Effects of competition on lifetime estimates of inbreeding depression in the outcrossing plant Crepis sancta (Asteraceae)


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Inbreeding depression was studied in two populations of a Mediterranean allogamous colonizing species Crepis sancta. In order to test the hypothesis that the magnitude of inbreeding depression can be modified by successional processes, the growth and survival of individuals resulting from two generations of inbred crosses including selfing were analysed with interspecific competition (in natural vegetation) and without interspecific competition (by removing natural vegetation). Inbreeding depression was weak for seed production. Germination was little affected by inbreeding but mortality and the number of capitula showed inbreeding depression, especially in the presence of competition. This suggests that inbreeding depression is very sensitive to variations in environmental conditions such as interspecific competition. As a consequence, inbreeding depression cannot be considered as constant in natural conditions.


Both genetical and ecological factors can influence the evolution of plant mating systems ( Barrett & Harder, 1996). Among genetic ones, the classical model is based on the balance between two opposing forces, the 50% transmission advantage of a gene causing selfing, and inbreeding depression ( Lloyd, 1979). Inbreeding depression is a general phenomenon defined as the reduced fitness of inbred offspring compared with noninbred ones. It is most likely the result of increased homozygosity at overdominant loci (overdominance hypothesis) and at loci with partially dominant deleterious mutations (partial dominance hypothesis) which are masked in the heterozygous state ( Charlesworth & Charlesworth, 1987). Among the ecological determinants affecting mating system evolution, the founding of a population in a new environment can generate a low density and isolation of individuals. Because selfing promotes reproductive assurance ( Jarne & Charlesworth, 1993), the classical view is that a good colonizer, in order to cope with recurrent situations of isolation, should be able to reproduce uniparentally through self-fertilization: this is called Baker’s law ( Stebbins, 1957). Yet not all colonizing plants conform to this model ( Price & Jain, 1981) and the existence of self-incompatibility in many colonizing plant species ( Abbott & Forbes, 1993; Imbert et al., 1996 ; Sun & Ritland, 1998) constitutes a paradox with regard to Baker’s law.

The ecological and genetical explanations are not mutually exclusive, and it has recently been emphasized that ecological and genetical determinants interact in a complex way ( Uyenoyama et al., 1993 ). Firstly, recent empirical or theoretical studies have shown that mating systems coevolve with inbreeding depression, in such a way that consanguineous mating contributes to purging deleterious mutations ( Lande & Schemske, 1985; Charlesworth et al., 1990 ), so that strong inbreeding depression is associated with allogamy, and vice versa. These models are based on a dynamic balance between the selfing rate and the number of deleterious alleles, which leads to the conclusion that allogamy and autogamy are the only stable states ( Lande & Schemske, 1985). These results are supported by experimental studies which show that inbreeding depression is often more pronounced in allogamous species than in autogamous ones, both in plants ( Barrett & Charlesworth, 1991; Husband & Schemske, 1996) and in animals ( Doums et al., 1996 ). During colonization, selfing rates can increase due to the lack of partners for cross-fertilization and, as a consequence of repeated selfing, inbreeding depression values can perhaps decrease. Furthermore, Lande & Schemske (1985) propose that recurrent bottlenecks increase homozygosity and probably contribute to purging deleterious alleles ( Kirkpatrick & Jarne, in press, have an alternative view). As a consequence, mating systems should evolve towards autogamy.

Secondly, although the interaction between inbreeding depression and environmental conditions has not been taken into account in theoretical models ( Charlesworth & Charlesworth, 1987), several empirical studies have shown that the magnitude of inbreeding depression can sometimes vary according to the environmental conditions ( Uyenoyama et al., 1993 ). There is a possibility that greenhouse experiments minimize inbreeding depression compared with more stressful natural conditions. Nevertheless, studies in natural conditions are scarce. Furthermore, no empirical study has yet considered that environmental variations in stressful conditions can occur in natural populations (see, however, Lloyd, 1980). Successional processes provide a natural situation in which environmental conditions change rapidly. Competition increases because the number and density of species augments dramatically in a few years with the development of the community, leading to the extinction of colonizing species ( Lepart & Escarré, 1983). To test the hypothesis that the expression of inbreeding depression can be modified by successional processes, it was measured in two simulated stages of a Mediterranean succession.

