequency change within 100 generations is small. Although purging selection is shown to be effective in regular systems of inbreeding, e.g. sib mating and selfing (Barrett and Charlesworth 1991), the extent to which purging selection saves populations from extinction risk by inbreeding is a future research subject.
Extinction vortex Some population trajectories by numerical simulations are shown in Figs. 2a and 2b. The parameter values are set as /~ = 4 • 10 -5, s = 1 and n = 5,000 for the case of recessive genes, and t~ -- 4 • 10 -5 , s = 0.01 a n d n -- 100 for the case of overdominant genes. In both cases some populations became extinct rapidly due to the interaction
(a) recessive g e n e s
I*10 5
10 6 10 5
1.104
o N r E 0
1"10 3
0 0..
Q J
100
10 4
"....
-.
10 3
.
.
.
.
.
.
.
10 2 10 1
J 10 0
10
11oo
2100
300
I
0
4100
___X_.___.L.
40
. . . . . . . .
80
_l_l.
. . . . . . .
120
_l
160
Generation
Generation
(b) o v e r d o m i n a n t gene 106 ~,,,,.,~..._._. ~ _
0. 006
105
.~
0. 004
t9
'--~ 103
.
~o
;, _ _
0
1~
~'~
...................................
.-%" q ...... ..........
....................
60
120
180
240
300
Generation
Generation Fig. 1. A population trajectory and gene frequency changes in an extinction vortex followed by an effective purging selection. The genetic mechanism is recessive deleterious genes. See the text for details and parameter values except for s (selection coefficient)
102
~ . . .~. .-.u : : : _ : : : : ~ . _
100
1oo
100
~
104
g
0. 002
K,,,,~.__._._.
2. Population trajectories after the onset of demographic disturbance for the case of recessive deleterious genes (a) and overdominant genes (b). The initial population size is at dcmographic and genetic equilibrium. The dotted lines represent trajcctories if there are no gcnctic factors (L ----0).
Fig.
EXTINCTION BY INBREEDING
L o c a l s t a b i l i t y a n a l y s i s a r o u n d t h e n e w e q u i l i b r i a is also e m p l o y e d t o c h e c k t h e a b o v e n u m e r i c a l r e s u l t s . It is postulated that a population becomes extinct if a new e q u i l i b r i u m is n o t s t a b l e . T h e n e w e q u i l i b r i u m p o p u l a t i o n size a n d i n b r e e d i n g coefficient a r e c a l c u l a t e d as N * = (1-6)/( and F* = --6(1 - i f ) } , w h e r e / ( a n d f l a r e t h e original equilibrium values before demographic disturb a n c e s a r e i n t r o d u c e d . It is a s s u m e d f o r s i m p l i c i t y t h a t the gene frequency does not change. The local stability a r o u n d t h e d i s t u r b e d n e w e q u i l i b r i a is e x a m i n e d w i t h t h e
b e t w e e n i n b r e e d i n g d e p r e s s i o n a n d d e m o g r a p h i c stresses. I n p a r t i c u l a r , t h e r a p i d d e c r e a s e o f p o p u l a t i o n size implies a synergistic interaction between decreasing population size a n d i n c r e a s i n g i n b r e e d i n g d e p r e s s i o n . However, f e a t u r e s o f e x t i n c t i o n differ b e t w e e n t h e t w o cases in t h e d e p e n d e n c e o n t h e e q u i l i b r i u m p o p u l a t i o n size. I n t h e case o f recessive g e n e s p o p u l a t i o n s w i t h i n t e r m e d i a t e e q u i l i b r i u m size b e c a m e e x t i n c t w h i l e in t h e case o f o v e r d o m i n a n t g e n e s s m a l l p o p u l a t i o n s b e c a m e extinct. T h e r e a s o n w h y t h e i n t e r m e d i a t e p o p u l a t i o n s in t h e case o f recessive g e n e s are p r o n e t o e x t i n c t i o n b y i n b r e e d i n g is t h a t s m a l l p o p u l a t i o n s c a n n o t m a i n t a i n recessive deleterious mutations enough to induce inbreeding depression ( T a n a k a 1997). Table l a .
