Researches on

Res. Popul. Ecol. 39(1), 1997, pp. 57-66. 9 1997 by the Society of Population Ecology

~_EOPULATION O N g i n a l

Extinction of Populations Due to Inbreeding Depression with Demographic Disturbances Yoshinari

TANAKA 1)

Laboratory of Theoretical Ecology, Institute of Environmental Science and Technology, Yokohama National University, Tokiwadai 79-7, Hodogaya-ku, Yokohama 240, Japan

The process of population extinction due to inbreeding depression with constant demographic disturbances every generation is analysed using a population genetic and demographic model. The demographic disturbances introduced into the model represent loss of population size that is induced by any kind of human activities, e.g. through hunting and destruction of habitats. The genetic heterozygosity among recessive deleterious genes and the population size are assumed to be in equilibrium before the demographic disturbances start. The effects of deleterious mutations are represented by decreases in the growth rate and carrying capacity of a population. Numerical simulations indicate rapid extinction due to synergistic interaction between inbreeding depression and declining population size for realistic ranges of per-locus mutation rate, equilibrium population size, intrinsic rate of population growth, and strength of demographic disturbances. Large populations at equilibrium are more liable to extinction when disturbed due to inbreeding depression than small populations. This is a consequence of the fact that large populations maintain more recessive deleterious mutations than small populations. The rapid extinction predicted in the present study indicates the importance of the demographic history of a population in relation to extinction due to inbreeding depression. Abstract.

Key words: extinction, inbreeding depression, deleterious mutation, genetic load, conservation biology.

Introduction There are two major genetic factors which may cause the extinction of a population. One is the erosion of genetic variances in adaptive characters, which limits the ability of a population to adapt to changing environments (Lande and Barrowclough 1987; Lynch and Lande 1993; Biirger and Lynch 1995). The other important factor is inbreeding depression (Frankel and Soule 1981; Schonewald-Cox et al. 1983; Soule 1986; Barret and Charlesworth 1991; Barret and Kohn 1991), which results from increasing homozygosity of recessive deleterious mutations in a rapidly decreasing population or from a regular system of inbreeding, e.g. sib mating and selfing (Charlesworth and Charlesworth 1987; Falconer 1989). In a wide range of animal and plant species, natural populations decrease size due to cost of inbreeding depression (Schoen 1983; 1) E-mail: tanaka@kan,ynu.ac.jp

Holtsford and Ellstrand 1990; Barret and Charlesworth 1991; Agren and Schemske 1993; van Treuren et al. 1993; Willis 1993). If inbreeding depression is caused by recessive deleterious mutations which are maintained as heterozygotes, the extent to which a population suffers inbreeding depression per unit decrease in heterozygosity due to inbreeding depends largely on the demographic history of the population. A large population that has experienced no bottlenecks may be more liable to inbreeding depression than is a small population because a large population with greater heterozygosity maintains more deleterious mutations. In a small population, however, most deleterious mutations have been purged by selection which is more effective with lower heterozygosity. Then, further decreases in population size due to any catastrophic events may not cause inbreeding depression (the rate of inbreeding depression is low). Thus the demographic history of a population may be

58

TANAKA

important in relation to extinction due to inbreeding induced by demographic disturbances. In a large population, which is at genetic equilibrium for many generations, higher heterozygosity is maintained than in a small populations. Demographic disturbances of such large populations decrease the mean fitness (the carrying capacity and the intrinsic rate of population growth) due to inbreeding depression induced by changes in heterozygosity. The decrease in mean fitness further diminishes the population size in the next generation. Hence there may be a synergistic interaction between decreases in population size and the process of inbreeding depression. Here I refer to such an accelerated process which results in the rapid extinction of a population as "extinction vortex by inbreeding depression" (c.f. Gilpin and Soule 1986). In this study, the process of extinction due to inbreeding depression is analysed numerically using a simple demographic and population genetic model. Conditions for rapid extinction due to inbreeding depression are evaluated in terms of mutation rate, equilibrium population size, and the rate of demographic disturbances. The possibility of rapid extinction highlights the importance of the interaction between genetics and demography for biological conservation.

Model We assumed a diploid sexual population with non-overlapping generations. Completely recessive deleterious mutations are assumed to arise at a number of autosomal loci. The number of loci is assumed to be 5000 throughout the analysis. In reality there may be more loci for mildly deleterious mutations, but such mutations are more incompletely recessive and contribute very little to inbreeding depression. The per-locus mutation rate and selection intensity against mutant homozygotes are assumed to be identical among loci. Lethal or semilethal mutations have very little ]:, +*rozygous effect on fitness while mildly deleterious or d ;imental mutations are more incompletely recessive dukai et al. 1972; Simmons and Crow 1977; Crow and oimmons 1983). Lethal and semilethal mutations are the major cause of inbreeding depression (Charlesworth and Charlesworth 1987; Lynch 1991). In Drosophila melanogaster, the diploid genomic mutation rate of almost completely recessive mutations is U~0.02 (Simmons and Crow 1977). Studies on Arabidopsis and some annual plant species suggest U-values from 0.01 to 0.02 while the red mangrove data based on chlorophyll deficiencies imply values of U ~ 0 . 2 (c.f. Willis 1992; Lande 1994). These values may be underestimation for mutation rates with regard to total fitness because in most studies only a few fitness components were measured. Organisms with

