Comparisons between exponential and Weibull survival models suggested that all extinction rates increased with time (see Additional file 1), so only Weibull models were considered in subsequent survival analysis. Comparisons among regression models indicated that (i) the models including all scenarios (survival models) and those including only the scenarios for which all trajectories were extinct after 1,000,000 generations (hereafter referred to as the subset of extinct trajectories) are in excellent agreement; (ii) the effects of ecological variables on demographic (T_D ) and genetic (T_G ) viabilities are qualitatively similar. In all cases, T_D and T_G were positively related to dispersal rate, fecundity and total carrying capacity, while they decreased with fragmentation, as well as the frequency of perturbations (but see below for more details on the effects of fragmentation). The spatial correlation of environmental perturbations (i.e., C_p = 1) always reduced demographic viability (as compared with the case where C_p = 0) but had weaker and more complex effects on genetic viability.

Although ecological variables had similar qualitative effects on the viabilities T_D and T_G , the strengths of these effects differed considerably between the demographic and genetic models. A hierarchical partitioning (HP) indicated that T_G was mainly related to metapopulation size and fragmentation (K_t , R² = 51%; N, R² = 18%) while T_D mostly depended on the basic growth rate and the perturbation regime (F, R² = 28%; P, R² = 17%) (All detailed results are provided in Additional file 1).

Therefore, when considering all extinct trajectories scenarios, demographic and genetic extinction times were only moderately correlated (Kendall's τ = 57%), but correlation increased when considering only scenarios with a constant environment (τ = 78%). In most situations, the demographic extinction time (T_D) was longer than the genetic extinction time T_G (median of T_G/T_D = 0.17) but the distribution of the T_G/T_D ratio was wide and skewed to the right (Mean = 2.1, Min = 7.10^-5, Max = 430). Further analysis indicated that this ratio strongly increases with total population size and the frequency of perturbations level and decreases with fragmentation (HP: R² were equal to 44%, 23% and 6% respectively for K_t, N and P, see Additional file 1). Hence, the ecological scenarios for which the genetic viability was higher than the demographic viability were those assuming large metapopulations and low levels of fragmentation (main effects are summarized on Figure 1a).

As expected, adding genetic constraints to the demographic model always decreased viability. However, the ratio (T_DG/T_D) varied strongly, from 10^-4 to 0.6 (Median = 0.15, Mean = 0.19, SD = 0.15). Multiple regressions and HP indicated that this ratio was mostly influenced by the perturbation regime and the basic fecundity (P, R² = 23%; F, R² = 7%, details in Additional file 1). The demographic and demo-genetic extinction times were very close to each other in slow growing metapopulations with a highly variable environment. In other situations, T_DG may be several orders of magnitude lower than T_D (Figure 1b).

A univariate regression model indicated that T_D explains about 68% of the variance in T_DG. The residuals of this regression were positively correlated with the metapopulation carrying capacity (HP, R² = 43%, Additional file 1).

Surprisingly, adding demographic constraints to the genetic model reduced viability in only 66% of the scenarios investigated (the ratio T_DG/T_G ranged from 7.10^-5 to 8.3 with a median value of 0.47). The analysis indicated that this ratio was mostly influenced by population size and fragmentation (K_t , R² = 9.1%, N, R² = 5.8%, Additional file 1). In small and highly fragmented metapopulations, the genetic extinction time was shorter than the demo-genetic extinction time while the reverse was generally true in other cases (Figure 1c, Additional file 1).

When considered as single predictors in extinct trajectories scenarios, T_D and T_G explained respectively 68% and 65% of the variance in the demo-genetic time to extinction. Using the maximum value (between T_G and T_D) did not improve model fit (R² = 64.6%), while using the minimum value increased the R² to 78%, a value close to the R² obtained by including both T_G an T_D in the model (without interaction). The interaction term between T_G and T_D was positive and highly significant but only marginally improved model fit (R² = 80%).

