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Evaluation of restricted-area culling strategies to control local red fox density Porteus, Thomas Allen 2015

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EVALUATION OF RESTRICTED-AREA CULLING STRATEGIES TO CONTROL LOCAL RED FOX DENSITY  by  Thomas Allen Porteus  B.Sc., University of Durham, 2001 M.Sc., University of Reading, 2002  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Zoology) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  April 2015  © Thomas Allen Porteus, 2015 ii  Abstract Lethal control of red foxes is often implemented on restricted areas where immigration from neighbouring sources is expected to make it difficult to keep local fox density low.  The justification of lethal wildlife control should include demonstrating its effectiveness.  To this end, population dynamics modelling may help to assess the performance of different control strategies in a range of real-world circumstances.  A Bayesian state-space model for within-year fox population dynamics was developed that could be fitted to data on daily culling effort and success obtained from gamekeepers on shooting estates in Britain.  The estimation model included parameters for key population processes within the culling area: immigration, cub recruitment and non-culling mortality.  A simulation-estimation study showed that given a minimum of three years’ data the estimation of fox density and demographic parameters was reliable.  Informative priors for the key model parameters were constructed using empirical data and meta-analysis.  Data from 22 estates were modelled on a two-weekly time-step.  Most estates achieved some suppression of the fox population relative to estimated carrying capacity, but few maintained consistently low densities.  The number of foxes killed was a poor indicator of culling effectiveness, highlighting the need for modelling.  Estimated immigration rates onto estates were typically high, indicating rapid replacement of culled foxes.  There was unexpectedly high spatial variation among estates in estimated carrying capacity and immigration rate.  There was evidence from a limited subset of estates that the variable density of released game birds may explain this.  The food requirement of the fox population during the nesting period was assumed to indicate predation pressure on wild birds.  Alternative culling strategies to reduce this requirement were evaluated using posterior parameter estimates from some estates.  Culling concentrated iii  in spring and summer only was more effective than culling uniformly throughout the year.  Autumn-only culling was not an effective strategy for wild birds.  Open-loop strategies were most effective as culling effort was used all the time.  However, closed-loop strategies, where culling effort was conditional on feedback from simulated field-sign searches, achieved similar effects on food requirements using less effort.  This revealed trade-offs between effectiveness, cost and animal welfare. iv  Preface This dissertation is original, unpublished work of myself, T.A. Porteus.  Several datasets were used in the analyses for which I was not involved in the survey design or data collection:  Data used in Chapter 2 and Chapter 7 resulted from the Fox Monitoring Scheme, a survey conceived by J.C. Reynolds and resourced by the Game & Wildlife Conservation Trust (GWCT).  All data entry and analysis was performed by myself.    Data used in Chapter 4 and Chapter 7 were obtained from the National Gamebag Census, a long-term annual dataset administered by GWCT.    Data used in Chapter 6 resulted from a distance sampling study conceived by J.C. Reynolds and M.J. Heydon, and resourced by GWCT.  Data were collected by J.C. Reynolds, M.J. Heydon and M.J. Short.  Data entry was performed by M.J. Heydon and M.J. Short.  All analysis was performed by myself.  Data used in Chapter 8 resulted from a study on snare use conceived by J.C. Reynolds and M.J. Short, and resourced by GWCT.  Data entry was performed by J.C. Reynolds and M.J. Short.  All analysis was performed by myself.  v  Table of Contents  Abstract .................................................................................................................................. ii Preface ................................................................................................................................... iv Table of Contents .................................................................................................................... v List of Tables ........................................................................................................................ xii List of Figures ..................................................................................................................... xiv Acknowledgements .......................................................................................................... xxxii Dedication ........................................................................................................................ xxxiii Chapter 1: General introduction ........................................................................................... 1 1.1 Management of wildlife populations ........................................................................ 1 1.2 Red fox control in Britain ......................................................................................... 3 1.3 Approaches to modelling .......................................................................................... 7 1.4 Evaluating management strategies ......................................................................... 10 1.5 Aims of the project and thesis structure ................................................................. 12 Chapter 2: The British red fox population and the Fox Monitoring Scheme ................. 15 2.1 Introduction............................................................................................................. 15 2.2 The British fox population: numbers and trends .................................................... 16 2.2.1 Numbers .............................................................................................................. 16 2.2.2 Trends ................................................................................................................. 17 2.2.3 National Gamebag Census .................................................................................. 18 2.3 Fox Monitoring Scheme ......................................................................................... 19 2.3.1 Survey methods .................................................................................................. 20 2.3.2 Results from the FMS dataset ............................................................................. 22 2.3.2.1 Type and location of contributors and estates ............................................ 22 2.3.2.2 Use of different methods ............................................................................ 23 2.3.2.3 Demographics of foxes killed ..................................................................... 24 2.4 Discussion ............................................................................................................... 26 2.5 Tables ...................................................................................................................... 33 2.6 Figures .................................................................................................................... 35 vi  Chapter 3: Modelling local-scale fox population dynamics: a simulation-estimation study ....................................................................................................................................... 45 3.1 Introduction............................................................................................................. 45 3.1.1 Why model fox populations? .............................................................................. 45 3.1.2 Fox ecology: background ................................................................................... 47 3.1.2.1 Fox mortality .............................................................................................. 47 3.1.2.2 Fox reproduction ......................................................................................... 49 3.1.2.3 Fox immigration ......................................................................................... 51 3.1.3 Suitable modelling approaches ........................................................................... 53 3.1.4 Determination of minimum data requirements ................................................... 55 3.1.5 Chapter aims ....................................................................................................... 56 3.2 Methods .................................................................................................................. 56 3.2.1 Data ..................................................................................................................... 56 3.2.2 Timing of fox breeding events ............................................................................ 58 3.2.3 Process model for population dynamics ............................................................. 59 3.2.4 Observation model for sighting data ................................................................... 61 3.2.5 Prior probability distributions for parameters ..................................................... 62 3.2.6 Simulation-estimation analysis ........................................................................... 63 3.2.6.1 Generation of lamping effort schedules ...................................................... 63 3.2.6.2 Simulation of data ....................................................................................... 64 3.2.6.3 Bayesian estimation .................................................................................... 67 3.2.6.4 Measurement of estimation bias ................................................................. 68 3.3 Results..................................................................................................................... 69 3.3.1 Timing of fox breeding events ............................................................................ 69 3.3.2 Influence of prior distribution specification ....................................................... 69 3.3.3 Effect of time series length ................................................................................. 71 3.3.4 Effect of effort variability ................................................................................... 72 3.3.5 Sensitivity to true parameter values .................................................................... 73 3.4 Discussion ............................................................................................................... 75 3.4.1 Choice of priors .................................................................................................. 75 vii  3.4.2 Minimum data requirements ............................................................................... 78 3.4.3 Information from variable effort ......................................................................... 80 3.4.4 Caveats and limitations ....................................................................................... 82 3.5 Tables ...................................................................................................................... 86 3.6 Figures .................................................................................................................... 89 Chapter 4: Estimating priors for immigration rate during restricted-area fox control .............................................................................................................................................. 100 4.1 Introduction........................................................................................................... 100 4.2 Methods ................................................................................................................ 104 4.2.1 Landscape categorisation .................................................................................. 104 4.2.2 NGC data .......................................................................................................... 105 4.2.3 Estimation model .............................................................................................. 106 4.3 Results................................................................................................................... 108 4.3.1 NGC data and land categorisation .................................................................... 108 4.3.2 Relationships of the fox cull to estate area ....................................................... 109 4.3.3 Immigration rate estimates ............................................................................... 109 4.4 Discussion ............................................................................................................. 109 4.5 Tables .................................................................................................................... 115 4.6 Figures .................................................................................................................. 116 Chapter 5: Meta-analytic methods to establish informative Bayesian priors for fox non-culling mortality rate .......................................................................................................... 124 5.1 Introduction........................................................................................................... 124 5.2 Methods ................................................................................................................ 130 5.2.1 McCarthy model ............................................................................................... 130 5.2.2 Hoenig model.................................................................................................... 133 5.2.3 Mortality rate-at-age assumption bias .............................................................. 136 5.2.4 Survivorship comparison .................................................................................. 139 5.3 Results................................................................................................................... 140 5.3.1 McCarthy model ............................................................................................... 140 5.3.2 Hoenig model.................................................................................................... 141 viii  5.3.3 Bias due to the constant mortality-at-age assumption ...................................... 143 5.3.4 Mortality rate prior ........................................................................................... 143 5.4 Discussion ............................................................................................................. 144 5.4.1 Credibility of alternative priors ........................................................................ 144 5.4.2 Other anthropogenic mortality factors .............................................................. 147 5.4.3 Reliability of data ............................................................................................. 149 5.4.4 Effect of senescence ......................................................................................... 151 5.4.5 Summary ........................................................................................................... 152 5.5 Tables .................................................................................................................... 153 5.6 Figures .................................................................................................................. 155 Chapter 6: An informative prior for the rate of successful search of foxes by gamekeepers from distance sampling data....................................................................... 169 6.1 Introduction........................................................................................................... 169 6.1.1 Sightability coefficient...................................................................................... 170 6.1.2 Rate of successful search .................................................................................. 173 6.1.3 Distance sampling ............................................................................................. 