Introduction

One of the main issues currently discussed in conservation genetics refers to the joint consequences of inbreeding depression and purging, which determine the minimum size of a population to be viable over time (Jamieson and Allendorf, 2012; Frankham et al., 2014). This minimum size is strongly dependent on the inbreeding load, that is, the load of deleterious recessive mutations concealed in heterozygotes (Morton et al., 1956). This load, when exposed by inbreeding, is responsible for at least part of the decline in fitness traits (inbreeding depression) ubiquitous in all captive populations and species at the verge of extinction (Hedrick and Kalinowski, 2000; Boakes et al., 2007; Charlesworth and Willis, 2009). Inbreeding depression and the fixation of deleterious mutations owing to the reduced efficiency of purifying selection in small populations (Lynch et al., 1995; Reed et al., 2003) are some of the main genetic factors which, combined with non-genetic ones, are likely to be responsible for the extinction of populations.

The inbreeding load, and thus the decline in fitness through inbreeding depression, can be partially restrained by genetic purging, the action of natural selection to remove recessive deleterious mutations exposed by inbreeding (see, for example, Leberg and Firmin, 2008). Although the effects of genetic purging are not systematically detected in empirical studies (Ballou and Lacy, 1998; Crnokrak and Barrett, 2002; Boakes et al., 2007; Leberg and Firmin, 2008), theoretical studies have shown that purging can be rather effective for moderate size populations (García-Dorado, 2012). Experiments specifically designed to quantify genetic purging confirm this expectation (Bersabé and García‐Dorado, 2013; López-Cortegano et al., 2016). Theoretical and empirical evidence also suggests that purging is effective under subdivision of a population into small subpopulations, bottleneck cycles or forced inbreeding by partial full-sib mating (Hedrick, 1994; Miller and Hedrick, 2001; Swindle and Bouzat, 2006; Fox et al., 2008; Ávila et al., 2010).

The consequences of inbreeding and purging are fundamental to determine the minimum effective size (Ne; Wright, 1931) of a population to avoid inbreeding depression and to survive in the short term. In this respect, a consensus rule arising from animal breeding programmes is that the minimum Ne is 50, which corresponds to a rate of increase in inbreeding of 1% per generation (Soulé, 1980; Franklin, 1980). Because of the arbitrariness of this rule, Frankham et al. (2014) proposed a more specific definition, based on a maximum decline of 10% in fitness over five generations. This minimum effective population size will obviously depend on the reproductive nature of the species considered and the total inbreeding load, usually estimated from the regression of the natural logarithm of survival on the inbreeding coefficient. The inbreeding load (B) is often measured in terms of the number of lethal equivalents carried by a genome (Morton et al., 1956). In a review by O’Grady et al. (2006), it was found that the total deleterious inbreeding load in vertebrates is about B=6 lethal equivalents per haploid genome, of which about 2 correspond to fecundity traits and 4 to survival traits. Similar or even larger estimates have been found by Kruuk et al. (2002), Grueber et al. (2010), Hedrick et al. (2015) and Hoeck et al. (2015), who obtained estimates of B ≈ 7.5, 8, 6 and 7, respectively.

In the light of this high total inbreeding load, Frankham et al. (2014) questioned the classical minimum Ne=50 to avoid short-term inbreeding depression. They estimated that the expected inbreeding depression accumulated in five generations for a population of effective size Ne=50 would be 1−exp(−0.05 × 6)=26%, a too large decline according to the above definition of minimum effective size. With the same calculation, they reckoned that Ne should be 142 to produce a maximum of 10% decline in fitness in five generations. Thus, Frankham et al. (2014) suggested revising the Ne=50 rule upwards to a minimum of Ne=100. This proposal has, however, generated some debate because the above rough estimations ignore purging (Franklin et al., 2014; García-Dorado, 2015). O’Grady et al., (2006) performed simulations to address the impact of an inbreeding load of six lethal equivalents on the time to extinction for a number of species, using the software Vortex (Miller and Lacy, 2003), which performs population viability analysis accounting for a wide number of genetic and demographic factors of extinction. They concluded that the median time to extinction of populations declines by an average of 37% with a mutational load of 6 lethal equivalents, relative to that with a mutational load of 1.57 lethal equivalents, which were usually assumed as default by the software and obtained from juvenile survival analysis of zoo populations (Ralls et al., 1988).