A complementary analysis indicated that the type of environmental perturbations considered does not influence the qualitative effects of ecological variables on T_D and T_G and T_DG, although the quantitative effects of F and P on T_D may vary with the type of perturbations (Additional file 1).

The effect of the level of fragmentation (N) on demographic, genetic and demo-genetic viabilities is illustrated in Figure 2. The comparison between demographic and genetic viabilities revealed contrasting patterns. With the demographic model, the optimal level of fragmentation increased with overall population size and dispersal rates and could reach high levels (> 10-15 patches). In contrast, genetic viability was maximized with low fragmentation levels (1 or 2 patches) in all situations. In this context, the consideration of demo-genetic interactions tended to buffer extinction times and all demo-genetic optimal fragmentation levels were intermediary between optimal demographic and genetic levels. These patterns remained qualitatively as long as environmental perturbations are spatially independent (sensitivity analyses presented in Additional files 2, 3 and 4).

Most ecological settings considered here had similar qualitative effects on demographic and genetic viabilities, in agreement with theoretical expectations. Large population sizes allow limiting demographic stochasticity 29 as well as the inbreeding and drift loads 30 ; high dispersal rates are associated with demographic rescue and recolonization 31 and reduced drift load 30 ; perturbations cause population bottlenecks that are directly associated with demographic extinction and inflate the genetic load 32 ; high fecundity rates allow counteracting the effect of demographic and environmental stochasticities 33 and elevate the genetic extinction load threshold. Only fragmentation (N) has less obvious effects (see below). However, despite these similarities, the processes leading to "demographic" and "genetic" extinctions are fundamentally different. Demographic viability is related to the mean and variance of the rate of increase 29 , which are primarily influenced by the perturbation regime and the basic growth rate (F and P). In contrast, genetic viability is associated with the inbreeding and drift loads, themselves primarily related to population size and fragmentation (K_t and N, 14 30 ). The causes and implications of this are further developed below.

While the genetic model presented here is a soft selection model (i.e., each local population contributes to the next generation independently of its mean relative fitness), the integration of demographic processes (i.e., the demo-genetic model) clearly adds a hard selection component (since the size and contribution of demes partly depend on their genetic loads). Theoretical work predicts that (1) With soft selection, population structure should reduce the efficiency of selection against mildly deleterious, nearly additive mutations 34 but will favor selection against severe and highly recessive mutations 35 ; (2) With hard selection, structure should lead to reduced loads even with additive or nearly additive mutations 14 .

While the present results are in agreement with predictions from soft selection models, the beneficial genetic effects of population structure expected under hard selection are not visible with the present demo-genetic model (i.e., increasing N and/or decreasing m has a negative effect on viability). This discrepancy is due to the fact that (i) the demo-genetic model is only a partial hard selection model, as local population size primarily depends on regulation processes (independent from genetic loads); (ii) potential genetic effects are masked by demographic effects (see, in particular, Figure 2c); (iii) the genetic and demo-genetic viabilities expressed in the present paper do not necessarily reflect expected equilibrium patterns. In particular, the purging processes leading to reduced inbreeding or drift load may be demographically costly, which implies that some metapopulations will have a low demo-genetic viability despite a small expected equilibrium load.

It is generally admitted in the theoretical literature that ecology-genetic interactions should strongly influence the persistence time of populations 15 16 17 18 36 37 . The demo-genetic interaction refers to the fact that the occurrence of genetic processes is affected by the demographic state of the population, and vice versa, leading to synergistic or antagonistic effects on viability. Here, the process of demo-genetic extinction may be decomposed in two phases: an accumulation phase, during which the overall genetic load increases while population size remains approximately equal to the carrying capacity, and an extinction phase during which the population declines to extinction. Demo-genetic interactions are important in both phases: (1) during the accumulation phase, stochastic variations in population size decrease the effective size and accelerate the rise of the genetic load; (2) during the extinction phase, the population decline increases the rate of mutation accumulation, which in turns accelerates population decline (the so-called mutational meltdown, formalized and discussed in refs 15 16 ).

