Willow grouse Lagopus lagopus population dynamics have interested Norwegian ecologists and hunters for decades and in spite of quite substantial research efforts starting as early as 1928 (B.J.F.F. 1928 reviewed in Myrberget 1988, Steen et al. 1988) the factors determining the changes in population size are poorly understood. Our aim was to evaluate which demographic factors have the strongest effect on the growth rate, λ, of breeding willow grouse hen numbers modelled assuming additivity in all demographic parameters. Based on this population dynamics model, we present a general harvest model for willow grouse and apply it to a data set covering 21 years assuming that hunting mortality is additive to natural mortality.

In mountainous regions of Fennoscandia, populations of small mammals such as lemmings Lemmus lemmus and voles Microtus spp. fluctuate strongly in size and are often characterised by cyclic population dynamics (Framstad et al. 1993). Predators feeding on small mammals consume willow grouse eggs and chicks as alternative prey. Thus, survival from egg laying to four weeks after hatching is very poor in small-mammal crash years (Steen et al. 1988). Microtine density, roughly classified as low, medium and high, predicted 41 % of the variation in number of chicks per pair in the autumn (Steen et al. 1988). Early autumn, chick production per pair ranged from 0.65 to 7.28 with a mean of 2.89 (Myrberget 1988). In spite of a 10-fold variation in chick production that correlated with microtine density (Steen et al. 1988), the evidence that breeding population size and breeding population fluctuations of willow grouse follow the microtine cycle is scanty. Total density and number of breeding hens are quite variable, but the effects of microtine density seem to be masked by variation in other factors such as winter survival (Myrberget 1988).

To resolve the effect of different demographic parameters on λ, key factor or sensitivity/elasticity analyses (Caswell 1989) may be used. Key factor analysis has been criticised for being unable to reveal the underlying processes in population dynamics due to an erroneous interpretation of population variance (Royama 1996). Sensitivity analysis evaluates the potential impact of a deviation in a parameter away from the mean parameter value. Sensitivity analyses cannot be used to directly compare the effects of demographic parameters on dynamics when the parameter values are expressed in different units. However, using elasticity analysis it is possible to compare the proportional change in one parameter with the proportional change in another. To get an impression of the degree to which a demographic parameter affects the population growth rate in the real world, elasticity coefficients must be evaluated after correcting, using the observed weighted mean and variance in the demographic parameters from natural populations. Since elasticity co efficients give the changes in population growth rate dependent on a percentage change in a parameter, we can multiply the elasticity coefficient with the corresponding coefficient of variation (CV) and get an estimate of the actual effect, the Actual-Elasticity-coefficient (AE-coefficient, van Tienderen 1995).

Knowing the elasticity coefficients of the population dynamics, it should be possible to identify which demographic parameters management efforts should focus on. The results from the elasticity analysis show that winter mortality is as important as survival from egg laying to four weeks after hatching in determining the growth in hen numbers. This means that hunting may be an important factor in determining population growth of willow grouse especially in years with low production. We therefore examine the harvest potential of willow grouse populations using data from a Norwegian island population.

Data background

The data originate from a 21-year study conducted on Tranøy, a 127 ha island situated in northern Norway, by the late Svein Myrberget and students (Myrberget 1988). Tranøy lies about 1 km from the nearest island and I km from the mainland. There was a dairy farm on the island from 1960 to 1965; apart from that there was no human activity on the island until 1976 when sheep were introduced. The general population trend of willow grouse as estimated by total counts of all birds on the island was declining (Fig. 1). Dogs were used to find nests which were monitored until hatching; later broods were flushed and juveniles and adults were counted at the appropriate times (for a more detailed description see Myrberget 1988).

Figure 1.

Population trend of the willow grouse population at Tranøy, 1960–1980. The broken line indicates the autulnn population density; the solid line indicates the total spring breeding population.

Raw data used in the analysis (Table 1) were read off the graphs of Myrberget (1988). The quality of the data is evident from the mean, maximum and minimum as well as sample sizes across years for the numbers of clutches and the numbers of birds on which survival estimates are based. Winter survival rates as presented by Myrberget do not represent the true survival, but a combination of natural mortality, hunting mortality, migration and immigration (see below for discussion). We therefore use ‘apparent winter survival’ instead of ‘winter survival’ throughout the paper when referring to this data set.

