Impact of unintentional selective harvesting on the population dynamics of red grouse

Authors


E-mail: n.bunnefeld06@imperial.ac.uk

Summary

1. The effect of selective exploitation of certain age, stage or sex classes (e.g., trophy hunting) on population dynamics is relatively well studied in fisheries and sexually dimorphic mammals.

2. Harvesting of terrestrial species with no morphological differences visible between the different age and sex classes (monomorphic species) is usually assumed to be nonselective because monomorphicity makes intentionally selective harvesting pointless and impractical. But harvesting of the red grouse (Lagopus lagopus scoticus), a monomorphic species, was recently shown to be unintentionally selective. This study uses a sex- and age-specific model to explore the previously unresearched effects of unintentional harvesting selectivity.

3. We examine the effects of selectivity on red grouse dynamics by considering models with and without selectivity. Our models include territoriality and parasitism, two mechanisms known to be important for grouse dynamics.

4. We show that the unintentional selectivity of harvesting that occurs in red grouse decreases population yield compared with unselective harvesting at high harvest rates. Selectivity also dramatically increases extinction risk at high harvest rates.

5. Selective harvesting strengthens the 3- to 13-year red grouse population cycle, suggesting that the selectivity of harvesting is a previously unappreciated factor contributing to the cycle.

6. The additional extinction risk introduced by harvesting selectivity provides a quantitative justification for typically implemented 20–40% harvest rates, which are below the maximum sustainable yield that could be taken, given the observed population growth rates of red grouse.

7. This study shows the possible broad importance of investigating in future research whether unintentionally selective harvesting occurs on other species.

Introduction

There is an increasing need to understand how populations react to alteration of their natural dynamics by human exploitation. Harvesting is a major cause of mortality in many wildlife populations, in some cases surpassing natural predation or disease (Langvatn & Loison 1999; Ballard et al. 2000; Sandercock et al. 2011). Harvesting can drive species to extinction (Rosser & Mainka 2002), can affect population growth rates (Ginsberg & Milner-Gulland 1994; Cameron & Benton 2004; Hutchings 2005; Milner, Nilsen & Andreassen 2007) and can have long-term evolutionary consequences (Coltman et al. 2003; Garel et al. 2007; Kuparinen & Merila 2007; Proaktor, Coulson & Milner-Gulland 2007). Harvesting often selects for or against certain age, sex and stage classes that carry specific traits (e.g., size in fisheries or horns and other trophies in mammals; Hutchings 2005; Garel et al. 2007; Milner, Nilsen & Andreassen 2007). While the effect of selective harvesting on fish and sexually dimorphic mammals has received increased attention (Festa-Bianchet 2003; Coltman 2008; Darimont et al. 2009), harvesting selectivity in monomorphic species, where a hunter does not consciously select for a certain age class or sex class, is less well explored, in both empirical and theoretical studies. In this study, we explore the effect of unintentional age- and sex-selective harvesting on the population dynamics of a monomorphic species that exhibits cyclic dynamics, the red grouse (Lagopus lagopus scoticus), using an age- and sex-structured, stochastic population model. We make use of field data to parameterize both the species’ vital rates and the form of the density-dependent harvesting selectivity.

Selective harvesting has been demonstrated in field studies of similar species to red grouse but the direction of selection can be difficult to determine, because of contradictory results. Studies on willow grouse (also called ptarmigan, Lagopus lagopus) found that harvesting selects for either juveniles (Bergerud 1970; Sandercock et al. 2011) or adults (Hörnell-Willebrand et al. 2006) and among adults for females (Sandercock et al. 2011). For red grouse, we showed in earlier work that the selectivity changed with population density such that at low density, harvesting selects for young birds and females, while with increasing density, harvesting selects for old birds and within the old birds for males (Bunnefeld et al. 2009).

Studies have demonstrated for some vertebrates that selective harvesting can create cycles, for example, in an empirical study on moose (Solberg et al. 1999). In a field study on freshwater fish, high age-selective harvesting rates changed the composition of the population and prolonged the length of the cycles (Huusko & Hyvärinen 2005). In this study, harvesting rate is defined as the percentage of total individuals harvested. A modelling study on the willow grouse showed that compensatory cyclicity in harvesting effort can create population cycles (Jonzén et al. 2003). Fryxell et al. (2010) also showed this for field data and a complementary model on moose (Alces alces) and white-tailed deer (Odocoileus virginianus).