The species studied, Crepis sancta, is a colonizing species widespread in the early stages of Mediterranean successions ( Imbert et al., 1996 ). It possesses a self-incompatibility system. Populations studied by Imbert et al. (1996 ) were strictly self-incompatible but a few self-compatible genotypes have been observed in the Mediterranean region. Unlike other populations of Mediterranean colonizing species, such as Conyza spp. which are very ephemeral, populations of Crepis sancta can persist for as long as 25 years. Populations at different stages of the successional process are demographically contrasted, from a few individuals in newly founded populations to several hundreds in 5- to 10-year-old fields.

This study was conducted from 1995 to 1998 and includes two generations of inbred matings. Because inbreeding depression can be expressed at different stages of the life cycle, analysis was conducted on the whole life cycle, including seed production (first stage), germination (second stage), mortality before reproduction (third stage) and reproduction (fourth stage), following the procedure of Husband & Schemske (1996).

Materials and methods


Crepis sancta (L.) Bornm. is an annual colonizing species in the family of Asteraceae. It occurs in the Mediterranean region. It is a diploid plant ( Babcock, 1947). It produces two types of achenes, a few peripheral ones (3–10 per head) which are heavy and have no pappus, and central ones (70–100 per head) which possess a pappus ( Imbert et al., 1996 ). The present study used central achenes only. Reproduction is strictly sexual, starts early in the spring (March) and lasts 5–8 weeks. Pollination is mostly entomophilous. Populations are frequently disturbed by natural perturbations or human activities. The dynamics of this species imply recurrent founding events and relatively low expected lifetime of the transient populations ( Imbert et al., 1996 ). Plants in young populations can produce more than 100 capitula (or heads) whereas plants in more mature populations (fields abandoned 20 years ago) may produce only two or three heads ( Imbert et al., 1996 ).

Field sampling

Two populations, each consisting of several hundred individuals, were sampled near Montpellier, southern France. Heads were harvested on several plants. Seeds from a single plant were considered half-siblings, i.e. we assume that plants were pollinated by several individuals. The violation of the hypothesis of multipaternity would lead to slight underestimation of the inbreeding coefficient, but will not affect the qualitative results. The mean distance between sampled plants was more than 10 m, to decrease the likelihood of sampling-related individuals. In the spring of 1995, seeds were harvested in a 4-year-old field at Vic-la-Gardiole (hereafter called V, 3°42′E 43°40′N). Seeds from five plants constituted the five families for experiment. The second population was a 10-year-old field sampled at St-Mathieu de Tréviers (hereafter called S, 3°48′E 43°47′N) 30 km from V. Nine individuals were collected in 1994 and were previously used for a quantitative genetic study (E.I., unpublished data). Therefore, the crossing design differed slightly between the two populations. Population S and a third population (Valence, 200 km from Montpellier) were used for between-population crosses.

In November (1995), seeds were germinated in Petri dishes, and then transplanted into a greenhouse at the beginning of 1996 in order to make controlled crosses.

Crossing design

In 1996, for the first generation of crosses in population V (G1, Fig. 1), four plants (half-siblings) in each of the five families were each crossed with three half-sibs (inbreeding coefficient, F=0.125), with a plant chosen at random in each of the four other families (F=0) and with four individuals (chosen at random) from the Valence population. Crosses between populations were made to test the possibility of heterosis due to deleterious alleles fixed by genetic drift in the populations ( Carr & Dudash, 1996). Each of these crosses was repeated twice for each maternal individual. Self-fertilization was also performed and produced a few seeds. This population is thus not strictly self-incompatible, but too few seeds were obtained by self-fertilization in G1 to allow analysis of the fitness effect of selfing on fitness over the whole life cycle.

Figure 1 Crossing design. Above (population V), seeds collected in the field from five individuals were used to generate five half‐sib families (G0). In G1, individuals from each family were crossed with half‐sibs, nonrelatives and individuals from another population. In G2, six types of crosses were made for each G1‐cross type, from selfing to between‐population crossing. Below (population S), crosses of nine individuals collected in the field were used to generate 10 full‐sib families (G0). In G1.

Figure 1 Crossing design. Above (population V), seeds collected in the field from five individuals were used to generate five half-sib families (G0). In G1, individuals from each family were crossed with half-sibs, nonrelatives and individuals from another population. In G2, six types of crosses were made for each G1-cross type, from selfing to between-population crossing. Below (population S), crosses of nine individuals collected in the field were used to generate 10 full-sib families (G0). In G1.

