The effective size of fluctuating populations

We consider a Wright-Fisher model whose population size is an autocorrelated stochastic process. Our interest is in the effects of autocorrelated fluctuations of the population size on the effective size. We define the inbreeding effective size and the variance effective size and show that these effective sizes are the same for this model. In the literature, it is said that the effective size is equal to the harmonic mean of population size when the size fluctuates. We will show, however, that the effective size is not the same as the harmonic mean of population size unless the fluctuations of population size are uncorrelated. The effective size is larger (resp. smaller) than the harmonic mean when the fluctuations of population size are positively (resp. negatively) autocorrelated. Further, we obtain some asymptotic expressions for effective size when the population size is very large and/or the autocorrelation of the fluctuation is very strong.     

Coalescent process with fluctuating population size and its effective size

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.     

A neutral model with fluctuating population size and its effective size

We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.     

The effective number of a population that varies cyclically in size. I. Discrete generations

We consider a dioecious population having numbers of males and females that vary over time in cycles of length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the effective population number (or size), N(e), is approximately equal to the harmonic mean of the effective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding effective population numbers or variance effective population numbers are involved in the calculation of N(e). If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding effective population number, N(eI), differs from the variance effective population number, N(eV). The mutation effective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue effective population number is the same as N(eV).     

Fixation probabilities when the population size undergoes cyclic fluctuations

Let us assume that there is a monoecious random mating population that changes cyclically in size. Then, the probability that a nonrecessive favorable mutant is ultimately fixed, if it is originally present in a single heterozygote, is approximately proportional to the harmonic mean of the effective population sizes in the cycle and inversely proportional to the population size when the mutant appears. This approximation works well if the selective advantage s of the mutant is small and the length k of a cycle is small in comparison with the population sizes in a cycle. If k is large the harmonic mean is, in general, replaced by a weighted harmonic mean that puts the largest weights on reciprocals of effective population sizes in the first few generations after the mutant appears.     

Genome size is negatively correlated with effective population size in ray-finned fish

A recent theory suggesting that genome size and complexity can increase as a passive consequence of small effective population size has generated much controversy. In this article, we demonstrate that freshwater fish species, which have smaller effective population sizes than marine fish species, have larger genomes. We show that genome size is negatively correlated with genetic variability, independent of phylogeny, body size and generation time. Genome duplication is also observed predominantly in freshwater fish. These results suggest that the raw materials of complexity originate under conditions of reduced selection efficiency.     

Effect of changes in population size on the correlation between mutation rate and heterozygosity

Summary The effect of changes in population size on the correlation between mutation rate and heterozygosity was studied by using two models: sudden change in population size and gradual change. It was shown that the results for the two models are close to each other, unless the rate of change for the latter is exceedingly slow. Thus, in many cases, the former model, which is much simpler than the latter, can be used to treat the present problem. Numerical computations showed that the correlation in a population that is expanding or has expanded in the recent past is stronger while the correlation in a population that is decreasing or has experienced a population reduction or bottleneck in the recent past is weaker than that for an equilibrium population with the same mean heterozygosity. However, regardless of whether the population is at equilibrium or not, the proportion of variation in heterozygosity that is attributable to variation in molecular weight over loci is rather small if the mean heterozygosity of the population is low, say of the order 0.05 or smaller.     

The effective size of bryophyte populations

Bryophytes with their dominant haploid stage conform poorly to the life cycles generally treated in population genetical models. Here we make a detailed analysis of what effective sizes bryophyte model populations have as a function of their breeding system. It is found that the effective size is rarely much smaller than the scored number of haploid gametophytic individuals, even when the limited number of diploids (sporophytes) formed is taken into account. The most severe decrease in effective size occurs when unisexual gametophytic females produce only a small number of fertile diploid sporophytes in male biased populations; this effect is due to the restricted sampling of male gametophytic individuals that then occurs. It is shown that the harmonic mean of diploid sporophytes formed per haploid gametophytic individuals is the relevant measure in these calculations and not the standard (and generally larger) arithmetic mean.     