Population cycles have been studied for many taxa, and drivers of these cycles include environmental effects, parasites and predation (Kendall, Prendergast & Bjørnstad 1998; Turchin 2003; Reuman et al. 2006). Recent modelling has focused on the two main hypotheses of host–parasite dynamics and territoriality as drivers of red grouse population dynamics. A continuous-time model by Dobson & Hudson (1992) demonstrated that parasites are an important factor in red grouse cycles. Empirical evidence for a role of parasites was supported by field experiments showing that cycles were less pronounced (though not eliminated) in treated populations (Hudson, Dobson & Newborn 1998; Redpath et al. 2006a). A second hypothesis has been proposed whereby density-dependent aggressive behaviour drives red grouse cycles. Red grouse males defend territories in autumn, and only territorial males are able to breed. The effect of territoriality on grouse cycles has been supported by field experiments (Moss, Watson & Parr 1996; Mougeot et al. 2003a,b; Mougeot, Evans & Redpath 2005; Mougeot et al. 2005) and by theoretical modelling (Matthiopoulos et al. 2003; Matthiopoulos, Halley & Moss 2005). The territorial models are based on time-lagged density-dependent aggressiveness of old males towards young males and lead to exclusion of young males from the next year’s breeding population. Recent studies show empirically (Redpath et al. 2006b) and theoretically (New et al. 2009) that parasites and territoriality interact such that high parasite loads decrease aggressive territorial behaviour and high aggressiveness increases susceptibility to parasites.

With up to 50% of the red grouse population harvested every year at some sites (Hudson 1985), it is surprising that harvesting is not often included explicitly in grouse models nor often considered for a possible role in the maintenance of grouse cycling. To the best of our knowledge, only three theoretical studies on red grouse and one on willow grouse have included harvesting. Aanes et al. (2002) explored constant and proportional harvesting under uncertainty and environmental stochasticity but did not include harvesting selectivity. Potts, Tapper & Hudson (1984) included harvesting in a discrete-time model to investigate the effect of parasites on red grouse cycles but the effect of harvesting on cycles was not further explored. In a continuous-time model, Hudson & Dobson (2001) reduced the survival rate by a factor proportional to the harvesting rate to show that, theoretically, harvesting should dampen or eliminate oscillations in red grouse populations. However, empirical data show that high-amplitude cycles continue on moors with regular and intensive red grouse shooting (Cattadori, Haydon & Hudson 2005). Hudson & Dobson (2001) introduced male-selective harvesting but did not study the effect on population cycles. A recent model incorporating harvesting by Chapman, Cornell & Kunin (2009) is based on aggressiveness as a driver of cycles (Matthiopoulos et al. 2003; Matthiopoulos, Halley & Moss 2005) and showed that increasing harvesting to 150% of current rates shortens cycles by about 1 year, but with a further increase in harvest rates, cycles returned to the initial state. The changes in cycle length found by Chapman, Cornell & Kunin (2009) are small compared with variation in cycle lengths among field data sets from different locations experiencing different harvesting rates. None of these studies considered density-dependent selective harvesting. Harvesting and its newly discovered selective structure in red grouse (Bunnefeld et al. 2009) is a largely unconsidered possible driver of, or contributor to, grouse cycles and may help explain the range of cycle lengths (3–13 years; Haydon et al. 2002) and cycle intensities found in the field.

This study is the first that explores the possible effects of the unintentional selectiveness of harvesting of red grouse or any monomorphic species. We develop a new model and use it to analyse the possible effect of selectivity at a range of harvest rates on the cyclic behaviour of red grouse population dynamics, on total off-take of red grouse and on the sustainability of red grouse harvesting.

Materials and methods

We modelled red grouse population dynamics with an age- and sex-structured model in discrete time. The model set-up captures the grouse age structure observed in the field, where the population consists of age-0 grouse (recruited less than a year ago), age-1 grouse and age-2+ grouse (Bunnefeld 2008). The dynamics were modelled using three phases of development per year to reflect the seasonality of events (Fig. 1; Matthiopoulos et al. 2003). Parameter values are in Table 1, and parameter definitions and their incorporation in the model are discussed later.

Figure 1.

 Representation of the red grouse model. The subscripts i,j represent the age and sex classes, respectively, with three age classes (0-year, 1-year and 2-year and older) and two sex classes. The parameters f, h, m describe the fecundity, harvest rate and over-winter mortality, respectively, in a given year t. Aggressiveness A depends on the number of grouse in July inline image and affects the mortality of young grouse of the year. Parasites W depend on the population size in the preceding July inline image and affect fecundity of all ages and survival of old grouse (1 year and more).