larger genomes may have higher genomic mutation rates (Crow 1993). In this study I employ p = 2 • 10 -4, 4 • 10 -5, and 4• 10 -6 as per-locus mutation rates for numerical simulations. Letting the number of loci be 5000, these values correspond to U = 1, 0.2, and 0.02, respectively. Due to deleterious mutations the mean fitness of a population decreases by a certain amount L, which is called the genetic load (Crow and Kimura 1970; Nei 1987). Here the load is assumed to influence linearly two demographic parameters (intrinsic rate of population growth r and carrying capacity K): r=rm~(1 - L ) and K = K,,+x(1-L), where rm~ and Kmaxare maximum r and K when no deleterious mutation is fixed or segregating in a population. In many cases the carrying capacity of a population represents competitive ability of organisms as well as quality and quantity of resources or habitats. For example, laboratory selection experiments under crowding condition increased K-values of Drosophila (Mueller 1988). And a few ecotoxicological experiments have shown that chemical pollutants decreased both carrying capacity and intrinsic rate of population growth (e.g. van Leeuwen et al. 1986). These experimental evidences suggest that the carrying capacity represents biological performance, which declines by inbreeding depression. A population is assumed to be at demographic and genetic equilibrium before demographic disturbances. Gene frequencies of deleterious mutations and inbreeding coefficients are maintained at equilibrium values which are determined by the per-locus mutation rate, the selection coefficient, and the population size. The equilibrium population size N is equal to the carrying capacity at demographic and genetic equilibrium: N=/(=Km~(1 - L ) , where i, is the equilibrium genetic load. There may be a substantial risk of population extinction (permanent reduction in mean fitness) due to fixation of mildly deleterious mutations in a relatively small asexual or a small sexual population (Lynch and Gabriel 1990; Gabriel et al. 1993; Lande 1994). However, in a fairly large population this process is extremely slow, and the risk due to inbreeding depression induced by erosion of heterozygosity among recessive lethal and semilethal mutations in a disturbed population is much larger than the risk due to fixation of mildly deleterious mutations. Thus, I assume that the process of fixation is disregarded in the process of extinction by inbreeding depression. The present analysis numerically simulates the population dynamics after demographic disturbances are introduced into the equilibrium state. The demographic disturbances are achieved by culling 15~ of individuals before reproduction occurs in each generation. Assuming maximal net reproductive rates larger than this culling rate, populations do not become extinct if no genetic effect decreases the demographic parameters. Several biotic or non-biotic factors can result in such demographic disturb-

E X T I N C T I O N BY I N B R E E D I N G

ances in natural populations. Hunting and destruction of habitats by human activities are one of the most likely causes.

Equilibrium population before demographic disturbances The mean inbreeding coefficient of each locus changes due to genetic drift, mutation and selection. In an inbred population the genotypic frequency of the mutant homozygote is q2(1-j~+qf, where q and f are the gene frequency of mutant and the inbreeding coefficient, respectively. Due to selection, a fraction of q2sis excluded from outbred individuals, and qs from inbred individuals (s: selection coefficient against mutant homozygote). Thus the proportion of inbred individuals to outbred individuals after selection is f(1 - qs)/(1-39(1 - q2s). Therefore, after selection, the inbreeding coefficient changes to f = f(1 -qs) _ f(1 -qs)

f(l_qs)+(l_f)(l_qZs)

l_fqs_(l_j)qEs ~ f(1-qs),

if the gene frequency and the inbreeding coefficient are small (i.e., q<__0.01 and f<_0.01). Due to mutation the inbreeding coefficient decreases at a rate of 2 p approximately. The inbreeding coefficient in the next generation due to mutation, selection and inbreeding is given by

f,+1~ [--~ + (1-~)f,](1-

2/t)(1 - qs),

(l)

where N is the effective population size (assumed to be equal to the actual population size throughout this study) and the subscript of t denotes generations. At equilibrium the coefficient reaches f~

1 4]~/~ +

59

Since at equilibrium the gene frequency of the deleterious mutation q is much lower than that of the wild-type allele 19 ~>>q'), the backward mutation is negligible, and the change in gene frequencies due to mutation is approximately A,,q~/~, where t~ is the mutation rate per locus per gamete per generation. According to Wright's formula, the change in gene freq(q- 1) quencies due to selection is derived from A~q-2W if" ' where IT"is the mean absolute fitness. Because the ~q marginal mean fitness of a locus is I~= 1 - s {q2(1 - j ) + qf},

aft

~q --s{2q(1-J)+f}. Substituting these expressions into Wright's formula, Asq+ Amq= 0 gives the equilibrium gene frequency as a quadratic equation: ( f - 1 ) t ] 2 - f t ] + P ~0. Using the equilibrium inbreeding coefficient (Eq. s 2), this equation yields the approximate gene frequency at equilibrium: 4~

1

12(~,

1

-

p)+~-~-~-2(N, /~)2+4p/s,

(3)

1-4pR

where ,~(~, ~)= 1 + 4 / ~ "

In actual numerical calculations, I evaluated the equilibrium gene frequency and the equilibrium inbreeding coefficient by iterative calculations using Eq. (2) and the quadratic

equation

[(fl--1)q2--~--C]@-~-~0] rather than b

i

using the above approximate solution of equilibrium gene frequency (Eq. 3).

