Journal of Applied Ecology 2011, 48, 1187–1194

doi: 10.1111/j.1365-2664.2011.02013.x

Hen harrier management: insights from demographic models ﬁtted to population data Leslie F. New1,2*, Stephen T. Buckland1, Stephen Redpath3 and Jason Matthiopoulos1,2 1

Centre for Research into Ecological and Environmental Modelling, The Observatory, University of St Andrews, St Andrews, Fife, Scotland, KY16 9LZ, UK; 2Scottish Oceans Institute, Department of Biology, University of St Andrews, St Andrews, Fife, KY16 8LB, Scotland, UK; and 3Aberdeen Centre for Environmental Sustainability, University of Aberdeen and the Macaulay Land Use Research Institute, Aberdeen AB24 2TZ, Scotland, UK

Summary 1. The impact of hen harriers Circus cyaneus on red grouse Lagopus lagopus scoticus populations has received much attention. However, little has been done to model the population dynamics of the hen harrier alone. Such a model is needed to help inform the diﬀering aims of conserving harriers and managing grouse moors, which serves as a reﬂection of human–wildlife conﬂicts around the globe. 2. On Langholm estate in Scotland, intensive studies have resulted in harrier numbers being known without error. We ﬁt a Bayesian population model to these data, using a super-population model to permit inference in the presence of demographic and environmental stochasticity and in the absence of observation error. 3. Hen harriers have a straightforward life history. After ﬂedging, juveniles show little natal site ﬁdelity, often dispersing long distances into breeding areas rich in their preferred prey, the ﬁeld vole Microtus agrestis and meadow pipit Anthus pratensis. Therefore, any increase in a local population is largely because of recruitment into the area as opposed to ﬂedging success. Once birds have settled in an area, harriers are generally site faithful, with year-to-year survival depending, in part, on the density of meadow pipits. 4. Our model suggests that temporal patterns in harrier numbers on managed grouse moors, in the absence of illegal persecution, are inﬂuenced by vole numbers, whereas meadow pipit density appears to have a limited eﬀect. 5. Synthesis and applications. Our modelling approach is a useful way to infer population processes, and the eﬀects of the environment on these processes, for populations censused without error. When used to predict future harrier numbers under alternate management scenarios, our model indicates that harrier numbers on Langholm estate, Scotland, could be reduced without any direct human intervention if the estate can be managed in a way that reduces vole populations. In contrast, there appears little to gain from managing meadow pipit densities. If these conclusions apply to other harrier populations, then management to reduce vole numbers, while maintaining grouse densities, may help alleviate the conﬂict between conservationists and managers of grouse moors. Key-words: Bayesian model, ﬁeld voles, meadow pipits, population dynamics, red grouse, super-population model, wildlife management

Introduction Much research, both internationally and locally, has focused on the conﬂict between the hen harrier Circus cyaneus (Linnaeus) and red grouse Lagopus lagopus scoticus (Latham) in Scotland (e.g. Redpath & Thirgood 1997; Redpath et al. 2004; Thirgood & Redpath 2008). The hen harrier is a red-listed *Correspondence author. E-mail: [email protected]

species that was almost completely extirpated from the UK in the nineteenth century because it preyed on red grouse, which lowered densities to the detriment of shooting bags (Marchant et al. 1990; Hull 2001). Eventually, the raptor was protected under UK law, but illegal persecution continues on shooting estates (Etheridge, Summers & Green 1997; Sotherton, Tapper & Smith 2009). Both species have been studied extensively because of ecological, conservation and economic concerns related to the two birds (e.g. Watson 1977; Hudson 1992;

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society

1188 L. F. New et al. Matthiopoulos et al. 2007; New et al. 2009). Hen harriers have been censused without error in long-term studies on some estates where there is no illegal killing (Redpath & Thirgood 1997). These data have been used to develop models of the impact of hen harriers on red grouse populations (Redpath 1991; Redpath & Thirgood 1997; Amar et al. 2004), and the hen harriers’ functional and numerical response (Redpath & Thirgood 1999; Redpath, Thirgood & Clark 2002; Asseburg et al. 2006), but have not previously been used to study the population dynamics of this bird of prey. Understanding hen harrier population dynamics is an important step in managing moorland habitat for the coexistence of harriers and grouse. While stakeholders agree that action needs to be taken to mitigate the eﬀects of harrier predation on red grouse, the exact nature of such actions remains undecided (Thirgood & Redpath 2008; Redpath & Thirgood 2009; Thompson et al. 2009). Conﬂict arises from several mitigation schemes that reduce revenue of moorland estates (Sotherton, Tapper & Smith 2009), from legal disputes, and uncertainty over the harriers’ responses to management actions, incurring risk for gamekeepers should mitigation plans fail. Hence better understanding of hen harrier population dynamics, beyond their impact on red grouse populations, will assist management of the harriers and their habitat. Hen harriers’ habitat choice is based on the presence of their main prey species, the meadow pipit Anthus pratensis (Linnaeus) and ﬁeld vole Microtus agrestis (Linnaeus) (Picozzi 1978); grouse are an opportunistic prey item and do not aﬀect habitat choice. An indirect approach to management through habitat manipulation or protection of other species (Thompson et al. 2009) requires that the bird’s main prey items are included in models of harrier dynamics. Diﬀerent management schemes’ viability cannot be assessed without an understanding of harriers’ reactions to alterations in habitat. In this paper, we develop a model for hen harrier population dynamics that could be used to help evaluate the eﬀectiveness of management plans for the conservation of the raptor and mitigation of its impact on the red grouse. In this study, the number of hen harriers breeding each year was known without error. This can be the case in small, wellstudied populations (e.g. killer whales Orcinus orca (Linnaeus) in the Paciﬁc north-west; Ford & Ellis 2006), or endangered species where individuals in the wild were originally captive bred (e.g. California condor Gymnogyps californianus (Shaw); Southwest Condor Review Team 2007). We allow demographic stochasticity by modelling the observed numbers of animals as a stochastic realisation from a theoretical superpopulation (Hartley & Sielken 1975), where stochasticity is modelled by specifying distributions for the numbers of survivors and recruits in each year. The survival and recruitment rates may also be modelled as functions of environmental covariates. This approach conceptualises the observed harrier population as one stochastic realisation of the population given the variability in survival and recruitment, allowing us to make inferences about other similar harrier populations. We begin by specifying the hen harrier’s population dynamics model. This model is ﬁtted in a Bayesian framework to data

from Langholm estate, a managed grouse estate in southern Scotland. We explain the ﬁtting process, which requires priors be placed on the model’s parameters. The adequacy of the model is checked by comparing the expected value of the realisations with the observed data. Finally, the models are used to investigate the impact on harrier numbers if management actions were taken to control the density of voles and pipits.

Materials and methods DATA COLLECTION

Data were collected on Langholm estate in the period 1992–1999. Prior to 1992, illegal persecution was taking place, so the data represent an establishing harrier population. Langholm is divided into six grouse beats (moorland areas managed by a speciﬁc keeper for grouse shooting). Meadow pipit abundance was estimated using line transects one km in length. A total of 18 one km2 squares were chosen at random, three in each of the six beats at Langholm. Each square contained two-line transects. In June, the same observer conducted all the counts, recording the number of individuals detected within each of six distance intervals (0–10, 10–20, 20–30, 30–50, 50–100 and 100– 200 m). An index of small mammal abundance was estimated through snap trapping in the spring. Two sites were chosen within each of the six beats to be representative of the beats’ habitat. At each site, 50 traps were set over two nights to give a total of 100 ‘trap nights’ per site (further details on data collection, methods and study site in Thirgood, Leckie & Redpath 1995; Redpath, Thirgood & Redpath 1995; Redpath & Thirgood 1997, 1999; Thirgood et al. 2000). Meadow pipit density estimates were obtained using Distance 5.0 (Thomas et al. 2010), with a truncation distance of 100 m from the line, as few pipits were recorded beyond 100 m. The recorded number of voles was converted to density using a scaling factor of 520 previously estimated to relate voles caught per 100 trap nights to density per km2 (Asseburg 2005). The number of nesting harriers was determined by observing the moor for displaying harriers in the early spring. Observation of the birds determined whether a male was bigamous, and if so, the relative status (primary or secondary) of the female was recorded. All nests were monitored closely through the breeding season, and regular checks ensured that no late breeders or other females were missed. There was no illegal persecution of hen harriers on the study area, so the eﬀect of illegal killing on hen harrier survival was not a concern in this case (Redpath & Thirgood 1997).