Table 1.   Parameters used for the baseline model. The units for ‘mortality threshold parasites’ are worms per bird and for ‘harvesting threshold’, number of birds. All other parameters are dimensionless
ParameterSymbolValueSourceSensitivity analysis range
Harvest rateh0·4Hudson 1985
Mean number of harvesting events in a seasond1·78Bunnefeld et al. 2009
Slope (fecundity)s−0·41Hudson, Newborn & Dobson 1992±0·1
Intercept (fecundity)u2·32Hudson, Newborn & Dobson 1992±0·2
Slope (parasites)p13·63Hudson, Newborn & Dobson 1992±4
Intercept (parasites)c−317·05Hudson, Newborn & Dobson 1992±50
Mortality threshold, parasitesWmax5000Hudson, Newborn & Dobson 1992±2000
Parasite aggregation 1-years old grousek20·5Bunnefeld 2008  0·2–6
Parasite aggregation 2+-years old grousek32·0Bunnefeld 2008  0·2–6
Intercept for the aggression modele0−3New et al. 2009±1
Grouse density time t−1 on aggressione10·001New et al. 2009±0·001
Grouse density time t−2 on aggressione20·04New et al. 2009±0·0175
Strength of density dependenceφ0·1New et al. 2009
Slope young-old selectiona0·25Bunnefeld et al. 2009±0·2
Intercept young-old selectionb−0·83Bunnefeld et al. 2009±0·7
Slope female–male selectionq−0·37Bunnefeld et al. 2009±0·3
Intercept female–male selectiong0·99Bunnefeld et al. 2009±0·8
Harvesting thresholdγ30Hudson, Dobson & Newborn 1998  0–100

Parasites and grouse density

Moss et al. (1993) and Hudson, Newborn & Dobson (1992) agree that grouse densities have a significant influence on Trichostringylus tenuis parasite intensities in the following year. T. tenuis has a simple life cycle with no intermediate host and has free-living larval stages that depend on favourable weather conditions to develop (Shaw, Moss & Pike 1989). Worms can be picked up both during the summer and during a second phase in autumn. The equation we used for the number of worms per grouse in a given year is estimated from a regression derived from field data by Hudson, Newborn & Dobson (1992). In our model, the average worm burden of the grouse population in year t depends on the total grouse population in July of year t−1, denoted inline image., where the subscripts i,j represent the age and sex classes, respectively, with three age classes (0-year, 1-year and 2-year and older) and two sex classes (Fig. 1):

image(eqn 1)

where Wt is the average number of worms per grouse in year t depending on the population in the preceding July, inline image, and two constants p and c that model the additive effect of summer and autumn pickup of worms.

The March-to-July model transition: ageing

All grouse are born during the March-to-July transition, so ageing is implemented in the model during that transition. Age structure is advanced before reproduction is implemented in the model.

The March-to-July model transition: reproduction

Parasite burdens in early spring (March) have been shown to affect fecundity in a number of studies (e.g., Hudson, Newborn & Dobson 1992; Moss et al. 1993; Newborn & Foster 2002). Moss, Watson & Parr (1996) found no correlation between breeding success and hen age in red grouse, while Mougeot, Redpath & Piertney (2006) did not find an effect of male age on the number of young fledged. Thus, fecundity ft in the model, here interpreted as the average number of offspring per breeding female grouse (see below for details), does not depend on parent age but on the worm burden Wt in a given year t and two constants s and u (Hudson, Newborn & Dobson 1992):

image(eqn 2)

The number of breeding females depends on males establishing territories, and although a few males have two females and some have none, male numbers largely govern the number of breeding females when females outnumber males (Watson & Jenkins 1968; Moss, Watson & Parr 1996; Mougeot et al. 2003b). The number of chicks added to the population is assumed to have an equal sex ratio:

image(eqn 3)

The figures inline image on the right of this equation are taken after the stage advancement is implemented. The right side is rounded to the nearest integer.

The July-to-October model transition: harvesting

The main harvesting season for grouse ranges from mid-August until mid-October (Hudson & Newborn 1995). In late October, most grouse harvesting has stopped. Harvesting mortality in the model is included as a proportional mortality:

image(eqn 4)

where ht,i,j is the harvest rate depending on time step t, age i and sex j. These are derived from the model parameter h, the overall harvesting mortality (across both sexes and all life stages) as specified below.