176 6.2 Methods ................................................................................................................ 177 6.2.1 Specification of a conceptual model for the rate of successful search ............. 177 6.2.1.1 Sighting probability .................................................................................. 177 6.2.1.2 Field of view ............................................................................................. 178 6.2.1.3 Speed of travel .......................................................................................... 178 6.2.1.4 Monte Carlo simulation ............................................................................ 179 6.2.2 Estimation of the rate of successful search from distance sampling data ......... 179 6.2.2.1 Survey procedure ...................................................................................... 179 6.2.2.2 Density estimation .................................................................................... 180 6.2.2.3 Empirical distribution of the rate of successful search ............................. 181 6.2.3 Obtaining the posterior predictive distribution ................................................. 181 6.3 Results................................................................................................................... 182 6.3.1 Conceptual priors .............................................................................................. 182 6.3.2 Distance sampling ............................................................................................. 183 ix  6.3.3 Posterior predictive distribution ....................................................................... 183 6.4 Discussion ............................................................................................................. 184 6.4.1 Summary ........................................................................................................... 190 6.5 Tables .................................................................................................................... 192 6.6 Figures .................................................................................................................. 195 Chapter 7: Population dynamics of foxes during restricted-area culling in Britain .... 202 7.1 Introduction........................................................................................................... 202 7.2 Methods ................................................................................................................ 206 7.2.1 Data ................................................................................................................... 206 7.2.2 Bayesian state-space model .............................................................................. 207 7.2.3 Prior probability distributions for parameters ................................................... 208 7.2.4 MCMC simulations .......................................................................................... 209 7.2.5 Culling vs. non-culling mortality ...................................................................... 210 7.2.6 Sensitivity analysis ........................................................................................... 211 7.2.6.1 Structural assumptions .............................................................................. 211 7.2.6.2 Prior probability distribution specification ............................................... 211 7.2.6.3 Process error specification ........................................................................ 212 7.2.6.4 Observation error specification ................................................................. 212 7.2.7 Relationships with gamebird release density .................................................... 214 7.3 Results................................................................................................................... 214 7.3.1 Parameter estimation ........................................................................................ 214 7.3.2 Density reconstruction ...................................................................................... 215 7.3.3 Mortality comparison........................................................................................ 218 7.3.4 Sensitivity analyses ........................................................................................... 218 7.3.5 Relationships with gamebird release density .................................................... 220 7.4 Discussion ............................................................................................................. 220 7.4.1 Parameter estimates .......................................................................................... 220 7.4.2 Effect of culling on fox density ........................................................................ 224 7.4.3 Relationships with gamebird releasing ............................................................. 228 7.4.4 Assumptions ..................................................................................................... 229 x  7.5 Tables .................................................................................................................... 232 7.6 Figures .................................................................................................................. 236 Chapter 8: Evaluating alternative fox management strategies for the benefit of wild game ..................................................................................................................................... 251 8.1 Introduction........................................................................................................... 251 8.2 Methods ................................................................................................................ 255 8.2.1 Data ................................................................................................................... 255 8.2.2 Operating model ............................................................................................... 256 8.2.2.1 Population dynamics model ...................................................................... 256 8.2.2.2 Observation model .................................................................................... 261 8.2.3 Culling control model ....................................................................................... 265 8.2.3.1 Lamping effort .......................................................................................... 266 8.2.3.2 Snaring effort ............................................................................................ 267 8.2.3.3 Cub removal.............................................................................................. 268 8.2.4 Food requirements ............................................................................................ 268 8.2.5 Validation of the culling control model ............................................................ 270 8.2.6 Operating model reference set .......................................................................... 270 8.3 Results................................................................................................................... 271 8.3.1 Reconstruction vs. projection ........................................................................... 271 8.3.2 Open-loop MSE ................................................................................................ 272 8.3.3 Closed-loop MSE.............................................................................................. 274 8.4 Discussion ............................................................................................................. 276 8.4.1 Strategy choice.................................................................................................. 276 8.4.2 Differences between culling methods ............................................................... 280 8.4.3 Trade-offs ......................................................................................................... 284 8.4.4 Extensions ......................................................................................................... 286 8.5 Tables .................................................................................................................... 288 8.6 Figures .................................................................................................................. 292 Chapter 9: Conclusions ...................................................................................................... 307 9.1 Future directions ................................................................................................... 313 xi  9.2 Concluding remarks .............................................................................................. 315 References ............................................................................................................................ 316 Appendices........................................................................................................................... 345 Appendix A Derivation of the Holling disc equation to partition search and handling time .......................................................................................................................................... 345 Appendix B Examination of the disc equation constant density assumption ................... 347 Appendix C Sequential searching through habitats of differing preference ..................... 352 Appendix D Development of an informative prior for per capita birth rate ..................... 358 D.1 Introduction....................................................................................................... 358 D.2 Methods ............................................................................................................ 359 D.3 Results............................................................................................................... 361 Appendix E Fox density and parameter estimates from modelled estates ....................... 366 Appendix F Sensitivity to observation error specification ............................................... 383 Appendix G Estimation of the probability of snaring success ......................................... 388 G.1 Introduction....................................................................................................... 388 G.2 Methods ............................................................................................................ 389 G.3 Results............................................................................................................... 389 Appendix H Open-loop scatterplots for other estates ....................................................... 393 Appendix I Effect of method on other estates .................................................................. 400 Appendix J Sensitivity to assumptions on other estates ................................................... 402 Appendix K Effect of feedback on other estates .............................................................. 404  xii  List of Tables Table 2.1.  The National Gamebag Census regional classification (Tapper 1992), detailing regional area and the proportion of each of these regions that Fox Monitoring Scheme estates covered. ................................................................................................................................... 33 Table 2.2.  Percentages of the total cull across FMS estates in each of the NGC regions made by different methods.  (N.B. some rows do not sum exactly to 100% due to rounding error)34 Table 3.1.  Prior probability distributions for estimated model parameters used in simulation-estimation analysis. ................................................................................................................. 86 Table 3.2.  Annual multiples of scaled mean weekly effort values used to generate the alternative lamping effort schedules. ...................................................................................... 86 Table 3.3.  True parameter values used to generate the base case simulated data. ................ 87 Table 3.4.  Mean posterior coefficient of variation (CV) for each parameter across 20 simulated populations under alternative annual lamping effort schedules. ............................ 87 Table 3.5.  Mean Percent Relative Bias in the medians of the marginal posterior probability distributions from 20 populations relative to different sets of true parameter values used to simulate them: the base set of values in Table 3.3 and two alternative sets of values; one giving a low fox density population (0.5[N0, v, r, M], 1.5[d]) and one giving a high fox population (1.5[N0, v, r, M], 0.5[d]).  K and σp were unchanged from the base case.  All estimation models used the same informative priors on v, r, M and d. .................................. 88 Table 4.1.  Summary statistics from the posterior probability distributions of immigration rate in different landscapes and the posterior predictive distribution.  Median values and credible intervals are all in fox km-2yr-1. .............................................................................. 115 Table 5.1. Mean percent relative bias (PRB) from 1,000 simulations where mortality rate estimated using catch curve analysis (Z) was compared to the mean mortality rate across ages (Mx) from a predicted population using alternative combinations of extrinsic mortality rate and rate of ageing under different models of ageing.  For each simulation, the stochastic error in Mx was assumed to follow a normal distribution with a mean of zero and a standard deviation of 0.2 yr-1............................................................................................................... 153 Table 5.2. Percent relative bias of Hoenig model Z predictions for foxes (maximum age of nine) made using estimates of constant mortality rate from catch curve analysis (Z) compared to using estimates of mean mortality rate across ages (Mx) from data simulated using different ageing models and assumed rate of ageing. ........................................................... 154 Table 6.1.  Symbols and description for model parameters and variables used in this chapter. .............................................................................................................................................. 192 xiii  Table 6.2.  Total survey effort, number of foxes sighted and densities of foxes estimated by line transect surveys in the three study regions, during 1995-1997.  The precision of the density estimates is shown by the coefficient of variation (CV). ......................................... 193 Table 6.3.  Difference between rate of successful search (d) conceptual prior median and CV values under alternative distributions placed on sighting probability (p), field of view radius (r) and speed of travel (v).  For each factor, these were either non-informative (flat) or informative.  The posterior predictive estimates of median and CV show the sensitivity to the conceptual prior specification. .............................................................................................. 194 Table 7.1.  Mean estimates of annual process error standard deviation σp from terrestrial species.  Ranges of values indicate estimates obtained either from more than one population or from using alternative estimation methods ...................................................................... 232 Table 7.2.  Estate area, number of contributed weeks’ data, number of weeks of lamping effort and annual lamping effort per km2 ............................................................................. 233 Table 7.3.  Prior probability distributions for estimated model parameters ......................... 234 Table 7.4.  Posterior median (and CV) for estimated parameters on each estate. ................ 235 Table 8.1. Symbols and description for model parameters and variables used in this chapter. .............................................................................................................................................. 288 Table 8.2. Posterior median parameter values for estates to which MSE was applied.  Low values relative to other estates are shown in italic and high values are shown in bold. ....... 290 Table 8.3. Environmental Zone and length of linear feature on each estate. ....................... 291  xiv  List of Figures Figure 2.1. Map of Great Britain showing the location (filled black circles) of the 75 rural estates that contributed data to the Fox Monitoring Scheme between January 1996 and August 2000.  