276
Yoshinari Tanaka
The mean fitness of a population decreases due to deleterious genes from the
maximum value 1 to 1 − L, where L is the genetic load (Crow and Kimura 1970;
Nei 1987). To evaluate the effects of inbreeding depression on extinction of
populations, the genetic load must be expressed as reductions of demographic
parameters. I assumed that the Malthusian fitness (W = e
r
, where r is the intrinsic
rate of population increase) decreases linearly with the genetic load. Because the
realized population growth rate under density dependence is e
r(1−N/K)
, where N is
population size and K is carrying capacity, the population growth rate of the mean
fitness of a genetically loaded population is λ = (1 − L)e
r max (1−N/K) where r max is the
maximum intrinsic rate of natural increase of a mutation-free population.
The population is assumed to be at demographic and genetic equilibrium.
Previous work has suggested the equilibrium is locally stable under realistic
parameter values (Tanaka 1998). External factors (e.g., habitat destruction or
hunting) that decrease population size from the equilibrium size are required to
trigger the inbreeding vortex. Reduction of the population size by such external
factors is called “demographic disturbance.” To model demographic disturbances,
I assume that the carrying capacity K decreases monotonically with time toward a
minimum value K min . An exponential function of time was used for K (i.e., K t =
(K 0 − K min )(1 − k)
t + K min , where k is the rate of decrease for K, and K 0 is the initial
carrying capacity).
At equilibrium, the expected inbreeding coefficient and the mean gene frequency over loci are determined by a balance between mutation, genetic drift, and
selection. The genetic load is also at equilibrium. The population growth rate at
equilibrium is denoted as = (1 − )e
r max (1−N/K)
, where and denote the equilibrium values of λ and L. Because of inbreeding depression, the population
growth rate is reduced. Denote the proportional reduction of λ as 1 − δ inb . The
population growth rate of any inbred population is λ = δ inb , where δ inb =
1 − L
1 −
.
The effect of the equilibrial genetic load on the population growth rate is
ignored in the present analysis to determine the extent to which inbreeding depression generated from demographic disturbances contributes to the process of extinction. The equilibrium population growth rate is assigned arbitrarily.
The per-locus genetic load at the ith locus l i is defined as l i =1–w i /w max , where w i
is the marginal mean fitness of the ith locus and w max is the maximum fitness of a
locus, assumed to be unity for all loci. Assuming multiplicative fitness without
epistatic interaction between loci, the total genetic load L is calculated as
L = 1 − ͟
i
(1 − l i )
= 1 − ͟
i
(w i )/W max
where W max is the maximum multilocus fitness, ∏w max = 1. Then L = 1 − ∏w i . The
i
i
marginal mean fitness at the ith locus is w i = 1 − sq i
2
, where q i is the gene frequency
of the recessive deleterious gene at the ith locus. The total genetic load is
approximately
Yoshinari Tanaka
The mean fitness of a population decreases due to deleterious genes from the
maximum value 1 to 1 − L, where L is the genetic load (Crow and Kimura 1970;
Nei 1987). To evaluate the effects of inbreeding depression on extinction of
populations, the genetic load must be expressed as reductions of demographic
parameters. I assumed that the Malthusian fitness (W = e
r
, where r is the intrinsic
rate of population increase) decreases linearly with the genetic load. Because the
realized population growth rate under density dependence is e
r(1−N/K)
, where N is
population size and K is carrying capacity, the population growth rate of the mean
fitness of a genetically loaded population is λ = (1 − L)e
r max (1−N/K) where r max is the
maximum intrinsic rate of natural increase of a mutation-free population.
The population is assumed to be at demographic and genetic equilibrium.
Previous work has suggested the equilibrium is locally stable under realistic
parameter values (Tanaka 1998). External factors (e.g., habitat destruction or
hunting) that decrease population size from the equilibrium size are required to
trigger the inbreeding vortex. Reduction of the population size by such external
factors is called “demographic disturbance.” To model demographic disturbances,
I assume that the carrying capacity K decreases monotonically with time toward a
minimum value K min . An exponential function of time was used for K (i.e., K t =
(K 0 − K min )(1 − k)
t + K min , where k is the rate of decrease for K, and K 0 is the initial
carrying capacity).
At equilibrium, the expected inbreeding coefficient and the mean gene frequency over loci are determined by a balance between mutation, genetic drift, and
selection. The genetic load is also at equilibrium. The population growth rate at
equilibrium is denoted as = (1 − )e
r max (1−N/K)
, where and denote the equilibrium values of λ and L. Because of inbreeding depression, the population
growth rate is reduced. Denote the proportional reduction of λ as 1 − δ inb . The
population growth rate of any inbred population is λ = δ inb , where δ inb =
1 − L
1 −
.
The effect of the equilibrial genetic load on the population growth rate is
ignored in the present analysis to determine the extent to which inbreeding depression generated from demographic disturbances contributes to the process of extinction. The equilibrium population growth rate is assigned arbitrarily.
The per-locus genetic load at the ith locus l i is defined as l i =1–w i /w max , where w i
is the marginal mean fitness of the ith locus and w max is the maximum fitness of a
locus, assumed to be unity for all loci. Assuming multiplicative fitness without
epistatic interaction between loci, the total genetic load L is calculated as
L = 1 − ͟
i
(1 − l i )
= 1 − ͟
i
(w i )/W max
where W max is the maximum multilocus fitness, ∏w max = 1. Then L = 1 − ∏w i . The
i
i
marginal mean fitness at the ith locus is w i = 1 − sq i
2
, where q i is the gene frequency
of the recessive deleterious gene at the ith locus. The total genetic load is
approximately