, genotypes of each full-sib family were crossed with full-sibs, half-sibs and nonrelated genotypes (in the same population).

In 1997, for the second generation of crosses (G2), due to the large number of crosses involved, we randomly chose the descendants of two plants (out of four) in each of the five families, and selected one offspring from each plant, corresponding to each of the three kinds of crosses in the first generation (with half-sibs, with nonrelatives of the same population and with individuals of the Valence population, Fig. 1). Thus for the new generation of crosses (G2), each family was represented by six plants (two plants per group). Each plant was crossed in order to produce six kinds of matings with different levels of consanguinity, namely self-fertilization cross, crossed with a full sib, with a half sib, with a cousin, with an individual of another family and with an individual from population S. Self-fertilization and crosses with full sibs were repeated at least four times per maternal plant to obtain a sufficient number of progeny of selfing to permit analysis over the whole life cycle. The other crosses were repeated twice.

In 1996, 10 full-sib families with two individuals per family were used for population S (G1, Fig. 1). These families studied by Imbert (unpublished data) came from nine sampled plants used in a diallel cross. In 1996, two plants per family were crossed (G1) with two full siblings (F=0.25), with two half-sibs (F=0.125) and with two nonrelatives (F=0). Each cross was repeated twice. Despite some differences in the experimental design, the inbreeding coefficients of the two populations were comparable, except for the absence of selfed progeny and between population outcrosses in the S population.


Hand-pollinations were performed in an insect-proof greenhouse, when all florets were receptive (50–100 florets). Pollen was collected from several heads of the donor plant using a paint brush, which was then brushed against recipient heads to achieve pollinations ( Kearns & Inouye, 1993). We measured the ratio of the number of viable seeds/total number of florets per head, by counting seeds and florets with a binocular microscope, each floret potentially producing one seed. The mass of 10 seeds per head was measured in G1 only. To check our crosses, nonpollinated heads were harvested at random in the greenhouse. No seeds were found, providing evidence that no pollen contamination had occurred and that autonomous self-fertilization ( Schoen & Lloyd, 1992) did not occur in these populations. Seeds were germinated in Petri dishes with 12 h artificial light (25 °C) and 12 h darkness (12 °C). Thirty seeds per cross and per individual were used. In G2 of population V, seed production was analysed in the three groups but in order to limit the number of individuals in our experiment, we used the offspring of group 1 only (see Fig. 1) for the germination, survival and reproduction stages. The percentage germination was measured every 2 days for 2 months. The final rate of germination, and the germination index ( Scott et al., 1984 ), defined as GI=[∑Ti.Ni]/S (where Ti is the number of days, Ni is the number of germinated seeds for the ith day and S is the total number of seeds), were analysed.

Before transplanting into the field, seedlings were grown in a greenhouse until rosettes reached 3 cm in diameter. No mortality occurred during this period. The ‘mortality before reproduction’ and ‘growth/reproduction’ stages were analysed at the experimental field at the CEFE-CNRS (Montpellier) simulating two contrasting stages of succession. The first treatment consisted in removing vegetation to simulate a newly abandoned field such as those occurring in the Montpellier region. In the second treatment, we left the natural vegetation, which is similar to that of an abandoned field 4 or 5 years old, with species such as Plantago lanceolata, Picris hieracioides, Scabiosa spp. and a few Poaceae. Rainfall was measured daily.

The experiment was conducted in a split-split-plot design with four blocks ( Snedecor & Cochran, 1967). The presence or absence of vegetation was recorded in the whole plot. Cross type was treated as split-plot and family as split-split-plot. Families were randomized in the split-split-plot. This experimental design was chosen for ease of statistical treatment and field measurements.

Rosettes grown in the greenhouse were transplanted into the field in February. In G1, due to an insufficient number of seedlings in some families, eight individuals (four families) from population S and 17 individuals (five families with unequal numbers of individuals) from population V per split-split plot were transplanted, for a total of 600 plants. The split-split plot of 25 plants (5 × 5) was surrounded with 24 other plants from which data were not taken, so that every plant had the same number of neighbours.

