16. Theoretical Properties of Extinction by Inbreeding Depression
275
extinct due to inbreeding depression. Rapid extinction by inbreeding depression
requires that the equilibrium population be perturbed by external nongenetic
factors and that the population size before the perturbation had been large enough
to maintain recessive deleterious mutations in the population (Tanaka 1997,
1998). In these studies, the perturbation of the equilibrium population was simulated by removal of individuals at a constant rate every generation (Tanaka 1997,
1998) or by assuming the carrying capacity to be exponentially decreasing in time
(Tanaka, 2000). Although rapid extinction by inbreeding depression does not
occur in a stable population, it is likely for a declining population.
The suggested sufficient conditions required for the extinction vortex to occur
are as follows. The equilibrium population size must be large (>10
5 ) so that
recessive deleterious genes are maintained with sufficient frequency. As inbreeding depression cannot alone induce extinction because of the local stability of
genetic equilibria, nongenetic demographic disturbances are required to reduce
the population size at some higher rate than purging selection can eliminate the
deleterious genes (and make inbreeding depression irrelevant). Also, the maximum (mutation-free) intrinsic rate of natural increase must be sufficiently small
so that reductions of population size positively feed back through the action of
inbreeding depression (Tanaka 1997, 1998). The overdominant genes are unlikely
to contribute to the extinction vortex by inbreeding depression because high
equilibrium segregation loads must precede the inbreeding vortex (Tanaka 1998).
The fluctuation of population size due to environmental stochasticity interacts
with inbreeding depression and reinforces the extinction vortex (Tanaka, 2000; cf.
van Noordwijk 1994). Real populations are subject to random fluctuations of
environmental factors, such as temperature, food availability, and predation pressure. This environmental stochasticity brings about random fluctuations in population size. That an extinction vortex due to inbreeding depression may be reinforced or enhanced by environmental stochasticity is important in evaluating the
extinction risk due to genetic factors.
In this chapter, I review the models used and show some analytical results on
extinction due to inbreeding depression.
Genetic Model
I assumed the genetic mechanism of inbreeding depression involves recessive
deleterious genes. The deleterious genes are assumed to be distributed among n
diallelic autosomal loci, which are identical with respect to the per-locus mutation
rate, the selection coefficient, and the degree of dominance. The model is a simple
extension of a one-locus, two-allele model. Linkage disequilibrium and epistatic
interaction between loci are neglected for simplicity. It is assumed that three
genotypes, AA, Aa, and aa, have mean fitnesses of 1, 1, and 1 − s, respectively,
where s denotes the selection coefficient. Thus all deleterious genes are assumed
to be completely recessive.
275
extinct due to inbreeding depression. Rapid extinction by inbreeding depression
requires that the equilibrium population be perturbed by external nongenetic
factors and that the population size before the perturbation had been large enough
to maintain recessive deleterious mutations in the population (Tanaka 1997,
1998). In these studies, the perturbation of the equilibrium population was simulated by removal of individuals at a constant rate every generation (Tanaka 1997,
1998) or by assuming the carrying capacity to be exponentially decreasing in time
(Tanaka, 2000). Although rapid extinction by inbreeding depression does not
occur in a stable population, it is likely for a declining population.
The suggested sufficient conditions required for the extinction vortex to occur
are as follows. The equilibrium population size must be large (>10
5 ) so that
recessive deleterious genes are maintained with sufficient frequency. As inbreeding depression cannot alone induce extinction because of the local stability of
genetic equilibria, nongenetic demographic disturbances are required to reduce
the population size at some higher rate than purging selection can eliminate the
deleterious genes (and make inbreeding depression irrelevant). Also, the maximum (mutation-free) intrinsic rate of natural increase must be sufficiently small
so that reductions of population size positively feed back through the action of
inbreeding depression (Tanaka 1997, 1998). The overdominant genes are unlikely
to contribute to the extinction vortex by inbreeding depression because high
equilibrium segregation loads must precede the inbreeding vortex (Tanaka 1998).
The fluctuation of population size due to environmental stochasticity interacts
with inbreeding depression and reinforces the extinction vortex (Tanaka, 2000; cf.
van Noordwijk 1994). Real populations are subject to random fluctuations of
environmental factors, such as temperature, food availability, and predation pressure. This environmental stochasticity brings about random fluctuations in population size. That an extinction vortex due to inbreeding depression may be reinforced or enhanced by environmental stochasticity is important in evaluating the
extinction risk due to genetic factors.
In this chapter, I review the models used and show some analytical results on
extinction due to inbreeding depression.
Genetic Model
I assumed the genetic mechanism of inbreeding depression involves recessive
deleterious genes. The deleterious genes are assumed to be distributed among n
diallelic autosomal loci, which are identical with respect to the per-locus mutation
rate, the selection coefficient, and the degree of dominance. The model is a simple
extension of a one-locus, two-allele model. Linkage disequilibrium and epistatic
interaction between loci are neglected for simplicity. It is assumed that three
genotypes, AA, Aa, and aa, have mean fitnesses of 1, 1, and 1 − s, respectively,
where s denotes the selection coefficient. Thus all deleterious genes are assumed
to be completely recessive.