MODEL DEVELOPMENT

We are ultimately interested in the relationship between hen harriers and red grouse in the UK. Therefore, we develop a general model for hen harrier population dynamics that could be ﬁtted to data from locations other than Langholm estate. The exception is Orkney, where diﬀerent processes drive the dynamics (Picozzi 1984; Rothery 1985). As harrier numbers primarily impact the red grouse through predation on grouse chicks, which hatch in the spring, data were collected to coincide with this time period. Therefore, the harrier model follows the timing of New et al. (2009)’s grouse population dynamics model and moves from spring in year t)1 to spring in year t. In addition, consumption rates are calculated per pair (Asseburg 2005), so we model the number of nests not the number of birds. High rates of polygamy in harriers make it diﬃcult to use the abundance of one sex as an accurate indicator for the abundance of the other (Hudson

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

A model for hen harrier population dynamics 1189 1992). A further complication is that the sex ratio can vary yearly (Picozzi 1984). Polygamy means males are not a limiting resource, so females are a better indicator of nest numbers. Females with bigamous partners compensate for the lower provisioning rate of the male (Redpath & Thirgood 1997). However, we may slightly overestimate predation by modelling nests instead of individuals. In year t, realisations of nest numbers from the super-population are determined through female recruitment and survival. We do not account for fecundity as it does not aﬀect harrier density in an area. This results from high rates of juvenile dispersal, with almost no natal site ﬁdelity. However, after dispersal, harriers are site faithful (Watson 1977; Picozzi 1978). Therefore, net movement into an area (recruitment) replaces fecundity in terms of increasing harrier numbers in an area. Given this information, the basic population model is as follows, lt ¼ HObs;t1 /t1 þ gt ;

eqn 1

where HObs,t)1 is the observed number of female hen harriers on the grouse moor at time t)1, lt is the number of harriers present at time t, /t)1 is the survival rate between time t)1 and t, and gt is recruitment at time t. Survival and recruitment are determined by other factors. Survival depends on abundance of the diﬀerent prey items. Only the consumption of pipits plays a role in hen harrier survival; the passerine aﬀects the between-site abundance of hen harriers, while year-to-year changes are related to voles (Redpath & Thirgood 1997). In Scotland, up to 80% of harriers’ diet can be passerines (Picozzi 1984; Redpath et al. 2001), implying a heavy dependence on pipits, not on voles. Males leave the moors as pipit numbers begin to fall, and most females do not stay on the moors over winter (Watson 1977). Yet, during the winter months, vole numbers often increase, while the decline frequently takes place during the summer (Lambin, Petty & MacKinnon 2000). If vole consumption was driving harrier survival, the raptor would remain on the moor when vole numbers are highest. Therefore, the fact that they leave the moors in the winter implies that vole densities are not driving survival. Furthermore, if both survival and movement were dependent on voles, we would expect harrier populations to cycle in stronger synchrony with the rodent, as years of low survival and recruitment would coincide. Yet the raptor can be found in the absence of voles and grouse, and harrier population dynamics do not display the extreme peaks and troughs of the vole populations, remaining at high densities even after vole populations crash (Redpath et al. 2002). Survival is therefore modelled as, /t ¼

expða0 þ a1 Pt Þ ; 1 þ expða0 þ a1 Pt Þ

eqn 2

where a0 and a1 are parameters to be estimated, Pt is pipit density per km2 in year t, and a0 encompasses all other sources of mortality. We assume that, with the exception of the eﬀect of pipits, per capita mortality is constant across years, so a0 is time-invariant. A management shift would change a0. Recruitment into an area depends on the density of the harriers’ primary prey species, ﬁeld voles and meadow pipits (Redpath & Thirgood 1999; Redpath et al. 2002; Redpath, Thirgood & Clark 2002), modelled as, gt ¼ expðb0 Vt þ b1 Pt b2 HObs;t1 /t1 Þ;

eqn 3

where b0, b1 and b2 are parameters to be estimated and Vt is vole density per km2 in year t. Recruitment is constrained to be positive and is limited by the density of surviving harrier nests, HObs,t)1/t)1. Without prey, this function is similar to the Ricker function for density dependence. In the absence of voles, pipits or

harriers the equation allows for the recruitment of a single young harrier. If vole and pipit densities are not available, eqn 3 could be modiﬁed to account for harriers’ habitat preferences, which is often correlated with prey density (Arroyo et al. 2009). To account for uncertainty in vole and pipit densities, the covariates were modelled as, Pt Nðpipitt ; r2p;t Þ Vt LNðlogðvolet Þ; r2v Þ;

eqn 4

where the yearly variance for pipits (rp,t2) was found as part of their density estimation and the variance for voles (rv2) had been previously estimated (New 2009). In constructing the model, realisations from the super-population, Ht, will be found for time t, using the observed number of harrier nests from time t)1, HObs,t)1. For a given realisation, we assume that the observed numbers will come from a distribution centred on the expected numbers of that realisation. Although the data are discrete, they are underdispersed, so the Poisson distribution is inappropriate. The underdispersion results from harrier numbers in time t being dependent on time t)1. In addition, as the observations are known without error, the realisations will be more tightly clustered around the mean. We account for the underdispersion using the double-Poisson distribution (Efron 1986; Ridout & Beabeas 2004). We model the mean (lt) and variance (rt2) of the harrier population size under our super-population model as follows: the expected number of birds surviving from the previous year is HObs,t)1/t)1, while the expected number of recruits is gt. Assuming a binomial model for survival and a Poisson model for recruits, we have, lt ¼ HObs;t1 /t1 þ gt r2t ¼ HObs;t1 /t1 ð1 /t1 Þ þ gt ;

eqn 5

giving the double-Poisson, HObs;t Poðlt ; ht Þ:

eqn 6

where ht = lt ⁄ rt2. However, the limitations in software used to ﬁt the model (WinBUGS) meant the double-Poisson could not be applied directly. Instead, we used a categorical distribution to approximate the double-Poisson (McCarthy 2007), resulting in, HObs;t catðpx Þ

px ¼

ht 1=2 expðht ½lt xÞðlt =xÞht x expðxÞxx WE x!

eqn 7

where WE is a normalising factor approximately equal to one, and x = 1, 2, ...,2HObs,t. See Appendix S1 for the relevant code.

MODEL FITTING

Our model was ﬁtted within a Bayesian framework, permitting the incorporation of previous biological knowledge through prior probability distributions. Priors are updated with data via the model, resulting in distributions representing posterior beliefs. Posteriors are used to make all inferences about the states and parameters, providing a direct measure of uncertainty around the estimates (Ellison 2004; McCarthy 2007). The equations presented earlier were ﬁtted to the data in WinBUGS 1.4.2, which uses Markov chain Monte Carlo (MCMC) methods (Spiegelhalter et al. 2007). MCMC approximates the posterior distribution by drawing samples via Monte Carlo integration using a Markov chain. If run for a suﬃcient time period, the chain eventually converges on a stationary distribution that is the distribution of

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

1190 L. F. New et al. interest and is no longer dependent on the starting values or the number of iterations. The expected values of the states and parameters of interest are found using the sample averages from the stationary distribution. Determining a chain’s convergence on the distribution of interest is carried out visually or with convergence diagnostics (Gilks, Richardson & Spielgelhalter 1996). The model parameters, a0, a1, b0, b1 and b2, are used to deﬁne harrier survival, /t (a parameters) and recruitment, gt (b parameters). Priors for /t and gt are not explicitly deﬁned. Instead, the priors for the a and b parameters determine implicit priors for /t and gt through eqns 1 and 2. Therefore, the priors for the a and b parameters should combine to form biologically realistic implicit priors on /t and gt, respectively. However, the priors need to remain sensible with regard to the individual parameters, as well as when part of the larger group. Priors were chosen for the parameter values to match their known behaviour. For survival (eqn 1), a0 can be either positive or negative, allowing survival estimates to range from zero to one. A prior of N (0,1) was chosen for a0 to allow for a uni-modal implicit prior on /t. The second parameter, a1, is constrained to be positive because pipits increase harrier survival (Redpath & Thirgood 1997). Given the estimated pipit densities (216Æ5–575Æ2 km)2), a prior was chosen to respect the scale of the data. This was achieved by back-transforming from a normal distribution with a small mean and a large variance, giving a log-normal prior with la1 = )5Æ29 and ra12 = 0Æ833. The a priors, when combined with the diﬀerent Pt values, result in a leftskewed implicit prior for survival, spiking at one. As harriers are known to have a high survival rate (Etheridge, Summers & Green 1997), the implicit prior is not unreasonable. Additionally, choosing a prior to give a seemingly more reasonable prior on /t)1 decreases the prior’s variance, restricting a1 and resulting in the failure of the posterior to move away from the prior. For recruitment (eqn 2), parameters b0 and b1 were constrained to be positive, as the presence of voles and pipits is known to cause harrier recruitment into an area. Priors were chosen based on previous estimates from a similar equation for the hen harriers’ aggregative response (Matthiopoulos et al. 2007), which indicated that values would be low given the possible high densities of both prey species. Therefore, a gamma prior with shape one and rate 50 was chosen for both parameters. The last parameter, b2, measures the eﬀect of density dependence on recruitment. The prior was constrained to be positive, as the negative eﬀect of density dependence is deﬁned in the model. Given harriers’ low territoriality (Redpath & Thirgood 1997), it seemed unlikely that potential recruitment would be reduced by more than the number of birds already present. However, it is possible, especially in limited habitats, that each bird present would stop more than one new bird from being recruited. Therefore, b2 was given a gamma prior with a mean and standard deviation of 0Æ2 (i.e. an exponential distribution with rate ﬁve); while values less than one are probable, the long tail allows higher values. The combination of the three priors, taking possible vole, pipit and harrier densities into account, creates a right-skewed implicit prior on recruitment. The implicit prior essentially allows for nearly inﬁnite recruitment into any given area. While not biologically realistic, allowing for the possibility of extremely high numbers of immigrants avoids limiting gt by being too informative (Fig. 1).