Another issue of relevance in relation with the above arguments is the type of genetic management appropriate for conservation programmes. Managements based on equalisation of contributions (ECs) from parents to progeny or minimum coancestry contributions (Ballou and Lacy, 1995; Wang, 1997; Caballero and Toro, 2000) are expected to maintain high levels of gene diversity but have the collateral effect of reducing the strength of natural selection, particularly on fecundity traits, implying a diminished strength of genetic purging. Analytical and simulation studies indicate that ECs is expected to produce higher fitness than random contributions for small populations (below about 50 individuals) in the short term (up to 10–20 generations) (Fernández and Caballero, 2001; Theodorou and Couvet, 2003; García-Dorado, 2012), and this is confirmed by empirical evidence (see Sánchez-Molano and García-Dorado, 2011 and references therein).

In contrast, other methods have been suggested to increase the strength of purging by applying different degrees of forced inbreeding (Crnokrak and Barrett, 2002), such as mating between full-sibs (Hedrick, 1994; Fox et al., 2008; Ávila et al., 2010) and crosses between lines of sibs (Wang, 2000). More recently, Theodorou and Couvet (2010, 2015) suggested the possibility of applying circular sib mating (Kimura and Crow, 1963) in conservation programmes. These authors showed that circular sib mating produces an efficient purging of the inbreeding load and maintains higher genetic diversity and higher population fitness than minimum coancestry contributions. The problem with any form of excessive inbreeding, as pointed out by Theodorou and Couvet (2010, 2015) and other authors (Reed et al., 2003; Fox et al., 2008), is the high short-term inbreeding depression and risk of extinction. Nevertheless, Theodorou and Couvet (2015) concluded that, in the case of circular mating (CM), this issue would not be too relevant for population sizes >30 individuals and species with high reproductive rate, so that CM could be advisable in conservation management in these scenarios.

Genetic extinction of small populations thus depends on the inbreeding load and the efficacy of genetic purging. Previous simulation results assumed purging of only lethal mutations (O’Grady et al., 2006), of only non-lethal ones (de Cara et al., 2013) or of both but with lower inbreeding loads than observed (Wang et al., 1999; Pérez-Figueroa et al., 2009; Theodorou and Couvet, 2010, 2015). Here we simulate small populations with different reproductive rates and allow purging of both lethal and non-lethal mutations affecting viability and fecundity to find the minimum population size used in conservation that aims to prevent fitness declining by >10% over a 5-generation period. We also evaluate the relative performance of ECs and CM in conservation programmes under the more realistic parameter values of inbreeding load.

Methods

Overview of the simulation design

Computer simulations were carried out of populations where deleterious alleles arise by mutations and are eliminated by selection and drift. The simulation procedure consisted of two steps. In the first, a large base population of size Nb=1000 monoecious diploid individuals was simulated for 10 000 generations to reach a mutation–selection–drift balance. The details of the simulation procedure to obtain the base population can be found in Pérez-Figueroa et al. (2009). In the second, samples were taken from this base population to found populations of small sizes, which were maintained under different management schemes for a further number of generations. Two scenarios were considered for managing the small populations, a wild population scenario, with random mating of parents and variable contributions of offspring according to their fitness, and a captive breeding scenario, where two different management methods were followed as detailed below.

Simulation model and mutational parameters

We assumed a model of deleterious mutations appearing with rate λ per haploid genome and generation with homozygous allelic effects obtained from a gamma distribution with mean and shape parameter β, such that the genotypic fitness values for locus i are 1, 1−hisi and 1−si, for the wild-type homozygote, the heterozygote and the mutant homozygote, respectively. Dominance coefficients (h) were assumed to have negative correlation with selection coefficients by using the joint s and h model of Caballero and Keightley (1994), where h values are taken from a uniform distribution ranging between 0 and exp(–ks), k being a constant to obtain the desired average value . Additionally, lethal mutations (sL=1) were assumed to occur with rate λL and dominance hL=0.02 (Simmons and Crow, 1977). The fitness of an individual was calculated as the product of genotypic fitness across all loci.