Response of population size to changing vital rates in random environments

Population size and population growth rate respond to changes in vital rates like survival and fertility. In deterministic environments change in population growth rate alone determines change in population size. In random environments, population size at any time t is a random variable so that change in population size obeys a probability distribution. We analytically show that, in a density-independent population, the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ _S while the standard deviation scales as $\sqrt t$ . Consequently, a vital rate with a larger elasticity of λ _S may produce a larger mean change in population size compared to one with a smaller elasticity of λ _S. But a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. Hence, the response of population size to perturbation of a vital rate depends not only on the elasticity of the population growth rate but also on the variance in change in population size. Our results provide a formula to calculate the probability that population size changes by a given percentage that works well even for short time periods.     

The consequences of the variance-mean rescaling effect on effective population size

The effective population size (N_e), and the ratio between N_e and census population size (N) are often used as measures of population viability. We show that using the harmonic mean of population sizes over time – a common proxy for N_e– has some important evolutionary consequences and implications for conservation management. This stems from the fact that there is no unambiguous relationship between the arithmetic and harmonic means for populations fluctuating in size. As long as the variance of population size increases moderately with increasing arithmetic mean population size, the harmonic mean also increases. However, if the variance of population size increases more rapidly, which existing data often suggest, then the harmonic mean may actually decrease with increasing arithmetic mean. Thus maximizing N may not maximize N_e, but could instead lower the adaptive potential and hence limit the evolutionary response to environmental change. Large census size has the clear advantage of lowering demographic stochasticity, and hence extinction risk, and under certain conditions large census size also minimizes the loss of genetic variation. Consequently, maximising census size has served as a useful dogma in ecology, genetics and conservation. Nonetheless, due to the intricate relationships among N_e, population viability and the properties of population fluctuations, we suggest that this dogma should be taken only as a rule of thumb.     

Genetic evidence for larger African population size during recent human evolution

Genetic evidence suggests that the long-term average effective size of sub-Saharan Africa is larger than other geographic regions. A method is described that allows estimation of relative long-term regional population sizes. This method is applied to 60 microsatellite DNA loci from a sample of 72 sub-Saharan Africans, 63 East Asians, and 120 Europeans. Average heterozygosity is significantly higher in the sub-Saharan African sample. Expected heterozygosity was computed for each region and locus using a population genetic model based on the null hypothesis of equal long-term population sizes. Average residual heterozygosity is significantly higher in the sub-Saharan African sample, indicating that African population size was larger than other regions during recent human evolution. The best fit of the model is with relative population weights of 0.73 for sub-Saharan Africa, 0.09 for East Asia, and 0.18 for Europe. These results are similar to those obtained using craniometric variation for these three geographic regions. These results, combined with inferences from other genetic studies, support a major role of Africa in the origin of modern humans. It is less clear, however, whether complete African replacement is the most appropriate model. An alternative is an African origin with non-African gene flow. While Africa is an important region in recent human evolution, it is not clear whether the gene pool of our species is completely out of Africa or predominately out of Africa.     

Evolution of variance in offspring number: The effects of population size and migration

It was shown by Gillespie [1974. Am. Nat. 108, 145–151], that if two genotypes produce the same average number of offspring on but have a different variance associated within each generation, the genotype with a lower variance will have a higher effective fitness. Specifically, the effective fitness is {ei65-1}, where w is the mean fitness, {ei65-2} is the variance in offspring number, and N is the total population size. The model also predicts that if a strategy has a higher arithmetic mean fitness and a higher variance than the competitor, the outcome of selection will depend on the population size (with larger population sizes favoring the highvariance, high-mean genotype). This suggests that for metapopulation sizes favoring the high-variance, high-mean genotype). This suggests that for metapopulations with large numbers of (relatively) small demes, a strategy with lower variance and lower mean may be favored if the migration rate is low while higher migration rates (consistent with a larger effective population size) favor the opposite strategy. Individual-based simulation confirms that this is indeed the case for an island model of migration, though the effect of migration differs greatly depending on whether migration precedes or follows selection. It is noted in the appendix that while Gillespie [1974. Am. Nat. 108, 145–151] does seem to be heuristically accurate, it is not clear that the definition of effective fitness follows from his derivation.     