Although the expected number of willow grouse arrived in 1971 and started establishing territories, half of the adult birds had left the island by mid June. Because Myrberget (1988) does not clearly state how summer survival and other demographic traits were treated for 1971, we excluded all data from 1971 from our main analysis. Note, however, that apparent winter survival can still be calculated since the birds arrived at the island in the spring.

Table 1.

Values of the demographic parameters used in the analysis: Clutch size (Clutch), survival of eggs from egg laying to hatching (egg survival, Egg_s), proportion of eggs that hatched (Hatch), proportion of chicks that survived from hatching to four weeks (chick survival, Chick_s), proportion of juveniles surviving the winter (juveniles apparent winter survival, Juvs), proportion of adults surviving the winter (adult apparent winter survival. Ad_s), adult survival from arrival at Tranøy until mid May (adult summer survival, SS), mean value of juvenile (Juv_s) and adult (Ad_s) apparent winter survival (Pooled winter survival), survival of an egg from egg laying to four weeks after hatching (PROD). The abbreviations are the same as the ones used in the formulas. Standard deviation in the clutch size is based on the total variance and is not weighted as in the other parameters.

As the data were gathered over many years, it might be possible to find time-dependent trends in the demographic parameter estimates. This was checked for by autoregression analysis with first order autoregressive error terms on the arcsine, transformed survival rates and untransformed clutch size. Only egg survival (from laying to hatching) showed a significant negative trend from 1960 to 1980. This was due to one year of extremely low egg survival, 1979, and was not a time-dependent trend in the egg survival rates. As there is no real trend in any of the apparent survival or reproductive parameter estimates, no parameters are detrended. Myrberget (1985, 1988) checked for density dependence in the demographic parameters for the Tranøy data and found that only clutch size was negatively correlated with willow grouse density.

In the data set, apparent winter survival is divided into the age-classes juveniles and adults. There is no division between first, second and third-year reproductive parameters. We therefore use the two apparent winter survival age-classes (first-year juveniles and adults) and one reproductive class in the model. Parameter descriptions used in the model are presented below with their model notation given in brackets.

Model and analysis

We chose to model the growth rate, X, as the change in the number of breeding female willow grouse (N_t) from one spring to the next spring (N_t+I). This is modelled by:

where λ is defined as:

In the model we assume an even sex ratio at hatching, identical survival rates between the sexes with no interaction/dependency between any of the mortality factors.

Sensitivity of λ with respect to any parameter, x_j; is defined as dλ/dx_j, i.e. the derivative of λ with respect to parameter λ; evaluated for the weighted mean values of the parameters (Caswell 1989). The elasticity coefficient is defined as dLnλ/dLn(x;) (Caswell 1989), and elasticity for the demographic parameter, i, is equal to:

Elasticity coefficients give the effect on X of a unit increase or decrease in the demographic parameter. Therefore, we can compare the effects of different demographic parameters directly. Elasticity coefficients only measure the theoretical effect, imposed by the model structure, of a change in a demographic factor and do not evaluate to which degree a given demographic parameter influences the X in real life. To examine the effect of a given demographic parameter we must incorporate the natural variation in the parameter (van Tienderen 1995). Between year coefficient of variation (CV) in the demographic parameters is a scaled measure of the between year variation. By multiplying the elasticity coefficients and their corresponding CV's we get an actual elasticity coefficient, the AE-coefficient:

A high AE-coefficient means that the elasticity coefficient is also high, that the potential influence on λ is high and there is large variation in the demographic trait. Demographic traits with the highest AE-coefficients have the largest influence on the growth rate, λ, and are probably the most important factor in determining the changes in population size. A high elasticity coefficient combined with a low variance will necessarily have less impact on λ.

Sensitivity, elasticity and AE-coefficients were calculated for all demographic parameters. To compare the role of survival from egg laying to four weeks after hatching and apparent winter survival we performed the same analysis.

Results

Sensitivity

Sensitivities evaluated for the weighted mean values of the demographic parameters from Myrberget (1988, Table 1) showed that juvenile apparent winter survival, during the period from late July to the following spring, is the most important factor (Table 2) in determining changes in λ; pooled apparent winter survival is also more important in determining changes in λ than survival from egg laying to four weeks after hatching (PROD).