The mutational models considered are shown in Table 1. Model A assumed mutational parameters in agreement with empirical data: λ and (Halligan and Keightley, 2009; Keightley, 2012), β (value appropriate to obtain a mutational variance of VM=0.002; García-Dorado et al., 1999, 2004), (García-Dorado and Caballero, 2000), λL (Simmons and Crow, 1977) and inbreeding load B (O’Grady et al., 2006). Model B considered a larger rate of mutations with effects smaller than those for model A, but keeping the same values of VM and B. Model C considered a much larger lethal mutation rate, thus providing a high lethal inbreeding load (about 2.5).

Table 1 Deleterious mutation models considered in simulations
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The above three models were used in the wild population scenario. For the captive breeding scenario, we assumed model A and an additional model with the parameters used by Theodorou and Couvet (model TC; Table 1), which implies an inbreeding load one-third of that of model A.

In all cases, up to 1000 neutral loci were also simulated in order to obtain estimates of neutral gene diversity. All neutral and selected loci were assumed to be generally unlinked, but some simulations were carried out assuming a genome of L (5–30) Morgans in genetic map length where recombination occurred at random positions of the genome without interference.

Simulated populations of small size and systems of mating

From the large base population, random samples were taken to found small populations of maximum size Nmax, ranging from 10 to 200 individuals, which were maintained for 50 generations under the evolutionary forces of mutation, selection and drift. One thousand replicates were carried out for each maximum population size and mutation parameter combination. In these small populations, fitness was partitioned into two traits, fecundity (Wf) and viability (Wv), which were assumed to be affected by 1/3 and 2/3 of the loci, respectively. At generation 0 of the small populations, fecundities and viabilities of individuals were scaled by their means, so that the initial average population fecundity and viability were at the maximum of Wf=Wv=1.

We assumed species with different reproductive rates by fixing a maximum number of progeny per individual of 2K, following a procedure similar to that used by Theodorou and Couvet (2015). The number of progeny from the mating between individuals i and j was then obtained as the product of 2K (wild scenario) or K (captive scenario; see Supplementary File S1) times truncated to the lower integer number, where Wf,x is the fecundity of individual x (x=i, j). Values of K were varied from 1.5 to 10 to consider a range of species from low to high reproductive abilities. An individual progeny x survived or died according to its viability, Wv,x. A random number was drawn from a uniform distribution in the range [0, 1] and was compared with Wv,x. If it was lower (higher) than Wv,x, the individual would survive (die). Thus, because of the limited fecundity and survival, a small population with a high inbreeding load could diminish in census size and become extinct. However, the maximum population size would always be Nmax, assuming a ceiling model where the maximum population size is limited by the resources. For some simulations, natural selection was assumed absent, what we call ‘no purging’ scenario.

The mating models are described in detail in Supplementary File S1. Briefly, a random contribution model was assumed in the wild population scenario. In this case, a polygamous mating system was assumed where parents were taken from the population with probability proportional to their fecundity values. In the captive scenario, two management systems were carried out, EC and CM. In this scenario, relaxation of selection in a benign environment (Fox and Reed, 2010) was taken into account by halving the selection coefficients of all mutations except lethal ones.