(2) s

1

where the tildes denote equilibrium values. Mean gene frequencies of deleterious mutations mostly change due to selection and mutation. For simplicity I disregard the dispersion of gene frequencies by random genetic drift. If the equilibrium population size is reasonably large (i.e., N_>10s), the dispersion does not markedly influence the mean gene frequency (Crow and Kimura 1970). However, in a small equilibrium population the dispersion is not negligible and the gene frequency distribution is skewed in the direction of loss of mutations (0 gene frequency). This reduces the mean gene frequency to a value noticeably lower than the deterministic expectation value. Therefore the present analysis may overestimate the mean gene frequencies in a small equilibrium population. Nonetheless, the mean gene frequency expected from the deterministic model in a small population is extremely near 0, and the dispersion of gene frequencies by genetic drift is very unlikely to alter the present conclusion concerning the rate of changes in genetic load.

Non-equilibrium dynamics Mutation and reproduction is followed by individual development until sexual maturity. I assume random mating within populations and linkage equilibria concerning deleterious mutations. Population growth per generation due to birth and death is expressed by the discr~e version of the logistic equation (see below). I n , equilibrium population the birth and the death cancel ea~. other (a pair of individuals produces two individuals Oh average) because the population size reaches the carrying capacity. The demographic disturbances decrease the population size to a value smaller than the carrying capacity, and the population tends to recover by growth. If the recovery by growth is depressed by inbreeding depression to the extent that the population cannot return to the original level, the population starts to decline generation by generation, inducing risk of extinction. Natural selection is assumed to operate during individual development before sexual maturity and reproduction occur. Changes in gene frequencies are calculated using Wright's formula

60

TANAKA

specified in the previous section as follows. The gene frequency in the next generation qt+t is qt +t= q~ + P, where q; is the gene frequency after selection, q ; -

qt-It l-It

and

It

is

per-locus genetic load, lt=s{q~(1-ft)+q~ft}. The total genetic load per individual is L t - n l t. Demographic disturbances cause random cullings of individuals before mating and reproduction so that the effective population size relevant for f-values (inbreeding coefficients) in the next generation should be counted after disturbances before reproduction. The inbreeding coefficients are calculated every generation using Eq. (1) from gene frequency and population size after selection and disturbance in the previous generation. Thus the f-value increases as population declines. Increases in f-value cause stronger inbreeding depression, which reduces population size in the next generation. This positive feedback process can cause rapid extinction. Recent experimental and theoretical studies have revealed that inbreeding depression depends on life history stages and such age specificity is important in purging deleterious mutations (Lande et al. 1994). For simplicity, however, I disregard the age specificity of inbreeding depression, and take account only of the net effect of inbreeding depression on reproduction. Due to demographic disturbances, a certain proportion of individuals dN (disturbance rate) was randomly excluded from a population after sexual maturity every generation. In reality the disturbance rate may change according to the population size. For example, due to hunting without proper management absolute numbers of excluded individuals may be more important than proportions (the disturbance rate increases as the population decreases). A numerical simulation based on an absolute number of excluded individuals did not qualitatively alter the conclusion concerning extinction time (data not shown). If the demographic disturbance depends on population density, however, the disturbance rate may decrease as population size decreases. The population size before disturbance N(t) (the t-th generation counted from the onset of disturbances) is calculated from the population size after disturbance in the previous generation N'(t- 1), using the discrete version of the logistic equation (Pielou 1977): N(t)=

N'(t-1)R(t)(1

N'(t-K(t)l)), where R(t)

and

K(t)

are the

net reproductive rate and the carrying capacity in the t-th generation, respectively. Both demographic parameters are functions of time (generations) because they decrease as genetic load increases. Hence R(t)=Rm~(1 -L(t)) and g(t)=Kmax(1-L(t)), where Rmax and ](,maxare the maximum net reproductive rate and the maximum carrying capacity without any deleterious mutation, respectively.

Numerical results Numerical calculations are repeated until a population becomes extinct or reaches a new asymptotic population size. Populations are regarded as extinct when the population size becomes less than 10. Alternative definitions (e.g., less than 2) do not considerably alter the conclusions. In contrast, a population is regarded to have escaped from extinction and reached a new equilibrium if the population size in integers does not change between generations. I assume Rma~= 1.2 and dN=0.15. Since the disturbance rate is set to be lower than Rm~- 1, populations do not become extinct due only to the direct effect of disturbances. Thus, all extinctions observed in this analysis are attributed to inbreeding depression which depresses the demographic parameters.