National Gamebag Census regions (dashed lines) are shown (1 = SW England; 2 = SE England; 3 = E England; 4 = C England; 5 = W Midlands; 6 = Wales; 7 = NW England; 8 = NE England; 9 = E Scotland; 10 = W Scotland.  Urban areas (grey 1-km squares) were identified based upon the dominant land classification in each 1-km square from the Countryside Survey 2000 dataset using the Countryside Information System (CEH 2005). ...................................................................................................................................... 35 Figure 2.2.  Monthly variation in mean annual number of foxes killed (pooled across estates) showing the composition of the cull by different control methods. ....................................... 36 Figure 2.3.  Monthly variation in mean annual lamping data pooled across estates, showing a) the number of hours spent lamping; b) the number of foxes seen; c) the number of foxes killed; d) the number of foxes seen per hour (sighting rate); e) the number of foxes killed per hour (efficiency). .................................................................................................................... 37 Figure 2.4.  Monthly sighting rate of foxes on FMS estates between January 1996 and August 2000.  Estates are labelled by NGC region code (see Figure 2.1), data are all shown on the same scale for comparison.  Blank months indicate either no contribution or zero lamping effort. ........................................................................................................................ 38 Figure 2.5.  FMS estates (filled black circles) showing the regional mean annual sighting rates (foxes seen per hour of lamping) across estates.  Greyscale shows sighting rate from low (light) to high (dark).  White indicates no data as foxes absent from these islands. ....... 39 Figure 2.6.  Timing of lamping hours used relative to the hours of darkness in each month. 40 Figure 2.7.  Sex ratio of the 4547 adult and juvenile foxes culled across 75 estates between January 1996 and August 2000............................................................................................... 41 Figure 2.8.  Comparison of the monthly distribution of the total fox cull and the sample made available for autopsy by contributors. .................................................................................... 42 Figure 2.9.  Comparison of the proportional use of different methods used to kill foxes between the total cull and the sample made available for autopsy by FMS contributors....... 42 Figure 2.10.  Percentage occlusion of fox canine teeth pulp cavities as a function of the date of death in days from an assumed mean birth date of 1 April.  Dashed line indicates the percentage occlusion below which canine teeth are yet to develop the first cementum annuli. ................................................................................................................................................ 43 Figure 2.11.  Data from the National Gamebag Census on the numbers of foxes killed on shooting estates in 1995. ......................................................................................................... 44 xv  Figure 3.1.  Data from two British fox populations on the percentage of female foxes that were pregnant at different times during the breeding season (Lloyd 1980) are summarised, with mean dates and percentages (red points) used to fit a cumulative logistic distribution to show the percentage of the female population that had conceived over time.  The distribution of births was obtained assuming a gestation period of 52 days, from which the distribution of weaned cubs was obtained by assuming that all cubs were weaned by eight weeks old.  The weaned distribution describes when cubs become vulnerable to lamping effort. .................. 89 Figure 3.2.  Fox density time series from 20 populations simulated on a weekly time step under a seasonal culling strategy (black).  The medians of the posterior probability distributions for weekly fox density estimated using an estimation model with either a) vague priors (blue) on all model parameters, or b) informative priors (red) on v, r, M and d, with vague priors on the other parameters, are plotted.  Blue and red shading shows the 80% credible intervals for vague prior and informative prior models, respectively.  Dotted line shows true value of K. ............................................................................................................ 90 Figure 3.3.  Bias in the medians of the marginal posterior probability distributions from 20 populations relative to the true parameter values used to simulate the populations on a weekly time step under a seasonal culling strategy.  In addition to the parameter bias, the mean bias in weekly Nt is shown.  The estimation model used either a) vague priors on all model parameters, or b) informative priors on v, r, M and d, with vague priors on the other parameters.  The median percent relative bias for each parameter is shown as a black bar, boxes represent the interquartile range (IQR), and whiskers represent the data range. ......... 91 Figure 3.4.  Profiles of the marginal posterior probability distributions of parameters estimated from 20 populations simulated on a weekly time step under a seasonal culling strategy.  The estimation model used either a) vague priors on all model parameters or b) informative priors on v, r, M and d, with vague priors on the other parameters.  Each coloured line represents one simulation; solid black line represents the prior, vertical dashed line represents the true parameter value used in the simulations. ........................................... 92 Figure 3.5.  Box plots showing bias in the medians of the marginal posterior probability distributions from 20 populations relative to the true parameter values used to simulate the populations on a weekly time step under a seasonal culling strategy.  In addition to the parameters bias, the mean bias in weekly Nt is shown.  Differences in parameter bias due to length of sighting rate time series that were a) two years or less, b) three years, c) four years, or d) five years long are shown.  The estimation model used informative priors on v, r, M and d, with vague priors on the other parameters.  The median bias for each parameter is shown as a black bar, boxes represent the interquartile range (IQR), and whiskers represent the data range. ...................................................................................................................................... 93 Figure 3.6.  Box plots showing bias in the medians of the marginal posterior probability distributions from 20 populations relative to the true parameter values used to simulate populations on a weekly time step under a seasonal culling strategy.  The amount of lamping effort was varied within a five-year time series by multiplying mean annual lamping effort by different values to give alternative schedules (S1 to S6).  In addition to the parameter bias, xvi  the mean bias in weekly Nt is shown.  The estimation model used informative priors on v, r, M and d, with vague priors on the other parameters.  The median bias for each parameter is shown as a black bar, boxes represent the interquartile range (IQR), and whiskers represent the data range. ......................................................................................................................... 94 Figure 3.7.  Differences between the median of the marginal posterior probability distribution for weekly fox density and the simulated ‘true’ weekly fox density from 20 populations to highlight the estimation error (as departure from zero) under variable annual lamping effort.  The annual effort variations were achieved by multiplying mean annual lamping effort by different values to give alternative schedules (plots a-f).  The estimation model used informative priors on v, r, M and d, with vague priors on the other parameters.  Each coloured line represents one simulation. ................................................................................ 95 Figure 3.8.  Fox density time series from 20 populations simulated on a weekly time step under a seasonal culling strategy (black).  Data were simulated using true values 0.5 times the base case values for N0, v, r and M, and 1.5 times the base case value for d in Table 3.3, giving lower fox densities relative to K (4 fox km-2) than the reference set in Figure 3.2.  The medians of the posterior probability distributions for weekly fox density estimated using an estimation model with informative priors (red) on v, r, M and d, with vague priors on the other parameters are plotted.  Red shading shows the 80% credible interval. ....................... 96 Figure 3.9.  Fox density time series from 20 populations simulated on a weekly time step under a seasonal culling strategy (black).  Data were simulated using true values 1.5 times the base case values for N0, v, r and M, and 0.5 times the base case value for d in Table 3.3, giving higher fox densities relative to K (4 fox km-2) than the reference set in Figure 3.2.  The medians of the posterior probability distributions for weekly fox density estimated using an estimation model with informative priors (red) on v, r, M and d, with vague priors on the other parameters are plotted.  Red shading shows the 80% credible interval.  Dotted line shows true value of K. ............................................................................................................ 97 Figure 3.10.  Profiles of the marginal posterior probability distributions of parameters estimated from 20 populations simulated on a weekly time step under a seasonal culling strategy.  Data were simulated using true values a) 0.5 times the base case values for N0, v, r and M, and 1.5 times the base case value for d in Table 3.3, or b) 1.5 times the base case values for N0, v, r and M, and 0.5 times the base case value for d.  The estimation model used informative priors on v, r, M and d, with vague priors on the other parameters.  Each coloured line represents one simulation; the solid black line represents the prior, vertical dashed line represents the true parameter value used in the simulations. .............................. 98 Figure 3.11.  Simulated fox density of 20 populations under different carrying capacity values.  All other parameter values were as defined in Table 3.3 (reference case is K=4 fox km-2). ....................................................................................................................................... 99 Figure 4.1.  Map of Britain (England, Scotland & Wales) showing landscape categories obtained from grouping the 40 land classes surveyed in Countryside Survey 2000 (following original grouping in Bunce et al. 1996a; Walsh & Harris 1996). ......................................... 116 xvii  Figure 4.2.  Location and landscape category of 535 British shooting estates that contributed data on foxes culled to the National Gamebag Census (NGC) between 1996 and 2000. .... 117 Figure 4.3.  Relationships between a) mean annual cull and estate area, and b) mean annual cull density and estate area within each landscape category. The Pearson correlation coefficient is displayed on each panel.  The numeral on the right of each row corresponds to the landscape category (I = arable a, II = arable b, III = arable c, IV = pastural a, V = pastural b, VI = marginal upland, VII = upland). ............................................................................... 118 Figure 4.4.  Frequency histograms for each landscape category showing 10,000 intercept (exp(a)) and slope (b) estimates from a randomisation test of the cull density and area relationship.  Locations of the empirically estimated values are shown by the dashed red lines.  The y-axes are not labelled for clarity........................................................................ 119 Figure 4.5.  Posterior probability density functions for immigration rate in each landscape category. The posterior predictive density function was determined by the hyper-parameters of the hierarchical model. ..................................................................................................... 120 Figure 4.6.  Relationship between posterior estimates of immigration rate in each landscape category with fox density estimated from faecal surveys along linear features in these landscapes (Webbon, Baker & Harris 2004).  Dashed line shows the fit of a linear model with the intercept fixed at zero to account for the assumption that immigration onto estates within a landscape cannot occur without foxes being present in the surrounding region. ... 121 Figure 4.7.  Relationship of estimated immigration rate to the mean density of pheasant released annually onto estates within each landscape.  Data were obtained from the National Gamebag Census from 535 estates controlling foxes during the period 1996-2000.  Dashed line shows the fit of a linear model. ...................................................................................... 122 Figure 4.8.  Relationship between the mean number of gamekeepers employed annually and estate area in each landscape category from those NGC estates contributing data on fox bags for the period 1996-2000. ..................................................................................................... 123 Figure 5.1.  Mammalian instantaneous mortality rate data (transformed from mean annual survival rate estimates) for 12 orders plotted against body mass.  The number of estimates within each order is indicated in parentheses.  Filled symbols relate to Carnivora data added to the McCarthy et al. (2008) dataset.  Also shown are linear models (fit by least squares) through the range of data for each order (solid lines relate to data shown by circle symbols, dotted lines relate to triangles and dashed lines relate to squares). ...................................... 155 Figure 5.2.  Instantaneous mortality rate data (transformed from mean annual survival rate estimates) for each species.  Species are grouped by taxonomic order, which are ranked by increasing body mass (L-R).  For species which had more than one estimate (i.e. from different sexes or studies), box plots indicate the variability in these mean estimates. ....... 156 xviii  Figure 5.3.  Carnivore instantaneous mortality rate data (transformed from mean annual survival rate estimates) for 8 families plotted against body mass.  The number of estimates within each family is indicated in parentheses.  Filled symbols relate to data added to the McCarthy et al. (2008) dataset, including all Canidae.  Also shown are linear models (fit by least squares) through the range of data for each family (solid lines relate to data shown by circle symbols and dashed lines relate to squares). .............................................................. 157 Figure 5.4.  Instantaneous mortality rate data for each of the taxonomic groups included in the Hoenig model plotted against the maximum age recorded for that species population (or the most geographically local population for which data existed – see 5.2.2).  The number of estimates within each group is indicated in parentheses.  Also shown are linear models (fit by least squares) through the range of data for each group. ...................................................... 158 Figure 5.5.  Carnivore instantaneous mortality rate data for 6 families included in the Hoenig model plotted against the maximum age recorded for that species population (or the most geographically local population for which data existed – see 5.2.2).  The number of estimates within each family is indicated in parentheses.  Also shown are linear models (fit by least squares) through the range of data for each family. ............................................................. 159 Figure 5.6.  Posterior probability distributions of the standard deviation in each of the random effects included in the McCarthy model to show the relative variation attributable to each effect. ............................................................................................................................ 160 Figure 5.7.  Posterior probability distributions for the random effects of each species included in the McCarthy model.  Species random effects are shown grouped by taxonomic order.  The number of species in each order is indicated in parentheses.  