R. Péroux, office National de la Chasse et de la Faune Sauvage, 11 Avenue de Fontmaure, F 63400 Chamalières, France (fax +33 473 196449; e-mail firstname.lastname@example.org).
1Hares are considered to be a valuable game species in most European countries. Hunting needs to be sustainable and sound management of hare populations requires some knowledge of the species’ demographic variability, especially regarding the breeding output, which is highly time- and space-dependent and may govern the population size and exploitability.
2Using shooting bag analysis and placental scar counts, mean fecundity and leveret survival were estimated at four study sites with contrasting hare numbers and density trends. These demographic parameters, harvest rates and adult natural survival rates (from the literature) were incorporated into a matrix projection model to analyse the population growth rate (λ) sensitivity and to derive indices of sustainable harvest rate (ISHR, i.e. rates compatible with λ≥ 1).
3The age structure comprised 48–69% young; fecundity varied between 12·2 and 15·0 leverets per breeding female; and 85–100% of adult females bred. These data combined gave a birth-to-autumn leveret survival index of 0·14–0·29 and, when loaded into the matrix model, resulted in simulated λs that matched the density changes observed in the study areas. The model structure, although simple, accounted for most of the relevant biological information.
4ISHR was 30% when derived iteratively as a function of mean values of those parameters with largest elasticities (leveret survival and doe fecundity). When environmental and demographic stochasticity were included in the model, the proportion of endangered trajectories where the density threshold was < 1 km2 over 50 km2 varied sharply, with small changes in harvest rate or initial population size. Small populations could only sustain ≤ 20% harvest rates.
5Demographic parameters derived from killed animals during year t can be used a posteriori to understand the species’ dynamics and as a baseline to modulate the shooting scheme in year t+ 1. An actual sustainable shooting scheme would require an additional real-time local process that, automatically, would take all sources of change in numbers into account. A two-stage management strategy (e.g. first computing a catch-effort estimator of population size based on numbers killed very early in the shooting period, then defining flexible harvest quotas) would help managers to cope with the unpredictable dynamics of the species and resulting fluctuations in hare numbers.
6Synthesis and applications. The results of this exploratory study suggest that sustainable shooting of hares is possible provided some local data about their dynamics are available and slightly conservative quotas are used. Modelling approaches have potential in assessing the latter, but also as a check on the coherence of the estimates of the former (comparison of observed and modelled λ).
In populations of European hare Lepus europaeus Pallas, changes in the growth rate (λ) following changes in the demographic parameters are dependent on age structure (Marboutin & Péroux 1995). This has major implications for any population exploitation. The hare is an important game species: more than 5 million are harvested annually in western Europe (Flux & Angermann 1990) and the 1998–99 French national bag there were about 920 000 hares (Péroux 2000). The demographic correlates of density patchiness and population trends must therefore be identified, particularly when trends indicate decline (Tapper 1992). Sustainable harvesting or conservation management must take the species’ and populations’ demographic characteristics into account (Tapper 1999; Macdonald & Tattersall 2001).
Adult hare survival is relatively constant apart from variations due to shooting conditions or disease epidemics (Marboutin & Péroux 1995; Lamarque, Barrat & Moutou 1996; Marboutin & Hansen 1998). Breeding success, when deduced from population age structure, may be highly variable and related to habitat characteristics and climate (Mauvy et al. 2002). Interpreting age structure may, however, be difficult without additional estimates of breeding parameters or increase in numbers (Caughley 1974). Changes in the fecundity and proportion of breeding females, or in apparent leveret survival (i.e. including dispersal), may influence breeding success and induce changes in population age structure. Pépin (1989) documented space-related changes in age structure as a function of changes in apparent leveret survival, using indirect fecundity estimations and ad hoc calculations. Recent research has shown that the proportion of breeding females and their fecundity varies over the geographical range of the species (Hansen 1992; Bensinger et al. 2000; Hackländer et al. 2001).
The objective of the present work was to explain small-scale patterns in age structures and population trends in French European hares as a function of changes in the underlying breeding parameters. The analysis of field data, obtained from shooting bags, was first undertaken separately on four contrasting study sites. Secondly, the demographic parameter estimates were averaged to define a ‘generic’ hare population (which would be likely to correspond to the field circumstances usually encountered by hunters and managers) and incorporated in a matrix projection model to simulate the species’ life cycle. We studied the effects of parameter variations on the generic population growth rate λ using elasticities (Caswell 2001). Indices of sustainable harvest rate (ISHR) were derived as a function of those parameters with larger elasticity. Finally, the uncertainty attached to ISHR due to environmental and demographic stochasticity, and the resulting management risk, was illustrated by Monte Carlo runs and population vulnerability analysis.