Extinction of populations Trajectories of population sizes after the onset of demographic disturbances in the case o f / ~ = 4 x 10 -5 are shown in Fig. 1. Whether a population becomes extinct depends on its initial size. With small or extremely large initial population sizes extinction is avoided while intermediate or large populations are liable to extinction. This is because a large population retains more recessive deleterious mutations than a small population in which mutations have been purged due to a lower degree of heterozygosity at equilibrium (see the next section). The final stage of extinction is characterized by extremely rapid decreases in population sizes in a relatively small lO 6

lO 5

.~ lO 4 10 3 _m ~. 10 2 0

101 100 0

20

40

60

80

I00

t

t

120

140

160

Generation

Fig. 1. Fate of populations with various initial equilibrium population sizes after the onset of demographic disturbances. The rate of demographic disturbances dN=0.15, the per-locus mutation rate p=4 • 10-5, the number of loci is 5000, and the potential net reproductive rate Rm~=l.2. The selection coefficient is unity (s= 1).

EXTINCTION

p=4X10-5 1 000

BY

61

INBREEDING

0.6

p=4Xl 0-6

-

N=104 800 -

"o 0.4

IJ

Q

0 .J O

!

E

I'-

600

-

cO .m U e-

I

,

400 -

I

/' /

IN=105

c

0.2

X

ILl

200 -

p=2Xl 0 -4

o

_Ll

I

102

103

0

I

I

104

I

105

Initial Population

I

106

107

Size

Fig. 2. Association between extinction time and initial equilibrium population size with various per-locus mutation rates. The rate of demographic disturbances du=0.15, the number of loci is 5000, and the potential net reproductive rate Rmo~ = 1.2.

I

I

I

I

I

I

I

20

40

60

80

100

120

140

160

Generation

Fig. 4. Changes in genetic loads after the onset of demographic disturbances. The changes with population sizes of N= l04 and 105 are exponentially increased, leading to rapid extinction of populations. All parameter values are the same as those in Fig. 1.

become extinct even when the initial size is small.

number of generations (i.e., 10~20). Since population sizes (the vertical axis in the figure) are presented on a logarithmic scale, the dropping curves indicate rapid decreases at hyper-geometric rates. Essentially the same pattern of association between extinction times and initial equilibrium population sizes is observed with different per-locus mutation rates (Fig. 2). There seems to be a threshold equilibrium population size above which the population will rapidly become extinct. With initial sizes larger than the critical size, the extinction time increases exponentially with initial population size and reaches infinity at the initial size of 106. With a high per-locus mutation rate ( p = 2 • 10 -4, U = I ) populations

N=102

u.. 102

e-

N=103 101

J-

.Q t,--

,,. 100

j/

N=104

O

"~ 10 -1

N=105 O

I

o 10 -2 0

20

~ 40

N=106 /

60

_.~_~_.-~~ I

I

I

I

80

100

120

140

160

Generation

Fig. 3. Changes in inbreeding coefficients (F) after the onset of demographic disturbances. Data obtained with population sizes of N=I04 and 105 are interrupted due to extinction. All parameter values are the same as those in Fig. 1.

Dynamics of inbreeding coefficient and genetic load Changes in the inbreeding coefficients are different between different initial population sizes (Fig. 3). Populations with small initial sizes (102 and 103) have very high f-values, approaching the upper limit (100%) at equilibrium, which increase very little after the onset of disturbances. The f-values in large populations increased markedly. In particular, the populations which became extinct ( N = 104 and 105) showed rapid increases in the f values in final generations. The very large population (/V= 106) did not show this final rapid increase in f-values although they increased initially at the same rate as those for the populations which became extinct. Because of the very large population size, the inbreeding depression caused by increasing f-values was not sufficient to cause populational extinction in the case of )V= 106. Genetic loads changed in accordance with the changes in f-values (Fig. 4). As expected from theories on equilibrium populations, the genetic load is considerably higher for small populations (about 10%) than for large populations. In the large populations ( N = 105 and 106) the genetic loads are very close to the equilibrium genetic load predicted for an infinite population (/S= 1 -e-'"; c.f. Crow and Kimura 1970). The increases in genetic loads with N = 104 and 105 coincided with the extinction of the populations (see Figs. 1 and 4). The genetic loads in the populations that survived (/~= 102, 103, and 106) did not increase markedly. The reasons for this constancy of load may differ between the two extreme population size cases. In the small popula-

62

TAmpA

105 104

0.5

O

0

-I

\ \ \ ',,

o ,m

102

0.3 0.2

0.2

10o 0

0.1

I

I

I

I

20

40

60

80

i

II ~, I 1O0 120 140 160

0

0

Generation

tions the equilibrium gene frequencies are extremely low (data not shown) and the f-values are close to the upper limit at equilibrium 0 r ~ 1). In the very large p o p u l a t i o n , although enough deleterious mutations are heterozygous, the assumed a m o u n t o f demographic disturbances cannot cause sufficient loss o f heterozygosity required for the initiation o f the extinction process.