In G2, the same design was employed with six cross types (i.e. inbreeding coefficients), five families and two individuals per family. Two rows of five plants with one nonstudied plant at each end were transplanted, giving a total of 480 plants because only individuals from group 1 were used. The survival of plants from transplantation to sexual maturity was checked every 2 days. Once all of the remaining plants had produced fruits, we measured the numbers of heads produced and the above-ground dry plant biomass for all individuals.

Total fitness

We calculated the total expected fitness as the product of germination rate, survival before reproduction and number of heads produced. Seed production was not included in the total fitness measure because low production of seeds can be due to self-incompatibility or to inbreeding depression (see discussion for further details). Total fitness for each cross was compared with the best fitness value in each analysis.

Statistical treatments

Statistical analyses were conducted using the generalized linear model, treating the two populations separately. Numbers of seeds/numbers of florets, germination rates and mortality were treated using a logistic model with binomial errors, logit link and a backward suppression of nonsignificant effects. Means were compared by t-tests ( Crawley, 1993), using the statistical package GLIM ( Payne, 1985). For the seed production and germination stages, the full model included a maternal family factor, a cross factor and maternal family–cross interaction. Residual deviance was overdispersed for the ratio of numbers of seeds/numbers of florets, and was therefore corrected using William’s procedure ( Crawley, 1993). Survival before reproduction was analysed as a binomial response including a vegetation factor, a cross factor and a vegetation × cross factor. The mass of 10 achenes and the germination index was analysed using ANOVA. Split-split-plot results were analysed with the GLM procedure of SAS ( SAS, 1989) using type III sums of squares. Block, vegetation, cross and family were treated as fixed factors. To satisfy ANOVA assumptions, the number of heads produced and biomass were log-transformed. Multiple comparisons of means were performed using Tukey-tests.


Seed production

In G1, population S showed a significant family effect, but no significant cross effect (likelihood ratio test (hereafter LRT) for family: χ2=32.2, d.f.=9, P < 0.001; Fig. 2). For population V, both cross and family effects were highly significant. The means of the half-sib and the between-population crosses differed significantly ( Fig. 2). The mass of 10 achenes was analysed in 1996 in a two-way ANOVA with the factors cross and family. The family effect was significant for population V only (F4;45=6.5, P < 0.01). Neither cross nor the interaction cross × family were significant.

Figure 2.

 Seed production estimates measured as the ratio of number of viable seeds/number of floret per heads, for G1 for the two populations. Means and standard errors are shown. Means with the same letter are not significantly different (t-test, P < 0.05).

In G2, the three groups of population V were analysed separately. For groups 1 and 2, cross and family effects were highly significant (LRT, group 1 for family: χ2=9.5, d.f.=4, P < 0.05; for cross: χ2=27.27, d.f.=5, P < 0.001; group 2 for family: χ2=12.68, d.f.=4, P < 0.05; for cross: χ2=25.18, d.f.=5, P < 0.001). For group 3, only the cross factor was significant (LRT, for cross: χ2=39.59, d.f.=5, P < 0.001). Comparisons of means showed a significant difference between selfing and all other treatments for all three groups, but no other significant differences were found. Because of the similar trends among the three groups, Fig. 3 shows the results for group 1 only.

Figure 3.

 Seed production estimates measured as the ratio of number of viable seeds/number of floret per heads, for G2 for population V (Group 1). Means and standard errors are shown. Means with the same letter are not significantly different (t-test, P < 0.05).

Germination stage

In generation G1, population S had 100% germination success and thus no effects could be tested. For population V, the logistic model applied to the final percentage of germination indicated a significant effect of cross only (LRT, for cross: χ2=9, d.f.=2, P < 0.05). Comparison of means showed only a significant difference between seeds from F=0 (96%) and seeds from between-population crossing (99%). An ANOVA on the quantitative variable germination index was not significant. This indicates that there were no differences in the germination time between the families or the crosses.

In G2, we found a significant effect of cross (LRT, for cross: χ2=15.56, d.f.=5, P < 0.01) for population V. Comparison of means showed a significant difference (P < 0.05) between F=0.56 (94%) and all other inbred crosses (98–99%), but a nonsignificant difference comparing between-population crosses and F=0.56. No differences were found in germination index.