Fig. 1. The implicit prior for harrier survival (/t) and recruitment (gt). Pipit density (Pt) is equal to 216 pipits km)2, vole density (Vt) is equal to 900 voles km)2, and the number of harrier pairs to remain in the area from time t)1 (Ht)1/t)1) is equal to 10.

The posterior means for the survival parameters, a0 and a1, were 0Æ211 [95% CI: ()0Æ7046, 1Æ013)] and 2Æ045*10)3 [95% CI: (4Æ87*10)4, 4Æ85*10)3)], respectively. The yearly variation in pipit density resulted in survival estimates ranging from 65% to 78% over the 8 years of the study (Fig. 2). This range of estimates includes the previous point estimate of 78% and falls within the 95% conﬁdence interval (57%, 90%) (Etheridge, Summers & Green 1997). However, the limits of the 95% credible intervals on survival can fall outside the previously estimated range (Fig. 2).

Results All parameters and states showed good mixing and convergence. The prior distributions were updated with the data, giving posteriors no longer unduly inﬂuenced by prior choice.

Fig. 2. A plot showing estimated survival (dashed line with ﬁlled circle) and recruitment (open circle), and their credible intervals, for the 8 years of the study.

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

A model for hen harrier population dynamics 1191 Table 1. Estimates of prey densities (km)2) with the corresponding 95% credible intervals

Voles (km)2)

Pipits (km)2)

Year

Data

Estimate

Data

Estimate

1993 1994 1995 1996 1997 1998 1999

1050Æ4 1060Æ8 280Æ0 535Æ6 1648Æ4 598Æ0 104Æ0

888Æ4 1045 282Æ5 652Æ2 1696 620Æ1 106Æ7

382Æ2 575Æ5 570Æ1 358Æ8 239Æ4 219Æ1 216Æ5

382Æ4 572Æ2 564Æ6 386Æ4 240Æ3 219Æ5 216Æ6

The parameter values for recruitment also fell within a range of realistic values. The parameters b0 and b1 have posterior means of 1Æ36*10)3 [95% CI: (5Æ59*10)4, 2Æ34*10)3)] and 9Æ54*10)4 [95% CI: (4Æ052*10)5, 2Æ48*10)3)], respectively, compared with previous estimates of 1Æ24*10)4 for the aggregative response to voles and 2Æ63*10)3 for pipits (Matthiopoulos et al. 2007). Given these values, an increase of 100 pipits km)2 would raise recruitment by approximately 9%, while the same increase in vole density raises gt by about 14%. The last parameter, b2 was estimated at 0Æ0431 [95% CI: (1Æ29*10)4, 0Æ151)], supporting the evidence of low territoriality among hen harriers. These parameters allow for the recruitment of between one and eight harriers a year, depending on prey densities (Fig. 2). All estimated prey densities were close to the observed values for voles and pipits (Table 1). Generally, the estimates of the expected number of successful hen harrier nests are in good agreement with the true values (Fig. 3), indicating that the super-population model is capable of producing realisations similar to the observed numbers. The greatest dissimilarity is in 1996, when the expected number is estimated as almost nine nests, while 14 nests were observed. This seems to be a result of the combination of low observed

(581Æ5, 1298) (687Æ9, 1025) (175Æ7, 430Æ5) (391Æ1, 1016) (1126, 2524) (387Æ7, 942Æ1) (65Æ60, 164Æ5)

(345Æ3, (508Æ0, (502Æ9, (343Æ5, (304Æ5, (178Æ7, (179Æ5,

419Æ2) 636Æ8) 627Æ0) 429Æ9) 276Æ7) 260Æ4) 253Æ9)

nest numbers in the previous year (eight) and low prey densities limiting the recruitment of harriers under our model. The estimates for recruitment (gt) varied signiﬁcantly from year to year, depending on prey densities (Fig. 2). Vole densities appeared to have the greatest eﬀect on recruitment, as evidenced by the normalised and exponentiated values of 0Æ244 and 0Æ0757 for b0 and b1, respectively. The greatest recruitment seems to occur at peak vole densities, even when the corresponding pipit density was low. The harriers that survive, remaining in the area from the previous year (HObs,t)1/t)1), have very little impact on recruitment (b2: 0Æ0431). We predicted harrier numbers one year into the future to investigate scenarios in which managers might manipulate habitat to control harrier numbers through their prey density. Using values close to the highest and lowest observed vole and pipit densities, predicted harrier numbers rise from 9Æ51 [95% CI: (7Æ67, 11Æ2)] to 14Æ3 [95% CI: (11Æ2, 19Æ4)] as prey densities increase in tandem. However, when vole density is low and pipit density high, there is no notable diﬀerence in harrier numbers [10Æ1, 95% CI: (8Æ02, 12Æ7)] from when the density of both species is low. When the reverse is true, the predicted number of harriers increases, although the credible intervals do overlap [12Æ5, 95% CI: (10Æ0, 16Æ2)] (Table 2). The model took 843 s to run two chains for 100 000 iterations on a 3Æ2-GHz processor. A burn-in of 10 000 iterations was used. Autocorrelation was low, and storage was not an issue, so the MCMC trials were not thinned. Convergence was assessed quantitatively using the Brooks–Gelman–Rubin statistic (Brooks & Roberts 1998), as well as through visual inspections of multiple chains with diﬀerent starting points. Trace plots were used to assess mixing. Over 60 variations of the model were run to investigate model structure and test the prior sensitivity of the parameter

Table 2. Estimates of the expected number of hen harriers and their 95% credible intervals, under hypothetical management scenarios that control the densities of voles and pipits (km)2)

Fig. 3. A plot showing the estimated harrier numbers (black line) in comparison with the observed numbers (ﬁlled circle), with the 95% prediction interval (dashed lines).

Voles (km)2)

Pipits (km)2)

Harriers

100 600 1100 100 1100

200 400 600 600 200

9Æ51 10Æ9 14Æ3 10Æ1 12Æ5

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

(7Æ67, (9Æ09, (11Æ2, (8Æ02, (10Æ0,

11Æ2) 12Æ7) 19Æ4) 12Æ7) 16Æ2)

1192 L. F. New et al. estimates, helping to address the informal way in which the priors were constructed. Under the restrictions we imposed using previous biological knowledge, only the a parameters showed prior sensitivity. The measure of the eﬀect of pipit density on harrier survival, a1, was sensitive to the choice of prior variance, resulting in a posterior equivalent to the prior if the variance was too small. However, a large variance resulted in a1 being less sensitive to the prior, giving a posterior markedly diﬀerent to the prior. The baseline for harrier survival, a0, showed sensitivity to restrictive priors that produced bi-modal implicit priors on survival. No other parameter showed prior sensitivity (Fig. 4). Only the draws for a0 and a1 were correlated, with higher values for a1 resulting in slightly lower values for a0. Some model variations included an intercept term (bI) in eqn 3. The term was almost completely dependent on the prior and did not aﬀect the estimates for the other b parameters. The presence of bI only aﬀected the ﬁnal recruitment term gt, as some intercept estimates could allow more than one harrier to be recruited in the absence of voles and pipits. It was therefore considered reasonable to exclude the term from the model. The prior sensitivity of bI is likely due to a combination of the constraints imposed by the limited amount of data available and the biological implausibility of hen harriers moving into an area where no signiﬁcant prey are available.