Effects of selection coefficient Differences in selection coefficients influence the time o f

s=l 0.80.60.4 I

2.5

2 -

.2

O

o (J

I

80

I

I

I

100 120 140 160

500

9

400

II/

,EE

1.5

0

I

60

extinction due to inbreeding depression (Fig. 5) while the equilibrium genetic load is independent o f the selection coefficients in an infinite p o p u l a t i o n (Fig. 7). W i t h smaller selection coefficients accumulation o f inbreeding and genetic load is slower (Figs. 6 and 7). But the exponential increase in genetic load is c o m m o n a m o n g cases with different selection coefficients. As long as mutations have noticeable effects (l>_s>_0.5, which is c o m m o n among recessive lethal or semi-lethal mutations), selection coefficients do not greatly influence the extinction time.

0.2

e-

--

I

40

Fig. 7. Effects of different selection coefficients on changes in genetic loads during the process of extinction. Values on the figure denote homozygous selection coefficients s. All parameter values except for the selection coefficients are the same as those in Fig. 1. Initial population size is N = 105.

I I

,D e-

I

20

Generation

Fig. 5. Effects of different selection coefficients on population trajectories during the process of extinction. Values on the figure denote homozygous selection coefficients. All parameter values except for the selection coefficients are the same as those in Fig. 1. Initial population size is N = 105.

II

o or//'

101 ~o.81 ',

A

,:

0.4

\ ,

103

0.2

/

/

O N

e~ O

S=I 0.80.60.4

0.6

II

I I

i ~=4XlO~

/

/z=4Xl045 /

p=2Xl 0 -4

~e- 300 .2

1

/ 0.5

=."

200

w

100 0

0

0

20

40

60

80

100 120 140 160

Generation

Fig. 6. Effects of different selection coefficients on changes in inbreeding coefficients during the process of extinction. Values on the figure denote homozygous selection coefficients s. All parameter values except for the selection coefficients are the same as those in Fig. 1. Initial population size is N = 105.

1

I 1.1

Net Reproductive

I 1.2

I 1.3

1.4

1.5

Rate before Disturbance

Fig. 8. Effects of potential net reproductive rates on extinction time. The rate of demographic disturbance is dN= 0.15. Values on the figure denote per-locus mutation rates. All parameter values except for the potential net reproductive rates are the same as those in Fig. 1. Initial population size is N = 105.

E X T I N C T I O N BY I N B R E E D I N G

(a) 500

Effects o f the net reproductive rates and disturbance rates on extinction time

105 102

N=104 103

I

/z = 2x10 -4

400

4)

E

p. tO 4-* r

300 -

200 -

4--'

)< UJ 100 -

I

I

I

I

I

I

I

I

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rate of Disturbance (b) 105 102103

N=104 500 -

/z = 4x10 -5 4)

400 -

E I- 300t,,O 200-

X IJJ

\

k

100 -

63

I

I

I

I

I

I

I

I

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rate of Disturbance

(c) N=105 102103104 500

/z = 4 x 1 0 -6

Extinction time is largely influenced by maximum net reproductive rates and disturbance rates (Figs. 8 and 9). With moderate or high per-locus mutation rates the net reproduction at equilibrium should be considerably higher than the disturbance rates for populations to avoid extinction. The extinction time seems to increase exponentially or at higher rates as the reproductive rate increases. Disturbance rates dN also determine the fate of populations (Figs. 9a, 9b, and 9c). In most cases the disturbance rate-extinction time curves are L-shaped, which indicates that there is a critical disturbance rate above which a population rapidly becomes extinct for a given set of parameters (initial population size, mutation rate, etc.) rather than a smooth correlation between disturbance rates and extinction time. In the above-mentioned figures shifts of the L-shaped curves toward the left represent higher liability to extinction. The relative rankings in extinction time among different initial population sizes changed between different mutation rates, making their interpretation more difficult. However, there are some general trends. In the extremely small population ( N = 102) and the largest population (.~=105) the critical disturbance rates (0.13~0.17) did not differ between different perlocus mutation rates compared with the populations with N---- 103 and N = 104, which implies that the fate of a very small or a very large population is less influenced by inbreeding depression. In contrast, in the populations with intermediate initial sizes the critical dN values varied considerably between different mutation rates. With /~= 4 • 10 -6, populations with different sizes have nearly the same critical dN value. Higher mutation rates result in much smaller critical dN values in the populations with N = 103 and 104. Thus their extinction time is sensitive to mutation rate, implying substantial risk of extinction due to inbreeding depression.

400 4)

E

Discussion

300 r,o "6 200

kI

t" .m

X

m

100

I

I

0.05

0.1

0.15

I

I

I

I

I

0.2

0.25

0.3

0.35

0.4

Rate of Disturbance

Fig. 9. Effects of rates of demographic disturbances on extinction time with various initial population sizes; (a) p = 2 • 10 4, f t : 4 • 10-5, and ( c ) / t : 4 • 10-6.