Mortality before reproduction

In G1, survivorship for the two populations was affected by the presence of competing vegetation ( Fig. 4). Both the cross and the vegetation effects were significant for population S (LRT, for cross: χ2=20.17, d.f.=2, P < 0.001; for vegetation: χ2=9.36, d.f.=1, P < 0.001). Inbreeding depression appeared greater in the presence of vegetation, but the cross–vegetation interaction for population S was not significant. For population V, only the vegetation factor was significant (LRT, for vegetation: χ2=15.44, d.f.=1, P < 0.001).

Figure 4.

 Means and standard errors of survival to reproduction for plants grown without vegetation (open bars) and with vegetation (shaded bars) for Populations S and V in G1 above and for population V in G2 below. Means with the same letter are not significantly different (t-test, P < 0.05).

In G2, mortality was less pronounced than in G1 for population V. The vegetation effect on survivorship was highly significant, and a nearly significant effect of cross was detected (LRT, for vegetation: χ2=15.44, d.f.=1, P < 0.001; for cross: χ2=10.95, d.f.=5, P < 0.052). The cross–vegetation interaction was not significant.


In G1, the block effect was significant for total biomass and the number of heads for population S, and for number of heads for population V (Table 1). This was probably due to the heterogeneity of soil fertility over the area of the experiment. The presence of vegetation significantly affected the number of heads and total biomass of the two populations ( Fig. 5). The cross effect was not significant. However, the cross–vegetation interaction was significant for number of heads in both populations, with lower performances for inbred individuals grown with competing vegetation. For population V, the family effect was significant, as were the cross–individual interaction and the vegetation × cross–family interaction.

Table 1.   Effects of vegetation, cross and family on the biomass and the number of heads for populations S and V. Mean squares were calculated from type III sums of squares in the split-split plot design (fourth stage). Block and vegetation are tested with a block × vegetation error term; cross and vegetation × cross with a block × vegetation × cross error; family, vegetation × family cross × family and vegetation × cross × family with the residual error term. Thumbnail image of
Figure 5 Biomass and number of heads for the different cross types and generations. Plants grown without vegetation (open bars) and with vegetation (shaded bars) for population S (G1) and V (G1 and G2). Means with the same letter are not significantly different (Tukey‐test, P < 0.05.

Figure 5 Biomass and number of heads for the different cross types and generations. Plants grown without vegetation (open bars) and with vegetation (shaded bars) for population S (G1) and V (G1 and G2). Means with the same letter are not significantly different (Tukey-test, P < 0.05.

). Scales differ between G1 and G2.

In G2, the experiment with population V confirmed the effect of vegetation (Table 1, Fig. 5) on both variables. The cross effect was significant for biomass and number of heads. Unlike G1, the cross–vegetation interaction was not significant. The family effect was significant, as in G1. The vegetation–family interaction was significant for biomass. The cross–family interaction was significant for biomass and for number of heads, due to the good performance of one family for F=0. In G2 plants growing without vegetation produced more biomass (×4) than individuals in G1. In contrast, plants in vegetation produced more biomass in G1 than in G2.

Total fitness

Table 2 gives the total fitness values. In G1, biparental inbreeding depression was clearly more pronounced in the presence than in the absence of vegetation, as shown by the cross–vegetation interaction. The pattern in G2 is less evident, as shown in our stage-by-stage analysis, but shows the same trend.

Table 2.   Relative effect of crosses on total fitness for two generations and for populations S (G1) and V (G1 and G2) calculated as germination rate × survival before reproducing × number of heads produced. Thumbnail image of


Inbreeding depression at the beginning of the life cycle

Crepis sancta is clearly an allogamous plant, as is often the case in the genus Crepis ( Babcock, 1947) but our results for population V prove that it is not strictly self-incompatible. Our findings do not permit us to define a class of self-compatible genotypes and a class of self-incompatible ones. Self-compatibility varies continuously, some plants being more self-fertile than others. This indicates that variability in self-compatibility exists in natural populations. At the seed production stage we found no significant biparental inbreeding depression either in G1 or G2. On the other hand, self-fertilization in the formation of G2 in population V produced significantly fewer seeds than other cross types. Lower seed production under selfing can be due either to the deleterious effects of increased homozygosity (inbreeding depression) or to the existence of a self-incompatibility system which prevents self-fertilization. Under the first hypothesis, seed production should also decrease with biparental inbreeding but this was not the case here. However, this relationship has been shown in Raphanus sativus ( Nason & Ellstrand, 1995) and in Picris hieracioides (P.-O.C., unpublished data), both self-incompatible species. Under the second hypothesis we expect a greater decrease in seed production by selfing because of the self-incompatibility. We showed this only in G2 with self-fertilization. Our results did not strongly support either hypothesis; however, the presence of a partially self-incompatibility system seems more probable than the existence of inbreeding depression which is not necessarily required and is probably low.