Discussion Our model has potential as a management tool, estimating the possible eﬀects of diﬀerent management plans or changing land-use practices on hen harrier numbers. Predicting the expected number of harriers under diﬀerent prey densities

showed that the greatest harrier numbers result from a combination of high vole and pipit densities. However, vole density had a larger impact, as high pipit density combined with low vole density resulted in a similar expected number of harriers as when the density of both species was low (Table 2). Given harriers’ response to voles, it may be possible to naturally decrease harrier settlement in an area without directly managing the raptors by reducing the size of vole populations through habitat manipulation. However, such changes in vegetation could detrimentally impact the wider biodiversity. Furthermore, even if implemented, the method’s eﬀect would not be felt immediately, as birds previously settled on the moorland are likely to remain for their life span. The remaining birds would need to compensate for the reduced prey availability, so there may also be a temporary increase in harrier predation on other prey. For the model to be generally applicable to any hen harrier population, illegal persecution should be taken into account (Etheridge, Summers & Green 1997). However, illegal killing was known to be absent on Langholm estate, so was not considered in this case. The parameter a0 can incorporate the eﬀects of persecution on /t; in the presence of illegal killing, the estimate of a0 should be lower in comparison with areas, such as Langholm, where the harrier is protected. As a0 is timeinvariant, we would be assuming that the eﬀort put into illegal persecution is also time-invariant, which is reasonable, provided the moor is managed as a shooting estate. Given that winter migration appears dependent on pipit densities (Watson 1977), the strong positive dependence of recruitment (gt) on vole densities in our model was somewhat unexpected. However, previous studies have found a relationship between the abundance of small mammals and harrier

Fig. 4. Prior (black) - posterior (dash) plots for the parameters of the hen harrier model. In the survival (/) plot, pipit density (Pt) is equal to 216 pipits km)2. 2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

A model for hen harrier population dynamics 1193 breeding densities (Redpath, Thirgood & Clark 2002). Therefore, it is not the relationship, but its strength, that was unexpected. While harrier recruitment will not halt in the absence of voles, it does decrease, limiting harrier numbers in an area. This might contribute to the reduced use by harriers of conifer plantations as the trees age (Thompson, Stroud & Pienkowski 1988). Harrier recruitment in young plantations is encouraged by the abundance of rodents. However, the density of rodents decreases as the trees grow (Usher & Gardner 1988), resulting in lower recruitment. While alternate prey may be enough to sustain the site-loyal harriers that initially nested in the plantation, prey densities would not be high enough to encourage young birds to settle in the area. Additionally, harriers are better adapted to hunting in open country (Watson 1977), so prey will become less available as conifer plantations become more closed. Therefore, recruitment would reduce to the point where the population size could not be maintained, as there would not be enough young harriers to replace the older birds lost to the population. Other aspects of the model behaved as expected. Given our prior choice, higher pipit densities resulted in greater harrier survival, but this did not adversely aﬀect recruitment because of low hen harrier territoriality. When pipits are absent or at low densities, but voles continue at high densities, relatively large numbers of harriers could remain in an area. While survival would be lower, recruitment would remain high, maintaining high harrier density even in the relative absence of pipits. This has implications for management because it means that the aggregative response of hen harriers to their prey may be a stronger determinant of population numbers within a given area than their numerical response. Therefore, although a combination of low vole and pipit densities minimises the estimate of future harrier numbers, there may be little beneﬁt managing moorlands for decreased pipit densities, especially given the mild decline in the bird’s densities since the 1980s (Marchant et al. 1990; Hull 2001). It is encouraging that the expected numbers from the superpopulation model are similar to the observed numbers of nests. This implies that the data are a realistic realisation from the super-population. The large diﬀerence between the ﬁtted and observed number of harriers in 1996 may be a result of both the modelling approach and possible diﬀerences in the population dynamics of an established versus establishing population. Amar et al. (2008) show that hen harriers may be limited, in part, by decreased meadow pipit densities, and there is some evidence that hen harriers can show synchrony with their prey (Hamerstrom 1979; Simmons et al. 1986). To some extent, our expected values from the super-population ﬁt with this hypothesis, while the observed values do not. It may be that while establishing, harriers move into areas in greater numbers. Once the population has established, the dynamics may change slightly, allowing for numbers more similar to those predicted by the super-population model. The harrier model is a straightforward application of Bayesian methodology and super-population modelling, providing estimates for the birds’ life history processes. The model could be modiﬁed to form a traditional state-space model (Newman

et al. 2006; Buckland et al. 2007) for data sets where harrier numbers are uncertain. In this case, eqns 5–7 would depend on the predicted, as opposed to the known, harrier numbers, and demographic stochasticity in the predicted numbers would be allowed. The observation equation might take the form of a binomial distribution, where n is equal to the expected value from the double-Poisson distribution in eqn 6 and p is to be estimated. Eqns 1–3 would continue to form the state process. In our study, issues do arise with the possible diﬃculty in separating survival and recruitment, as the population’s age structure can only be weakly inferred from the data. Had it been possible to individually identify harriers, or to identify newly recruited birds, then the data would have allowed stronger inference on survival and recruitment. Additionally, the limited data availability combined with the relatively large number of parameters being estimated required the use of informative priors based on previous biological knowledge. A longer time series of data could make it possible to relax the restrictions on our priors, in addition to determining whether there is a diﬀerence in behaviour between established and establishing harrier populations. Overall, our model is a step towards improving the management of hen harriers. A combination of high vole and pipit densities leads to greater harrier numbers, raising the possibility of moorland management strategies to indirectly inﬂuence harrier populations. The stronger than anticipated coupling between vole densities and harrier recruitment, conﬁrming the importance of voles as a food item, suggests focusing the implementation of such a strategy on vole, not on pipit, densities. Widespread use of non-persecutory harrier management will lead to increased harrier densities on a national scale, as the grouse moors will no longer serve as a sink for the raptor. The reduced prey availability in these areas will encourage juveniles to settle elsewhere, such as on young conifer plantations or unmanaged moorland, where persecution does not occur, thus increasing the survival of the wider population.

Acknowledgements The work presented in this manuscript was carried out as part of L. New’s PhD thesis at the University of St Andrews.

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Redpath, S.M., Arroyo, B.E., Etheridge, B., Leckie, F., Bouwnam, K. & Thirgood, S.J. (2002) Temperature and hen harrier productivity: from local mechanisms to geographical patterns. Ecography, 25, 533–540. Redpath, S.M., Arroyo, B.E., Leckie, F.M., Bacon, P., Bayﬁeld, N., Guitierrez, R.J. & Thirgood, S.J. (2004) Using decision modelling with stakeholders to reduce human-wildlife conﬂict: a raptor-grouse case study. Conservation Biology, 18, 350–359. Ridout, M.S. & Beabeas, P. (2004) An empirical model for underdispersed count data. Statistical Modelling, 4, 77–89. Rothery, P. (1985) Estimation of age-speciﬁc survival of hen harriers (Circus cyaneus) in Orkney. Statistics in Ornithology (eds P. North & B. Morgan), pp. 341–355. Springer-Verlag, Berlin. Simmons, R., Barnard, P., MacWhirter, B. & Hanseon, G. (1986) The inﬂuence of microtines on polygyny, productivity, age and provisioning of breeding Northern Harriers: a 5-year study. Canadian Journal of Zoology, 64, 2447– 2456. Sotherton, N., Tapper, S. & Smith, A. (2009) Hen harrier and red grouse: economic aspects of red grouse shooting and the implications for moorland conservation. Journal of Applied Ecology, 46, 955–960. Southwest Condor Review Team (2007) A review of the second ﬁve years of the California condor reintroduction program in the southwest. U.S. Fish and Wildlife Service, California. Spiegelhalter, D., Thomas, A., Best, N. & Lunn, D. (2007) WinBUGS User Manual, 1.4.2 edn. Thirgood, S.J., Leckie, F.M. & Redpath, S.M. (1995) Diurnal and seasonal variation in line transect counts of moorland passerines. Bird Study, 42, 257– 259. Thirgood, S. & Redpath, S. (2008) Hen harrier and red grouse: science, politics and human-wildlife conﬂict. Journal of Applied Ecology, 45, 1550–1554. Thirgood, S.J., Redpath, S.M., Newton, I., Rothery, P. & Aebischer, N. (2000) Raptor predation and population limitation in red grouse. Journal of Animal Ecology, 69, 504–516. Thomas, L., Buckland, S.T., Rexstad, E.A., Laake, J.L., Strindberg, S., Hedley, S.L., Bishop, J.R.B., Marques, T.A. & Burnham, K.P. (2010) Distance software: design and analysis of distance sampling surveys for estimating population size. Journal of Applied Ecology, 47, 5–14. Thompson, D.B.A., Stroud, D.A. & Pienkowski, M.W. (1988) Aﬀorestation and upland birds: consequences for population ecology. Ecological Change in the Uplands (eds M.B. Usher & D.B.A. Thompson), pp. 237–259. Blackwell Scientiﬁc Publications, Oxford. Thompson, P.S., Amar, A., Hoccom, D.G., Knott, J. & Wilson, J.D. (2009) Resolving the conﬂict between driven-grouse shooting and conservation of hen harriers. Journal of Applied Ecology, 46, 950–954. Usher, M.B. & Gardner, S.M. (1988) Animal communities in the uplands: how is naturalness inﬂuenced by management?. Ecological Change in the Uplands (eds M.B. Usher & D.B.A. Thompson), pp. 75–92. Blackwell Scientiﬁc Publications, Oxford. Watson, D. (1977) The Hen Harrier. Poyser, Berkhamsted, UK. Received 24 December 2010; accepted 6 May 2011 Handling Editor: Mick McCarthy