The extinction phase (i.e., phase 3 in Lynch and colleagues papers) is generally assumed much shorter than the accumulation phase in theoretical genetic models, and the extinction time is assumed equal to the duration of the accumulation phase (i.e., the time necessary to obtain a deterministically decreasing population). Here, in all scenarios, the initial decrease in fitness was more rapid with the demo-genetic model as compared with the genetic model (see Additional file 5). This led to T_DG < T_G in two thirds of the scenarios investigated, in agreement with previous findings on the demo-genetic interaction. However, in the other third of the scenarios, demo-genetic viability was higher than genetic viability (Figure 1c). This comes from the fact that populations may be demographically resilient and persist several generations after their growth rate has become negative (Additional file 5). Therefore, fitness-based estimations of viability may lead to strong over- or under estimations of the risk of extinction, depending of the relative weights of the demo-genetic interaction and the demographic resilience on viability.

As expected, incorporating genetic considerations to the demographic model was unequivocally deleterious. However, the magnitude of the reduction in viability was highly variable. It is generally admitted that the proportional effect of genetic deterioration on viability should be larger for populations with a high demographic viability 17 37 38 39 . While the present results are in clear agreement with this expectation, they highlight the critical need to distinguish the intrinsic and extrinsic ecological threats to population viability to estimate the effect of genetic problems on extinction. In some of the scenarios investigated, the demo-genetic extinction times were about half the demographic extinction times, while in other cases they were 10,000 times shorter. Typically, the first situation corresponds to large populations with highly variable, low quality environments (environmental stochasticity is the primary cause of extinction, with or without genetic deterioration), while the later corresponds to small, fragmented populations in stable, good quality environments (genetic deterioration is the primary cause of decline and extinction). In the context of the debate over the environmental versus genetic causes of species extinction, the present results demonstrate that the arguments of the partisans of the environment 1 2 40 and genetic 9 10 11 hypotheses are both theoretically justified. The net impact of genetic deterioration processes on extinction may be strongly variable among and within species, since it strongly depends on the ecological conditions faced by metapopulations (although it is clear that some variation in other factors such as mating systems, life history traits, dispersal pattern or demographic history, will further increase this variability). Importantly, however, many human induced environmental changes are likely to engender situations where genetic deterioration is the primary extinction cause, in particular where available habitats have been reduced in quantity, and not in quality.

Contrary to other ecological settings (K_t, m,...), fragmentation has non obvious (and potentially contradictory) qualitative effects on demographic and genetic viabilities. From a genetic view-point, most theoretical studies agree that subdivision has detrimental effects on fitness (i.e., few large patches perform better than many small patches 17 19 30 ). In the ecological literature, conclusions are less clear 41 . The optimal level of fragmentation depends on overall population size and dispersal ( 42 43 44 , see upper panels of Figure 2) as well as on the regime of perturbations 45 46 . In the absence of fragmentation (single population), strong perturbations may rapidly drive the whole population to extinction, even if population size is large 33 47 . In contrast, with very high levels of fragmentation (many small populations), environmentally induced extinction risk is spread over several units, but the probability of local extinction due to demographic stochasticity increases as local population sizes decrease. Thus, in many realistic situations, an intermediate level of fragmentations will be optimal. Theoretical work indicates that this general result remains true under a wide range of realistic conditions, provided that (i) the cost of dispersal is not too strong; (ii) environmental variations are not fully correlated among patches of habitats 43 45 46 .

Consequently, (i) demographically optimal levels of fragmentation may be much higher than genetically optimal levels; (ii) demo-genetic optimal levels are, in most cases, intermediary between demographic and genetic optimal levels (Figure 2). This implies that optimal levels of fragmentations (e.g., in the context of reserve design) may be either under- or over-estimated when based on simple demographic or genetic approaches.