Surprisingly for an allogamous species, the germination rate showed no evidence of biparental inbreeding depression in G1 or G2 for either population. Thrall et al. (1998 ) found that germination rates for progeny from full-sib crosses were approximately half those of noninbred progeny of Silene alba. Inbreeding depression after self-fertilization is present at the germination stage but is weak (less than 5%).

Environmental effects on inbreeding depression

In G1, the only biparental inbreeding depression detected for population S was for survival before reproduction. Biparental inbreeding was not significant over the whole experiment but the cross–vegetation interaction was significant in the field experiment for number of heads for both populations. Clearly, this shows that inbreeding depression is enhanced under competition. In G2, the effect of inbreeding depression showed no clear pattern for population V. Pre-reproduction survival was higher than in G1, with no evidence of inbreeding depression in spite of higher inbreeding coefficients. Inbreeding depression was significant for biomass and number of heads but less pronounced than in G1 if we compare similar inbreeding coefficients.

The vegetation effect was more pronounced in G2 than in G1, i.e. the differences in biomass and number of heads between the two treatments were greater in G2 ( Fig. 5). The uniformity of the experimental design for the two consecutive years suggests that variation in an unmanipulated factor modified the results. It is unlikely that selection against deleterious alleles occurred between the two generations because genotypes used for the second generation of crosses were grown in a greenhouse with no selection after germination, since there was no mortality of seedlings before reproduction. On the other hand, environmental conditions could have differed between G1 and G2. This is supported by the stronger development of plants without competition in the second year. Environmental differences between G1 and G2 could be due to differences in rainfall. The precipitation during April and May was high in G2 (164.2 mm) compared with 51 mm in G1. This period corresponds to the growth and bolting period for this species. Consequently, competitor species were bigger in G2 than in G1 (P.-O.C., personal observation) suggesting more intense competition in G2. We can hypothesize that rainfall enhanced the survival of C. sancta during this period but also increased interspecific competition, leading to decreased growth of plants in the presence of competing vegetation, and reducing the possibility of differences between inbred and noninbred plants. On the other hand, in the absence of vegetation there was stronger development of C. sancta in G2 than in G1.

Nevertheless, our study demonstrates that the magnitude of inbreeding depression is sensitive to environmental conditions such as competition. To date, inbreeding depression in theoretical models has been assumed to be determined only by the genotypes of the individuals ( Hauser & Loeschke, 1996). In a study of Lobelia cardinalis and L. siphilitica, no variation in inbreeding depression was detected with stress in field condition but without competition ( Johnston, 1992). No variation with fertilizer levels was found for Schiedea lydgatei ( Norman et al., 1995 ). Other empirical studies have shown that this assumption is not always true. The effect of vegetation in our study is in accordance with results obtained for Sabatia angularis ( Dudash, 1990) and for Plantago coronopus ( Koelewijn, 1998) which found that a stressful environment increases the magnitude of inbreeding depression. Intraspecific competition with noninbred genotypes has also been found to enhance inbreeding depression ( Wolfe, 1993).

Considering the whole life cycle, our results suggest that inbreeding depression is more pronounced in the two last stages of the life cycle compared with the two first stages. This trend is inconsistent with other results found in allogamous species. In a review article, Husband & Schemske (1996) found that outcrossers show a higher inbreeding depression at the beginning of the life cycle than inbreeders.

It is generally admitted that measures of inbreeding depression in greenhouses are lower than in field conditions (see Introduction). Here we show experimentally that two simulated natural situations give different inbreeding depression values indicating that fluctuations of ecological conditions cannot be neglected in experimental studies because it can provide an important source of variation in the expression of inbreeding depression.


We thank P. Jarne and J. Ronfort for helpful discussions and constructive criticism during this study. We thank the three anonymous reviewers for helpful comments. We also thank C. Collin for assistance in the field. This research was supported by the Centre National de la Recherche Scientifique, in addition to a grant to P.-O.C. from the Ministère de l’Enseignement Supérieur et de la Recherche.