Supporting Information Additional Supporting Information may be found in the online version of this article: Appendix S1. WinBUGS code for the hen harrier model. As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials may be re-organized for online delivery, but are not copy-edited or typeset. Technical support issues arising from supporting information (other than missing ﬁles) should be addressed to the authors.

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

doi: 10.1111/j.1365-2664.2011.02013.x

Hen harrier management: insights from demographic models ﬁtted to population data Leslie F. New1,2*, Stephen T. Buckland1, Stephen Redpath3 and Jason Matthiopoulos1,2 1

Centre for Research into Ecological and Environmental Modelling, The Observatory, University of St Andrews, St Andrews, Fife, Scotland, KY16 9LZ, UK; 2Scottish Oceans Institute, Department of Biology, University of St Andrews, St Andrews, Fife, KY16 8LB, Scotland, UK; and 3Aberdeen Centre for Environmental Sustainability, University of Aberdeen and the Macaulay Land Use Research Institute, Aberdeen AB24 2TZ, Scotland, UK

Summary 1. The impact of hen harriers Circus cyaneus on red grouse Lagopus lagopus scoticus populations has received much attention. However, little has been done to model the population dynamics of the hen harrier alone. Such a model is needed to help inform the diﬀering aims of conserving harriers and managing grouse moors, which serves as a reﬂection of human–wildlife conﬂicts around the globe. 2. On Langholm estate in Scotland, intensive studies have resulted in harrier numbers being known without error. We ﬁt a Bayesian population model to these data, using a super-population model to permit inference in the presence of demographic and environmental stochasticity and in the absence of observation error. 3. Hen harriers have a straightforward life history. After ﬂedging, juveniles show little natal site ﬁdelity, often dispersing long distances into breeding areas rich in their preferred prey, the ﬁeld vole Microtus agrestis and meadow pipit Anthus pratensis. Therefore, any increase in a local population is largely because of recruitment into the area as opposed to ﬂedging success. Once birds have settled in an area, harriers are generally site faithful, with year-to-year survival depending, in part, on the density of meadow pipits. 4. Our model suggests that temporal patterns in harrier numbers on managed grouse moors, in the absence of illegal persecution, are inﬂuenced by vole numbers, whereas meadow pipit density appears to have a limited eﬀect. 5. Synthesis and applications. Our modelling approach is a useful way to infer population processes, and the eﬀects of the environment on these processes, for populations censused without error. When used to predict future harrier numbers under alternate management scenarios, our model indicates that harrier numbers on Langholm estate, Scotland, could be reduced without any direct human intervention if the estate can be managed in a way that reduces vole populations. In contrast, there appears little to gain from managing meadow pipit densities. If these conclusions apply to other harrier populations, then management to reduce vole numbers, while maintaining grouse densities, may help alleviate the conﬂict between conservationists and managers of grouse moors. Key-words: Bayesian model, ﬁeld voles, meadow pipits, population dynamics, red grouse, super-population model, wildlife management

Introduction Much research, both internationally and locally, has focused on the conﬂict between the hen harrier Circus cyaneus (Linnaeus) and red grouse Lagopus lagopus scoticus (Latham) in Scotland (e.g. Redpath & Thirgood 1997; Redpath et al. 2004; Thirgood & Redpath 2008). The hen harrier is a red-listed *Correspondence author. E-mail: [email protected]

species that was almost completely extirpated from the UK in the nineteenth century because it preyed on red grouse, which lowered densities to the detriment of shooting bags (Marchant et al. 1990; Hull 2001). Eventually, the raptor was protected under UK law, but illegal persecution continues on shooting estates (Etheridge, Summers & Green 1997; Sotherton, Tapper & Smith 2009). Both species have been studied extensively because of ecological, conservation and economic concerns related to the two birds (e.g. Watson 1977; Hudson 1992;

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society

1188 L. F. New et al. Matthiopoulos et al. 2007; New et al. 2009). Hen harriers have been censused without error in long-term studies on some estates where there is no illegal killing (Redpath & Thirgood 1997). These data have been used to develop models of the impact of hen harriers on red grouse populations (Redpath 1991; Redpath & Thirgood 1997; Amar et al. 2004), and the hen harriers’ functional and numerical response (Redpath & Thirgood 1999; Redpath, Thirgood & Clark 2002; Asseburg et al. 2006), but have not previously been used to study the population dynamics of this bird of prey. Understanding hen harrier population dynamics is an important step in managing moorland habitat for the coexistence of harriers and grouse. While stakeholders agree that action needs to be taken to mitigate the eﬀects of harrier predation on red grouse, the exact nature of such actions remains undecided (Thirgood & Redpath 2008; Redpath & Thirgood 2009; Thompson et al. 2009). Conﬂict arises from several mitigation schemes that reduce revenue of moorland estates (Sotherton, Tapper & Smith 2009), from legal disputes, and uncertainty over the harriers’ responses to management actions, incurring risk for gamekeepers should mitigation plans fail. Hence better understanding of hen harrier population dynamics, beyond their impact on red grouse populations, will assist management of the harriers and their habitat. Hen harriers’ habitat choice is based on the presence of their main prey species, the meadow pipit Anthus pratensis (Linnaeus) and ﬁeld vole Microtus agrestis (Linnaeus) (Picozzi 1978); grouse are an opportunistic prey item and do not aﬀect habitat choice. An indirect approach to management through habitat manipulation or protection of other species (Thompson et al. 2009) requires that the bird’s main prey items are included in models of harrier dynamics. Diﬀerent management schemes’ viability cannot be assessed without an understanding of harriers’ reactions to alterations in habitat. In this paper, we develop a model for hen harrier population dynamics that could be used to help evaluate the eﬀectiveness of management plans for the conservation of the raptor and mitigation of its impact on the red grouse. In this study, the number of hen harriers breeding each year was known without error. This can be the case in small, wellstudied populations (e.g. killer whales Orcinus orca (Linnaeus) in the Paciﬁc north-west; Ford & Ellis 2006), or endangered species where individuals in the wild were originally captive bred (e.g. California condor Gymnogyps californianus (Shaw); Southwest Condor Review Team 2007). We allow demographic stochasticity by modelling the observed numbers of animals as a stochastic realisation from a theoretical superpopulation (Hartley & Sielken 1975), where stochasticity is modelled by specifying distributions for the numbers of survivors and recruits in each year. The survival and recruitment rates may also be modelled as functions of environmental covariates. This approach conceptualises the observed harrier population as one stochastic realisation of the population given the variability in survival and recruitment, allowing us to make inferences about other similar harrier populations. We begin by specifying the hen harrier’s population dynamics model. This model is ﬁtted in a Bayesian framework to data

from Langholm estate, a managed grouse estate in southern Scotland. We explain the ﬁtting process, which requires priors be placed on the model’s parameters. The adequacy of the model is checked by comparing the expected value of the realisations with the observed data. Finally, the models are used to investigate the impact on harrier numbers if management actions were taken to control the density of voles and pipits.