Dashed vertical lines indicate zero effect. ............................................................................................................... 161 Figure 5.8.  Predicted allometric relationship between instantaneous adult mortality rate and body mass for species from the Carnivora order.  Predictions are shown from an analysis using the McCarthy et al. (2008) dataset (grey lines) and from an analysis including the additional data (black lines).  Solid lines show the posterior median prediction and the dashed lines the 95% credible interval.  Each mean mortality rate estimate from the database is shown by a circle............................................................................................................... 162 Figure 5.9.  Posterior probability distributions of predicted natural mortality rate for red foxes obtained from sensitivity analyses using a) the McCarthy model, and b) the Hoenig model.  To determine the influence of each taxonomic order/group, the analysis proceeded with data from each of the orders/groups dropped sequentially.  The legend indicates which order/group was dropped in each analysis. ........................................................................... 163 Figure 5.10.  Posterior probability distributions for the standard deviation in ln(Z) deviates from a model that allowed estimation error in total instantaneous mortality rate to differ between taxonomic groups compared to a model that pooled estimation error. .................. 164 xix  Figure 5.11.  Predicted relationship between instantaneous adult mortality rate and maximum age for species within the carnivore group.  Predictions are shown from analyses with both non-hierarchical and hierarchical model structures.  Solid lines show the posterior median prediction and the dashed lines the 95% credible interval.  Each mean mortality rate estimate from the database is shown by a circle. ................................................................................ 165 Figure 5.12.  Posterior probability distributions of the slope and intercept parameters for each taxonomic group from a) a non-hierarchical model and b) a hierarchical model.  The hierarchical model panel also shows the posterior predictive distribution for each parameter that is described by the hyperparameters. ............................................................................. 166 Figure 5.13.  Comparison of the red fox non-culling mortality rate prior probability distributions established using the Hoenig and McCarthy models.  Histograms of the posterior samples are shown by dashed lines.  The legend shows the parameter values of the lognormal distribution, i.e. the mean and standard deviation of the natural logarithm of mortality rate. ........................................................................................................................ 167 Figure 5.14.  Survivorship of different rural British fox populations from observed adult age distributions, showing the comparison with the predicted survivorship that would occur if a population was under either the Hoenig or McCarthy mortality rate estimate.  All fox ages were determined from incremental lines in the teeth cementum.  aHeydon and Reynolds (2000b), bKolb and Hewson (1980), cLloyd (1980). ............................................................ 168 Figure 6.1.  Mean number of hours spent lamping per week by gamekeepers on each of the 75 FMS estates. ..................................................................................................................... 195 Figure 6.2.  A gamekeeper searching for foxes across an estate.  The area searched is determined by the gamekeeper’s velocity and their field of view, with the fraction of the population seen within this area determined by the sighting probability (adapted from Case 2000). .................................................................................................................................... 196 Figure 6.3. Estimated values (histogram) and best fit probability distribution (red line) of a) sighting probability (n = 6 estimates) and b) speed of travel (n = 69) from reanalysis of the distance sampling data from Heydon et al. (2000). .............................................................. 197 Figure 6.4.  Histograms of grouped distance data with fitted detection functions fits to distance sampling data, showing χ2 goodness-of-fit P-value. a) Wales – autumn survey (n = 101 fox sightings); b) Midlands – autumn (n = 232); c) East Anglia – autumn (n = 92); d) Wales – spring (n = 72); e) Midlands – spring (n = 81); f) East Anglia – spring (n = 39). .. 198 Figure 6.5. Histogram of empirical rate of successful search estimates (n = 69) obtained from distance sampling analysis. ................................................................................................... 199 Figure 6.6.  Posterior probability density functions for a) the median rate of successful search and b) the standard deviation in the natural logarithm of the rate of successful search estimates. .............................................................................................................................. 200 xx  Figure 6.7.  Posterior predictive distribution for the rate of successful search, the empirical rate of successful search data obtained from distance sampling, and the least informative conceptual prior. ................................................................................................................... 201 Figure 7.1.  Marginal posterior probability distributions for the estimated parameters N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (non-culling mortality rate), and d (rate of successful search) from the 22 estates.  Each line type and colour combination represents the same estate in each panel.  The prior probability distribution for each parameter is shown by bold black line. ............................................... 236 Figure 7.2.  Spatial variation in posterior median estimates for a) carrying capacity (K), b) initial fox density (N0), c) immigration rate (v), d) per capita birth rate (r), e) non-culling mortality rate (M), and f) the rate of successful search (d). .................................................. 237 Figure 7.3.  Results for CUL showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 238 Figure 7.4.  Results for DLQ showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 239 Figure 7.5.  Results for HIR showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 240 Figure 7.6.  Results for NOG showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per xxi  capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 241 Figure 7.7.  Results for NYP showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 242 Figure 7.8.  Results for VAR showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 243 Figure 7.9.  Comparison of the estimated culling mortality and non-culling mortality implied by the posterior median of M on each estate over the range of the contributed data.  Posterior median fox density and carrying capacity of each estate are shown.  The scale of mortality differed so estates are ordered by row from top-left to bottom-right by total mortality. ..... 244 Figure 7.10. Sensitivity of marginal posterior distributions for parameters (columns) from six estates (rows) to different structural and prior distribution assumptions.  The reference case distributions shown are from the informative prior model.  The y-axes are not labelled for clarity. ................................................................................................................................... 245 Figure 7.11. Sensitivity of posterior median fox density on six estates to a) structural and b) prior distribution assumptions.  Reference case shown is from the informative prior model.  The dashed lines show the posterior median carrying capacity for the reference case. ....... 246 Figure 7.12.  Histogram of the mean estimates of the empirical process error standard deviation from the 22 estates.  These realised values were obtained by calculating the standard deviation in process errors across the time series for each MCMC chain iteration and summarising the mean of those.  The fixed value used in the model was 0.2. .............. 247 Figure 7.13.  Results for DLQ showing sensitivity of a) posterior median fox density and b) marginal posterior parameter estimates to specification of the process error standard deviation, σp.  The values of σp were either fixed at 0.05, 0.1, or 0.2, or were estimated using either a lognormal prior distribution with median of ln(0.05) and CV of 0.2 or a uniform xxii  prior with lower and upper bounds of 0.001 and 1.0, respectively.  The reference case, where σp is fixed at 0.2, is shown in bold. ....................................................................................... 248 Figure 7.14.  Results for VAR showing sensitivity of a) posterior median fox density and b) marginal posterior parameter estimates to specification of the process error standard deviation, σp.  The values of σp were either fixed at 0.05, 0.1, or 0.2, or were estimated using either a lognormal prior distribution with median of 0.05 and CV of 0.2 or a uniform prior with lower and upper bounds of 0.001 and 1.0, respectively.  The reference case, where σp is fixed at 0.2, is shown in bold. ............................................................................................... 249 Figure 7.15.  Relationships of fox immigration rate (v) with density of a) game birds released (pheasants and red-leg partridges), b) game birds shot, and c) game birds not shot from those released on five estates for which NGC data were available.  Game bird bag and release data were obtained as mean values for years 1996-2000. ............................................................ 250 Figure 7.16.  Relationships of fox carrying capacity (K) with density of a) gamebirds released (pheasants and red-leg partridges), b) gamebirds shot, and c) gamebirds not shot from those released on five estates for which NGC data were available.  Game bird bag and release data were obtained as mean values for years 1996-2000. ............................................................ 250 Figure 8.1.  The main model components in a management strategy evaluation.  Note the different pathways of open- and closed-loop feedback control and model-free and model-based culling control models. ............................................................................................... 292 Figure 8.2.  Cumulative distribution function of the geometric distribution describing the probability that one fox is captured by snares given a number of snare nights in a week. .. 293 Figure 8.3.  Weekly food requirement of individual fox cubs during their first year.  Adult male and female fox food requirements are shown as a reference.  Curves are based upon data from Sargeant (1978), adjusted for British foxes. ......................................................... 294 Figure 8.4.  Results from open-loop management strategy evaluation showing the food requirement of the fox population on DLQ under maximum levels of culling effort for each strategy: 1) year-round uniform, 2) year-round with seasonal lamping effort, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Food requirements under each strategy (left) are compared with the food requirement of foxes given no culling effort (right).  Red and orange bands indicate the timing of culling effort and relative level of lamping effort.  The nesting period is shown in grey as a reference.  Numbers shown within the nesting period indicate the total food requirement (in kg km-2) of the fox population during this period. ................................................................................................................. 295 Figure 8.5.  Scatterplots showing results of open-loop management strategy evaluation for DLQ under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each xxiii  strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 296 Figure 8.6.  Scatterplots showing results of open-loop management strategy evaluation for VAR under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 297 Figure 8.7.  Scatterplots showing results of open-loop management strategy evaluation for OCS under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 298 Figure 8.8.  Trade-offs between nesting period food requirements and food required during the rest of the year under different combinations of culling methods on the DLQ estate.  Letters refer to each culling method (or combination thereof): L = lamping, S = snaring, E = cubs killed at earths, A = all methods.  Results are from open-loop MSE and are shown under the maximum levels of effort for each method. .................................................................... 299 Figure 8.9.  Trade-offs between the effectiveness of different combinations of culling methods at reducing the food requirement during the nesting period and the number of foxes which must be killed annually to achieve this level of control on the DLQ estate.  Letters refer to each culling method (or combination thereof): L = lamping, S = snaring, E = cubs killed at earths, A = all methods.  Results are from open-loop MSE and are shown under the maximum levels of effort for each method. .......................................................................... 300 Figure 8.10.  Sensitivity of results on the DLQ estate to different assumptions about the standard deviation in process errors (σp) on a two-weekly time-step; the effect of assuming that immigration is a constant process throughout the year or is a seasonal process that occurs during the fox dispersal period; the effect of a higher or lower carrying capacity; and the effect of lower or zero immigration.  Results are from open-loop MSE and are shown under the maximum levels of effort for all methods.  Points are jittered around the x-values in each panel so that underlying points can be seen. ......................................................................... 301 Figure 8.11.  Trade-offs between nesting period food requirements and food required during the rest of the year under open-loop (O) and closed-loop (C) MSE on the DLQ estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. .......................................... 302 xxiv  Figure 8.12.  Trade-offs between the food requirement during the nesting period and the number of foxes which must be killed annually to achieve this level of control under open-loop (O) and closed-loop (C) MSE on the DLQ estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. ............................................................................................. 303 Figure 8.13.  Trade-offs between nesting period food requirements and the number of snares used annually under open-loop (O) and closed-loop (C) MSE on the DLQ estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. ................................................ 304 Figure 8.14.  Trade-offs between nesting period food requirements and the number of lamping hours used annually under open-loop (O) and closed-loop (C) MSE on the DLQ estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. ................. 305 Figure 8.15.  Trade-offs between nesting period food requirements and the number of hours spent searching for scats annually under open-loop (O) and closed-loop (C) MSE on the DLQ estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. ................. 306 Figure B.1. Relationship between the total number of foxes killed on a 5 km2 estate (area A) and the area searched (area a) showing the bias at each successive pass of 𝑨 from making the assumption that fox density is constant over a time interval compared to if fox density at the start of each pass is reduced by the number of foxes killed on the previous pass.  See text for input parameters.  Each panel (a-f) shows the effect of differing levels of lamping efficiency [sighting probability (p) × killing probability (k)].  Panel c) is the scenario most likely relevant to the FMS data (*indicates that the decrease in density occurs at multiples of A) 351 Figure C.1.  A hypothetical estate of 10 km2, divided into 10 ha blocks, comprised of different habitat types (numbered 1-4) which have increasing size (1.5 km2, 2.5 km2, 3 km2, 3 km2).  Each habitat type contains different numbers of foxes (weighted: ×4, ×2, ×1, ×0.5) based upon decreasing prey availability in a given season.  Gamekeepers are assumed to start in the area of highest fox density (1) and search sequentially through each habitat type, searching the area of lowest fox density (4) last. .................................................................. 