283
if~{1
standard method of local stability analysis for discrete d y n a m i c s ( A p p e n d i x B). G i v e n t h a t a n e w e q u i l i b r i u m is n o t s t a b l e , t h e r e s u l t i n d i c a t e s t h a t t h e n e w d i s t u r b e d e q u i l i b r i u m is n o t a t t a i n a b l e f o r t h e a n a l y z e d p o p u l a t i o n i f
Eigenvalues in a local stability analysis on the disturbed equilibrium in the case of recessive deleterious genes s=l
s=0.8
s=0.6
s=0.4
s=0.2
21
--0.0047
--0.00424
--0.00371
--0.00306
--0.00219
22
-- 0.024
-- 0.0238
-- 0.0235
-- 0.0233
-- 0.0229
21 2z
0.00141 - 0.0492
0.00127 -- 0.0483
0.00112 - 0.0474
0.000924 - 0.0462
0.000662 -- 0.0445
21
22
0.0476 --0.178
0.0425 --0.172
0.0365 --0.165
0.0291 --0.155
0.019 --0.141
21
-- 0.00403
-- 0.00403
-- 0.00403
-- 0.00403
-- 0.00403
22
-0.0366
--0.0366
-0.0366
-0.0366
--0.0366
21
-- 0.0207 -- 0.0438
-- 0.0207 -- 0.0438
-- 0.0207 -- 0.0438
-- 0.0207 -- 0.0438
-- 0.0207 -- 0.043
N = 1,000,000
N = 100,000
N = 10,000
N = 1,000
N=100 22
Table l b .
Eigrnvalues in a local stability analysis on the disturbed equilibrium in the case of over-dominant genes s=0.05
s=0.01
s=0.005
s=0.001
s=0.0005
s=0.0001
N = 1,000,000 21 22
--0.017 --0.028
--0.005 --0.02
--0.0025 --0.02
--0.0005 --0.02
--0.00025 --0.02
--0.000054 --0.02
0.0077 --0.038
- 0.0047 --0.021
- 0.0024 --0.021
- 0.00052 --0.02
- 0.00028 --0.02
- 0.000091 --0.02
0.022 --0.072
- 0.0028 -0.027
- 0.002 -0.025
- 0.00083 -0.022
- 0.00064 -0.021
-- 0.00047 -0.02
0.122 --0.207
-- 0.00026 --0.052
- 0.0039 --0.038
- 0.0046 --0.023
- 0.0045 --0.021
- 0.0043 --0.02
N = 100,000 21 22
-
N = 10,000 21
22 N = 1,000 21 22
284
TANAKA
the gene frequency change due to selection is slow, and the p o p u l a t i o n will become extinct. The local stability analysis provided results compatible to those by the numerical simulation (Tables l a and lb). In the case o f recessive deleterious genes, only intermediate p o p u l a t i o n s (37" = 104, 105) result in unstable equilibria,
Table 2a.
Equilibrium genetic load before and after onset of demographic disturbance in the case of recessive deleterious genes. N and s represent the original equilibrium population size and the selection coeffieient, respectively. s=l
s=0.8
s=0.6
s=0.4
s=0.2
N = 1,000,000 L before stress L after stress
0.182 0.197
0.182 0.197
0.182 0.197
0.182 0.197
0.182 0.197
N = 100,000 L before stress L after stress
0.191 0.325
0.191 0.325
0.191 0.325
0.191 0.324
0.191 0.322
N = 10,000 L before stress L after stress
0.274 0.858
0.274 0.855
0.274 0.85
0.273 0.842
0.271 0.824
N = 1,000 L before stress L after stress
0.33 0.411
0.33 0.411
0.33 0.411
0.33 0.411
0.33 0.41
N = 100 L before stress L after stress
0.33 0.337
0.33 0.337
0.33 0.337
0.33 0.337
0.33 0.337
which suggests rapid extinction. In the case o f overd o m i n a n t genes only small populations that have large selection intensities (N = 104, 103, and s = 0.05) are unstable. C o m p a r i s o n between the equilibrium genetic loads before and after the onset o f d e m o g r a p h i c disturbances is shown in Tables 2a and 2b. In the case o f recessive deleterious genes, the equilibrium genetic l o a d is determined by mutation-selection balance. The equilibrium l o a d increases as the equilibrium p o p u l a t i o n size decreases because inbreeding in small populations decreases heterozygosity and exposes the effects o f recessive deleterious genes. L o a d b y recessive genes is increased greatly after the demographic disturbance is introduced, especially for populations that became extinct. As for the overd o m i n a n t genes, the equilibrium genetic load is mainly caused by segregation and is largely determined b y selection intensity. The genetic l o a d augmented by inbreeding is only a small part o f the total genetic l o a d in the case o f overdominance. The extremely large genetic load (L _> 0.4) exists before the onset o f d e m o g r a p h i c disturbances in the case o f f q = 104, 103, 102 a n d s _> 0.01. Nonetheless, we assumed ? = 0.2 by setting rmax as large values. However, such a large equilibrium genetic load is not realistic in natural populations. All other cases in the o v e r d o m i n a n t genes showed very little increase in the genetic load. These results imply that o v e r d o m i n a n t genes are not likely to cause an extinction vortex by inbreeding depression in nature.