All study areas were in central France (Auvergne region) but differed in their landscape characteristics (Table 1) and population trends. The first two were lowlands: Chareil-Montord (CM) was a mixed arable and farmland area, and Chauriat (CH) was more intensively cultivated. The other two were highlands mainly supporting cattle breeding. Planèze 1 (P1) bordered Planèze 2 (P2) but they were separated by elevation (an 1100-m contour line). P1 included some cereal crops whereas P2 was more elevated and almost pure grassland. Based on night counts and distance sampling models (Péroux et al. 1997; Langbein et al. 1999), hare density was estimated in March and November in CM and CH. Night counts were performed in P1 and P2 only from November 2001, and the trend in hare numbers during the study period was deduced from the changes in the shooting bags.
Table 1. Main landscape features of the study areas, located in the Auvergne region (central part of France): CM and CH were lowland areas, P1 and P2 were mountainous areas
Agricultural area (AA) (%)
Cereal (% of AA)
Grassland (% of AA)
data collection and analysis
Annual shooting bags were considered as a random population sample. Shooting took place from mid-September to late November in P1 and P2, and during October–November in CM and CH (i.e. always after the breeding season). Data from animals shot were collected in 1993–99 in CM, 1994–99 in CH and in 1996–99 in P1 and P2. The eyes of every collected hare were kept in 10% formalin and female uteri were frozen. Animals were aged by the eye-lens mass technique: from a standard curve (Broekhuizen & Maaskamp 1979) hares were classified as adults or young of the year, for which bimonthly cohorts were established according to the estimated date of birth. The following analyses were undertaken on all sites individually. The age structure was defined as the proportion of young in the sample. The proportion of breeding females (i.e. those having delivered at least one litter) and fecundity data were estimated by placental scar analysis, a method based for the detection and classification of embryo implantation sites on uterine walls late in the breeding season (Bray et al. 2003). As scars fade over winter, those seen in animals shot in autumn represent all breeding in the preceding breeding season and none from the season before. Implantation sites were counted to estimate last breeding-season's production, then pooled by visual similarities to define litter numbers and sizes. Because the limited sample sizes did not allow the analysis of yearly variations within each site the data were pooled over years by population trend.
Leveret survival was estimated by comparing age structures to fecundity data. Let PYi be the proportion of young in the shooting bag from study site i, then the number of young per adult in the bag is PYi/(1 − PYi) (equation 1). The adult sex ratio is likely to be imbalanced because males usually survive better than females: let k be the ratio of sex-dependent annual survival rates (k = SYm/SYf = 0·50/0·40 in yearlings, then SAm/SAf = 0·55/0·50 in adults, model 13 in Marboutin & Hansen 1998) and let nt = (SM)t + (SF)t, with t= 1, 2, 3, … , n years. The adult asymptotic sex ratio was calculated from a horizontal life table convergence, based on simulated annual cohorts of yearlings and adults, as lim.(Σktnt)/Σnt = 1·41. Let PBFi be the proportion of breeding females in population i, then the number of young per breeding female in bag i is NYi = [PYi/(1 – PYi)] × (1 + 1·41)/PBFi (equation 2), to be weighed next for female survival during the time elapsed between the breeding chronology median and the opening of the shooting season (mi = 3, 4, 3·5, 3 and 2·5 months, respectively, in CM, CH1, CH2, P1, P2). The resulting number of surviving young per female then alive is (equation 3), with an averaged monthly survival rate φi = [0·40 × PYi + 0·50 × (1 − PYi)]1/12 to account for the mothers’ age structure (yearling, adult). Let Fi be doe fecundity, and the intuitive leveret survival index is NY′i/Fi. Young females that bred between birth and the shooting season, however, produced some young in the bag. By eye-lens mass, we defined five bimonthly age classes among the PYi leverets (i.e. those born in January–February, March–April, etc.). Let ACi,j be the frequency of age class j in population i, and PBYi,j the corresponding proportion of breeding young females, and let fi be their fecundity (independent of age class). The contribution of young females relative to adults, could roughly be estimated as Ci = [(fi × PYi/2) ×Σ(ACi,j × PBYi,j)]/[Fi × (1 − PYi) × PBFi/(1 + 1·41)] (equation 4). The leveret survival index therefore is (equation 5).