356 Figure C.2.  Functional response of gamekeepers to foxes depending upon the assumptions made about the habitat types within the estate being searched over. ................................... 357 Figure D.1.  Simulated litter size per female data from mid-Wales and East Anglia obtained by generating random numbers from a gamma distribution parameterised using the sample xxv  mean and standard deviation in litter size from each region.  Data are shown binned into integer values. ....................................................................................................................... 362 Figure D.2.  Distribution of ‘observed’ per capita birth rate.  Gamma probability density functions fitted to these data by maximum likelihood and by analytical calculation of the shape and rate parameters are shown for comparison. ......................................................... 363 Figure D.3.  Prior and posterior probability distributions for the shape and rate parameters of the gamma distribution. ........................................................................................................ 364 Figure D.4.  Posterior predictive distribution for per capita birth rate specified by gamma distribution shape and rate parameters obtained by updating the prior with the FMS litter size data. ....................................................................................................................................... 365 Figure E.1.  Results for BMM showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a)-c) the bird nesting period is shown as a reference. ........................ 367 Figure E.2.  Results for CHU showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 368 Figure E.3.  Results for CIP showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 369 Figure E.4.  Results for DWS showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per xxvi  capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 370 Figure E.5.  Results for EWE showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 371 Figure E.6.  Results for FAH showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 372 Figure E.7.  Results for FHC showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 373 Figure E.8.  Results for GDE showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 374 Figure E.9.  Results for GHT showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), xxvii  and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 375 Figure E.10.  Results for HUS showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 376 Figure E.11.  Results for LEL showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 377 Figure E.12.  Results for MAH showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 378 Figure E.13.  Results for OCS showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 379 Figure E.14.  Results for RAM showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), xxviii  and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 380 Figure E.15.  Results for VDL showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 381 Figure E.16.  Results for YEM showing a) posterior fit of the model to sighting rate (Yt/Et); b) posterior estimates of bi-weekly fox density (Nt) in relation to the cull removed by different methods; c) estimated Nt as a proportion of carrying capacity (blue line denotes median population at posterior median K); d) priors (or post-model-pre-data distribution) and marginal posteriors of N0 (initial density), K (carrying capacity), v (immigration rate), r (per capita birth rate), M (instantaneous non-culling mortality rate), d (rate of successful search), and fox density in the final time-step.  Histograms in panel d) show posterior medians from all estates.  In panels a-c) the bird nesting period is shown as a reference. .......................... 382 Figure F.1.  Results for DLQ showing sensitivity of a) posterior median fox density and b) marginal posterior parameter estimates to specification of the observation error.  Process error standard deviation, σp, was fixed at 0.2.  Observation errors were either assumed to be Poisson, negative binomial (NB), or lognormal (LN).  The dispersion parameter, φ, of NB errors was either fixed at the maximum likelihood value estimated from the data prior to running of the estimation model or was estimated using a uniform prior with lower and upper bounds of 0 and 25, respectively.  The standard deviation, σobs, in LN errors was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively.  The reference case with Poisson observation errors is shown in bold......................................................... 384 Figure F.2.  Results for VAR showing sensitivity of a) posterior median fox density and b) marginal posterior parameter estimates to specification of the observation error.  Process error standard deviation, σp, was fixed at 0.2.  Observation errors were either assumed to be Poisson, negative binomial (NB), or lognormal (LN).  The dispersion parameter, φ, of NB errors was either fixed at the maximum likelihood value estimated from the data prior to running of the estimation model or was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively.  The standard deviation, σobs, in LN errors was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively.  The reference case with Poisson observation errors is shown in bold......................................................... 385 Figure F.3.  Results for DLQ showing sensitivity of a) posterior median fox density and b) marginal posterior parameter estimates to specification of the observation error.  Process error standard deviation, σp, was estimated using a uniform prior with lower and upper bounds of 0.001 and 1.0, respectively.  Observation errors were either assumed to be Poisson, negative binomial (NB), or lognormal (LN).  The dispersion parameter, φ, of NB xxix  errors was either fixed at the maximum likelihood value estimated from the data prior to running of the estimation model or was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively.  The standard deviation, σobs, in LN errors was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively. ..................... 386 Figure F.4.  Results for VAR showing sensitivity of a) posterior median fox density and b) marginal posterior parameter estimates to specification of the observation error.  Process error standard deviation, σp, was estimated using a uniform prior with lower and upper bounds of 0.001 and 1.0, respectively.  Observation errors were either assumed to be Poisson, negative binomial (NB), or lognormal (LN).  The dispersion parameter, φ, of NB errors was either fixed at the maximum likelihood value estimated from the data prior to running of the estimation model or was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively.  The standard deviation, σobs, in LN errors was estimated using a uniform prior with lower and upper bounds of 0 and 1, respectively. ..................... 387 Figure G.1.  Data from individual snare locations showing the time-to-termination from all causes and the time-to-termination from fox captures only. ................................................ 390 Figure G.2.  Kaplan-Meier survival function showing the cumulative probability that a snare will not have been terminated due to any cause (grey) or due to fox capture (brown).  Dashed lines indicate the 95% confidence limits. ............................................................................. 391 Figure G.3.  Cumulative incidence functions showing time-to-termination of snares from all causes (grey) and from fox captures (brown).  Solid lines are empirical CIFs obtained as the complement of Kaplan-Meier survival functions.  Dashed lines are best-fit parametric CIFs based upon the daily probability of fox capture per snare f. ................................................. 392 Figure H.1.  Scatterplots showing results of open-loop management strategy evaluation for GDE under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 394 Figure H.2.  Scatterplots showing results of open-loop management strategy evaluation for GHT under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 395 Figure H.3.  Scatterplots showing results of open-loop management strategy evaluation for HIR under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by xxx  strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 396 Figure H.4.  Scatterplots showing results of open-loop management strategy evaluation for HUS under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 397 Figure H.5.  Scatterplots showing results of open-loop management strategy evaluation for NYP under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 398 Figure H.6.  Scatterplots showing results of open-loop management strategy evaluation for YEM under different levels of culling effort.  Colour scale ranges from high food requirements (red) to low food requirements (green).  Figures are arranged in sets of four by strategy 1) year-round uniform, 2) year-round with seasonal lamping variation, 3) spring (Feb-Jul) control only, and 4) autumn (Jun-Nov) control only.  Figures a)-d) within each strategy set refer to different percentages of cubs killed at earths.  Dashed horizontal lines show the mean observed level of lamping effort on this estate. ........................................... 399 Figure I.1.  Trade-offs between the effectiveness of different combinations of culling methods at reducing the food requirement during the nesting period and the number of foxes which must be killed annually to achieve this level of control on the VAR estate.  Letters refer to each culling method (or combination thereof): L = lamping, S = snaring, E = cubs killed at earths, A = all methods.  Results are from open-loop MSE and are shown under the maximum levels of effort for each method. .......................................................................... 400 Figure I.2.  Trade-offs between the effectiveness of different combinations of culling methods at reducing the food requirement during the nesting period and the number of foxes which must be killed annually to achieve this level of control on the NYP estate.  Letters refer to each culling method (or combination thereof): L = lamping, S = snaring, E = cubs killed at earths, A = all methods.  Results are from open-loop MSE and are shown under the maximum levels of effort for each method. .......................................................................... 401 Figure J.1.  Sensitivity of results on the VAR estate to different assumptions about the standard deviation in process errors (σp); the effect of assuming that immigration is a constant process throughout the year or is a seasonal process that occurs during the fox xxxi  dispersal period; the effect of a higher or lower carrying capacity; and the effect of lower or zero immigration.  Results are from open-loop MSE and are shown under the maximum levels of effort for all methods.  Points are jittered around the x-values in each panel so that underlying points can be seen. .............................................................................................. 402 Figure J.2.  Sensitivity of results on the NYP estate to different assumptions about the standard deviation in process errors (σp); the effect of assuming that immigration is a constant process throughout the year or is a seasonal process that occurs during the fox dispersal period; the effect of a higher or lower carrying capacity; and the effect of lower or zero immigration.  Results are from open-loop MSE and are shown under the maximum levels of effort for all methods.  Points are jittered around the x-values in each panel so that underlying points can be seen. .............................................................................................. 403 Figure K.1.  Trade-offs between the food requirement during the nesting period and the number of foxes which must be killed annually to achieve this level of control under open-loop (O) and closed-loop (C) feedback on the VAR estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. ............................................................................................. 404 Figure K.2.  Trade-offs between the food requirement during the nesting period and the number of foxes which must be killed annually to achieve this level of control under open-loop (O) and closed-loop (C) feedback on the NYP estate.  For closed-loop options, numbers 2 or 8 refer to the number of search hours per week, letters M or H refer to the probability of finding scats if present (Medium = 0.5, High = 0.8).  Results are shown under the maximum levels of effort for all methods. ............................................................................................. 405  xxxii  Acknowledgements I could not have completed this research without the help, encouragement and support of many people.  I would sincerely like to thank my supervisor, Dr. Murdoch McAllister, for providing me with the opportunity to study at UBC.  He introduced me to Bayesian statistics and his expertise and enthusiasm for research were invaluable to me in developing this thesis.  I would like to express my gratitude to Dr. Jonathan Reynolds (Game & Wildlife Conservation Trust), whose patience and mentorship has enabled me to reach this stage in my career.  I would also like to thank the other members of my committee, Dr. Carl Walters (UBC Fisheries Centre), Dr. Steve Martell (International Pacific Halibut Commission) and Dr. Villy Christensen (UBC Fisheries Centre), for providing valuable insights into population dynamics modelling and for feedback on my thesis. Funding and data for this thesis were provided by the Game & Wildlife Conservation Trust (GWCT).  I must thank them for their continued financial support and I look forward to my future work there.  I thank many GWCT staff for their assistance, but specifically I would like to acknowledge Mike Short for sharing his understanding of practical fox control issues and for reminding me about the joys of fieldwork. I have been fortunate to have a talented group of fellow Zoology graduate students to learn from during my time at UBC, many of who have become good friends.  I specifically thank Shannon Obradovich for keeping my computer in the Fisheries Centre running while I was in the UK so I could finish my simulations.  I also need to thank the unconditional support of a fantastic group of friends (the “Gong Show”) away from my department who have kept me sane by making me laugh with their exploits.  Most importantly, thank you to my mum for emotional and financial support, and to my brother Ed for moral support and inspiring me to keep a balance between work and exercise. Finally, I owe my deepest thanks to my wife, Cosima.  She has been a constant source of love and laughs and without her patient support and encouragement, especially through difficult times, I could not have been able to do this.  Thank you for sharing your life and the doctoral experience with me. xxxiii  Dedication      In memory of my Dad, who sparked my interest in wildlife biology and sadly passed away before the end of my doctoral journey.  1  Chapter 1: General introduction 1.1 Management of wildlife populations Populations of wildlife species are managed for three main reasons: 1) conservation, which aims to maintain and, where appropriate, increase populations of threatened species; 2) harvesting, which aims to maintain populations of exploited species at productive levels; and 3) population control, which attempts to decrease populations of pest species below some critical level (Shea et al. 1998).  Management of wildlife populations can achieve these aims by controlling the human activities affecting them (Hilborn 2007).  Management of a population requires the development of strategies prescribing actions that strike a suitable balance between effectiveness, efficiency and humaneness (Reynolds & Tapper 1996).  Each alternative strategy will differ with respect to these considerations, leading to a decision-making problem as managers must choose between alternative strategies that may prescribe very different actions (Dorazio & Johnson 2003).  The ability to make effective strategic decisions depends upon having suitable information on the state of the population under management.  The task for wildlife managers is to understand their local situation and balance the trade-offs to determine the most efficient and humane strategy that is likely to achieve the best balance between conflicting objectives. Conservation, harvesting, and population control are often interrelated.  