Effective population size,genetic diversity,and coalescence time in subdivided populations

A formula for the effective population size for the finite island model of subdivided populations is derived. The formula indicates that the effective size can be substantially greater than the actual number of individuals in the entire population when the migration rate among subpopulations is small. It is shown that the mean nucleotide diversity, coalescence time, and heterozygosity for genes sampled from the entire population can be predicted fairly well from the theory for randomly mating populations if the effective population size for the finite island model is used.     

Effective size of a fluctuating age-structured population

Previous theories on the effective size of age-structured populations assumed a constant environment and, usually, a constant population size and age structure. We derive formulas for the variance effective size of populations subject to fluctuations in age structure and total population size produced by a combination of demographic and environmental stochasticity. Haploid and monoecious or dioecious diploid populations are analyzed. Recent results from stochastic demography are employed to derive a two-dimensional diffusion approximation for the joint dynamics of the total population size, N, and the frequency of a selectively neutral allele, p. The infinitesimal variance for p, multiplied by the generation time, yields an expression for the effective population size per generation. This depends on the current value of N, the generation time, demographic stochasticity, and genetic stochasticity due to Mendelian segregation, but is independent of environmental stochasticity. A formula for the effective population size over longer time intervals incorporates deterministic growth and environmental stochasticity to account for changes in N.     

The effective number of a population that varies cyclically in size. II. Overlapping generations

We consider haploid and dioecious age-structured populations that vary over time in cycles of length k. Results are obtained for both autosomal and sex-linked loci if the population is dioecious. It is assumed that k is small in comparison with numbers of haploid individuals (or of numbers of males and females) in any generation of a cycle. The inbreeding effective population size N(e) is then approximately given by the expression [T summation operator (k-1)(j=0)1/[N(e)(j)T(j)]](-1), where N(e)(j) and T(j) are, respectively, the effective population size and generation interval that would hold if the population was at all times generated in the same way as at time j. The constant T, which is the effective overall generation interval, is defined to be k times the harmonic mean of the quantities T(j). Our expressions for T and N(e), in terms of N(e)(j) and T(j), are general, but the N(e)(j)s are derived under the assumption that offspring are produced according to Poisson distributions.     

Diversification rates increase with population size and resource concentration in an unstructured habitat

Understanding the mechanisms controlling the generation and maintenance of biodiversity provides some of the planet's greatest and most pressing challenges. Variation in resource concentration, which varies widely at multiple scales, may cause biodiversity to increase, decrease, or exhibit a unimodal response and underlying mechanisms remain obscure. We established experimental cultures of long-term stationary phase (LTSP) Escherichia coli to test whether per capita heterozygosity varies with resource concentration, and, if so, whether population sizes associated with different resource concentrations contributed to these patterns. Our results provide the clearest example to date of increasing per capita heterozygosity with increasing resource concentration. Further, our experimental manipulations of population size, independent of resource concentration, provide the first unequivocal evidence that population size is one of the underlying factors controlling per capita heterozygosity along such resource gradients. Specifically, we show that cultures with higher maximum population sizes, associated with higher resource concentrations, have higher per capita heterozygosity. These experiments provide the first experimental evidence for an underappreciated factor controlling biodiversity along resource gradients--population size. This direct evidence of population size influencing diversification rates has implications for regional and global scale patterns of biodiversity.     

Population genetic structure and effective population size of ayu (Plecoglossus altivelis), an amphidromous fish