We used likelihood-ratio statistics (G2; Agresti 1990) to analyse changes in the proportion of young, breeding females and frequency distributions of the numbers of females with 1, 2, 3, … litters between study areas. Scar and litter numbers, and litter sizes, were modelled with generalized linear models (GLM) using site (s), body mass (m) or age (a), and site × body mass (or age) interaction, as predictors. Age was expressed as a continuous variable by eye-lens mass in order to be comparable with other published data. The model best supported by the data at hand was selected by minimizing Akaike's information criterion (AIC), which balances errors of under- and overfitting while providing a robust basis for inference, especially from exploratory investigations (Anderson, Burnham & Thompson 2000). Backward selection started from a global model including all factors and first-order interactions, down to the simplest one with only the grand mean µ. Analyses performed with GLIM 3·77 (Crawley 1993) gave a scaled deviance of each model [DEV = −2 ln(θ), ln(θ) being the log-likelihood function evaluated at its maximum]. DEV is a relative measure of the model's fit to the data, given the number of estimated parameters (np), and minimizing AIC = DEV + 2np balanced the quality of fit and parsimony.
Because model structure richness is limited by the amount of available robust biological information (Burnham & Anderson 1998), a simple two-sex, age-structured, density-independent model was chosen. The species life cycle was simplified to a graph based on three age classes (leveret, yearling, adult; see the Appendix), and a projection matrix model was run with the corresponding transition probabilities and reproductive parameters (Caswell 2001). Fecundities (Fi), proportion of breeding females (PBFi) and apparent leveret survival (i.e. including dispersal; SL) were from the present work. Yearling and adult survival rates for each sex class (SYm = 0·50, SAm = 0·55, SYf = 0·40, SAf = 0·50) were taken from Marboutin & Hansen (1998). Shooting mortality was considered additively to the natural mortality rates: harvest rates were estimated as HR = (number of hares killed km−2)/(number of hares killed km−2+ post-shooting density) and survival during hunting (SH) was 1 − HR. Unified life models software (ULM; Legendre & Clobert 1995) was used to (i) simulate the population growth rate λ, (ii) analyse its elasticity to parameter changes and (iii) estimate indices of sustainable harvest rates (ISHR, i.e. the largest values of HR that gave λ≥ 1). The λ were compared with mean growth rates obtained directly as averaged ratios of yearly density estimates based on distance sampling (whenever available, i.e. in CM and CH). Harvest rates were estimated indirectly in P1 and P2 (no distance sampling data) by adjusting them iteratively in the model until the simulated growth rates matched the declining trends in the bag statistics.
In order to provide more conclusions on sustainable harvesting of hare populations generally, the elasticity analysis of λ and the estimation of ISHR were then conducted on a ‘generic’ population (i.e. with averaged demographic parameters) by modelling within the known bounds of the key demographic variables as estimated from a pooling of all of the data. Elasticity coefficients assess the effects of model parameter variations on λ. Parameter pi's elasticity, ei, is such that if pi is changed by γ percentage, λ is changed by γ × ei percentage, and is calculated as the slope of log (λ) plotted against log (pi). ei can be rescaled with a coefficient of variation [CV = (1 + 1/4n) × (100σ/x̄)] to incorporate the possible natural variation in the demographic parameters. The resulting actual elasticity coefficient AEi = ei × CVi, combines ‘the proportional sensitivity of the growth rate to the parameter with the observed empirical variation in that parameter’ (Haydon et al. 1999). CV of leveret survival and fecundity parameters were based on the between-study standard deviation. Yearling and adult survival were set constant over study areas in the present modelling; only the basic elasticity coefficients were therefore calculated (no CV available).