Equilibrium genetic load before and after the onset of demographic disturbance in the case of overdominant genes. N and s represent the original equilibrium population size and the selection coefficient, respectively. Table 2b.
s=0.05
s=0.01
s=0.005
s=0.001
s=0.0005
s=0.0001
N - - 1,000,000 L before stress L after stress
0.918 0.918
0.393 0.394
0.221 0.222
0.049 0.049
0.025 0.025
0.005 0.005
N = 100,000 L before stress L after stress
0.918 0.918
0.394 0.396
0.222 0.224
0.049 0.052
0.025 0.027
0.005 0.006
N = 10,000 L before stress L after stress
0.918 0.921
0.396 0.416
0.225 0.248
0.053 0.069
0.028 0.038
0.006 0.009
N = 1,000 L before stress L after stress
0.922 0.942
0.42 0.517
0.252 0.333
0.071 0.09
0.039 0.047
0.009 0.01
N = 100 L before stress L after stress
0.945 0.982
0.518 0.609
0.33 0.383
0.086 0.094
0.045 0.048
0.0092 0.0099
EXTINCTION BY INBREEDING
285
I thank Y. lwasa and M. Shimada for helpful discussion on the topic. This work is supported in part by CREST (Core Research for Evolutional Science and Technology) of Japan Science and Technology Corporation (JST) (Principal investigator is J. Nakanishi).
Acknowledgments:
Discussion Both the numerical simulation and the local stability analysis imply that rapid extinction due to interaction between demographic stresses and inbreeding depression (extinction vortex by inbreeding depression, c.f. Gilpin and Soule 1986) can occur with either recessive deleterious mutations or o v e r d o m i n a n t genes. However, the o v e r d o m i n a n t genes sufficiently detrimental so as to induce an extinction vortex must generate extremely high segregation load at equilibrium, which is not realistic in nature. The present analysis assumed that the equilibrium genetic load is not i m p o r t a n t and the m a x i m u m intrinsic rate o f natural increase (the intrinsic rate o f a mutation-free population) and carrying capacity to be arbitrary values such that the equilibrium values are equivalent to assumed constants. By assumption, even if the equilibrium genetic load is unrealistically large, the equilibrium values o f r and K are not influenced by the genetic load. This assumption m a y be justified if genetic b a c k g r o u n d s evolve to compensate long-term genetic loads in an evolutionary time scale so that many organisms p e r f o r m at nearly the same mean fitness regardless o f the number o f deleterious mutations they possess. In this study genetic effects on p o p u l a t i o n growth are caused by increments o f genetic load after the onset o f demographic stresses (inbreeding depression). Nonetheless, the large segregation load in the case o f overdominance is obviously important. The results concerning the extinction vortex by o v e r d o m i n a n t genes are likely to be a mathematical artifact. If o v e r d o m i n a n t genes are prevailing, very small populations always suffer an extremely high extinction risk. This implication m a y not be the case for recessive deleterious genes, which are not effective in long-term small equilibrium populations owing to purging selection. Our analysis dealt with recessive deleterious mutations and o v e r d o m i n a n t genes as two different genetic mechanisms o f inbreeding depression. However, the two genetic mechanisms are not exclusive of each other, and real genetic systems m a y be mixtures o f them. A l t h o u g h o v e r d o m i n a n t genes are not likely to be the m a j o r genetic system o f inbreeding depression (c.f. M u k a i 1978; Charlesworth and Charlesworth 1987), there m a y be a substantial extinction risk due to o v e r d o m i n a n t genes. There is some experimental evidence that supports the overdominance hypothesis (Mitton and Grant 1984; Mitton 1993). The present and the previous results, which suggested that long-term small p o p u l a t i o n s are free from extinction risk by inbreeding depression if its m a j o r factor is recessive deleterious genes ( T a n a k a 1997), should be interpreted cautiously.