In both game management for harvesting and in species conservation, control of predator populations has long been recognised as one of the key management actions available to managers (Leopold 1933).  Control of predator and pest species for goals of game management and threatened species conservation is a widespread practice and there is plentiful evidence of the positive effect predator removal can have on prey populations (Côté & Sutherland 1997; Holt et al. 2008; Salo et al. 2010).  Nevertheless, the net benefit of predator control is increasingly subject to public scrutiny, and scientific evaluation of the potential effectiveness and ecological outcomes of culling is required (Fall & Jackson 2002).  Definition of measurable management objectives is therefore a fundamental step in determining management strategy – otherwise performance of the strategy cannot be evaluated.   2  It has been recommended that success of population control strategy should be measured not in terms of predator density, but by a change in prey density or prey breeding success (Sinclair, Fryxell & Caughley 2006).  However, in many situations – particularly at local scales – evaluating objectives relating to prey density is not straightforward.  There may not be any data on prey populations, or it may be difficult to monitor them accurately.  The relationship between predator density and prey density may not be well defined, as demonstrating the size of the effect of removing predators typically requires detailed study (Hone 1994), e.g. eight years for predator removal (Tapper, Potts & Brockless 1996; Fletcher et al. 2010) or predator exclosure studies (Krebs et al. 1995).  The detection of effects must also be at the relevant scale.  There may not be any observable response in the prey population to culling effort due to compensatory mortality caused by other predator species (Salo et al. 2010), due to overwhelming stochastic events, or perhaps most importantly, because the predator control was ineffectually implemented.  In the event of small or absent responses to control by prey populations, managers judging their strategy solely in terms of prey-related metrics are thus left with a dilemma.  Their options for future action include stopping culling and directing effort elsewhere (e.g. habitat management); continuing as before in the belief that the control effort was having an effect; or adopting an alternative predator control strategy in the belief that the previous strategy wasn’t working.  Measuring the performance of a strategy directly in relation to predator density could reduce these decision-making problems.  Wildlife research (e.g. a predator removal study) is designed to generate data and infer ecosystem relationships using models that describe the system, with statistical tests determining the significance of hypotheses about the unknown parameters of these relationships (Prato 2005).  The conventional approach managers have used to make strategic decisions is to take generalisations (rules-of-thumb) from such quantitative research together with previous personal or collective experiences and combine these into a strategy.  For example, if a study found that culling during the spring using a particular method was the most effective, a manager might adopt this strategy in their local situation, particularly if they have previous successful experience at this time of year.  Actions are typically not designed to be responsive to changing field conditions.  While this is a practical and pragmatic 3  approach to developing a management strategy, the location and scale of the research will often have been very different to the situation where the strategy is to be applied.  Consequently, management actions within the strategy may not be appropriately targeted to the new situation, where the dynamics of the controlled population may be different.  This can result in management that is ineffective in both the short- and long-term and which comes with excessive economic and welfare cost, leading to greater risk of mismanagement and conflicts of interest among different stakeholders.  Effective management can be reliably achieved only by tailoring strategy closely to the objectives for each local situation. 1.2 Red fox control in Britain The red fox Vulpes vulpes is commonly managed as a pest throughout its range (Macdonald & Reynolds 2004).  Foxes are widely distributed throughout the Northern hemisphere and Australia, having been introduced to the eastern USA and Australia for sport hunting in the 1800s (Baker & Harris 2008).  Given this wide distribution, many alternative strategies and methods to control fox populations have been used.  In Britain, foxes are frequently brought into conflict with game and conservation interests as they are predators of both game (e.g. pheasant Phasianus colchicus, redleg partridge Alectoris rufa, grey partridge Perdix perdix; Reynolds & Tapper 1995a, 1996; Tapper, Potts & Brockless 1996; Draycott et al. 2008) and other threatened species (e.g. brown hare Lepus europaeus, lapwing Vanellus vanellus, stone curlew Burhinus oedicnemus; Reynolds & Tapper 1995b; Fletcher et al. 2010).  The generalist nature of foxes as predators means they have the potential to drive prey species into declines via depensatory predation, particularly when fox abundance is enhanced by human practices that enhance one or more of the prey species (Reynolds & Tapper 1996; Sinclair et al. 1998).  Foxes can cause problems to livestock farming through predation of lambs, poultry and piglets, though the economic scale of these losses is debated (Heydon & Reynolds 2000a; Moberly et al. 2003).  Foxes are also vectors for pathogens and parasites, e.g. the mite Sarcoptes scabiei which causes scabies in humans and sarcoptic mange in canids; and (though not currently in Britain) rabies and the tapeworm Echinococcus multilocularis which can cause human alveolar echinococcosis (Craig 2003; Soulsbury et al. 2007). 4  Britain has a long history of predator control, and regional efforts to reduce fox numbers for the benefit of animal husbandry have taken place at least since Saxon times (Reynolds & Tapper 1996).  Foxes are native to Britain, but their ecological status is heavily altered by the historical activities of man.  These include changes in habitat following agricultural intensification; elimination of natural predators (e.g. wolf Canis lupus and lynx Lynx lynx); introduction of new prey species (e.g. rabbit Oryctolagus cuniculus and pheasant); and the provision of other new food resources in urban areas (Tapper 1999).  These changes have generally been advantageous to the fox, generating conflicts with human interests and new ecological relationships with other species.  Predator control to benefit small game species in Britain began in the early nineteenth century on restricted areas of land privately managed for game hunting (Tapper 1992).  At this time, intensive control by gamekeepers led to regional extinction of some mammalian predator species, though the fox was afforded some protection due to its status as sporting quarry for fox hunts (Reynolds & Tapper 1996).  Following the Second World War, social change and agricultural intensification resulted in a fall in the numbers of gamekeepers in full-time employment (Tapper 1992).  The consequent release from culling pressure resulted in the fox population increasing through the second half of the twentieth century (Tapper 1992; Whitlock, Aebischer & Reynolds 2003). The paradigm that common generalist predators including the fox can limit the numbers of some wild prey species while being supported by a much broader resource base underlies the use of predator control in game management and threatened species conservation in Britain.  It is evidenced in Britain by formal predator removal experiments, well-documented case examples, and quantitative studies, but remains a much debated topic (Reynolds & Tapper 1996; Tapper, Potts & Brockless 1996; Stoate & Leake 2002; Fletcher et al. 2010; Reynolds et al. 2010b; Ewald, Potts & Aebischer 2012; Potts 2012; White et al. 2014).  Nevertheless, it is not the purpose of this thesis to question the motives for culling foxes, but rather to study the process itself and the extent to which it is successful in controlling fox density. 5  The rural British landscape is a patchwork of relatively small landholdings such as farms, shooting estates, and nature reserves.  Most landholdings are smaller than 10 km2, but some large estates may be several times this size.  Most predator management occurs locally within these restricted areas by gamekeepers, but a large proportion of landholdings (57%) are not under any form of fox control (Defra 2012).  This proportion does vary by region, reflecting general differences in management aims (Heydon & Reynolds 2000a).  There is little co-ordinated large-scale regional fox control due to excessive cost; but where this is achieved by collaboration between many neighbouring gamekeepers it can result in suppressed regional fox density (e.g. East Anglia, Heydon & Reynolds 2000a; b).  Fox control is particularly associated with shooting estates with the aim of benefitting gamebird populations for harvesting (Tapper 1992; Reynolds & Tapper 1996; Heydon & Reynolds 2000a).  The other main aims of fox control are elimination or reduction of losses to fox predation of livestock or threatened wildlife species (Reynolds 2000).   Methods of culling foxes in Britain have changed over time with changes in legislation and advances in technology.  The current legal methods are 1) shooting with a rifle or shotgun, 2) restraint in non-locking neck snares, and 3) live-capture cage traps.  Foxes may also be flushed from underground by terriers to be shot above ground.  Foxes captured alive must be dispatched humanely with a firearm.  Each method has potential welfare costs and these must be balanced with effectiveness and efficiency.  There is no statutory closed season so foxes of any age and sex may be killed (Heydon & Reynolds 2000a).  Outlawed methods include leg-hold traps, self-locking neck snares, poison and fumigants (Reynolds & Tapper 1996).  The Hunting Act 2004 (www.hmso.gov.uk) recently outlawed hunting with scent hounds and the use of terrier dogs to kill cubs at breeding earths.  Though there is a regional difference in prevalence of culling method used (Heydon & Reynolds 2000a), the majority of culling is done by night shooting using a rifle and spotlight, a method known as lamping. There are many decisions a gamekeeper must make when determining a fox control strategy.  These are related to the objectives of control which on a shooting estate will be to improve the harvest of game.  Shooting estates can be broadly split into two extreme types: 6  wild bird estates and released bird estates.  Wild bird estates aim to achieve a harvestable surplus from wild stocks, while released bird estates hand-rear birds in pens and release them into the wild prior to the start of the shooting season (Tapper 1999).  There are many questions that gamekeepers must decide upon, including:  When in the year should culling effort be used?  How intensive does the culling effort need to be?  Is there a threshold when culling can be stopped?  When does continuing to cull become inefficient?  Which culling method is best, and does this change with the time of year?  Are there non-target capture or animal welfare costs with the choice of method?  What monitoring data should be collected to learn from and improve the culling program? For each decision to be made there will be a set of possible management actions, a set of uncertain events associated with each action, and a set of outcomes that must be measured (Ellison 1996). There are potentially a number of performance measures for fox control on shooting estates: 1) annual harvest (i.e. the gamebag), which can be monitored relatively easily from bag returns; 2) breeding productivity of gamebirds, which requires both spring and autumn counts; 3) survival of released gamebirds, which requires some level of tagging effort; and 4) return rate of released gamebirds (i.e. the proportion of released gamebirds that are shot during the shooting season).  But fox control is just one aspect of the management on an estate.  Foxes are not the only managed predator species in Britain, with corvids, e.g. carrion crow Corvus corone and magpie Pica pica, and mustelids, e.g. stoat Mustela erminea and weasel Mustela nivalis, often being controlled concurrently (Tapper, Potts & Brockless 1996).  Habitat management is also an action that can affect game species, and is often applied for this reason, especially to compensate the effects of agricultural intensification.  It is therefore very difficult to ascribe a change in these performance measures to fox control alone.  In any case, few estates regularly conduct spring and/or autumn game counts, and 7  even fewer estates monitor the survival of released or wild birds, so there are usually limited data.  In absence of data on prey responses which can be attributed to fox control, the gamekeeper can judge the effect of culling only through some measure of local fox population density itself. Monitoring within-year changes in local fox density is a major challenge.  There are a range of methods available for monitoring predator populations (Gese 2001; Wilson & Delahay 2001; Long et al. 2008).  However, established field methods to estimate fox density, e.g., using mark-recapture methods, require a level of effort typically unavailable to gamekeepers and are inappropriate in the fast-changing context of intensive culling.  Nevertheless, gamekeepers do have local and time-specific information available to them.  Field signs and sightings can be used as indices of abundance, as can catch-per-unit-effort.  One method, the use of a spotlight and rifle, has much in common with scientific survey methods, and by recording the number of hours of effort and the number of foxes sighted and/or shot it generates data indicative of relative local abundance.  Mathematical models can make use of such culling data to produce a quantitative understanding of local fox population dynamics during the culling process, and thereby evaluate its effectiveness.  Exploratory models can also be used as “What if…” tools to predict the likely impact of different culling actions (Buckland et al. 2007) and identify ways to increase effectiveness. 1.3 Approaches to modelling In complex ecological systems, predictions from models on the effect of management decisions on populations are subject to several uncertainties including: 1) process uncertainty, resulting from demographic stochasticity and environmental variability; 2) observation uncertainty, resulting from errors in measurement and sampling of ecological systems; 3) structural uncertainty, resulting from incomplete knowledge about how the system should be modelled; 4) parameter uncertainty, resulting from lack of knowledge about the parameter values within a model structure; and 5) implementation uncertainty, resulting from the incomplete control of management actions  (Ellison 1996; Williams 1996; Parkes et al. 2006).   8  Formerly, the approach to strategic decision making relied heavily on frequentist statistical methods (Wade 2000; Prato 2005).  A fundamental aspect of frequentist statistics is that inference is based on the expected frequency that the observed data are likely to be obtained with hypothetical replicates of sampling (McCarthy 2007).  The assumption that a study is representative of the system and can be repeated independently is usually violated in most ecological studies due to process uncertainty making each study unique (Ellison 1996; Prato 2005).  Frequentist methods such as sensitivity analysis or bootstrapping can be used to evaluate the effect of parameter uncertainty on decision outcomes, and have previously been used to evaluate fox management strategies at the landscape scale in Britain (Rushton et al. 2006).  However, these methods can fail to reveal the effects of uncertainty and are difficult to interpret (McAllister & Kirkwood 1998).  Relationships between population density and relative abundance indices are also often non-linear (Parma et al. 1998) and may only be known for a certain population density and under a limited set of habitat and environmental variables.  This makes it unsafe to generalise to individual locations with other circumstances.  Frequentist methods are further limited by independent treatment of the results of different experiments or studies, which does not allow synthesis of those results (Ellison 2004; Prato 2005).   