The genome of each individual was explicitly represented as L = 1000 different diploid loci that could carry either a wild-type or a deleterious allele. Following the approach of Jaquiéry et al. 19 , I considered variable selection (s) and dominance (h) coefficients among loci and a negative correlation between them 50 . Selective coefficients were exponentially distributed among the L loci, with an average severity set to s = 0.05 51 . At each locus k, the dominance coefficient h_k was computed as h_k = exp(-cs_k )/2, where c was a constant computed so that the expected dominance of all mutations in the genome equaled h = 0.35 19 52 . During fertilization, the probability of transmission of each allele at each locus was given by Mendelian rules. New deleterious mutations stochastically occurred in each zygote (Poisson distributed, with mean U = 1.0, 53 ). I assumed no epistasis, no linkage and no reverse mutations. Importantly, the mutation parameters used in the present study derive from model organisms, such as Drosophila, nematodes or bacteria. These estimates are used in most theoretical studies, and should be viewed as conservative estimates (in particular, per zygote mutation rates are likely to be higher in most species of conservation concern than in Drosophila species, 51 54 ).

At each locus k, the initial frequency of deleterious alleles was given by the mutation-selection balance 55 : q _0k = U/(L s_k h_k ). The genome of the initial population considered at t = 0 was implemented according to these frequencies (Bernoulli process). Thus, the individuals present at t = 0 derived from a large panmictic population (for all ecological scenarios investigated, I assumed that sudden reduction and fragmentation of the habitat occurred at t = 0).

I assumed that deleterious alleles acted on offspring viability. The proportional reduction in survival of the individual i was then given by

w i = 1 w 0 ∏ L w k i

Where w_ki was equal to 1, 1-h_ks_k or 1-s_k , for a locus k without mutation, with a heterogeneous deleterious mutation, or a homozygous mutation, respectively. w ₀ was the expected initial reduction in fitness due to deleterious alleles present at time zero, given by

w 0 = ∏ L ( 1 - h k s k ) q 0 k

In each generation t, the average reduction in survival of the metapopulation (W _(t)) was computed.

In each generation t, in each patch j, all individuals paired randomly with possibility of self-fertilization at the random rate. The fecundity of each individual was determined by a Poisson trial of parameter F. Punctual negative environmental perturbations resulted in a reduction (punctual in time) of local offspring survival. The survival of each offspring i present in patch j at generation t was drawn from a Bernoulli function of expectation s _0i = (1-p _j(t) )w_i , that depended on the genetic characteristics w_i of the individual (see above) and on the occurrence of a local perturbation. The occurrence of a perturbation was determined by a Bernoulli trial of parameter P (= the general per generation frequency of perturbations). If no perturbation occurred in patch j at time t, p _j(t) was set to zero. If a perturbation occurred, the value of p _j(t) was drawn from an empirical severity distribution 47 . The parameter C_p denoted the degree of spatial correlation of perturbations. I considered two extreme situations where perturbations were either independent (C_p = 0) or fully correlated in time (C_p = 1) across patches. An alternative method to model perturbations (in which perturbations temporally reduce local carrying capacity) is presented in results and Additional file 1.

I assumed that all patches had the same maximum local carrying capacity (K = K_t/N), and were initially at their carrying capacity. All parents died after reproduction. Local regulation consisted in a truncation of population size of the offspring to K in each generation for each patch. Truncation was made independently of the genetic qualities of individuals.

Dispersal occurred by assuming that a proportion m (Bernoulli process) of the individuals present in each patch after the selection and regulation steps (but before the reproduction step) emigrated (island model) in each generation. Extinction occurred when there were no more individuals in the metapopulation.