If most inbreeding depression results from recessive deleterious mutations (Mukai 1978; Charlesworth and Charlesworth 1987; Lynch 1991), it is caused by decreases in heterozygosity due to regular systems of inbreeding or mating in a small population. Since inbreeding depression reduces the growth rate of a population, it depresses recovery of the population from demographic disturbances. This reduces the population size and accelerates inbreeding in the next generation. Hence there is synergistic interaction between the process of inbreeding depression and decreases in population sizes due to demographic disturbances, which is suggested by the rapid

64

TANAKA

increases in inbreeding coefficients and genetic loads (Figs. 3 and 4). Another analysis has shown that the extinction vortex due to inbreeding depression can start with a single demographic disturbance (a founder flush) (Tanaka in preparation) while the present analysis assumes that demographic disturbances occur at a constant rate every generation. There is a critical range of initial equilibrium population sizes within which a population will become extinct due to inbreeding depression. A large population may be more liable to extinction due to inbreeding depression than a small population because a large population suffers less prior inbreeding and maintains deleterious mutations at higher frequencies. Since equilibrium levels of population sizes vary enormously among species (It6 1981; Royama 1992), the liability to extinction may be specific to species or populations. Positive correlations between cost of inbreeding (rates of inbreeding depression) and equilibrium population sizes or heterozygosities could strengthen such a hypothesis (Ouborg and Treuren 1994; Latta and Ritland 1994; Fitzsimmons et al. 1995). The hypothesis is only weakly supported by empirical studies. Most studies are based on plant populations with varying selfing rates, and the assumption of equilibria for heterozygosities H of marker loci and gene frequencies of deleterious mutations is not convincing. Cost of inbreeding may not be positively correlated with f (or H) values among local populations that share an ancestral population. Because the purging of deleterious mutations by selection is very time-consuming, relatively recent declines in population sizes may not change gene frequencies of deleterious mutations noticeably while f values decrease. If so, the cost of inbreeding (the increment of genetic load by a per-unit increase in f values) is kept rather constant with various f values among local populations, and the correlations between heterozygosities and the cost of inbreeding disappear. This interpretation of the absence of correlation betweenfvalues and rates of inbreeding depression might be supported by the widespread positive correlations between heterozygosity and fitness among plant local populations (Mitton and Grant 1984; AUendorf and Leary 1986; Zouros 1987; Fitzsimmons et ah 1995). At equilibrium, genetic loads differ very little with considerably different equilibrium population sizes andfvalues (see Fig. 4). The differences in f values which are negatively correlated with fitness may represent transient states of populations decreasing or increasing from an ancient common population. A comparative study on major taxonomic groups of animals has implied negative association between f values and population sizes (Rails et al. 1988). Predicted cost of inbreeding varies largely among mammalian populations. Carnivores, including cheetahs, whose population size is considerably smaller than that of herbivorous animals,

maintain much smaller numbers of lethal equivalents per genome. Some large mammalian species may not be liable to inbreeding depression because they have an extremely low degree of heterozygosity probably due to ancient bottlenecks (Bonnel and Selander 1974; O'Brien et al. 1983, 1987; Sage and Wolff 1986; Merola 1994). A very small population is likely to suffer great risk of extinction from other sources. Demographic and environmental stochasticity may severely endanger small populations (Leigh 1981; Goodman 1987; Iwasa and Mochizuki 1988; Gabriel and Biirger 1992; Lande 1993). Those stochastic effects which are important in very small populations may interact with inbreeding depression in the final stage of extinction. There is some empirical evidence that inbreeding depression is reinforced under deteriorated environments (Templeton and Read 1983, 1984; Soule and Kohn 1989; Pray et al. 1994). Extinction due to inbreeding depression may be more accelerated under deteriorated environments than under ideal environments. The most important suggestion from the present analysis is that the effective population size should not be reduced rapidly if the heterozygosity is high and the intrinsic rate of population growth is insufficiently high to compensate the inbreeding depression. Rapid decreases in effective sizes in an originally large population induce extinction vortexes due to inbreeding depression, which may lead to rapid extinction. There are several reasons for decreases in effective sizes other than loss of actual population size. If a population is geographically subdivided, habitat segregation or restricted migration between subpopulations can reduce the effective population size rapidly without decreasing the actual size (c.f. Crow and Aoki 1984; Lacy 1987). Changes in the mating system may also reduce the effective population size. Excluding premature males from a population may not reduce or even increase actual size (Watt 1968), but it may impose a risk of extinction due to inbreeding depression. Since inbreeding depression occurs whenever effective population sizes decrease rapidly with enough previous heterozygosities, the extinction vortex due to inbreeding depression can accompany any type of extinction process. Under environmental changes, populations must evolve at a sufficient rate in order to adapt to the changing environment (Lynch and Lande 1993; Bfirger and Lynch 1995). Limits of adaptive evolution due to limited additive genetic variances in adaptive characters reduce effective sizes and may enhance inbreeding. Synergistic interaction between genetic factors of extinction (e.g. inbreeding depression and erosion of additive genetic variances in adaptive characters) and consequent loss of proliferation of populations may be important for populational extinction (c.f. Gabriel et al. 1991). The rate of demographic disturbances is assumed to be constant in this study for simplicity. However, actual

EXTINCTION BY INBREEDING

disturbances m a y fluctuate due to stochastic factors. T h e r a n d o m f l u c t u a t i o n o f the d e m o g r a p h i c disturbances is likely to e n h a n c e e x t i n c t i o n due to i n b r e e d i n g d e p r e s s i o n since the effective p o p u l a t i o n size is closer to the g e o m e t r i c m e a n t h a n the a r i t h m e t i c m e a n o f p o p u l a t i o n sizes o v e r generations. N o n e t h e l e s s , h o w the fluctuating disturbances alter the present c o n c l u s i o n needs f u t u r e analyses. Acknowledgments: I thank Y. Iwasa, H. Matsuda, M. Morgan, and T. Yahara for discussion on this topic. I am very grateful to D. Schoen for introducing me to empirical research on inbreeding depression in plants. This work is supported by a postdoctoral grant of the National Sciences and Engineering Research Council of Canada to Y.T.