The degree of unintentional selectivity in the harvest has been shown to be a function of total harvest mortality (h) in field studies (Bunnefeld et al. 2009):

image(eqn 5)
image(eqn 6)

Here, Fbag is the ratio of young-to-old harvested grouse during a single harvesting event, F(J) is the ratio of young-to-old living grouse in July, Gbag is the ratio of old-females-to-old-males among grouse harvested during a single harvesting event and G(J) is the ratio of old-females-to-old-males among living grouse in July. Here, ‘young’ refers to 0-age grouse, and ‘old’ refers to other grouse. In model runs, the quantity h*N(J)t determines the number of grouse harvested in year t, which is divided by the mean number of harvesting events in a season, denoted d (Table 1), because the eqns 5 and 6 from Bunnefeld et al. (2009) are calibrated for a single harvesting event. The number of harvesting events is the number of times the same area is harvested per year. The resulting number of harvested grouse is entered in eqns 5 and 6 to obtain Fbag and Gbag for a single harvesting event. Assuming for simplicity that the harvesting events occurring in a season are equivalent, the same ratios apply to the entire harvesting take of the year. The ratios are used in two linear equations to constrain the ht,i,j. Another linear equation constraining the ht,i,j follows because total harvesting mortality is the sum of class-specific mortalities. Three more equations assumed for simplicity and for lack of data to the contrary are ht,0,m = ht,0,f, ht,1,m = ht,2,mand ht,1,f = ht,2,f, which mean that there is no sex selectivity in the age zero class and no difference in age selectivity between 1- and 2-year-old grouse. The complete set of six independent linear equations describing the selectivity in six unknowns, shown in Appendix S1, determines a unique solution for the ht,i,j. The constants a, b, q and g are taken from field studies (Table 1; Bunnefeld et al. 2009). For very large harvesting mortalities, some ht,i,j calculated in this way were >1. In such cases, the corresponding stage was set to 0, and excess harvesting mortality was redistributed to other stages while still enforcing eqns 5 and 6 as nearly as possible. Populations were rounded to the nearest integer after imposing harvesting mortality.

Harvesting was introduced as threshold-proportional harvesting, where a certain proportion of the population is taken only when the population size is above a certain threshold. Threshold-proportional harvesting was suggested as best strategy for willow grouse hunting under uncertainty by Aanes et al. (2002). We used overall harvesting rates, h, between 0·1 and 0·9 that did not vary with time. For the threshold γ, we followed observations from field studies indicating that grouse harvesting tends to be stopped when July density inline image falls below 30 grouse km−2 (Hudson, Dobson & Newborn 1998).

The October-to-March model transition: parasite-related mortality

Parasites have been shown to affect over-winter survival of grouse, with the proportion of grouse dying dependent on parasite intensity (Hudson, Newborn & Dobson 1992). Parasite burdens in a grouse population show a negative binomial distribution (Hudson, Newborn & Dobson 1992; Bunnefeld 2008). In the model, parasite burdens for each grouse in year t were assumed to come independently from a negative binomial distribution with mean Wt and dispersion parameter k1 for 1-year-old grouse and k2 for 2+-year-old grouse. Young (i.e., 0-year-old) grouse were not affected. The parameters ki differ between age classes but not sex classes, as found in a recent field study (Bunnefeld 2008). Although there is evidence for interannual variation in ki (Hudson, Newborn & Dobson 1992), we assume a constant value over time for simplicity. Given a maximum survivable parasite load Wmax beyond which grouse are taken to die, the probability of each grouse dying in a given year was calculated from the negative binomial cumulative distribution function. The number of grouse of each class that actually died was simulated from a binomial distribution.

The October-to-March model transition: exclusion by aggressiveness

Young (0-year-old) male grouse numbers are limited by the number of territories. Territoriality is mediated by aggressiveness and population density in the current and previous year (Mougeot, Evans & Redpath 2005; Mougeot et al. 2003b; Matthiopoulos et al. 2003). Here, we adopt the function of aggressiveness from New et al. (2009) as:

image(eqn 7)

where e0, e1 and e2 are constants and j = 1 is for males. Here, inline image.

Young male grouse mortality is then modelled by the equation

image(eqn 8)

where φ is a constant and j = 1 is for males. Young female mortality mt,0,2 was taken to parallel young male mortality, following the results of (Watson & Jenkins 1968; Moss, Watson & Parr 1996; Mougeot et al. 2003b), so that mt,0,2 was set equal to mt,0,1.

Stochasticity

Rainfall in the preceding July (Moss et al. 1993) and the preceding minimum July temperature (Hudson, Newborn & Dobson 1992) explain a considerable amount of the variation in spring and autumn parasite intensities, respectively. Cattadori, Haydon & Hudson (2005) show that climate interacts with parasites in red grouse, and thereby synchronizes population fluctuations in areas subject to similar climatic conditions. Stochasticity was introduced with a standard deviation of two on the slope p. Values were drawn from a normal distribution and reflect the variability derived from Hudson, Newborn & Dobson (1992). Values were independent among years.