ISHR were estimated as a function of parameters with larger elasticity. ISHR uncertainty due to environmental and demographic stochasticity was investigated as a function of hare numbers (N) by a population vulnerability analysis and Monte Carlo runs. Basically, a population is projected over many years and projections are repeated many times to simulate different possible population trajectories. Fecundity and survival rates are randomly chosen for each time step from underlying distributions, and used in matrix calculations to project the population size at the next census. Environmental stochasticity was modelled by sampling the parameter values from log-normal distributions. The distributions’ means, variances and truncation values were derived from the parameter estimates from the present work. Demographic stochasticity was added using a binomial distribution for survival. In the French context, a 50-km2 hypothetical shooting ground was considered. Initial numbers and harvest rates were increased step-by-step (N0 from 125 to 5000, i.e. density (D) from 2·5 to 100 hares km−2, HR from 0% to 50%). A 25-year vulnerability risk was simulated for each HR ×N0 combination, estimated by the proportion of endangered trajectories from 1000 Monte Carlo runs (i.e. those where N went through a < 50 minimum over 50 km−2, i.e. D < 1). The threshold level for sustainability was arbitrarily set at 5% of endangered trajectories.
patterns in hare numbers
The post-breeding density was stable in CM during the study period (on average 21 hares km−2). In the CH site, a 3-year phase of increase (CH1, 4–13 hares km−2) was followed by a 4-year decrease (CH2, 13–3 hares km−2). In P1 and P2, night counts were performed only in November 2001 and yielded a mean density of 1 hare km−2 in both sites. Using the shooting bag statistics, the declining trend of the bag during the study period, however, was sharper in P2 than in P1.
Age structure varied significantly between data sets (CM = 0·66, CH1 = 0·69, CH2 = 0·49, P1 = 0·63, P2 = 0·48; G2 = 25·4, d.f. = 4, P < 0·001). The frequency distribution of bimonthly cohorts showed a peak in May–June (Fig. 1). Some with an early shooting closure were right-censored because very young hares born late in the season could rarely be harvested due to their hiding behaviour. Others (CH2 and P2) were left-censored, i.e. with few specimens from early cohorts of birth. The January–August frequency distributions differed significantly between areas (five study areas and four bimonthly cohorts; G2 = 28·7, d.f. = 12, P < 0·01). The January–February and March–April cohort patterns in CH2 and P2 areas contributed about 60% to this chi-square value, although involving only 20% of the contingency table cells.
In adults, the proportion of breeding females was lower in P2 than in any other study area (Table 2). The overall difference was not significant (G2 = 5·09, d.f. = 4, P= 0·28) but the statistic was concentrated in a single degree of freedom (contribution of P2 cells; partial G2 = 3·4). The number of scars per female varied both with body mass and site (Table 2; model F in Table 3 and Fig. 2a). Litter number was best modelled as a function of body mass (model J in Table 3 and Fig. 2b), contrary to litter size (model K, Table 3). Mean litter size varied between 1·3 and 3·4 by litter rank, and the study area–rank interaction was non-significant (GLM analysis: χ2 = 1·00, 4 d.f., NS). Averaged litter size according to increasing ranks was 2·1, 3·2, 3·1, 2·8, 2·7, 2·1 and 1·3.
Table 2. Breeding parameters in adult hares estimated from placental scar analysis. Due to limited annual sample sizes, data were pooled over the study period (sample sizes in parentheses)
Proportion of breeding females
Mean no. of scars
Mean litter size
Mean no. of litters
0·95 (n = 82)
12·8 (n = 63)
2·6 (n = 302)
4·9 (n = 63)
1·00 (n = 15)
15·0 (n = 14)
2·7 (n = 61)
5·4 (n = 14)
0·92 (n = 37)
13·9 (n = 30)
2·9 (n = 146)
4·9 (n = 30)
0·94 (n = 82)
13·8 (n = 46)
2·8 (n = 227)
4·9 (n = 46)
0·85 (n = 34)
12·2 (n = 21)
2·6 (n = 92)
4·6 (n = 21)
Table 3. Generalized linear modelling of changes in the number of scars, litters and litter size, as a function of study area (s), body mass (m) and age (a). µ is the grand mean. Generalized linear models and AIC-based selection (AIC = DEV + 2 np) were used to evidence the most relevant structure in the data (minimum AIC in bold)
The proportion of young females that bred between their birth and the following shooting period was low and constant over study areas (G2 = 0·65, d.f. = 4, P= 0·96; x̄ = 0·13, n= 236 uteri). Other breeding parameters were not compared due to very limited sample sizes. The mean number of scars per breeding young female was 4 (n = 29 uteri), mean litter size was 2 (n = 57 litters), and the mean number of litters was 2 (n = 29 uteri). The proportion of young breeding females depended on the date of birth, those born early in the year being more prone to breed than those born later on (Fig. 3).