References Agren, J. and D.W. Schemske (1993) Outcrossing rate and inbreeding depression in two annual monoecious herbs, Begonia hirsuta and B. semiovata. Evolution 47: 125-135. Allendorf, F. W. and R. F. Leary (1986) Heterozygosity and fitness in natural populations of animals, pp. 57-76. In M. E. Soule (ed.) Conservation biology: The science of scarcity and diversity. Sinauer Associates, Sunderland, Mass. 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. Barrett, S. C. H. and J. R. Kohn (1991) Genetic and evolutionary consequences of small population size in plants: implications for conservation, pp. 3-30. In. D. A. Falk and K. E. Holsinger, (eds.) Genetics and conservation of rare plants. Oxford University Press, New York. Bonnel, M. I. and R. K. Selander (1974) Elephant seals: genetic variation and near extinction. Science 184: 908-909. Bfirger, R. and M. Lynch (1995) Evolution and extinction in a changing environment: a quantitative-genetic analysis. Evolution 49: 151-163. Charlesworth, D. and B. Charlesworth (1987) Inbreeding depression and its evolutionary consequences. Annual Review of Ecology and Systematics 18: 237-268. Crow, J. F. (1993) Mutation, mean fitness, and genetic load. Oxford Series in Evolutionary Biology 9: 3-42. Crow, J. and K. Aoki (1984) Group selection for a polygenic behavioral trait: Estimating the degree of population subdivision. Proceedings of National Academy of Sciences of the USA 81: 6073-6077. Crow, J.F. and M. Kimura (1970) An introduction to population genetics theory. Harper and Row, New York. Crow, J.F. and M . J . Simmons (1983) The mutation load in Drosophila. pp. 1-35. In M. Ashburner, H. L. Carson, and J. N. Thompson, Jr. (eds.) The genetics and biology of Drosophila, Vol. 3c. Academic Press, New York. Falconer, D. S. (1989) Introduction to quantitative genetics. 3rd ed. Longman, New York. Fitzsimmons, N. N., S. W. Buskirk and M. H. Smith (1995) Population history, genetic variability, and horn growth in bighorn sheep. Conservation Biology 9:314-323. Frankel, O. H. and M. E. Soule (1981) Conservation and Evolution. Cambridge University Press, Cambridge. Gabriel, W. and R. Burger (1992) Survival of small populations

65

under demographic stochasticity. Theoretical Population Biology 41: 44-71. Gabriel, W., R. Burger and M. Lynch (1991) Population extinction by mutational load and demographic stochasticity, pp. 49-59. In A. Seitz and V. Loeschcke (eds.) Species conservation: a population-biological approach. Birkhauser, Berlin. Gabriel, W., M. Lynch and R. Burger (1993) Muller's ratchet and mutational meltdowns. Evolution 47: 1744-1757. 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. Goodman, D. (1987) The demography of chance extinction, pp. 1134. in M. E. Soule (ed.) Viable populations for conservation. Cambridge University Press, Cambridge. Holtsford, T.P. and N.C. Ellstrand (1990) Inbreeding effects in Clarkia tembloriensis (Onagraceae) populations with different natural outcrossing rates. Evolution 44: 2031-2046. It5, Y. (1981) Comparative ecology. 2nd ed. Cambridge University Press, Cambridge. Iwasa, Y. and H. Mochizuki (1988) Probability of population extinction accompanying a temporary decrease of population size. Researches on Population Ecology 30: 145-164. Lacy, R. C. (1987) Loss of genetic diversity from managed populations: interacting effects of drift, mutation, immigration, selection, and population subdivision. Conservation Biology 1: 143-158. Lande, R. (1993) Risk of population extinction from demographic and environmental stochasticity, and random catastrophes. American Naturalist 142:911-927. Lande, R. (1994) Risk of population extinction from fixation of new deleterious mutations. Evolution 48: 1460-1469. Lande, R. and G. F. Barrowclough (1987) Effective population size, genetic variation, and their use in population management, pp. 87123. In M. Soule (ed.) Viable populations for conservation. Cambridge University Press, New York. Lande, R., D.W. Schemske and S.T. Schultz (1994) High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations. Evolution 48: 965-978. Latta, R. and K. Ritland (1994) The relationship between inbreeding depression and prior inbreeding among populations of four Mimulus taxa. Evolution 48: 806-817. Leigh, E. G. (1981) The average lifetime of a population in a varying environment. Journal of Theoretical Biology 90: 213-239. Lynch, M. (1991) The genetic interpretation of inbreeding depression and outbreeding depression. Evolution 45: 622-629. Lynch, M. and W. Gabriel (1990) Mutation load and the survival of small populations. Evolution 44: 1725-1737. Lynch, M. and R. Lande (1993) Evolution and extinction in response to environmental change, pp. 234-250. In P. M. Kareiva, J.G. Kingsolver and R.B. Huey (eds.) Biotic interactions and global change. Sinauer Associates, Sunderland, Mass. Merola, M. (1994) A reassessment of homozygosity and the cost for inbreeding depression in the cheetah, Acinonyx jubatus: Implications for conservation. Conservation Biology 8: 961-971. Mitton, J. B. and M. C. Grant (1984) Associations among protein heterozygosity, growth rates, and developmental homeostasis. Annual Review of Ecology and Systematics 15: 479-499. Mueller, L. D. (1988) Evolution of competitive ability in Drosophila by density-dependent natural selection. Proceedings of National Academy of Sciences of the USA 85: 4383-4386. Mukai, T. (1978) Population Genetics. Kodansha Scientific, Tokyo. (in Japanese)