Pedigrees were saved in all cases. The following parameters were calculated each generation and averaged over replicates: (1) the probability of extinction, computed as the percentage of replicated populations extinct at a given generation; (2) the time to extinction, computed as the average generation number when populations become extinct; (3) the number of individuals (N) in the population at each generation (excluding extinct populations); (4) the average fecundity (Wf) and the average viability (Wv) of the population; (5) the inbreeding load for fecundity (Bf) and viability (Bv), calculated as B=Σsi(1−2hi)piqi (Morton et al., 1956), where qi is the mutant frequency, pi=1−qi and the summation is for all loci affecting fecundity or viability; (6) the inbreeding coefficient estimated from pedigrees (F); (7) the average expected frequency of heterozygotes at generation t for neutral loci; and (8) the average effective size (Ne) calculated from the inbreeding coefficient as Ne=1/(2ΔF), where ΔF is the average rate of change in inbreeding between generations 5 and 10, that is, ΔF=(F10−F5)/(5 × (1−F5)).

Results

Minimum population size to avoid excessive inbreeding depression in a wild population scenario

Figure 1 illustrates the evolution of populations over 50 generations with a maximum size Nmax=50 and three reproductive rates (K) under mutation model A (see Table 1). A model with no purging is also displayed as a reference. Extinctions and reductions in population size were more extensive with a lower reproductive rate of the species (K), as expected (Figures 1a and b). Some recoveries in census size were observed for all K values, but these still finally led to extinction in most cases. The average drop in fecundity and viability was considerably lower under selection than that expected with no purging (Figures 1c and d), clearly showing the impact of purging in removing inbreeding load (Figures 1e and f). Supplementary Figure S1 gives the corresponding results for Nmax=100, with the same qualitative results.

Figure 1
figure 1

Evolution over generations of different parameters for populations of maximum size Nmax=50 and maximum reproductive rate K (2K is the maximum number of progeny per individual) under mutational model A (see Table 1). A case with no selection (no purging) is shown as a reference. (a) Probability of extinction. (b) N: average population size (excluding extinct populations). (c) Wf: average population fecundity. (d) Wv: average population viability. (e) Bf: inbreeding load for fecundity. (f) Bv: inbreeding load for viability.

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The drop in fitness during the first five generations is given in Figure 2a for different maximum population sizes (Nmax) and K values under mutation model A. The horizontal line at 10% drop indicates the limiting value suggested for a population to avoid severe short-term fitness decline owing to inbreeding depression. With no purging, the drop in fitness was substantially >10% for all Nmax values (Figure 2a). However, when purging selection was taken into account the drop became <10% for maximum population sizes above Nmax ≈ 130.

Figure 2
figure 2

(a) Drop in fitness (fecundity × viability) after five generations for populations of maximum size Nmax under mutational model A (see Table 1) for different reproductive rates (K; where 2K is the maximum number of progeny per individual) and for a model without selection (no purging). (b) Ratio of the effective population size (Ne, estimated from the change in pedigree inbreeding between generations 5 and 10) to the maximum population size (Nmax).

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The ratio of the effective population size (Ne) computed in the initial generations to the maximum population size (Nmax) is given in Figure 2b, showing a tendency towards a value of Ne/Nmax ≈ 0.5 with no purging and a bit over with purging. Therefore, the populations showing a drop in fitness <10% were those with effective size above Ne ≈ 70.

The average time until extinction and the probability of extinction at generations 10, 25 and 50 for the different K values are shown in Figure 3. Except for species with very low reproductive rate (K=1.5, a maximum of 3 progeny per individual), extinction rate in the first 50 generations was low for maximum population sizes over Nmax ≈ 130 (Ne ≈ 70). From Figure 3, and assuming the asymptotic ratios of Ne/Nmax observed in Figure 2b, the average times until extinction for K=1.5, 2.5, 3.5 and 10 were about 0.3Ne, 0.8Ne, 1.2Ne and 2.9Ne generations.

Figure 3
figure 3

Time to extinction (average generation number when populations become extinct) and probability of extinction at generations 10, 25 or 50 (percentage of replicates extinct at a given generation) for populations of maximum size Nmax under mutational model A (see Table 1) and different reproductive rates (K).