In each generation t, in each patch j, two parents were randomly selected to reproduce. Fertilization and new mutations occurred as described above to create the new individual i, that survived or not, according to its relative fitness w_i (Bernoulli process). This operation (random selection of parents with replacement) was repeated until there was K _j(t) surviving offspring. Then all parents died. K_{j (t)}, the carrying capacity of each patch j at generation t, depended on the occurrence of a perturbation in patch j. It was computed as the rounded value of K(1-p _j(t)), where p _j(t) was as described above for the demo-genetic model. In case where all local carrying capacities equaled zero, one patch was randomly drawn and its carrying capacity was set to one, so that the overall carrying capacity was always higher than zero. Here, contrary to the demo-genetic model, (i) population dynamics were independent from W _(t) and F. Local populations were always at their carrying capacity, which could vary for environmental reasons as in the demo-genetic model: perturbations were modeled by the parameters P (= frequency of perturbations) and C_p (= spatial correlation of perturbations) and acted by temporally reducing local carrying capacities (same distribution of severity as in the demo-genetic model); (ii) "demographic extinction" could not occur (there was always at least one individual in the metapopulation). Environmentally-driven variations of population size (i.e., environmental perturbations) tended to reduce the long term effective size and increase the inbreeding level, with subsequent effects on selection. The metapopulation was assumed "genetically extinct" when the average offspring survival W _(t) dropped below the threshold W* = F ^-1, where F was the average basic fecundity (corresponding to the theoretical population's replacement rate in the absence of genetic load). Dispersal occurred as for the demo-genetic model.

The demographic model worked similarly to the demo-genetic model, but the effect of selected genetic variation on population dynamics was removed (i.e., w_i was set to one for all individuals). The similarities and differences between the demographic, genetic and demo-genetic models are summarized in Table 1.

Demographic, genetic and demo-genetic metapopulation viabilities were examined by assuming that time zero corresponded to sudden environmental changes (reduction and fragmentation of the habitat). I used Monte Carlo simulations for 1476 combinations of the ecological input parameters (N, K_t, m, F, P, C_p , see Table 2 for details). For each combination, 250 population trajectories were drawn. In each case, the median times to demographic, genetic and demo-genetic extinction were computed (respectively noted T_D, T_G and T_DG ). I was interested in investigating the viability of small to medium metapopulations (K_t ranged from 50 to 2000) with a wide range of persistence times. With the demo-genetic model (i.e., the most realistic one), typical persistence times were a few tens or hundreds of generations (median value was less than 100 generations); however, in some scenarios assuming large populations with high and stable growth, times to extinction could be very long, especially with the demographic or genetic sub-models. Therefore, for technical reasons, all trajectories were stopped after 1,000,000 generations. In scenarios for which one or several trajectories were non extinct after 1,000,000 generations, extinction times could not be computed. These extinction times were treated either as right-censored data (i.e., data for which extinction was not observed, 56 ) or as lacking data in statistical analyses (see details below).

I used survival models, generalized linear models (GLM) and hierarchical partitioning (HP) to examine the relationships between ecological input parameters and extinction times.

As a first step, exponential and Weibull survival models assuming right censored extinction events were fitted to examine the relationship between extinction times (T_D, T_G and T_DG ) and ecological input parameters, considering all 1476 scenarios. The Weibull and exponential models were compared to describe whether extinction rates were constant over time or not 57 .

As a second step, GLM and HP were applied to those scenarios for which extinction times were available (i.e., excluding scenarios for which one or several trajectories were non extinct after 1,000,000 generations) to examine variations in the ratios of extinction times (e.g., T_D /T_G ) and their relationships with ecological variables. HP uses all models in a regression hierarchy to distinguish those variables that have high independent correlations with the dependent variable. At all steps, results from the survival models, GLM and HP were compared to help interpretation. Although most extinction events occurred within the range 10-10,000 generations (> 80% of scenarios), the variance was high and the distribution of extinction times was non-normal. Thus, graphical results are presented using logarithmic scales and extinction times were log-transformed (Neperian logarithm) in all statistical analysis. Other dependent and independent variables were transformed using usual functions to achieve the best linearity (transformations provided in results). The aim of these analyses was not to test the significativity of regressions but rather to examine the direction and strength of relationships between ecological variables and viabilities 20 . All qualitative results were compared to theoretical expectations and in cases where results were different from expectations, additional simulations were run to determine the underlying causes of observed patterns. Metapopulation models were developed in Pascal language (source codes are available upon request). All statistical analysis was performed with R 2.10.1 58 , specifically using the Survival 57 and Hier.part 59 packages.