The temporal and spatial population genetic structure of ayu Plecoglossus altivelis (Salmoniformes: Plecoglossidae), an amphidromous fish, was examined using analysis of variation at six microsatellite DNA loci. Intracohort genetic diversities, as measured by the number of alleles and heterozygosity, were similar among six cohorts (2001–2006) within a population (Nezugaseki River), with the mean number of alleles per cohort ranging from 11·0 to 12·5 and the expected heterozygosity ranging from 0·74 to 0·77. Intrapopulational genetic diversities were also similar across the three studied populations along the 50 km coast, with the mean number of alleles and the expected heterozygosity ranging from 11·33 to 11·67 and from 0·75 to 0·76, respectively. The authors observed only one significant difference in pair-wise population differentiation ( F _ST-value) between the cohorts within a population and among three populations. Estimates of the effective population size ( N _e) based on maximum-likelihood method yielded small values (ranging from 94·8 to 135·5), whereas census population size ranged from c. 4800 to 24 000. As a result, the ratio of annual effective population sizes to census population size ( N _e/ N ) ranged from 0·004 to 0·023. These estimates of N _e/ N agree more closely with estimates for marine fishes than that of the larger estimates for freshwater fishes. The present study suggests that ayu which is highly fecund and shows low survival during the early life stages is also characterized by having low value of N _e/ N , similar to marine species with a pelagic life cycle.     

Extinction risk of a density-dependent population estimated from a time series of population size

Environmental threats, such as habitat size reduction or environmental pollution, may not cause immediate extinction of a population but shorten the expected time to extinction. We develop a method to estimate the mean time to extinction for a density-dependent population with environmental fluctuation. We first derive a formula for a stochastic differential equation model (canonical model) of a population with logistic growth with environmental and demographic stochasticities. We then study an approximate maximum likelihood (AML) estimate of three parameters (intrinsic growth rate r, carrying capacity K, and environmental stochasticity sigma(2)(e)) from a time series of population size. The AML estimate of r has a significant bias, but by adopting the Monte Carlo method, we can remove the bias very effectively (bias-corrected estimate). We can also determine the confidence interval of the parameter based on the Monte Carlo method. If the length of the time series is moderately long (with 40-50 data points), parameter estimation with the Monte Carlo sampling bias correction has a relatively small variance. However, if the time series is short (less than or equal to 10 data points), the estimate has a large variance and is not reliable. If we know the intrinsic growth rate r, however, the estimate of K and sigma(2)(e)and the mean extinction time T are reliable even if only a short time series is available. We illustrate the method using data for a freshwater fish, Japanese crucian carp (Carassius auratus subsp.) in Lake Biwa, in which the growth rate and environmental noise of crucian carp are estimated using fishery records.     

Distribution of nucleotide differences between two randomly chosen cistrons in a population of variable size

The distribution of the number of nucleotide differences between two randomly chosen cistrons in a finite population is studied here when the population size changes from generation to generation. When genetic variability is measured by heterozygosity (i.e., the probability that two cistrons are different), by the probability that two cistrons differ at two or more nucleotide sites, or by mean number of site differences between cistrons, it is seen that in a population going through a small bottleneck all of these measures decline rapidly but, as soon as population size becomes large, they start to increase owing to new mutations. The amount of reduction in these measures depends not only on the size of bottleneck but also on the rate of population growth. The implications of this study explaining the observed variations in the rates of amino acid substitutions during the evolutionary process are also discussed.     

Effective number of breeders and maintenance of genetic diversity in the captive bearded vulture population

We combined pedigree data with data derived from 14 microsatellite loci to investigate genetic diversity and its maintenance in the captive source population for the reintroduction of the bearded vulture into the Alps. We found the captive population to be genetically more variable than the largest natural population in Europe, both in terms of mean number of alleles per locus and mean observed and expected heterozygosity. Allelic diversity of the captive population was higher than, and mean heterozygosity measurements were comparable with the ones found in two large, extinct populations from Sardinia and the Alps represented by museum specimens. The amount of genetic variability recruited with the founders was still present in the captive population of the year 2000, mainly because the carriers of rare alleles were still alive. However, the decline in expected heterozygosity and the loss of alleles over generations in captivity was significant. Point estimates of effective population size, N(e), based on pedigree data and estimates of effective number of breeders, N(b), based on allele frequency changes, ranged from 20 to 30 and were significantly smaller than the census size. The results demonstrate that the amount of genetic variability in the captive bearded vulture population is comparable or even larger than the amount present in natural populations. However, the population is in danger to lose genetic variability over time because of genetic drift. Management strategies should therefore aim at preserving genetic variability by minimising kinship, and at increasing N(e) by recruiting additional founders and enhancing gene flow between the released, the captive and natural populations.