leveret survival estimations
Combining age structures andFig. 1 to fecundity data (Table 2 and Fig. 3) gave a leveret index of survival that varied between 0·14 and 0·29 (Table 4). Detailed calculations for the CM site are given hereafter as an example. The 66% young age structure was equivalent to 0·66/(1 − 0·66) = 1·94 young per adult in the bag. The corrected number of young per breeding female in the bag was 1·94 × (1 + 1·41)/0·95 = 4·92, and the mean number of surviving young per female alive at the breeding chronology median was 4·92 × (0·932)3 = 3·98 [based on averaged φ = (0·40 × 0·66 + 0·50 × 0·34) = 0·43 = 0·93212 to account for the age structure of mothers]. The contribution of young breeding females relative to that of adults was 0·08: 100%, 26% and 6% of young females born in January–February, March–April and May–June could breed (Fig. 3). Combining these figures with the age structure, 66% young, and with the age class frequency distribution (Fig. 1) gave (4 scars × 66 young/2) × [(100% × 2·8%) + (26% × 18·8%) + (6% × 43·4%)] = 13·6 leverets; in adults, the production of young was 12·8 scars × (34/2·41) × 95% of breeding females = 171·6 leverets, and 13·6/171·6 = 0·08. The leveret index of survival in CM was 3·98/(1·08 × 12·8) = 0·29.
Table 4. Estimation of the leveret index of survival by means of comparison of age structures to fecundity data from the study areas (columns)
Equations as given in the Methods.
Number of young per adult in the bag, equation 1
Corrected number of young per breeding female in the bag, equation 2
Number of young per breeding female alive at the breeding chronology median, equation 3
Contribution of young breeding females, equation 4
Leveret index of survival, equation 5
Pooling of all of the data characterized the generic average hare population: the age structure was 62% young, fecundity 13·4, frequency of breeding females 0·93 and leveret survival 0·22.
modelling the population growth ratesλ
Constraining the model parameters to the sets of values from each study area gave λCM = 1·02 (with an observed harvest rate HR = 41%) vs. Nt+1/Nt = 1·01 on average in distance sampling-based density, and when HR was set at zero; λCH1 = 1·65 (with HR = 8%) vs. Nt+1/Nt = 1·66, ; λCH2 = 0·87 (with HR = 21%), vs. Nt+1/Nt = 0·78, = 1·09. In CH1, λ was ≥ 1 as long as HR < 44%, whereas λCH2 was ≥ 1 as long as HR < 9%. No harvest rate data and night counts were available in P1 and P2. Without shooting mortality, the model gave =1·57 and =1·08. The bag statistics, however, suggested a population decline in both sites: λ < 1 was achievable with a simulated HR as high as ≥ 36% in P1 and HR ≥ 8% in P2.
sensitivity analysis and sustainable exploitation of the generic population
Based on AE coefficients, leveret survival could cause the greatest proportional perturbations in λ (Table 5). Fecundity was possibly the second most important parameter, especially if considering its great variability over the species’ geographical range (between 5 and 15 leverets per breeding female). With the averaged hare population values, λ was 1·44 in the absence of additional shooting mortality, and λ was ≥ 1 as long as HR was ≤ 30%. When using only data from study sites where harvest rate estimates were available, the elasticity coefficient of survival during shooting was among the largest (AESH= 23·5; AESL= 26·7).
Table 5. Demographic parameters ranked according to their actual elasticity coefficient (AE = elasticity × CV; see the Methods, Dynamics modelling). The AE coefficient is a relative measure of the importance of the parameters on λ
Estimated in present work but considered constant over study sites.
ISHR were then estimated as a function of combinations between values of leveret survival and doe fecundity. Without any kind of stochasticity (Fig. 4), low values of both parameters would lead to a conservation strategy, i.e. λ ≥ 1 would only be achieved with HR ≤ 5%, whereas combining the highest ones observed in the present work (SL = 0·29, F1 = 15, PBF1 = 1) would lead to a ±50% ISHR. Once stochasticity was added to account for the unpredictable changes in vital rates and low population sizes, a variable and potentially large vulnerability risk was attached to ISHR (Fig. 5). Population trajectories starting from very low numbers (N ≤ 250, i.e. D≤ 5 km−2) could safely sustain only low harvest rates (i.e. ≤ 20%) whereas moderate harvest rates (c. 30%) were sustainable once N was ≥ 500 (i.e. D≥ 10 hares km−2). Trajectories starting from N≥ 1250 (i.e. D≥ 25 hares km−2) could sustain harvest rates as high as 35%, but higher ones were not sustainable even in much larger populations.