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The drop in fitness in the first five generations was slightly lower for mutational models B and C than for model A, and the effective population sizes were very similar for the three models (Supplementary Figure S2). In addition, extinction rates were similar for models A and B but somewhat lower for model C (Supplementary Figure S3), as this latter implied many more lethal genes than models A and B and, therefore, a more efficient purging was allowed. Thus, in what follows we will focus on model A as the most conservative one in terms of purging efficiency. The above results assumed free recombination among mutations. Assuming a short genome length (L=5 Morgans), the probability of extinction was somewhat larger than for free recombination or L=20, but the effect was small (Supplementary Figure S4).

Conservation management in captive breeding populations

Figure 4 illustrates the evolution of a population over 50 generations for Nmax=40 under two conservation management methods, EC and CM. Extinction and decline in population size was larger for the CM scheme (Figures 4a and b) than for EC scheme owing to the sharp decline in fitness occurred during the first generations with CM (Figures 4c and d). CM method resulted in a substantial amount of purging of inbreeding load (Figures 4e and f) because of the large initial increase in inbreeding (Figure 4g). This elimination of inbreeding load produced a higher fecundity than EC in the final generations (Figure 4c). Expected heterozygosity for neutral genes declined more under CM than under EC (Figure 4h).

Figure 4
figure 4

Evolution over generations of different parameters for populations of maximum size Nmax=40 and maximum reproductive rate K=2.5 (2K is the maximum number of progeny per individual) under mutational model A (see Table 1) in a conservation management scenario. Selection coefficients of deleterious non-lethal mutations were halved. (a) Probability of extinction. (b) N: average population size (excluding extinct populations). (c) Wf: average population fecundity. (d) Wv: average population viability. (e) Bf: inbreeding load for fecundity. (f) Bv: inbreeding load for viability. (g) F: average pedigree inbreeding coefficient. (h) H: average expected heterozygosity for neutral genes.

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The probability of extinction with the two management methods is given in Figure 5 for different values of Nmax and K. For low reproductive rates, method CM showed substantial extinction even during the first 10–25 generations. It also showed a substantially larger extinction rate than for EC with higher values of K (see also Supplementary Table S1). This method produced a remarkable drop in fitness in the early generations (Supplementary Figure S5). The effective population size for CM was larger than that for EC under a no purging model, but the opposite occurred when selection was taken into account (Supplementary Figure S5). Thus the amount of diversity maintained by CM was always lower than that maintained by EC when purging was accounted for (Supplementary Figure S6). Simulations with restricted recombination did not alter qualitatively the above results (Supplementary Figure S7).

Figure 5
figure 5

Probability of extinction at generations 10, 25 or 50 (percentage of replicates extinct at that generation) for populations of maximum size Nmax and two reproductive rates (K; where 2K is the maximum number of progeny per individual) under mutational model A (see Table 1) in a conservation management scenario. Selection coefficients of deleterious non-lethal mutations were halved.

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These results contrast with those obtained using mutational models with lower inbreeding loads, such as model TC (Table 1), for which extinction rates were considerably lower and CM maintained a higher neutral expected heterozygosity than EC under some scenarios (Supplementary Table S1). In fact, for this model, EC incurred more extinctions than CM on some occasions. This did not occur, however, for the model with a higher inbreeding load of B ≈ 6 (Supplementary Table S1).

Discussion

Minimum population size to avoid excessive inbreeding depression and extinction in the short term

When the minimum effective population size in conservation is defined as the one which results in the rate of fitness decline being below 10% in the first five generations (Frankham et al. 2014), our results suggest that this number is of the order of Ne ≈ 70 for a range of species’ reproductive rates. The above result is based on a genetic model accounting for an inbreeding load of the order of B=6. Thus our results are midway between the classical rule of Ne=50, originally suggested by Soulé (1980) and Franklin (1980), and the latest proposal from Frankham et al. (2014), who suggested to double that figure. Assuming absence of purging, Frankham et al. (2014) calculated that an effective size >142 was necessary to limit fitness decline to <10% in five generations. This agrees with our simulations, which show that, without purging a drop in fitness below 10% would be reached with Nmax>200 (Figure 2a), corresponding to Ne>100. However, the full consideration of purging of both lethal and non-lethal genes reduces these values (Figure 2a).