66

TANAKA

Mukai, T., S. I. Chigusa, L. E. Mettler and J. F. Crow (1972) Mutation rate and dominance of genes affecting viability in Drosophila melanogaster. Genetics 72: 335-355. Nei, M. (1987) Molecular evolutionary genetics. CambridgeUniversity Press, New York. O'Brien, S. J., D. E. Wildt, D. Goldman, D. R. Merril and M. Bush (1983) The cheetah is depauperate in genetic variation. Science 211: 459-462. O'Brien, S. J., D. E. Wildt, M. Bush, T. M. Caro, C. Fitzgibbon, I. Aggundey and R.E. Leakey (1987) East African cheetahs: evidence for two population bottlenecks? Proceedings of National Academy of Sciencesof the USA 84:508-511. Ouborg, N. J. and R. V. Treuren (1994) The significance of genetic erosion in the process of extinction. IV. Inbreeding load and heterosis in relation to population size in the mint Salviapratensis. Evolution 48: 996-1008. Pielou, E. C. (1977) Mathematical ecology. 4th ed. John Wiley & Sons, New York. Pray, L. A., J. M. Schwart, C. J. Goodnight and L. Stevens (1994) Environmental dependency of inbreeding depression: implications for conservation biology. Conservation Biology 8: 562-568. Rails, K., J. Ballou and A. Templeton (1988) Estimates of lethal equivalents and the cost of inbreeding in mammals. Conservation Biology 2: 185-193. Royama, T. (1992) Analytical population dynamics. Chapman & Hall, New York. Sage, R.D. and J.O. Wolff (1986) Pleistocene glaciations, fluctuating ranges, and low genetic variability in a large mammal (Ovis dalh). Evolution 40: 1092-1095. Schoen, D. J. (1983) Relative fitnesses of selfed and outcrossed progeny in Gilia achilleifolia (Polemoniaceae). Evolution 37: 292-301. Schonewald-Cox, C.M., S.M. Chambers, B. MacBryde and L. Thomas (1983) Genetics and conservation: a referencefor managing wild animal and plant populations. Benjamin-Cummings, London. Simmons, M. and J. F. Crow (1977) Mutations affecting fitness in

Drosophila populations. Annual Reviews of Genetics 11: 49-78. Soule, M. (1986) Conservation biology: the science of scarcity and diversity. Sinauer Associates, Sunderland, Mass. Soule, M. E. and K. A. Kohm (1989) Research priorities for conservation biology. Island Press, Washington, D.C. Templeton, A. R. and B. Read (1983) The elimination of inbreeding depression in a captive herd of Speke's gazelle, pp. 241-261. In C . M . Schonewald-Cox, S.M. Chambers, B. MacBryde and L. Thomas, (eds.) Genetics and conservation. Benjamin-Cummings, Menlo Park, California. Templeton, A. R. and B. Read (1984) Factors eliminating inbreeding depression in a captive herd of Speke's gazelle. Zoo Biology 3: 177199. van Leeuwen, C. J., M. Rijkeboer and G. Niebeek (1986) Population dynamics of Daphnia magna as modifed by chronic bromide stress. Hydrobiologia 133: 277-285. van Treuren, R., R. Bijlsma, N.J. Ouborg and W. van Delden (1993) The significance of genetic erosion in the process of extinction. IV. Inbreeding depression and heterosis effects caused by selfing and outcrossing in Scabiosa columbaria. Evolution 47: 1669-1680. Watt, K. E. F. (1968) Ecology and resource management. McGrawHill, New York. Willis, J. H. (1992) Genetic analysis of inbreeding depression caused by chlorophyll-deficient lethals in Mimulus guttatus. Heredity 69: 562-572. Willis, J H. (1993) Effects of different levels of inbreeding on fitness components in Mimulus guttatus. Evolution 47: 864-876. Zouros, E. (1987) On the relationship between heterozygosity and heterosis: an evolution of evidence from marine mollusks, pp. 255270. In M.C. Rattari, J.G. Scandalios and G.S. Whitt (eds.) lsozymes: current topics in biological and medical research, Vol. 15. Liss, New York. Received 31 October'1996; Accepted 23 May 1997