Materials and methods DATA COLLECTION

Data were collected on Langholm estate in the period 1992–1999. Prior to 1992, illegal persecution was taking place, so the data represent an establishing harrier population. Langholm is divided into six grouse beats (moorland areas managed by a speciﬁc keeper for grouse shooting). Meadow pipit abundance was estimated using line transects one km in length. A total of 18 one km2 squares were chosen at random, three in each of the six beats at Langholm. Each square contained two-line transects. In June, the same observer conducted all the counts, recording the number of individuals detected within each of six distance intervals (0–10, 10–20, 20–30, 30–50, 50–100 and 100– 200 m). An index of small mammal abundance was estimated through snap trapping in the spring. Two sites were chosen within each of the six beats to be representative of the beats’ habitat. At each site, 50 traps were set over two nights to give a total of 100 ‘trap nights’ per site (further details on data collection, methods and study site in Thirgood, Leckie & Redpath 1995; Redpath, Thirgood & Redpath 1995; Redpath & Thirgood 1997, 1999; Thirgood et al. 2000). Meadow pipit density estimates were obtained using Distance 5.0 (Thomas et al. 2010), with a truncation distance of 100 m from the line, as few pipits were recorded beyond 100 m. The recorded number of voles was converted to density using a scaling factor of 520 previously estimated to relate voles caught per 100 trap nights to density per km2 (Asseburg 2005). The number of nesting harriers was determined by observing the moor for displaying harriers in the early spring. Observation of the birds determined whether a male was bigamous, and if so, the relative status (primary or secondary) of the female was recorded. All nests were monitored closely through the breeding season, and regular checks ensured that no late breeders or other females were missed. There was no illegal persecution of hen harriers on the study area, so the eﬀect of illegal killing on hen harrier survival was not a concern in this case (Redpath & Thirgood 1997).

MODEL DEVELOPMENT

We are ultimately interested in the relationship between hen harriers and red grouse in the UK. Therefore, we develop a general model for hen harrier population dynamics that could be ﬁtted to data from locations other than Langholm estate. The exception is Orkney, where diﬀerent processes drive the dynamics (Picozzi 1984; Rothery 1985). As harrier numbers primarily impact the red grouse through predation on grouse chicks, which hatch in the spring, data were collected to coincide with this time period. Therefore, the harrier model follows the timing of New et al. (2009)’s grouse population dynamics model and moves from spring in year t)1 to spring in year t. In addition, consumption rates are calculated per pair (Asseburg 2005), so we model the number of nests not the number of birds. High rates of polygamy in harriers make it diﬃcult to use the abundance of one sex as an accurate indicator for the abundance of the other (Hudson

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

A model for hen harrier population dynamics 1189 1992). A further complication is that the sex ratio can vary yearly (Picozzi 1984). Polygamy means males are not a limiting resource, so females are a better indicator of nest numbers. Females with bigamous partners compensate for the lower provisioning rate of the male (Redpath & Thirgood 1997). However, we may slightly overestimate predation by modelling nests instead of individuals. In year t, realisations of nest numbers from the super-population are determined through female recruitment and survival. We do not account for fecundity as it does not aﬀect harrier density in an area. This results from high rates of juvenile dispersal, with almost no natal site ﬁdelity. However, after dispersal, harriers are site faithful (Watson 1977; Picozzi 1978). Therefore, net movement into an area (recruitment) replaces fecundity in terms of increasing harrier numbers in an area. Given this information, the basic population model is as follows, lt ¼ HObs;t1 /t1 þ gt ;

eqn 1

where HObs,t)1 is the observed number of female hen harriers on the grouse moor at time t)1, lt is the number of harriers present at time t, /t)1 is the survival rate between time t)1 and t, and gt is recruitment at time t. Survival and recruitment are determined by other factors. Survival depends on abundance of the diﬀerent prey items. Only the consumption of pipits plays a role in hen harrier survival; the passerine aﬀects the between-site abundance of hen harriers, while year-to-year changes are related to voles (Redpath & Thirgood 1997). In Scotland, up to 80% of harriers’ diet can be passerines (Picozzi 1984; Redpath et al. 2001), implying a heavy dependence on pipits, not on voles. Males leave the moors as pipit numbers begin to fall, and most females do not stay on the moors over winter (Watson 1977). Yet, during the winter months, vole numbers often increase, while the decline frequently takes place during the summer (Lambin, Petty & MacKinnon 2000). If vole consumption was driving harrier survival, the raptor would remain on the moor when vole numbers are highest. Therefore, the fact that they leave the moors in the winter implies that vole densities are not driving survival. Furthermore, if both survival and movement were dependent on voles, we would expect harrier populations to cycle in stronger synchrony with the rodent, as years of low survival and recruitment would coincide. Yet the raptor can be found in the absence of voles and grouse, and harrier population dynamics do not display the extreme peaks and troughs of the vole populations, remaining at high densities even after vole populations crash (Redpath et al. 2002). Survival is therefore modelled as, /t ¼

expða0 þ a1 Pt Þ ; 1 þ expða0 þ a1 Pt Þ

eqn 2

where a0 and a1 are parameters to be estimated, Pt is pipit density per km2 in year t, and a0 encompasses all other sources of mortality. We assume that, with the exception of the eﬀect of pipits, per capita mortality is constant across years, so a0 is time-invariant. A management shift would change a0. Recruitment into an area depends on the density of the harriers’ primary prey species, ﬁeld voles and meadow pipits (Redpath & Thirgood 1999; Redpath et al. 2002; Redpath, Thirgood & Clark 2002), modelled as, gt ¼ expðb0 Vt þ b1 Pt b2 HObs;t1 /t1 Þ;

eqn 3

where b0, b1 and b2 are parameters to be estimated and Vt is vole density per km2 in year t. Recruitment is constrained to be positive and is limited by the density of surviving harrier nests, HObs,t)1/t)1. Without prey, this function is similar to the Ricker function for density dependence. In the absence of voles, pipits or

harriers the equation allows for the recruitment of a single young harrier. If vole and pipit densities are not available, eqn 3 could be modiﬁed to account for harriers’ habitat preferences, which is often correlated with prey density (Arroyo et al. 2009). To account for uncertainty in vole and pipit densities, the covariates were modelled as, Pt Nðpipitt ; r2p;t Þ Vt LNðlogðvolet Þ; r2v Þ;

eqn 4

where the yearly variance for pipits (rp,t2) was found as part of their density estimation and the variance for voles (rv2) had been previously estimated (New 2009). In constructing the model, realisations from the super-population, Ht, will be found for time t, using the observed number of harrier nests from time t)1, HObs,t)1. For a given realisation, we assume that the observed numbers will come from a distribution centred on the expected numbers of that realisation. Although the data are discrete, they are underdispersed, so the Poisson distribution is inappropriate. The underdispersion results from harrier numbers in time t being dependent on time t)1. In addition, as the observations are known without error, the realisations will be more tightly clustered around the mean. We account for the underdispersion using the double-Poisson distribution (Efron 1986; Ridout & Beabeas 2004). We model the mean (lt) and variance (rt2) of the harrier population size under our super-population model as follows: the expected number of birds surviving from the previous year is HObs,t)1/t)1, while the expected number of recruits is gt. Assuming a binomial model for survival and a Poisson model for recruits, we have, lt ¼ HObs;t1 /t1 þ gt r2t ¼ HObs;t1 /t1 ð1 /t1 Þ þ gt ;

eqn 5

giving the double-Poisson, HObs;t Poðlt ; ht Þ:

eqn 6

where ht = lt ⁄ rt2. However, the limitations in software used to ﬁt the model (WinBUGS) meant the double-Poisson could not be applied directly. Instead, we used a categorical distribution to approximate the double-Poisson (McCarthy 2007), resulting in, HObs;t catðpx Þ

px ¼

ht 1=2 expðht ½lt xÞðlt =xÞht x expðxÞxx WE x!

eqn 7

where WE is a normalising factor approximately equal to one, and x = 1, 2, ...,2HObs,t. See Appendix S1 for the relevant code.