The aim of this study was to (i) estimate the breeding parameters and indices of leveret survival in contrasted hare populations; (ii) identify which parameters might account for the patterns observed in their population dynamics; and (iii) provide general guidelines for the sustainable exploitation of hunted populations. Analyses of points (ii) and (iii) were based on matrix population models that, like any model, are simplified versions of what is actually happening in the wild and as such the results need to be considered in the context of the underlying assumptions.
In adults, fecundity and the proportion of breeding females, although variable (12–15 scars, 85–100%), ranged in the upper part of the data analysed by Bray et al.'s (2003) staining method: in Austria and Germany Hackländer et al. (2001) and Bensinger et al. (2000) found 82–84% of breeding females, delivering 10–11 leverets. Other studies used the original simple staining method, i.e. bleaching in 70% alcohol, and some populations were studied at the northern edge of the species’ range. Mean fecundity per breeding doe was 9 in Hungary and Slovakia (Kovacs 1983; Slamecka, Hell & Jurcik 1997), 8 in Sweden (Frylestam 1980) and only 5 in Denmark (Hansen 1992), with a varying proportion of breeding does of 68% to 100%. All other studies (reviewed in Hansen 1992) provided lower fecundities, but using even less robust methods. At least part of the variability observed in fecundity may therefore be due to some methodological artefacts. In the present work, fecundity did not vary so much as a function of absolute density than as a function of population trends: in the Chauriat population does delivered a few more leverets during the phase of increase than during the phase of decline. Fecundity also increased with body mass, because of changes in litter numbers rather than of changes in litter size. Hackländer et al. (2001) found a negative relation between fecundity and age (R2 = 0·21) that was evidenced neither in our data nor that of Bensinger et al. (2000) and Frylestam (1980). However, harvest rates were higher in the latter studies and the population age structure could be shaped in favour of younger animals. Sampling errors may also account for the differences observed between these studies, and the ageing method is not accurate for very old hares due to a large variability in the eye-lens mass of old adults within a given age cohort.
In females younger than 9 months, only those born early in the year (January–April) bred before the shooting season, a result of physiological constraints in the young does’ ovarian function (Caillol et al. 1992; Ruf, Hackländer & Arnold 2001). Due to small sample sizes, the variability of this demographic trait between study sites could not be investigated. Its elasticity coefficient, however, suggested that changes in this variable would change the population growth rate λ only very slightly (eF0 = 0·05).
The leveret survival index estimated herein (SL = 0·14–0·29) partially overlapped with those in Pielowski (1981; 0·23), Pépin (1989; 0·25–0·50), and Hansen (1992; 0·20–0·30). This could stem from methodological differences, e.g. in fecundity estimation, and could also point at the natural variability in leverets’ survival, which had the largest impact on λ as deduced from the elasticity analysis. Survival was lowest in the declining populations from open field and mountainous grassland habitats. Hansen (1997; K. Hansen, unpublished data) reported that leveret survival rates could be lowered (0·33–0·11) in modern and simplified farming systems, contrary to more diverse systems. Although in the Chauriat site the farming practices did not change during the study period, the population displayed a 3-year phase of increase and then a 4-year phase of decline, with a parallel decrease in leveret survival (0·25–0·14). Such peak-and-low patterns in animal numbers may suggest some kind of parasite–host or predator–prey relationship. Foxes Vulpes vulpes may be a common predator of hares (Reynolds & Tapper 1995) and so they were recorded in the present study when detected while counting hares at night. The mean number of foxes detected per sampling point (i.e. fox encounter rate) was used as a crude index of fox abundance. There was no significant correlation between the increase in hare density following breeding (Dautumn/Dspring) and (i) fox encounter rates in spring (r = 0·41, n = 7 years) or autumn (r = 0·63), and (ii) the increase in fox encounter rates during the same period (r = 0·11). This spring-to-autumn increase in fox numbers averaged 2·5 in 1993–95 (i.e. when SL was 0·25), and only 1·7 in 1996–99 (i.e. when SL was 0·14). Therefore, the data did not support a possible link between hare dynamics and fox dynamics in Chauriat. McLaren, Hutchings & Harris (1997) suggested that ‘grassland management changes, such as the increase in grass cutting for silage, could enhance leveret mortality’. The eye-lens frequency distribution observed in the present work showed that the leveret cohorts from January to April were underrepresented in the highest study site with the largest proportion of grassland, although no grass silage was cut there by that time. Explanations for the lower leveret survival rate (0·17) could be related to climate, food availability and impaired energy budgets (Hackländer, Arnold & Ruf 2002a; Hackländer, Tataruch & Ruf 2002b).