Bayesian statistical methods offer an alternative approach to dealing with the uncertainty inherent in ecosystem management, with Bayesian inference differing from frequentist inference in several epistemic ways.  Bayesian probability allows ranking of the credibility of different hypotheses (e.g., about a given parameter value or model structure) in light of the sample data.  In contrast, frequentist inference provides probabilities for the data arising given a particular hypothesis (such as a null hypothesis).  The definition of probability is thus different, with the frequentist approach allowing only probability statements about observed data.  In contrast, the Bayesian approach allows probability to be defined as an individual’s degree of belief in the likelihood of an unobservable event or hypothesis, and as such can be used to make probabilistic predictions about the state of the system.  The Bayes posterior probability distribution therefore allows for a formal incorporation of uncertainty in management models (Walters & Ludwig 1994), and management questions are able to be answered directly as the result is a probability statement 9  about the credibility of different hypotheses given the data obtained.  This stands in contrast to frequentist probability statements, such as the commonly used P-value that relates to the expected long-run frequency distribution of potential results as extreme or more extreme than the set of results obtained if the null hypothesis happened to be correct (Reckhow 1990).  This makes it easier for managers to communicate predictions about strategic decisions to decision-makers and stakeholders. The principal advantage of Bayesian inference is that it explicitly incorporates prior knowledge in the form of a prior probability distribution (Ellison 1996, 2004; Wade 2000; Prato 2005; McCarthy 2007).  The prior is used along with sample data to compute the posterior on which inference is performed.  Allowing inference to take an iterative progression means that as new data are collected the model interpretation is updated and improved.  Informative priors can reduce uncertainty and improve estimation performance by constraining the model within reasonable biological limits.  The concerns with use of informative priors in Bayesian models are their subjectivity and potential to overwhelm posterior estimates if they are too informative relative to the data (Dennis 1996).  Provided care is taken to ensure that prior information is used in a logical and sensible manner, these concerns must be weighed against the advantage of making inferences about populations on which data are scarce (Martin et al. 2013).  The value of using suitable informative priors has been shown whether they are derived from expert knowledge (Martin et al. 2005; Kuhnert, Martin & Griffiths 2010), published data (McCarthy & Masters 2005; Martin et al. 2013), or other analytical methods (McAllister, Pikitch & Babcock 2001; McAllister, Stanley & Starr 2010).  Bayesian methods are hence well suited to the sparse and noisy data typically available to wildlife managers and uncertainty from biologically important but unknown parameters can be rigorously incorporated into models describing the system (Wade 2000).  Many ecological models are complex, with inference being made on multiple parameters describing different populations.  Bayesian hierarchical modelling is a tool for probabilistically sharing information among populations to improve parameter estimates and their precision (Gelman et al. 2004; Clark 2005; Cressie et al. 2009).  The approach uses data from numerous populations, assuming that some hierarchical or nested structure relates them.  10  This may either be spatial, e.g. within a region (Su, Peterman & Haeseker 2004), or temporal, e.g. within a year (McAllister et al. 2004).  Parameter estimates for populations that are closer to one another in either space or time may also be assumed to be more similar than for populations that are further away.  For these situations it is possible to extend the hierarchical approach to explicitly incorporate the spatial or temporal correlation structure into the model, e.g. using a spatially correlated prior distribution (Su, Peterman & Haeseker 2004). Bayesian hierarchical modelling is thus a powerful approach for complex problems where data are sparse and noisy (Clark 2005).  The application of Bayesian inference to ecological questions has grown significantly in the past two decades (McCarthy 2007; Gimenez et al. 2009; King et al. 2010; Kéry 2010).  Despite being widely applied in ecology and fisheries science (McAllister et al. 1994; Walters & Ludwig 1994; Punt & Hilborn 1997), Bayesian methods have only recently been applied to the management of vertebrate pest species (Chee & Wintle 2010). 1.4 Evaluating management strategies The adaptive management approach seeks to learn how to manage an ecosystem under uncertainty about the effects of alternative management strategy decisions (Walters 1986).  The concepts of adaptive management were proposed in the context of harvesting for fish (Walters & Hilborn 1976; Smith & Walters 1981; Parma & Deriso 1990) and were developed for waterfowl (Williams & Johnson 1995; Williams, Johnson & Wilkins 1996; Johnson et al. 1997).  More recently the approach has been applied to vertebrate pest control (Parkes et al. 2006).  The approach proceeds by managing according to a plan in which decisions are made and modified as a function of what is known and learned about the system, including information about the effect of previous management actions (Walters 1986; Parma et al. 1998).  The first step is to define explicit management objectives and identify performance indicators.   Next is to develop alternative management strategies and identify management actions that can be implemented under each strategy to achieve the objectives.  Following implementation, the effectiveness of management is evaluated by examining the monitoring data.  The management actions are then modified iteratively to 11  improve management outcomes.  In this way, adaptive management involves an iterative Bayesian learning process (Walters 1986; Ellison 1996). However, despite the intuitive approach to managing real-world problems there are very few examples where it has been applied in its entirety.  Among the reasons for this are a failure to embrace uncertainty and difficulties in funding (Walters 2007; Keith et al. 2011).  The management strategies that are most likely to be informative about a system are also those that are the most risky, and large-scale monitoring effort required to evaluate them is expensive.  Such reasons can be understandable under real-world constraints.  For example, for a gamekeeper to apply an adaptive management approach would not only require an estate large enough that two or more strategies could be applied concurrently, but also one where economic and conservation costs to wild game populations could be tolerated under a strategy that potentially required a drastic reduction in fox control effort.  Few estates have budgets large enough handle these risks and the loss of harvestable game to shoot may result in unemployment for the gamekeeper. Management strategy evaluation (MSE) is a stochastic computer simulation-based approach to testing a number of alternative management strategies using quantifiable performance measures derived from the operational objectives (Milner-Gulland 2011).  It may be used to design adaptive and non-adaptive robust strategies, depending on how the learning process is represented.  MSE makes identification and modelling of uncertainties central to the approach, allowing the robustness of alternative management strategies in meeting objectives to be examined.  Originally developed as an approach to whale harvest and fisheries management (Kirkwood 1997; Butterworth & Punt 1999; Sainsbury, Punt & Smith 2000), MSE has recently been proposed as a useful approach to solving terrestrial conservation and pest control issues where real-world experimentation is not feasible (Chee & Wintle 2010; Milner-Gulland et al. 2010; Bunnefeld, Hoshino & Milner-Gulland 2011).  This makes MSE a useful approach in the evaluation of fox control strategies. MSE makes use of an operating model to simulate the true state of the population dynamics, parameterised using knowledge of the biological processes of the species being managed.  To account for process uncertainty the model parameters are usually in the form 12  of probability distributions, e.g. posterior distributions from Bayesian analyses.  It is possible to use a variety of operating models to account for structural uncertainties.  Monitoring of the population can then be simulated using an observation model that generates an abundance measure, typically under some simulated observation error and bias to account for observation uncertainty.  The observation data are then passed through one of two alternative options to determine a control rule that is either model-based or model-free (McAllister et al. 1999).  In the model-free approach the control rule is based directly on observed data, while the model-based approach incorporates an assessment model that is used to estimate model parameters and the subsequent control rule is based on the output.  Management actions are rarely implemented perfectly, with error coming from two sources: 1) managers not complying with the rules, and 2) individual dynamics of managers (e.g. when and where control occurs; Bunnefeld, Hoshino & Milner-Gulland 2011).  An implementation model can account for this uncertainty by simulating the application of the control decisions on the system and the output is the number of animals removed from the population during that time step, which is passed to the operating model as the final phase in the iterative process. 1.5 Aims of the project and thesis structure The problems faced by managers of fox populations in Britain are shared by all pest and predator control operations attempting to improve the effectiveness of control strategies on restricted areas.  These problems relate to the difficulty in understanding the population and how it responds to control effort, namely:  What effects have past control efforts had on the fox population dynamics within a given area?  What is the most effective time to use fox control effort to reduce the potential impacts of predation?  Are all control methods similarly effective at reducing fox density?  What are the trade-offs between effective and efficient control? This thesis will attempt to answer these questions using Bayesian modelling methods.  The main goals of this thesis are to 1) develop a local-scale population dynamics model that can 13  be used to estimate within-year fox density on restricted areas such as shooting estates from data which can be collected by gamekeepers, 2) use meta-analysis and empirical data to construct informative prior probability distributions for key model parameters to reduce parameter uncertainty, 3) apply these priors within the developed model to estimate demographic parameters and within-year fox density on a number of estates across Britain; and 4) evaluate the ability of different strategies to meet management objectives on these estates.  The thesis is organised into seven data analysis chapters and a summary:  Chapter 2 presents a description of the data sources that are available on British fox populations and summarises the results from a dataset on the daily culling effort and success of gamekeepers culling foxes on restricted areas.  Chapter 3 introduces a Bayesian state-space model for within-year population dynamics that incorporates the key population processes at the local scale and which can be fitted to data obtained from gamekeepers.  The reliability of the estimations from the model is evaluated using simulation-estimation analysis.  Chapter 4 presents a methodology for estimating the rate of immigration into culled fox populations and uses a meta-analysis with data from different landscapes to construct an informative prior.    Chapter 5 uses meta-analysis to construct informative priors for non-culling mortality rate based upon life history invariant and allometric relationships with mortality rate.  The effect of senescence in the fox population on the estimated non-culling mortality rate is examined.    Chapter 6 provides a description of the detection process of foxes by gamekeepers using the lamping culling method.  This applies the predation mechanics theory behind Holling’s disc equation (Holling 1959a) to define the model parameter which scales the number of foxes sighted to the fox density as the rate of successful search.  Distance sampling analysis of fox sighting data and expert judgement is then used to construct an informative prior for the rate of successful search of foxes by gamekeepers.    Chapter 7 explores the application of the model developed in Chapter 3 to estimate the model parameters and fox density on a number of shooting estates in Britain.  The 14  reconstruction of fox density allows the effect of previous culling efforts to be determined.  Local variation in parameter estimates is also examined in relation to data on the density of released gamebirds from a subset of the modelled estates.    Chapter 8 presents a management strategy evaluation to evaluate alternative fox control strategies that aimed to reduce the food requirement of the fox population during the bird nesting period.  This is performed on a subset of the modelled estates which represented the range of ecological conditions experienced on these shooting estates.  The control strategies considered differed with respect to the timing of control, e.g. seasonal or year-round, and the culling methods used, e.g. lamping, snaring or removal of cubs at earths.  Chapter 9 provides a synopsis of the main findings and future avenues of research.15  Chapter 2: The British red fox population and the Fox Monitoring Scheme 2.1 Introduction Control of red foxes (Vulpes vulpes) in Britain has different aims, from gamekeepers attempting to ensure a harvestable surplus of either wild or reared game on a sporting estate, to farmers attempting to reduce their livestock losses, and to wildlife reserve wardens attempting to conserve threatened species.  Those who attempt to control fox numbers must – consciously or otherwise – adopt a culling strategy that they feel will achieve their aims at their operating scale.  For gamekeepers, this is typically a restricted area of <10 km2.  Strategic decisions include when to time the cull, how long to carry on for, and what method(s) to use.  The long tradition of fox management in Britain has led to gamekeepers having their own preferred strategies for controlling foxes, based upon accumulated experiences of which method(s) suits their local situation, and at what particular time of year (e.g. Frain 2006).  While this is undeniably useful knowledge, making a poor decision can have consequences, in terms of effectiveness in achieving the aim of control, economic costs to the estate, and welfare costs to the foxes. The use of population dynamics modelling and management strategy evaluation could potentially help gamekeepers fine-tune their decisions based upon quantitative data on local fox populations.  However, inference using models will only ever be as good as the data being used to estimate model parameters – the garbage in-garbage out principle – which makes the use of suitable local scale data critically important.  While there is a great deal of published information about British fox populations (Harris et al. 1995; Reynolds 2000; Macdonald & Reynolds 2004; Baker & Harris 2008; Aebischer, Davey & Kingdon 2011), much of this is based upon data collected at spatial and temporal scales that are not necessarily appropriate for modelling populations on restricted areas in which culling is performed throughout the year.  Estimation of fox density must be on shorter-than-annual time-steps to allow decisions on seasonal control strategy to be made, requiring data on these time-scales.  Data specific to individual estates are necessary to obtain parameter estimates representative of the conditions on those estates.  This chapter briefly outlines the existing 16  data on British fox populations and summarises the results of a large survey on local scale fox control effort across Britain that attempted to fill a gap in the available knowledge of how culling effort varies both spatially and temporally. 