Elasticity, E_i

The elasticity coefficient evaluated for the weighted mean values gave the same elasticity for all reproductive parameters except for adult survival and adult summer survival (Table 3). This means that a 10% change in Clutch, Egg_s, Hatch, Chick_s and Juv_s has the same impact on X. Adult apparent winter survival, Ad_s, had a slightly larger impact than Clutch, Egg_s, Hatch, Chick_s and Juv_s. Adult summer survival, S_s, had twice the impact of the same change in any of the other parameters. Pooled apparent winter survival had an elasticity equalling 1.1. Thus a 10% reduction in winter survival caused, for instance by hunting leads to an 11 % reduction in the number of breeding hens relative to what should have been expected from the chick production and pooled winter survival. Changes in pooled apparent winter survival will have an impact 2.3 times greater than the impact of the same change in survival from egg laying to four weeks after hatching.

Table 2.

Ranked sensitivity index for all the demographic parameters mentioned in Table 1. Survival from egg laying to four weeks after hatching is the product of egg survival (Egg_s), hatching success (Hatch) and chick survival (Chick_s) and is analysed as one value. All notations as in Table 1.

Table 3.

Demographic factors ranked according to their actual elasticity coefficients (elasticity x CV). The AE-coefficient is a relative measure of the importance of the different demographic factors on λ. All notations as in Table 2.

Actual elasticity coefficient, AE

AE-coefficients given in Table 3 show that juvenile survival (Juv_s) is the most important parameter. Both Egg_s and Chick_s have almost identical influence on changes in λ. Summer adult survival illustrates the use of AE-coefficients well. The sensitivity and elasticity analyses show that adult apparent summer survival could have a large impact on the changes in λ, but since there is hardly any between year variance in adult summer survival this contributes little to changes in λ. Comparisons between the AE-coefficients for survival from egg laying to four weeks after hatching (0.18) and pooled apparent winter survival (0.2) show that the two factors have an almost identical impact on λ.

Sensitivity and elasticity analyses demand that there is no covariance between the different parameters in the model nor any density dependence in the parameters. To check this we correlated all demographic parameters. The only significant correlation (P ≤ 0.05) was between clutch size and egg survival (r = -0.45, P = 0.048) and juvenile apparent winter survival and the proportion of eggs that hatched (r = 0.5, P = 0.044). After a Bonferoni correction the P-value for acceptance was 0.002, so none of the correlations should be regarded as significant. As stated earlier, the only density-dependent factor found earlier was clutch size (Myrberget 1985, 1988).

Managing a willow grouse population.

Managing a game population with a 10-fold stochastic variation in chick production and a randomly varying adult mortality is fraught with problems (Lande et al. 1995). For willow grouse in Norway the harvest must either vary with intensity or, optimally, use a threshold harvest strategy (Lande et al. 1995). In the following section we develop a general management model to investigate how high a harvest a willow grouse population can sustain. We demonstrate a density and chick production dependent (PROD) variable management strategy and apply it to the Tranøy willow grouse population.

Management models for wildlife populations should only include input parameters available to the managers/hunters before the hunting season. Since chick production of willow grouse is dominated by the impact of predation and random factors such as weather conditions, the final decision as to how many willow grouse can be harvested, can only be made after chick production is known. We focus on two properties, i.e. the proportion of the population harvested and the consequences of this to the population. To estimate the harvest parameter, H_mort, we need four figures, i.e. the apparent winter survival of juveniles and adults, chick production and the number of breeding hens. Chick production (PROD) is expressed as: PROD = ((Clutch x 0.5) x Egg_s x Hatch x Chick_s).Traditionally, production is expressed as ‘Chicks per pair’ in autumn, which equals two times PROD.

To find the harvestable proportion of the population, hunting mortality, H_mort, is included in equation 2 above as natural survival multiplied with (l-H_mort). Total compensation of natural survival through hunting is equivalent to assuming perfect density-dependent survival (Ellison 1991). Partial or total compensation of survival to hunting mortality has been observed in some tetraonids, but density dependence has not been shown in willow grouse (Myrberget 1985, Ellison 1991), even though breeding densities during 1971–1980 were only half the pre-1971 densities. A given level of compensation (partial or total) can be obtained either by reducing the effect of hunting (lowering H_mort) or increasing winter survival. Including H_mort in equation 2 gives a growth rate, λ, expressed as:

If λ = 1 the following equation can be used to calculate H_mort:

The two apparent winter survival parameters are given in Table 1, and chick production can be estimated in late August when the number of chick and adults in flushed broods can be counted. On Tranøy almost every displaying cock will get a hen. Thus, the number of displaying cocks yields a good estimate of the number of breeding hens. Numbers can be estimated using line transect techniques (Buckland et al. 1993). At Tranøy the breeding population has been followed closely by searching for nests, and the number of breeding pairs is known through total counts.