Caswell (2001) warns that population matrix simulation results are projections rather than predictions, because they rely both on model structure quality and demographic data. Some kind of field validation of these projections is therefore necessary: our model demonstrated good agreement between the simulated λ and the population changes actually observed in those study areas where night counts and distance sampling analysis were conducted. The assumptions underlying the model should, nevertheless, be considered, especially regarding the robustness of the population vulnerability analysis (Beissinger & Westphal 1998).
First, yearling and adult survival rates were set constant over study areas and stemmed from mark–recapture data in the literature, whereas local estimates may have been more relevant. Most changes observed in adult survival rates from different areas, however, can usually be traced back to changing harvest-induced mortality (Marboutin & Péroux 1995), an explicit parameter in the present modelling. Local and patchily distributed disease epidemics (Rattenborg 1997) may induce some heterogeneity in survival rates as well. In France, the SAGIR national network collates the pathological findings from dead bodies, collected on a voluntary basis mostly by hunters (Lamarque, Barrat & Moutou 1996). No pathological disease outbreak was detected in our study areas, although on CH2 some abnormal over-wintering mortality may have occurred (e.g. due to pseudotuberculosis; Barré, Louzis & Tufféry 1977), as suggested by the density changes observed between November year t and March year t + 1 (on average −40%). With a corresponding additional mortality rate in the matrix model, λsimulated/λobserved was 0·64/0·78 as opposed to 0·86/0·78 without it. Using CH2-specific yearling and adult survival rates did not improve the fit of the model (λ underestimated by 18% with vs. overestimated 10% without the additional mortality). The use of constant survival rates over study areas was therefore a parsimonious preliminary approach, but this assumption would be better relaxed as it may bias the estimate of survival in leverets (equations 2 and 3) and overestimate the impact of changing survival in leverets (sensitivity analysis of λ).
Secondly, the model did not include explicit information on dispersal but the leveret survival rates were apparent ones (i.e. survival rate × philopatry rate), and the population was simulated over an area large enough (50 km2) to encompass most of the dispersal phenomenon [in Bray (1998) only 7% of young hares moved farther than 12·5 km, a 50-km2 radius].
Thirdly, the model was density-independent, whereas the ability of hares to persist in the presence of sustained exploitation may be evidence for the occurrence of density dependence. Hare spatial distribution is usually aggregated (Marboutin & Aebischer 1996; Marboutin & Péroux 1999) and the mating system is hierarchy-dominated (Holley 1986, 1992). As the population's spatial structure may change with density (Jezierski 1972), one may reasonably expect that both under- and overcrowding may affect the population dynamics. Frylestam (1980) suggested a negative density dependence on breeding efficiency, whereas some robust data combined from Eskens et al. (1999) and Hackländer et al. (2001) suggested an inverse relationship. As shooting always reduces population size, Allee effects (Courchamp, Clutton-Brock & Grenfell 1999) may also occur in small populations. In the CH site, where D was varying by a threefold factor, plotting the rate of increase over two successive years Nt+1/Nt against the number in the first year Nt did not yield any strong relationship (r = 0·32, n= 8 years; with Nt+1 being the autumn density in year n+ 1 before shooting, and Nt the autumn density in year n after shooting). Our density-independent modelling may therefore be considered as a parsimonious ‘makeshift’.
Fourthly, shooting mortality was totally additive to natural mortality. Compensatory mortality is not that easy to demonstrate (Smith & Willebrand 1999) but omitting it is, management-wise, a conservative approach. Modelling assumed that a given proportion of hares was always harvestable, irrespective of density, whereas hunter efficiency will decrease with declining hare numbers (hunting may even favour some escape behaviour; Hutching & Harris 1995).
The theoretical principles for sustainable exploitation are still under debate (reviewed by Saether 2001), and increasingly powerful but complex models have not yet been fully evaluated with field data. None the less, the present modelling, whilst rough, illustrated the relative impact of demographic parameters on the population growth rate and on exploitability.