References Barrett, S. C. H. and D. Charlesworth (1991) Effects of a change in the level of inbreeding on the genetic load. Nature 352: 522-524. Bulmer, M. (1994) Theoretical evolutionary ecology. Sinauer, Sunderland. Charlesworth, D. and B. Charlesworth (1987) Inbreeding depression and its evolutionary consequences. Annual Review of Ecology and Systematics 18: 237-268. Crow, J. F. and M. Kimura (1970) An introduction to population genetics theory. Harper & Row, New York. Frankel, O. H. and M. E. Soule (1981) Conservation and evolution. Cambridge University Press, Cambridge. Gilpin, M. E. and M. E. Soule (1986) Minimum viable populations: the processes of species extinction, pp. 13-34. In M. E. Soule (ed.) Conservation biology: the science of scarcity and diversity. Sinauer Associates, Sunderland, Mass. Iwasa, Y. (1990) Introduction to mathematical biology. HBJ, Tokyo (in Japanese). Mitton, J. B. and M. C. Grant (1984) Associations among protein heterozygosity, growth rate, and developmental homeostasis. Annual Review of Ecology and Systematics 15: 479-499. Mitton, J. B. (1993) Theory and data pertinent to the relationship between heterozygosity and fitness, pp. 17-41. In N. W. Thornhill (ed.) The natural history of inbreeding and outbreeding. The University of Chicago Press, Chicago. Mukai, T. (1978) Population genetics. Kodansha Scientific, Tokyo (in Japanese). Nei, M. (1987) Molecular evolutionary genetics. Cambridge University Press, Cambridge. Roughgarden, J. (1979) Theory of population genetics and evolutionary ecology: an introduction. Macmillan, New York. Simmons, M. and J. F. Crow (1977) Mutations affecting fitness in Drosophila populations. Annual Review of Genetics 11: 49-78. Tanaka, Y. (1997) Extinction of populations due to inbreeding depression with demographic disturbances. Researches on Population Ecology 39: 57-66. Received 22 January 1998; Accepted 26 November 1998
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TANAKA
Appendix A In this appendix we examine local stability of the dynamics of population size and inbreeding coefficient around equilibria without demographic stress. Under the assumption that gene frequency change is so slow that perturbation around the equilibrium is negligible, the dynamics of population size and the inbreeding coefficient is N AN= rN(1----~-)
where J = Os for the case of a recessive deleterious mutation, and J = s/2 for the case of an overdominant gene. Here r and K are functions of F and q. If small perturbations of N and F from the equilibria are written as v = N - .g" and r = F - F, the dynamics of v and r is
:,(;) w h e r e A = I ( S-A - ~F-) ~.;
0(_~)~,
. The standard method oflocal stability analysis evaluates stability around the equilibria by
means of eigenvalues of the linear dynamical system (Roughgarden 1979; Iwasa 1990; Bulmer 1994). The eigenvalues are evaluated as 2 = where B = rmax(1--ns~F~ C
rmax [ 1 = T ~
-B_+
B24~STC--4C 2
1
+ 2gmax(1-nsglff) i- s# and
+ 2s# (1 --ns[lf)
ns4(1--f) ] Kmax(1 - ns#f) for the case of 1
a recessive deleterious mutation, and
s
B : rmax{1--12ns(l+ff)} -f 2Kma~{l_lns(l+f) } + ~ - a n d C:
rn~ax[ Klmax
+s{1-- 89
2gmax{lnS(1-F+ns )- (1 +F)}J] for the case of an overdominant gene.
In both cases B > 0 and C > 0 if s _< 1 and Kmax > 0. Then both eigenvalues are negative, and the equilibria are locally stable.
Appendix B In this appendix we examine the local stability of N and F around the disturbed equilibria under demographic stress. Unstable equilibria imply rapid extinction due to interaction between the demographic stress and the inbreeding depression. The disturbed equilibria with dN demographic stress are N* = (I - 8 ) / ( and F* =f/{ 1 - ~ (1 - F ~ } , where ~ - r(1 --dN) and tin is the rate of demographic disturbance (proportional reduction of population size due to a non-genetic factor). Under the assumption that the gene frequency change is so slow that perturbation around the equilibrium is negligible, the dynamics of population size and the inbreeding coefficient is
AN =r(1--dN)N(1--~ )--dNN AF~2N where J = ~s for the case of a recessive deleterious mutation, and J = s/2 for the case of an overdominant gene. The partial derivatives in the elements of the transition matrix A must be evaluated at the disturbed equilibria N* and F*. The elements of A are calculated as 0 ,AN[ ~*, F* ---- --rm~x(1 --L*)(1 --8)(1 --2dN)-- ~ aN
~rr a F, l ~, z ,
OL : -(~-)rmaxKmax(1-L*)(1-~)(1-dN) l-F* -- _ 2 {Kmax(1--L*)(1 __~)}2
1
~F-AFI ~,F* = --( 2Kma~(I--L*)(1--~) ~Unfortunately, the eigenvalues cannot be solved in a simple algebraic form. parameter values. Some representative results are shown in Tables la and lb.
j). They were numerically evaluated with reasonable
