Short-term threats to persistence

Drift in small populations has many of the same properties as inbreeding. In fact,

$$f_{t+1} = f_t + \left(1 - {1\over 2N_e}\right)(1 - f_t)$$ (3)

$${1 - f_{t+1} \over {1 - f_t}} = {1 \over 2N_e}$$ (4)

$$f_t = 1 - \left(1 - {1\over 2N_e}\right)^t$$ (5)

where $f$ is the inbreeding coefficient, a measure of the degree of inbreeding. By trial and error animal breeders have discovered how much inbreeding can be tolerated by domestic animals before the lines begin to decline in performance and fecundity [3,14]. Their rule of thumb is that the per generation rate of inbreeding ${1 - f_{t+1} \over {1 - f_t}} =~ {1 \over 2N_e}$ should not be greater than 2%-3%. If we want to be a little conservative, we might reduce this to 1%. Thus, we require⁷

$$N_e > 50$$

An alternative approach is to observe that animal breeders have found a obvious effect on fecundity in small populations when the inbreeding coefficient approaches 0.5. If we want our population to be viable for 100 generations

$$0.5 > 1 - \left(1 - {1 \over 2N_e}\right)^{100}$$ (6)

$$\left(1 - {1 \over 2N_e}\right)^{100} > 0.5$$ (7)

$$1 - {1 \over 2N_e} > (0.5)^{0.01} = 0.993$$ (8)

$$0.007 > {1 \over 2N_e}$$ (9)

$$N_e \ge 73$$ (10)

Still another approach is to assume that the deleterious effects noted in these small populations is primarily the result of the expression of recessive deleterious alleles. Then using the results of drift theory we can calculate the probability that a population has a particular allele frequency, given assumptions about the strength of selection and mutation rates. For a broad range of strentgths of selection and for what are thought to be typical per locus mutation rates ($10^{-6}$ per generation) it is possible to calculate the effect on population mean fitness. This calculation suggests that populations will suffer noticeably (mean fitness reduction greater than 10%) if $N_e < 100\hbox{--}300$ [8].

Finally, we can consider a population-level version of Muller's ratchet. As a result of genetic drift, there's always a chance that a deleterious allele can be fixed as a result of genetic drift. If it does and if it reduces the reproductive capacity of the population, the population size may get smaller, making it easier for new deleterious mutations to fix, which will reduce the population size further, and so on, and so on, and so on. Mike Lynch has called this process the ``mutational meltdown'' [4,12]. The expected time to extinction increases rapidly in mutational meltdown models such that in populations with an effective size greater than a few hundred persistence times are well into the hundreds or thousands of generations (4).

Figure 4: Mean time to extinction for monoecious populations with a constant carrying capacity (open circles and dashed line) and monogamous populations with variable carrying capacities (lognormal with the indicated coefficients of variation). Mutational parameters are close to those estimated for Drosophila melanogaster: genomic mutation rate = 1.5, selection coefficient against recessive homozygotes = 0.015, dominance coefficient = 0.35. Ten offspring are produced per individual (from [12].

In addition to the effects of inbreeding depression, which have been found in virtually every outbreeding organism - plant or animal - that has ever been studied⁸, there has been some suggestion that loss of genetic diversity may have an immediate impact on short-term survival. My (admittedly biased) reading of the evidence is that the evidence on this point is at best equivocal.

The bottom line? Recall that to buffer the effects of environmental stochasticity we require populations consisting of thousands or tens of thousands of reproductive individuals. $N_e \over N$ is unlikely to be less than 0.1, except in rare circumstances, so populations large enough to buffer environmental stochasticity are almost certainly large enough to buffer genetic stochasticity. On the other hand, when populations are critically endangered, genetic changes may pose an additional threat to population persistence.

An example: Populations of the Flordia panther are extremely small. The high proportion of kinked tails and cowlicks among individuals in the population (about 90% in both cases) coupled with the poorest semen quality ever recorded suggests that the population is suffering from substantial inbreeding depression. As a result,⁹Texas cougars were released into Florida in the hope that individual fitness would rapidly improve. No kinked tails have been recorded from $F_1$ and $F_2$ offspring, and the proportion of $F_1$ and $F_2$ individuals with the cowlick are far less frequent (about 14%), suggesting that efforts to reduce the effects of inbreeding depression may have been successful [7].

An exception: loss of self-incompatibility alleles in plants with genetically determined self-incompatibility. Hymenoxys acaulis: Illinois populations have a single compatibility type, Ohio populations have only 3-9. Long-term persistence of Illinois populations requires import of genotypes from Ohio. Reproductive capacity of all populations limited by availability of compatible mates.