Figure 2.

Theoretical harvest percentage of the autumn population demanding zero growth rate in number of breeding hens dependent on chick production. The line furthest to the right is where adult apparent winter survival is 0.51 and juvenile survival apparent winter survival is 0.26 (from Table 1). From right to left each line represents the harvest percentage assuming that survival is underestimated by 5, 10, 15 and 20%, respectively

Figure 3.

Number of willow grouse harvestable at Tranøy assuming mean adult apparent winter survival (0.51) and juvenile apparent winter survival (0.26), the yearly observed spring densities and year specific chick production (solid line). The dotted line represents the harvest if Myrberget (1988) underestimated apparent winter survival by 10% or willow grouse show a 100% compensation in natural survival rates for 10% increased mortality due to hunting. Negative values indicate that no harvest is possible and that the population declines regardless of hunting

We calculated the possible harvest percentage of willow grouse in the autumn (Fig. 2) as a function of different chick production levels, with λ =1. The different curves in Figure 2 represent scenarios where the survival estimates from Tranøy underestimated the true survival by 5, 10, 15 and 20%. Consequently, the four different curves also describe possible hunting yields at different levels of 100% compensation (see above).

When using the model with the Tranøy willow grouse population data, the yearly yield expressed as a function of the number of hens (see Fig. 1), chick production (PROD) and the pooled apparent winter survival estimates (see Table 1), is given in Figure 3. The striking result is that during 1960–1980 the population could not be hunted in four of the 21 years. This was also the case when assuming 100% compensation of 10% losses due to hunting or 10% underestimation of winter survival rates.

Discussion

In his review, Myrberget (1988) concluded that the most important factor regulating breeding population size in willow grouse is annual chick production (PROD). This conclusion was based on two observations: 1) that there was a positive highly significant correlation between the numbers of adults and juveniles in autumn and the number of breeders the following year, and 2) that there was a positive correlation between the proportion of juveniles in the autumn census and the increase in the number of breeding hens.

In contrast to Myrberget's conclusions, our analysis shows that variation in survival from egg laying to four weeks after hatching has slightly less impact on the variation in breeding population size growth rate than pooled winter survival (see Table 3). Survival from egg laying to four weeks after hatching multiplied by clutch size equals chick production. The between year variation in chick production is caused by variation in survival from egg laying to four weeks after hatching and not by variation in clutch size.

Our analysis ranks the different parameters according to their potential importance in determining changes in the number of breeding willow grouse hens. First-year willow grouse apparent winter survival is about 1.5 times more important than both egg and chick survival, which have similar impacts on breeding hen population growth rate. The fourth most important parameter is adult apparent winter survival. Juvenile apparent winter survival is the most important factor in determining changes in the breeding population, λ, as sensitivity (unsealed measure of the effect on λ), elasticity (scaled measure of the effect on λ) and AE-coefficient (relative importance measured by adjusting the elasticity coefficient with the actual variability in the parameter) all give the same results. Further research aimed at determining the most important factors affecting population size changes should therefore be devoted to finding the causes of the variation in first-year apparent winter survival.

Judging from the sensitivity analysis, adult apparent winter survival is quite important in determining λ. In the elasticity analysis it is the second most important parameter after adult summer survival. When accounting for between year variation in adult winter survival, the AE-coefficient, it drops to fourth place. This is due to little between year variation in adult winter survival. Egg survival is the fifth most important parameter judging from the sensitivity analysis, but after accounting for the large variation in egg survival between years it moves to second place. These two cases illustrate an important point: elasticity and sensitivity analyses can only indicate the possible impact of one parameter and we can only judge which factor can change growth rate in a natural population after accounting for the between year variance in the demographic parameters, AE-coefficient (van Tienderen 1995).