The potential impact of shooting exploitation on λ was, logically, important, as evidenced by its large elasticity value. In the Chauriat site, the harvest rate increase (CH1, 8%; CH2, 22%) was not sustainable and occurred 1 year after the population had entered the phase of decline. Overshooting may thus be regarded as an aggravating factor but not as the primary cause of population decline. In the deterministic environment, the index of sustainable harvest rate [i.e. max(HR) compatible with λ≥ 1] in the generic hare population averaged 30% of the post-breeding population size, a value also provided as a rule of thumb by Broekhuizen (1976). From Stoate & Tapper (1993), the mean shooting rate was 49% with an averaged density of 31 hares km−2. Macdonald & Tattersall (2001) reported much larger ISHR but warned that their matrix model, first developed for deer, was probably not suitable for hares. In the present work, breeding parameter variability could induce large variations in ISHR (0–50%) but one can estimate them only post-shooting on the basis of an analysis of killed hares. Any current information on breeding success in a real-time management schedule is thus impossible, and using only mean values could be tentative. Monte Carlo simulations of endangered population trajectories, including both environmental and demographic stochasticity in the model, illustrated the resulting management risk. Given our threshold levels (≤ 5% of trajectories with N≤ 50, i.e. D≤ 1 hare km−2 over 50 km2), ISHR ranged between 20% and 35% depending on the population post-breeding size, but the actual variance could be larger if using density-dependent modelling. McLaren et al. (1997) found that a hare population would be unlikely to go extinct when density was above 3 hares km−2 over 100 km2. With our model parameterization and extinction criteria, a hare population of D= 3 hares km−2 did not go extinct as long as HR was ≤ 20%.
Constant offtake rates over such large areas are likely in those countries with large hunting estates (eastern and part of western Europe) but the actual harvest impact in smaller estates may be even greater (smaller populations are likely to undergo local extinctions). Contrasted harvesting regimes, e.g. constant effort harvesting or constant yield harvesting, may, however, have contrasting impacts on population sizes and extinction risks (Letty, Reitz & Mettaye 1998; Stevens & Sutherland 1999). More complicated strategies, e.g. those including profitability objectives, must be compared with competing scenarios using much more elaborate models than the present one (Lande, Engen & Saether 1995). In such cases, strategies that rely on population growth rates rather than on population size are more robust to biased estimates, and simple strategies based on harvesting a small proportion of the population (i.e. under exploitation) would be most likely to be sustainable (Milner-Gulland et al. 2001). Knowing both the population size and its dynamics would, however, be the safest basis on which to define an optimal strategy that would balance the risks of under- and overexploitation.
synthesis and applications
The model needs data (age structures, uteri) that can only be collected after breeding and shooting. Because hare dynamics are both time- and space-dependent, the mean values of demographic parameters can be used as a baseline and real-time shooting management is required, taking into account all local sources of change in population size. Such a flexible approach is currently being tested in some French pilot hunting grounds, with modulated harvest quotas based on more shooting days or more hares to be bagged. They are defined with the information collected in the first shooting days of the season (Péroux 1995; Nichols, Lancia & Lebreton 2001). This is not ‘adaptive management’ (sensuNichols, Johnson & Williams 1995) but rather a rough real-time management. Shooting-based data can provide both demographic information and an index of population size (based on catch per unit-effort methods) that may be integrated further into simple management plans: harvesting the resource in turn allows for a data-based management programme to modulate harvest pressure. As quoted in Nichols, Lancia & Lebreton (2001), ‘data from hunting programs … can contribute strongly to improve our understanding of the dynamics of hunted populations; in that sense, the phenomenon under study brings some information in itself ’.
Hunters kindly provided the samples. E. Bro and S. Legendre gave helpful advice for modelling with ULM. K. Hackländer, F. Reitz, E. Taran, anonymous referees, and the Journal's editorial board improved the first version of the manuscript.
european hare population life cycle and parameters
Population sampling was based on post-breeding autumn shooting. From the shooting bag analysis, two age classes could be distinguished by eye-lens mass: juveniles born in the year (N1), and adults (N2). A third, transient, age class (leverets N0) was produced by N1 and N2 individuals, and a small part of N0 also bred during the time span between their birth and the start of shooting season. The final matrix population projection model, however, contained only N1 andN2 age classes: the self-loop of early breeding in weaned leverets was added to yearling and adult fertilities as the model was run on a yearly basis (from autumn year n to autumn year n+ 1). Yearlings and adults had to survive from mid-October to mid-May before breeding (i.e. a birth pulse 1 month later, and corresponding to the eye-lens mass frequency distribution peak). The model is therefore an age-stratified, two-sex, density-independent one.