Several caveats should be mentioned regarding these results. First, our model assumes absence of stochastic demographic changes in the population other than the reduction in population size owing to inbreeding depression and fixation of mutations. If other factors occur, such as biases in the sex ratio or sudden reductions in population size owing to genetic or non-genetic causes, the deduced minimum effective size could be larger. Second, we have assumed that the whole inbreeding load is due to recessive or partially recessive deleterious mutations. It is possible that part of this load is due to overdominant loci or other types of balancing selection that would not be purged by inbreeding. However, whereas partial recessive deleterious mutations are ubiquitous and are considered the predominant cause of inbreeding depression (Charlesworth and Willis, 2009), heterozygote advantage should be a minority (Hedrick, 2012). Third, we have assumed a model with inbreeding load B=6, but other recent experimental data suggest larger values of B=7–8 (Kruuk et al., 2002; Grueber et al., 2010; Hoeck et al., 2015). The impact of inbreeding depression on the fitness decline, extinction risk and loss of genetic variation in small populations could thus be larger, which calls for a higher minimum effective population size.

We have focussed in this paper on short term-inbreeding and survival of small–medium-sized populations. There are other aspects regarding survival of populations that are out of the scope of the paper. For example, it has been proposed that Ne=500 is the minimum population size to retain evolutionary potential in perpetuity (Franklin, 1980). This figure has been recently suggested to be twice as large as that by Frankham et al. (2014). In addition, a long-term minimum viable population size is also frequently discussed, which refers to the number of individuals required for a high probability of population persistence over the long run (Shaffer, 1981; Soulé, 1987), for example, 80% persistence over >20 years (Shaffer, 1981) or 99% persistence over 40 generations (Frankham et al., 2014). The inference obtained by the review of population viability analyses carried out by Traill et al. (2007) suggests a median of the minimum viable population size of about 4200 individuals (95% confidence interval of 3600–5100). Ratios of Ne/N are of the order of 0.1–0.2 (Frankham, 1995; Palstra and Fraser, 2012), so considering a conservative value of 0.1 this would imply a minimum effective size (Ne) around 400. These inferences, however, cannot be taken as general rules because of the huge variation in the estimates of Ne/N across species and other factors (Flather et al., 2011; Jamieson and Allendorf, 2012).

Our results are not directly addressing the issue of the long-term survival and minimum viable population sizes because they ignore environmental stochastic factors and only consider genetic issues. We assumed a ceiling model with a maximum population size (Nmax), so that populations cannot increase over that ceiling, and this would be a factor increasing the chances of extinction of the population. Therefore, any inference on long-term survival arising from our results should be taken with caution. Even so, our results suggest that, unless the reproductive rate of the species is very low, the chances of extinction for genetic reasons after 50 generations are low when maximum population sizes are around Nmax=130 (Ne=70) (Figure 3). Assuming again a ratio Ne/N=0.1, this would imply census sizes of N=700 individuals. However, for low reproductive rate species (K=1.5) we found that the probability of extinction at generation 50 was nearly 100% even for maximum census sizes of Nmax=200 (Figure 3). Therefore, genetic extinction for low reproductive rate species could be avoidable only with very large population sizes, perhaps of the order of thousands, in agreement with the meta-analysis results of Traill et al. (2007).

We found that the average times until extinction for species with reproductive rates of K=1.5, 2.5, 3.5 and 10 were about 0.3Ne, 0.8Ne, 1.2Ne and 2.9Ne generations. The averages obtained for intermediate values of K are therefore consistent with those estimated by Soulé (1980) based on the experience of animal breeders (1.5Ne generations) and by Reed and Bryant (2000) for populations in captivity (Ne generations). In addition, the median values of time to extinction estimated by simulation by O’Grady et al. (2006) for 30 species were 5.7, 11.9 and 21.8 generations for population sizes of N=50, 250 and 1000, respectively (averages obtained from O’Grady et al., 2006). Assuming Ne/N=0.1, this implies times to extinction of 1.1Ne, 0.5Ne and 0.2Ne generations, which are similar to our results for K values of 1.5–3.5.