Uncertainty in harvesting rate was modelled by varying the overall harvest rate by a standard deviation, independently in each year and following a normal distribution (but constrained to lie between 0 and 1). The uncertainty reflects the unknown difference between the harvest rate set by the managers and the actual harvest rate that is realized by the hunters. The degree of uncertainty was fixed for all years by fixing the standard deviation at 0·05.

Output measures

Five hundred simulations of the model of length 250 years were run for each harvesting rate for rates between 0·1 and 0·9, separately for selective and nonselective harvesting. Several output measures were generated from these time series: mean total population in July, July young-to-old ratio, yield, fecundity, winter mortality and extinction probability. July young-to-old ratio is commonly computed in field studies (Newborn & Foster 2002; Hudson, Newborn & Dobson 1992; Redpath et al. 2006a); hence, it was calculated from model runs for comparison. The ratio was computed separately for each year, and the mean over simulations was used. Yield was defined as the mean number of grouse harvested. Fecundity was computed as the mean number of births per breeding pair in a given year and was considered 0 when no breeding pairs existed. Winter mortality rates were computed for all birds and separately for specific classes, for year transitions in which any birds remained in the focal classes. The per cent of simulations going extinct by year 75 was assessed; 75 was an arbitrary but reasonable choice as a time horizon on which managers might reasonably consider extinction risks. A simulated population was considered extinct if only one sex remained.

Spectral analysis and cycle length

Time series generated by the model were assessed with spectral analysis to estimate the nature of cycles. For all combinations of parameters examined, 200 simulations of 500 years length were computed, after a 100-year burn-in period to eliminate transients. The spectrum was then averaged over the simulations for each frequency. Averaged spectra were plotted for harvest rates ranging from 0·01 to 0·5 (Reuman et al. 2006) to characterize how harvesting affected spectra differently under selective versus nonselective harvesting models.

Sensitivity analysis

Sensitivity analysis involved varying all key parameters of the model simultaneously within the observed range from the literature (see Table 1 for key parameters and observed ranges) and testing whether the difference in predicted extinction rate between models incorporating and not incorporating selectivity of harvesting was stable to the parameter changes, at a harvesting rate of 70%. One thousand sets of parameter values were generated independently from uniform distributions, and 25 simulations were carried out using each model (i.e., with and without harvest selectivity) for each parameter set. After 75 years, extinction/nonextinction was assessed for each simulation. The extinction rate at 70% harvesting was chosen for the sensitivity analysis because extinction risk increases sharply with increasing harvest rates at that harvest level for the selective harvesting model using central parameters, and extinction risk is of practical importance. Logistic regression models were fitted to the simulation results to explain the probability of extinction and how it was affected by parameter values for each harvesting scenario (generalized linear model with binomial error distribution and logit link; McCarthy, Burgman & Ferson 1995). Independent variables (the parameters) were scaled to have a mean of zero, and a standard deviation of unity to be able to compare parameters on different scales (Saltelli, Tarantola & Campolongo 2000). Extinction risks for each parameter set, as assessed by logistic regression, were compared to find out whether risks for the selective harvesting model were consistently greater than those for the nonselective model, as they were for the central parameters used (Results), or whether the difference in extinction risk by harvesting strategy was sensitive to parameters.

Results

Baseline model

The baseline model incorporates a selective proportional harvest, where 40% of the population is removed annually (a typical value on moors). The model produces a mean population size of 122 grouse per km2 in July, also called autumn population in the literature, and 55 grouse per km2 in March, also called spring population. The mean July young/old ratio is 2·38, and the dominant cycle length is 12 years. Thus, the output of the model is within the ranges observed in field studies for these measures (Table 2). Appendix S2 shows a figure of an example model time series. Model output appeared to reasonably represent data on grouse dynamics when selective harvesting was incorporated. These results support the realism of the model.

Table 2.   Outputs of the baseline selective harvesting stochastic model as compared to field-observed values of the same measures to assess the realism of the model (see main text). Population density is in grouse per km2
Population measureStochastic modelField studiesSource for field data
Population density (July)122Range: 83–252Redpath et al. 2006a
Population density (March)51Range: 29–193Redpath et al. 2006a
July young/old2·38Mean: 2·3
Range: 0·4–3·4
Newborn & Foster 2002
Cycle length 12Mean: 8·3
Range 3–13
Haydon et al. 2002

Effect of selective harvesting on yield

Maximum yield is reached at an annual harvest rate of 0·35 for the selective harvesting model and 0·40 for the unselective model (Fig. 2a). The yield increases up to the maximum yield of 49 grouse shot/year for selective and 52 for nonselective harvesting, for each km2 of moor. Yield curves are characterized by a relatively steep increase in yield up to the maximum compared with a slower decrease at harvest rates above the maximum yield. The shape of the yield curve is mostly driven by the yield curve of young birds, which contribute the largest proportion of all age classes to the total yield (Fig. 2). The selectivity of harvesting varies with density but in general more old males are shot than would be expected based on the overall harvest mortality (Appendix S3).