MODEL FITTING

Our model was ﬁtted within a Bayesian framework, permitting the incorporation of previous biological knowledge through prior probability distributions. Priors are updated with data via the model, resulting in distributions representing posterior beliefs. Posteriors are used to make all inferences about the states and parameters, providing a direct measure of uncertainty around the estimates (Ellison 2004; McCarthy 2007). The equations presented earlier were ﬁtted to the data in WinBUGS 1.4.2, which uses Markov chain Monte Carlo (MCMC) methods (Spiegelhalter et al. 2007). MCMC approximates the posterior distribution by drawing samples via Monte Carlo integration using a Markov chain. If run for a suﬃcient time period, the chain eventually converges on a stationary distribution that is the distribution of

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

1190 L. F. New et al. interest and is no longer dependent on the starting values or the number of iterations. The expected values of the states and parameters of interest are found using the sample averages from the stationary distribution. Determining a chain’s convergence on the distribution of interest is carried out visually or with convergence diagnostics (Gilks, Richardson & Spielgelhalter 1996). The model parameters, a0, a1, b0, b1 and b2, are used to deﬁne harrier survival, /t (a parameters) and recruitment, gt (b parameters). Priors for /t and gt are not explicitly deﬁned. Instead, the priors for the a and b parameters determine implicit priors for /t and gt through eqns 1 and 2. Therefore, the priors for the a and b parameters should combine to form biologically realistic implicit priors on /t and gt, respectively. However, the priors need to remain sensible with regard to the individual parameters, as well as when part of the larger group. Priors were chosen for the parameter values to match their known behaviour. For survival (eqn 1), a0 can be either positive or negative, allowing survival estimates to range from zero to one. A prior of N (0,1) was chosen for a0 to allow for a uni-modal implicit prior on /t. The second parameter, a1, is constrained to be positive because pipits increase harrier survival (Redpath & Thirgood 1997). Given the estimated pipit densities (216Æ5–575Æ2 km)2), a prior was chosen to respect the scale of the data. This was achieved by back-transforming from a normal distribution with a small mean and a large variance, giving a log-normal prior with la1 = )5Æ29 and ra12 = 0Æ833. The a priors, when combined with the diﬀerent Pt values, result in a leftskewed implicit prior for survival, spiking at one. As harriers are known to have a high survival rate (Etheridge, Summers & Green 1997), the implicit prior is not unreasonable. Additionally, choosing a prior to give a seemingly more reasonable prior on /t)1 decreases the prior’s variance, restricting a1 and resulting in the failure of the posterior to move away from the prior. For recruitment (eqn 2), parameters b0 and b1 were constrained to be positive, as the presence of voles and pipits is known to cause harrier recruitment into an area. Priors were chosen based on previous estimates from a similar equation for the hen harriers’ aggregative response (Matthiopoulos et al. 2007), which indicated that values would be low given the possible high densities of both prey species. Therefore, a gamma prior with shape one and rate 50 was chosen for both parameters. The last parameter, b2, measures the eﬀect of density dependence on recruitment. The prior was constrained to be positive, as the negative eﬀect of density dependence is deﬁned in the model. Given harriers’ low territoriality (Redpath & Thirgood 1997), it seemed unlikely that potential recruitment would be reduced by more than the number of birds already present. However, it is possible, especially in limited habitats, that each bird present would stop more than one new bird from being recruited. Therefore, b2 was given a gamma prior with a mean and standard deviation of 0Æ2 (i.e. an exponential distribution with rate ﬁve); while values less than one are probable, the long tail allows higher values. The combination of the three priors, taking possible vole, pipit and harrier densities into account, creates a right-skewed implicit prior on recruitment. The implicit prior essentially allows for nearly inﬁnite recruitment into any given area. While not biologically realistic, allowing for the possibility of extremely high numbers of immigrants avoids limiting gt by being too informative (Fig. 1).

Fig. 1. The implicit prior for harrier survival (/t) and recruitment (gt). Pipit density (Pt) is equal to 216 pipits km)2, vole density (Vt) is equal to 900 voles km)2, and the number of harrier pairs to remain in the area from time t)1 (Ht)1/t)1) is equal to 10.

The posterior means for the survival parameters, a0 and a1, were 0Æ211 [95% CI: ()0Æ7046, 1Æ013)] and 2Æ045*10)3 [95% CI: (4Æ87*10)4, 4Æ85*10)3)], respectively. The yearly variation in pipit density resulted in survival estimates ranging from 65% to 78% over the 8 years of the study (Fig. 2). This range of estimates includes the previous point estimate of 78% and falls within the 95% conﬁdence interval (57%, 90%) (Etheridge, Summers & Green 1997). However, the limits of the 95% credible intervals on survival can fall outside the previously estimated range (Fig. 2).

Results All parameters and states showed good mixing and convergence. The prior distributions were updated with the data, giving posteriors no longer unduly inﬂuenced by prior choice.

Fig. 2. A plot showing estimated survival (dashed line with ﬁlled circle) and recruitment (open circle), and their credible intervals, for the 8 years of the study.

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

A model for hen harrier population dynamics 1191 Table 1. Estimates of prey densities (km)2) with the corresponding 95% credible intervals

Voles (km)2)

Pipits (km)2)

Year

Data

Estimate

Data

Estimate

1993 1994 1995 1996 1997 1998 1999

1050Æ4 1060Æ8 280Æ0 535Æ6 1648Æ4 598Æ0 104Æ0

888Æ4 1045 282Æ5 652Æ2 1696 620Æ1 106Æ7

382Æ2 575Æ5 570Æ1 358Æ8 239Æ4 219Æ1 216Æ5

382Æ4 572Æ2 564Æ6 386Æ4 240Æ3 219Æ5 216Æ6

The parameter values for recruitment also fell within a range of realistic values. The parameters b0 and b1 have posterior means of 1Æ36*10)3 [95% CI: (5Æ59*10)4, 2Æ34*10)3)] and 9Æ54*10)4 [95% CI: (4Æ052*10)5, 2Æ48*10)3)], respectively, compared with previous estimates of 1Æ24*10)4 for the aggregative response to voles and 2Æ63*10)3 for pipits (Matthiopoulos et al. 2007). Given these values, an increase of 100 pipits km)2 would raise recruitment by approximately 9%, while the same increase in vole density raises gt by about 14%. The last parameter, b2 was estimated at 0Æ0431 [95% CI: (1Æ29*10)4, 0Æ151)], supporting the evidence of low territoriality among hen harriers. These parameters allow for the recruitment of between one and eight harriers a year, depending on prey densities (Fig. 2). All estimated prey densities were close to the observed values for voles and pipits (Table 1). Generally, the estimates of the expected number of successful hen harrier nests are in good agreement with the true values (Fig. 3), indicating that the super-population model is capable of producing realisations similar to the observed numbers. The greatest dissimilarity is in 1996, when the expected number is estimated as almost nine nests, while 14 nests were observed. This seems to be a result of the combination of low observed

(581Æ5, 1298) (687Æ9, 1025) (175Æ7, 430Æ5) (391Æ1, 1016) (1126, 2524) (387Æ7, 942Æ1) (65Æ60, 164Æ5)

(345Æ3, (508Æ0, (502Æ9, (343Æ5, (304Æ5, (178Æ7, (179Æ5,

419Æ2) 636Æ8) 627Æ0) 429Æ9) 276Æ7) 260Æ4) 253Æ9)

nest numbers in the previous year (eight) and low prey densities limiting the recruitment of harriers under our model. The estimates for recruitment (gt) varied signiﬁcantly from year to year, depending on prey densities (Fig. 2). Vole densities appeared to have the greatest eﬀect on recruitment, as evidenced by the normalised and exponentiated values of 0Æ244 and 0Æ0757 for b0 and b1, respectively. The greatest recruitment seems to occur at peak vole densities, even when the corresponding pipit density was low. The harriers that survive, remaining in the area from the previous year (HObs,t)1/t)1), have very little impact on recruitment (b2: 0Æ0431). We predicted harrier numbers one year into the future to investigate scenarios in which managers might manipulate habitat to control harrier numbers through their prey density. Using values close to the highest and lowest observed vole and pipit densities, predicted harrier numbers rise from 9Æ51 [95% CI: (7Æ67, 11Æ2)] to 14Æ3 [95% CI: (11Æ2, 19Æ4)] as prey densities increase in tandem. However, when vole density is low and pipit density high, there is no notable diﬀerence in harrier numbers [10Æ1, 95% CI: (8Æ02, 12Æ7)] from when the density of both species is low. When the reverse is true, the predicted number of harriers increases, although the credible intervals do overlap [12Æ5, 95% CI: (10Æ0, 16Æ2)] (Table 2). The model took 843 s to run two chains for 100 000 iterations on a 3Æ2-GHz processor. A burn-in of 10 000 iterations was used. Autocorrelation was low, and storage was not an issue, so the MCMC trials were not thinned. Convergence was assessed quantitatively using the Brooks–Gelman–Rubin statistic (Brooks & Roberts 1998), as well as through visual inspections of multiple chains with diﬀerent starting points. Trace plots were used to assess mixing. Over 60 variations of the model were run to investigate model structure and test the prior sensitivity of the parameter

Table 2. Estimates of the expected number of hen harriers and their 95% credible intervals, under hypothetical management scenarios that control the densities of voles and pipits (km)2)

Fig. 3. A plot showing the estimated harrier numbers (black line) in comparison with the observed numbers (ﬁlled circle), with the 95% prediction interval (dashed lines).