Forced excessive inbreeding as a management method in captive breeding populations

We have compared the performance of a conservation method restraining inbreeding, EC, and a method forcing it, CM. Theodorou and Couvet (2010, 2015) compared CM with minimum coancestry contributions and matings (Fernández and Caballero, 2001), a method which makes use of pedigree or molecular marker information to minimize genetic drift and inbreeding in conservation management. This method is more powerful than EC in the initial generations, but we used EC instead as a more simple method that can be applied in all circumstances, thus giving conservative results in terms of amount of variation maintained.

Theodorou and Couvet (2010, 2015) showed that CM maintains higher levels of gene diversity than EC in the long-term, and because it is more effective in removing the inbreeding load, it leads to higher population fitness. They also acknowledged that the main drawback of CM is the possibly high short-term inbreeding depression and high extinction risk. However, they concluded that, with the mutational model they investigated, this would be only a relevant issue for small populations (Nmax<30) and low reproductive ability (F<3, where F is analogous to our K value). We cannot make a direct comparison between their simulations and ours because the models followed by Theodorou and Couvet (2010, 2015) are not exactly the same as those used here. First, as mentioned above, Theodorou and Couvet (2010, 2015) used minimum coancestry contributions with minimum coancestry matings, whereas we just applied ECs with only avoidance of full-sib mating. Second, Theodorou and Couvet (2010) used a model of mutations of fixed effects, rather than a model with variable mutational effects. Furthermore, whereas Theodorou and Couvet (2015) considered such variable mutational effects, they not only considered deleterious mutations for fitness but also stabilising selection on a quantitative trait, so that overall fitness is a compound of deleterious mutations and selection on the trait. Thus our results for extinction rates are rather different from those of Theodorou and Couvet (2015). For example, the probability of extinction with Nmax=20 and F=2.4 from Table 1 of Theodorou and Couvet (2015) was 66% and 10% for CM and minimum coancestry contributions and matings, respectively, at generation 25, whereas our corresponding probabilities for K=2.5 were 0%. Our simulations give, therefore, lower extinction rates.

Our simulations using the mutational parameters of Theodorou and Couvet (2015) (model TC in Supplementary Table S1), which account for deleterious inbreeding loads of the order of two lethal equivalents, show that extinction risk under CM is negligible for K>2.5 and Nmax>20 even at generation 50 (Supplementary Table S1). However, assuming inbreeding loads of the order of those observed experimentally, CM may imply an extinction risk too high to be an advisable management method in captive breeding programmes.

CM can be considered a type of population subdivision where each individual is considered a subpopulation and each subpopulation contributes equally to the next generation. Without selection, this mating scheme is expected to produce a higher rate of decrease in expected heterozygosity in the early generations but a final lower rate than under random mating (Kimura and Crow, 1963; Robertson, 1964; Wang and Caballero, 1999). Thus it is expected that CM would produce a larger effective population size and higher gene diversity than EC after a number of generations if a model with no selection is assumed (see Supplementary Figure S5). When selection is taken into account, this may still be the case if mutational models with low inbreeding load are assumed, as the model considered by Theodorou and Couvet (2015) (model TC). Under these parameters, expected heterozygosity (at later generations) is sometimes larger than that obtained with EC (Supplementary Table S1). However, for models with an inbreeding load of the order of six lethal equivalents, this is no longer the case; both the effective size and the maintained neutral heterozygosity for CM are lower than those for EC. Hence, the possible advantage of CM in further increasing the effective size and maintaining larger neutral diversity than EC clearly disappears when models involving high inbreeding loads are assumed.

Overall, we can therefore conclude that CM should not be generally advised as a management method in captive breeding conservation programmes. This is in accordance with other studies which suggest that purging the detrimental variation through forced inbreeding is a risky strategy in the genetic management of endangered species (Miller and Hedrick, 2001) and the common practice of avoiding inbreeding should be followed (Boakes et al., 2007; Leberg and Firmin, 2008).

Data archiving

There were no data to deposit.