Figure 2.

 The yield at different overall proportional harvest rates for the stochastic baseline model without selective harvesting (dashed) and with selective harvesting (solid). See Appendix S3 for the strength of the selection on different age–sex classes.

Effect of selective harvesting on fecundity

There is a slow increase in fecundity at low harvest rates (<0·3) and a steeper increase at medium harvest rates (0·3 to 0·6) under both selective and nonselective harvesting (Fig. 3a). The observed pattern is because of the relationship between parasite loads and fecundity. The decrease in population size with more intense harvesting decreases parasite burdens and therefore increases fecundity. The skewed sex ratio created by selective harvesting affects the number of fledglings in a monogamous bird, where the less abundant sex determines the number of breeding pairs. At higher harvest rates (>0·6), fecundity drops dramatically in the selective harvesting model but much less so in the nonselective model because selective harvesting allows the sex ratio to become much more skewed (Fig. 3a). Hence, the effect of a skewed sex ratio caused by selective harvesting is a reduced overall population size (Fig. 3b) and yield (Fig. 2a).

Figure 3.

 Fecundity (a), mean total July population (b), winter mortality rate for young males (c) and probability of extinction within 75 years (d) for harvest rates ranging from 0·1 to 0·9.

Effect of selective harvesting on over-winter mortality

The effect of harvesting on over-winter mortality rates varies with age class and differs for young grouse depending on whether harvesting is selective or nonselective. The mortality rate of young males is driven by aggressive exclusion by old males, which is in turn governed by the grouse density in July in the current year and in the year before. There is a decline in mortality in young males as harvest rate increases, with a steep decline between 0·3 and 0·5 (Fig. 3c). Mortality is lower for selective harvesting than for nonselective harvesting because mortality depends on population size (through aggressiveness), and this has already dropped because of missing breeding partners. Mortality of other age-stage classes is about the same regardless of the selectivity of harvesting (Appendix S4).

The effect of selective harvesting on extinction probability

At an overall harvest rate of 0·55, the population goes extinct under selective harvesting in a small proportion of years, whereas the first extinction events for the nonselective harvest occur at an overall harvest rate of 0·7 (Fig. 3d). At an overall harvest rate of 0·8, the probability of extinction has reached almost 100% under selective harvesting, while for unselective harvesting, even at high harvest rates of 0·9, there is only a small chance of extinction. The difference between selective and nonselective harvesting here is because of the uneven sex ratios that selective harvesting causes. Thus, at small population size, no breeding can occur under selective harvesting and the population goes extinct. Therefore, whether harvesting is selective or nonselective, even though this selectivity is unintentional, may have large practical implications for population management.

Cycling

Spectral estimates for the selective model show that cycles between 3 and 13 years occur under realistic total harvest rates of between 10% and 40% and are stronger than another component of variation that occurs at shorter cycle lengths (2 years). Beyond 40% harvesting, cycles increase to more than 13 years duration (Fig. 4a), and below 10% harvesting, there is about as much power in the short, 2-year cycles as in the 3- to 13-year cycles. The observation of 3- to 13-year cycles in the model results is consistent with field observations, where the dominant period of cycling is between 3 and 13 years for typical harvest levels. Very few or no data are available on red grouse populations experiencing harvesting rates less than 10%, so we cannot validate or refute the model prediction that short cycles will predominate under those conditions. Under visual examination of output from the selective harvesting model for realistic harvest rates, 3- to 13-year periodicity is dominant and 2-year periodicity is less noticeable (Appendix S2). In contrast, power spectra for the unselective model show substantially weaker peaks in the 3- to 13-year range, as compared to the selective harvesting model. For harvest rates between 10% and 40%, shorter cycle lengths are also unrealistically more dominant for this model than the 3- to 13-year cycle period (Fig. 4b). Visual examination of time series confirmed these results: 2-year oscillations appear visually to be stronger than longer-term oscillations (Appendix S2). The occurrence of 3- to 13-year cycles under selective but not under nonselective harvesting suggest that the unintentional selectivity of harvesting may play a role in the maintenance and characteristics of grouse cycling, because, when selectivity is removed from the model, cycling patterns become unrealistic.