Voles (km)2)

Pipits (km)2)

Harriers

100 600 1100 100 1100

200 400 600 600 200

9Æ51 10Æ9 14Æ3 10Æ1 12Æ5

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

(7Æ67, (9Æ09, (11Æ2, (8Æ02, (10Æ0,

11Æ2) 12Æ7) 19Æ4) 12Æ7) 16Æ2)

1192 L. F. New et al. estimates, helping to address the informal way in which the priors were constructed. Under the restrictions we imposed using previous biological knowledge, only the a parameters showed prior sensitivity. The measure of the eﬀect of pipit density on harrier survival, a1, was sensitive to the choice of prior variance, resulting in a posterior equivalent to the prior if the variance was too small. However, a large variance resulted in a1 being less sensitive to the prior, giving a posterior markedly diﬀerent to the prior. The baseline for harrier survival, a0, showed sensitivity to restrictive priors that produced bi-modal implicit priors on survival. No other parameter showed prior sensitivity (Fig. 4). Only the draws for a0 and a1 were correlated, with higher values for a1 resulting in slightly lower values for a0. Some model variations included an intercept term (bI) in eqn 3. The term was almost completely dependent on the prior and did not aﬀect the estimates for the other b parameters. The presence of bI only aﬀected the ﬁnal recruitment term gt, as some intercept estimates could allow more than one harrier to be recruited in the absence of voles and pipits. It was therefore considered reasonable to exclude the term from the model. The prior sensitivity of bI is likely due to a combination of the constraints imposed by the limited amount of data available and the biological implausibility of hen harriers moving into an area where no signiﬁcant prey are available.

Discussion Our model has potential as a management tool, estimating the possible eﬀects of diﬀerent management plans or changing land-use practices on hen harrier numbers. Predicting the expected number of harriers under diﬀerent prey densities

showed that the greatest harrier numbers result from a combination of high vole and pipit densities. However, vole density had a larger impact, as high pipit density combined with low vole density resulted in a similar expected number of harriers as when the density of both species was low (Table 2). Given harriers’ response to voles, it may be possible to naturally decrease harrier settlement in an area without directly managing the raptors by reducing the size of vole populations through habitat manipulation. However, such changes in vegetation could detrimentally impact the wider biodiversity. Furthermore, even if implemented, the method’s eﬀect would not be felt immediately, as birds previously settled on the moorland are likely to remain for their life span. The remaining birds would need to compensate for the reduced prey availability, so there may also be a temporary increase in harrier predation on other prey. For the model to be generally applicable to any hen harrier population, illegal persecution should be taken into account (Etheridge, Summers & Green 1997). However, illegal killing was known to be absent on Langholm estate, so was not considered in this case. The parameter a0 can incorporate the eﬀects of persecution on /t; in the presence of illegal killing, the estimate of a0 should be lower in comparison with areas, such as Langholm, where the harrier is protected. As a0 is timeinvariant, we would be assuming that the eﬀort put into illegal persecution is also time-invariant, which is reasonable, provided the moor is managed as a shooting estate. Given that winter migration appears dependent on pipit densities (Watson 1977), the strong positive dependence of recruitment (gt) on vole densities in our model was somewhat unexpected. However, previous studies have found a relationship between the abundance of small mammals and harrier

Fig. 4. Prior (black) - posterior (dash) plots for the parameters of the hen harrier model. In the survival (/) plot, pipit density (Pt) is equal to 216 pipits km)2. 2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194

A model for hen harrier population dynamics 1193 breeding densities (Redpath, Thirgood & Clark 2002). Therefore, it is not the relationship, but its strength, that was unexpected. While harrier recruitment will not halt in the absence of voles, it does decrease, limiting harrier numbers in an area. This might contribute to the reduced use by harriers of conifer plantations as the trees age (Thompson, Stroud & Pienkowski 1988). Harrier recruitment in young plantations is encouraged by the abundance of rodents. However, the density of rodents decreases as the trees grow (Usher & Gardner 1988), resulting in lower recruitment. While alternate prey may be enough to sustain the site-loyal harriers that initially nested in the plantation, prey densities would not be high enough to encourage young birds to settle in the area. Additionally, harriers are better adapted to hunting in open country (Watson 1977), so prey will become less available as conifer plantations become more closed. Therefore, recruitment would reduce to the point where the population size could not be maintained, as there would not be enough young harriers to replace the older birds lost to the population. Other aspects of the model behaved as expected. Given our prior choice, higher pipit densities resulted in greater harrier survival, but this did not adversely aﬀect recruitment because of low hen harrier territoriality. When pipits are absent or at low densities, but voles continue at high densities, relatively large numbers of harriers could remain in an area. While survival would be lower, recruitment would remain high, maintaining high harrier density even in the relative absence of pipits. This has implications for management because it means that the aggregative response of hen harriers to their prey may be a stronger determinant of population numbers within a given area than their numerical response. Therefore, although a combination of low vole and pipit densities minimises the estimate of future harrier numbers, there may be little beneﬁt managing moorlands for decreased pipit densities, especially given the mild decline in the bird’s densities since the 1980s (Marchant et al. 1990; Hull 2001). It is encouraging that the expected numbers from the superpopulation model are similar to the observed numbers of nests. This implies that the data are a realistic realisation from the super-population. The large diﬀerence between the ﬁtted and observed number of harriers in 1996 may be a result of both the modelling approach and possible diﬀerences in the population dynamics of an established versus establishing population. Amar et al. (2008) show that hen harriers may be limited, in part, by decreased meadow pipit densities, and there is some evidence that hen harriers can show synchrony with their prey (Hamerstrom 1979; Simmons et al. 1986). To some extent, our expected values from the super-population ﬁt with this hypothesis, while the observed values do not. It may be that while establishing, harriers move into areas in greater numbers. Once the population has established, the dynamics may change slightly, allowing for numbers more similar to those predicted by the super-population model. The harrier model is a straightforward application of Bayesian methodology and super-population modelling, providing estimates for the birds’ life history processes. The model could be modiﬁed to form a traditional state-space model (Newman

et al. 2006; Buckland et al. 2007) for data sets where harrier numbers are uncertain. In this case, eqns 5–7 would depend on the predicted, as opposed to the known, harrier numbers, and demographic stochasticity in the predicted numbers would be allowed. The observation equation might take the form of a binomial distribution, where n is equal to the expected value from the double-Poisson distribution in eqn 6 and p is to be estimated. Eqns 1–3 would continue to form the state process. In our study, issues do arise with the possible diﬃculty in separating survival and recruitment, as the population’s age structure can only be weakly inferred from the data. Had it been possible to individually identify harriers, or to identify newly recruited birds, then the data would have allowed stronger inference on survival and recruitment. Additionally, the limited data availability combined with the relatively large number of parameters being estimated required the use of informative priors based on previous biological knowledge. A longer time series of data could make it possible to relax the restrictions on our priors, in addition to determining whether there is a diﬀerence in behaviour between established and establishing harrier populations. Overall, our model is a step towards improving the management of hen harriers. A combination of high vole and pipit densities leads to greater harrier numbers, raising the possibility of moorland management strategies to indirectly inﬂuence harrier populations. The stronger than anticipated coupling between vole densities and harrier recruitment, conﬁrming the importance of voles as a food item, suggests focusing the implementation of such a strategy on vole, not on pipit, densities. Widespread use of non-persecutory harrier management will lead to increased harrier densities on a national scale, as the grouse moors will no longer serve as a sink for the raptor. The reduced prey availability in these areas will encourage juveniles to settle elsewhere, such as on young conifer plantations or unmanaged moorland, where persecution does not occur, thus increasing the survival of the wider population.

Acknowledgements The work presented in this manuscript was carried out as part of L. New’s PhD thesis at the University of St Andrews.

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Supporting Information Additional Supporting Information may be found in the online version of this article: Appendix S1. WinBUGS code for the hen harrier model. As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials may be re-organized for online delivery, but are not copy-edited or typeset. Technical support issues arising from supporting information (other than missing ﬁles) should be addressed to the authors.

2011 The Authors. Journal of Applied Ecology 2011 British Ecological Society, Journal of Applied Ecology, 48, 1187–1194