Figure 4.

 Power spectra of grouse population dynamics (total populations in July). Dominant cycle periods (years) for a range of overall harvesting intensities for a) selective and b) nonselective harvesting are given by the locations of power spectrum peaks. Cycle intensity for these peaks is shown on the y-axis. The range of observed cycle periods from field studies is given by the two dashed lines. Peaks at the same overall harvest rate in the 3- to 13-year range are higher for selective harvesting than for nonselective harvesting and are realistically more dominant than the peaks at the 2-year period, whereas peaks at the 2-year period are, unrealistically, as dominant as the 3- to 13-year peaks for the unselective harvesting model.

Sensitivity analysis

The sensitivity analysis shows that the extinction rate at a harvest of 0·7 (compare Fig. 3d) is consistently higher for the selective harvesting than for the nonselective harvesting model; this model prediction is relatively insensitive to uncertainty in the model parameters. Of 1000 random parameter simulations within the range specified in Table 1, selective harvesting has a higher extinction risk than nonselective harvesting in 82% of cases.

Discussion

This study is the first that explores the effects of the unintentionally selective harvesting of a monomorphic species in a sex- and age-specific population model. New et al. (2009) found that a model for red grouse incorporating both parasites and territoriality fitted the data better than either mechanism alone. Our objective was to extend the model to the three main processes affecting grouse population dynamics, harvesting, territoriality and parasitism, and carry out a detailed age–sex-specific investigation of the effect of selective and nonselective harvesting on population dynamics. The main findings of this study are that selective harvesting reduces the annual yield in red grouse compared with nonselective harvesting, increases the chances of local extinction at harvest rates above 0·6 and strengthens the observed cycle lengths of 3–13 years observed in field studies. Therefore, this study extends the single-mechanism approaches used in many theoretical studies and contributes to understanding of the effects of unintentional harvesting selectivity on population dynamics.

Selective harvesting reduces the annual yield by around 6% in red grouse, compared with the levels expected from unselective harvesting. In the selective harvesting case, the highest physical yield is reached at lower harvest rates (35%) than in nonselective harvesting (40%). While other studies have looked at the effect of static sex-biased harvesting on populations of geese (Hauser et al. 2007) and mammals (Milner, Nilsen & Andreassen 2007), we take into account the recent finding that in red grouse the direction of age- and sex-specific harvest selectivity changes with population density, although the direction of the selectivity is male-biased over a large range of harvesting intensities (Appendix S3) in line with empirical data (Bunnefeld et al. 2009). The previous study by Hudson & Dobson (2001) included selective harvesting for old males, but not density-dependent shifts in selectivity. They found that selectivity reduced the overall parasite burden of the population and therefore allowed for a higher population and a higher yield, contrary to the results we generate with a more empirically grounded selectivity function.

The risk of overexploitation and local extinction increases rapidly for harvesting rates higher than 50% under the selective harvesting model, but not under the nonselective model. Most grouse managers have adopted the precautionary strategy of rarely harvesting more than 40% of the population in a given year (Hudson 1985) and a threshold of 30 grouse per km2, a harvesting level and threshold that our model suggests provide high yield at minimal risk of local extinction. Sandercock et al. (2011) showed that harvest rates above 15–20% were most likely to be additive to natural mortality. In red grouse, high harvesting rates up to 40% are likely to be sustainable only on moor managed for grouse using predator control and habitat management. These results are in line with more general modelling showing that for fluctuating populations under large uncertainty, proportional threshold harvesting is the optimal strategy for high yield and low extinction risk (Lande, Saether & Engen 1997).

Cycle lengths for our selective harvesting model ranged from about 5 to about 13 years between 10% and 40% harvesting. This is within the range of the most common cycle lengths observed in the field (3–13 years), with the strongest cycles observed in the field showing a length of 8·3 years (Haydon et al. 2002). The same study reported that 25% of the populations they examined had cycles longer than 15 years. This may correspond with the findings of our study that harvest rates above 40% can produce cycle lengths longer than 15 years. We also show that longer cycles generally occurred at higher harvest rates. Jonzén et al. (2003) also demonstrated that increasing harvest rates from zero to 0·35 lead to longer cycles in willow grouse. Under the red grouse model of Chapman, Cornell & Kunin (2009), increasing harvest effort led first to shorter cycles and then to longer cycles, but changes were small, ranging between 9 and 10·5 years, compared with the wide variation in cycle rates observed in field studies and produced by our model. Therefore, our model suggests that selective harvesting might help explain the range of cycle periods observed in the field (Cattadori, Haydon & Hudson 2005).

Theoretical studies on the general interplay between harvesting and population fluctuations show that harvesting accounts for an equivalent amount of variance in population density and yield as environmental effects and that the interplay between the two factors is dependent on the population growth rate (Jonzén, Ripa & Lundberg 2002). It is known that cycle length in red grouse increases with latitude, which negatively affects growth rates (Haydon et al. 2002; Shaw et al. 2004). Other factors, such as temperature, rainfall, distance from the Atlantic coast and distance to the next population, explain synchrony in red grouse population dynamics (Kerlin et al. 2007). Similarly to the predictions by Jonzén et al. (2003), our model predicts that harvest rates and cycle period lengths should be related and that harvest rates might be another contributing factor in addition to those previously implicated in changes in cycle lengths. Long time series including both harvest rates and population counts or their proxies (for determining cycle lengths) have so far not been available for analysis, so this model prediction cannot yet be tested. Experimental changes in harvest intensities might lead to new insights on the effects of harvesting on population dynamics, if changes are persistently implemented for at least 10 years.

Red grouse are highly sedentary with an average dispersal distance of less than 900 metres and <350 metres for females and males, respectively (Warren & Baines 2007). Thus, in this modelling study, we assume a closed population with no immigration or emigration. Metapopulation dynamics and movement between patches are important factors for population dynamics generally and, together with sex- and age-selective harvesting and dispersal, have been shown to be crucial for management of highly structured populations such as red deer (Clutton-Brock et al. 2002). However, as a result of the sedentary nature of red grouse, it was not necessary to account for dispersal in our model.

Predation plays a major role for many fluctuating populations (Turchin 2003) but red grouse populations managed for shooting show very low predation pressure because of predator removal by game keepers (Hudson 1992). Thus, we have assumed that predation is negligible in our model. However, this assumption would not be valid for other harvested populations, for example willow grouse, where predation is a considerable source of mortality (Steen & Haugvold 2009). Sandercock et al. (2011) showed for willow grouse that harvest mortality under 15% was partially compensated, but at rates above 15–20%, harvest mortality was additive to natural mortality. Some studies have combined predation and movement by addressing harvesting in resource–consumer metapopulations (Strevens & Bonsall 2011). These approaches could be coupled with our approach on grouse to investigate the role of selective harvesting in spatially explicit environments under predation pressure, for application to willow grouse or another species for which predation is significant.

Studies have clearly shown that territoriality to be a driving force in red grouse population dynamics, but the functional form of aggressiveness to defend territories at various grouse densities has not been fully explored (Mougeot et al. 2005; New et al. 2009). Aggressive interactions have been found to affect red grouse population fluctuations, and experimental removal of old males has been shown to prevent population crashes (Moss, Watson & Parr 1996). Selective harvesting focuses mostly on old males, as assumed in the model of this study and suggested by field studies at higher densities (Bunnefeld et al. 2009). Selective harvesting might allow more young males to establish territories, with the latter carrying fewer parasites. The interaction between aggressiveness and parasites, and their relative roles in driving red grouse population dynamics, has recently been shown (Redpath et al. 2006a,b), but the effect of removing old males on both the aggressiveness and parasite burden of the population remains to be tested in field studies. The role of harvesting in these processes is still largely overlooked in empirical and theoretical studies, despite its potential importance in maintaining cyclicity, as shown in this study.

Management relevance

Small game hunting, which is widespread in Europe, North America and Australia, encompasses many species of mammals and birds and contributes significantly to the economy of local communities (Sharp & Wollscheid 2009). For example, red grouse shooting provides year-round rural employment and additional customers for hotels and restaurants in rural areas of the northern United Kingdom (Sotherton, Tapper & Smith 2009). This study shows that the selectivity of harvesting in red grouse can affect yield, local extinction risk and cycle length, three measures of paramount interest to managers of harvested species. The aims of minimizing the probability of local extinction and increasing yield under selective harvesting require lower harvesting rates compared with unselective harvesting. Monitoring the number of potential breeding pairs after selective harvesting might increase the predictability of postbreeding population estimates and thus promote more sustainable and consistent management of exploited resources. Given the economic relevance of small game hunting, the results of this study suggest useful new directions for management, trading off yield and local extinction risk.

Acknowledgements

The field work part of this study was funded by The Game and Wildlife Conservation Trust. We are grateful to Nicholas Aebischer, Emily Nicholson, Tom Ezard and Lynsey McInnes for their advice and comments. NB gratefully acknowledges the support of a John Stanley Scholarship and EJMG of a Royal Society Wolfson Research Merit award.

Ancillary