2.2 The British fox population: numbers and trends 2.2.1 Numbers Few estimates of absolute density are available because foxes, like many carnivores, are difficult to census directly due to their cryptic behaviour (Reynolds 2000; Gese 2001).  Most of the published estimates are from intensive local studies, e.g. using telemetry, earth (den) counts.  However, to provide estimates most of these still make assumptions about fox demography, i.e. the numbers of non-breeding females and itinerant foxes, which vary locally, as well as territory size (Reynolds & Tapper 1995a).  Density at this scale is highly variable, ranging from 0.025 foxes km-2 in upland Scotland to over 30 foxes km-2 in some urban areas where food is superabundant (Macdonald & Reynolds 2004).  Extensive regional (>1000 km2) surveys, e.g. using distance sampling, show that density is also variable at this scale, with pre-breeding density estimated to be 0.16 foxes km-2 in East Anglia, 0.41 foxes km-2 in mid-Wales and 1.17 foxes km-2 in the East Midlands (Heydon, Reynolds & Short 2000).  Such surveys are expensive, and it is often more practical to estimate relative density (e.g. using indirect methods such as faecal counts, culling records), particularly over larger areas (Sadlier et al. 2004).  Combination of pre-breeding faecal density counts with a measurement of defecation rates in captive foxes enabled absolute density estimation in arable, pastural, marginal upland and upland landscapes, which was 0.79-2.23, 1.39-1.88, 0.82 and 0.21 foxes km-2 respectively (Webbon, Baker & Harris 2004).  This distribution of densities broadly matches the annual fox bag density (foxes killed km-2 yr-1) on shooting estates in different regions (Tapper 1992). National fox population estimates have previously been made by extrapolating from local studies based upon land classification, and suggested a pre-breeding population of 252,000 (95% CI, 204 000-300 000 in 1981 (Macdonald, Bunce & Bacon 1981) and 240,000 (95% CI not reported but authors gave a low reliability rating) in 1995, with annual cub 17  production of 425,000 (Harris et al. 1995).  However, the lack of demographic data on foxes in rural habitats made both these population estimates highly uncertain (Harris et al. 1995).  Density estimates from faecal density counts were combined with the land area in each rural landscape type to give a pre-breeding population estimate of 225,000 (95% CI, 179,000-271,000; Webbon, Baker & Harris 2004), in addition to 33,000 foxes in urban areas (Harris et al. 1995). 2.2.2 Trends Fox population trends over time can be examined using relative abundance indices, especially when these come from long-term surveys.  There are around half-a-dozen survey schemes that regularly collect data on foxes as part of multi-species monitoring programmes in Britain.  The methods of these various schemes are detailed in the Tracking Mammals Partnership Report  (Battersby 2005), and they include the National Gamebag Census (NGC), the Breeding Bird Survey (BBS), the Waterways Breeding Bird Survey (WBBS), Winter Mammal Monitoring scheme (WMM), Mammals on Roads survey (MOR), and the urban-specific Living with Mammals (LWM) and Garden Bird Watch (GBW) surveys.  Data from most of these schemes are suitable only for describing national or regional scale yearly relative abundance trends as they do not contain sufficient spatial or temporal resolution to examine trends at local scales over shorter time periods.  The data from these survey schemes indicate that there has been a long-term upward trend in the fox population since WWII that stabilised during the mid-1990s, a period during which there has been a concurrent expansion of range into areas from where foxes were historically absent or scarce.  There are numerous factors that might explain this increasing trend, but a key one is likely to be the decrease in number of gamekeepers during the twentieth century that resulted in less culling pressure on foxes than in the early 1900s (Tapper 1992).  The stabilisation of the trend seen at the end of this period may be the result of a sarcoptic mange epidemic in the mid-late 1990s that heavily affected urban foxes and rural foxes to a lesser extent (Soulsbury et al. 2007).  Some of these survey schemes provide coarse indices of abundance but suggest that fox populations may show quite large between year variation and regional differences that can change over time (Battersby 2005). 18  2.2.3 National Gamebag Census Foxes have been managed on sporting estates across Britain for many centuries (Reynolds & Tapper 1996), meaning there was huge potential for long-term information to be gathered on local fox populations from gamekeepers’ records.  Established in 1961 by the Game & Wildlife Conservation Trust (GWCT), the NGC is the longest-running survey scheme and was established to provide a central repository of records from participating estates across Britain.  The data comprise the numbers of game and predator species killed annually on an estate, known as ‘bag data’ (Tapper 1992).  Sample size of the NGC is large compared to other survey schemes: between 1961 and 2000 the total number of estates contributing data on fox bags to the NGC was 1,051, with an annual mean of 264 ± 18.8 estates (1 s.e., n=40).  The geographical coverage is also wide due to a substantial proportion of rural GB being managed for game shooting (Tapper 1992).  NGC estates cover 5% of the UK by area, though this varies regionally from 15% in eastern Scotland to under 1% in the Midlands (Whitlock, Aebischer & Reynolds 2003).  Relative abundance trend analysis using a generalised additive model showed that in the period 1975 to 2000 fox bags increased significantly in south-east England, East Anglia, east and west Midlands, north-east England and east and west Scotland; though no significant change was detected in south-west England, Wales or north-west England (Whitlock, Aebischer & Reynolds 2003).  Since 2000, the population has seen an increase in England and Wales, but not in Scotland (Aebischer, Davey & Kingdon 2011). The utility of the NGC data in fox population dynamics modelling is low, despite the local scale at which data are recorded.  The temporal resolution of the data (total annual culls) precludes modelling within-year changes in density, which would be necessary in order to provide advice to managers on seasonal strategy.  A further issue is that bag records do not usually contain data on culling effort.  A change in bag size may therefore be due to variations in intensity of effort or the control method used over time, rather than changes in fox density per se.  Observed trends in bag data can be misleading if effort is not taken into account (McDonald & Harris 1999; Reynolds 2000; Sadlier et al. 2004).  In some regions the number of foxes killed is also a poor predictor of fox density because of variable 19  immigration of juveniles from neighbouring areas (Reynolds 1994), which means the number culled is dependent upon both effort and the regional pattern of productivity and availability of dispersing juveniles (Sadlier et al. 2004).  The trends in the NGC are however supported by relative abundance indices from other survey schemes (Battersby 2005), and given that there are also problems with the landscape-based approach to estimating density, particularly in converting faecal counts to fox density, there is no definite way of knowing which population trend is correct without knowledge on culling effort. 2.3 Fox Monitoring Scheme In the early 1990s, it was recognised by the GWCT that there was a need to understand how NGC fox bags arose and how the cull on each estate might influence (or not) the within-estate fox population dynamics (J. C. Reynolds, pers. comm.).  No suitable data on rural fox populations at the spatial and temporal scale necessary to undertake such an analysis existed at that time, but the value of obtaining data from farmers, gamekeepers, pest controllers and other individuals or groups that were involved in fox control was understood through previous experience (e.g. from the NGC).  In addition to the data such managers typically provided to the NGC, i.e. the number of foxes killed and estate size, they could potentially record the date, effort and method used to cull foxes on their estate or farm, the sex, age class and productivity of any foxes culled on a given date, and other information from field signs and direct sightings about fox presence during the year, e.g. the number of active breeding earths.  The Fox Monitoring Scheme (FMS) aimed to gather a more informative dataset than was available through the NGC by collecting some of this more detailed data from managers.  In later chapters this thesis will use the FMS dataset to model within-year fox population dynamics on individual estates, but to provide suitable background to the dataset the rest of this chapter provides a brief summary of the dataset by detailing how data were collected and providing an analysis and discussion of some general results. 20  2.3.1 Survey methods Legislation and Local conditions across Great Britain constrain the choice of fox control methods available to managers in rural estates, and regardless of regional traditions each manager will have a personal preference for particular methods and a range of abilities in implementing them.  Fox density differs across Britain (Heydon, Reynolds & Short 2000; Webbon, Baker & Harris 2004), which likely reflects the history and intensity of control in different regions as culling is the chief cause of mortality in rural foxes (Heydon & Reynolds 2000a; b).  Together with differing regional management practices this leads to large regional differences in the number of foxes killed per unit area, as shown in results from the NGC (Tapper 1992).  Some form of regional representation would be desirable in a survey of fox control effort to illustrate the range of different fox control practices in different regions. There is no statutory requirement to keep or submit records of fox control efforts, so any survey would be restricted to volunteers who were both willing to record and submit honest details of their fox control effort and success over a long period of time, and permitted to do so by the respective landowners of the estate(s) on which they managed foxes.  Potential contributors to the FMS were solicited from an advertisement in a quarterly GWCT membership publication, making the survey self-selecting, and therefore non-random.  This limited how regionally representative it could be, with the number of willing volunteers reflecting the prevalence of shooting estates in different regions.  For this reason, additional contributors from regions not represented were identified from existing contacts within the GWCT who were likely to be willing volunteers.  This gave some representation in each of the regions classified by the NGC, which divides Britain into ten approximately similar sized regions built up using administrative county boundaries (Tapper 1992).  Previous experience in handling voluntary survey data from the NGC showed that contributions from individual estates were likely to change over time as both gamekeepers and estate owners changed.  As this was likely to be a problem in the FMS, additional contributors were found if others dropped out in an attempt to keep the number of ‘active’ contributors relatively constant.  In total, 122 gamekeepers and professional pest controllers from across Britain registered an 21  interest in participating in the survey, of which 74 became active contributors to the FMS during the survey. Each participating contributor was provided with record books in the form of a daily diaries running from 1 January 1996 to 31 August 2000 in which to record quantitative data about their fox control efforts on an estate in.  Record books covered either a ‘spring-summer’ season (April-August) during which cubs recruit into the fox population, or an ‘autumn-winter’ season (September-March) during which, in absence of immigration, fox numbers can only decrease.  This meant 10 different record books were supplied to each contributor to cover this time period, with individual record books returned to GWCT once the period covered was finished.  By gathering data on a day-by-day basis, it was hoped the risk of any contributor being able to convincingly falsify data would be small, but to help ensure data reliability all contributors were interviewed by telephone at approximately 6-month intervals during the scheme, i.e. following the return of the most recent record book, at which time any unusual data from previously submitted books were queried.  17 contributors controlled foxes on more than one estate, resulting in the return of 106 separate sets of record books. It was necessary to cleanse the data as some record book sets were incomplete regarding certain key information necessary for modelling local fox population dynamics.  17 record book sets, mostly from pest controllers, returned data relating to fox control on separate estates within a much larger geographical area without information on the area over which the control effort was applied on a daily basis.  These data were considered unsuitable as they could not be used to reconstruct fox density within a restricted-area.  These record books were removed from the dataset.  19 record book sets contained less than one full calendar year of data and were also removed as knowledge on seasonal differences in effort was important.  Only 11 record book sets were completed for the entire 1996-2000 period, with an average of 48 books (range 40-54) being completed during any one season as contributors came and left the scheme.  Out of the 106 record book sets contributed from different estates, 75 remained following the cleansing process from 54 different contributors. 22  For each day of a completed FMS record book, the data were: 1) the start and finish time of any lamping session completed; 2) the number of foxes seen and number killed by lamping; 3) the number of occupied breeding earths located and the number of adult foxes and cubs killed at them; 4) the number of foxes killed by a) snaring, b) vermin drives, c) using a sit-and-wait method from a high-seat, d) deer stalking, e) using live-capture traps, or f) any other method.  These data fields were all unambiguous.  Contributors were additionally asked to categorise foxes killed as either 1) adult or juvenile (sub-adult) males, 2) adult or juvenile females, 3) adult or juvenile of unknown sex (for foxes killed but not recovered), or 4) cubs of either sex not yet independent of the breeding earth.  By the end of the summer cubs become undistinguishable from adult foxes and so contributors were not asked to judge the age of juvenile or adult foxes killed away from breeding earths.  Contributors were asked to submit either recovered carcasses or jaw bones for autopsy by the GWCT.  This would enable accurate aging of foxes by tooth sectioning, and if carcasses were recovered, provide a measure of reproductive performance (estimated litter size) by examining uteri for evidence of postpartum placental scars.  All autopsy and tooth sectioning procedures were conducted by GWCT research staff, while data entry, cleaning and analysis was performed by the author. 2.3.2 Results from the FMS dataset 2.3.2.1 Type and location of contributors and estates The 54 contributors to the FMS were predominantly gamekeepers working full-time (59%) or part-time (30%).  The remaining 11% of contributors were either professional fox controllers or farmers.  The FMS estates covered 0.25% of the total land area of Britain (Table 2.1).  The regional distribution of the 75 estates (Figure 2.1) highlights the non-random nature of the sample.  As expected, the majority of estates were from east England (35%) and south-east England (21%), but there were at least two estates sampled from each of the 10 NGC regions.  The area of FMS estates in each region highlights this variation (Table 2.1).  Most estates were located in very rural areas rather than on the edge of urban areas that might be large enough to support urban fox populations.  Though urban foxes tend 23  to disperse towards rural areas (Page 1981) they disperse over shorter distances than rural foxes, with the mean dispersal distance usually <10 km (Trewhella, Harris & McAllister 1988).  This data is therefore likely to be representative of rural foxes with few foxes killed that were born in urban areas. 2.3.2.2 Use of different methods Between January 1996 and August 2000 a total of 5,655 foxes were killed.  The FMS contributors were self-selected as keen users of lamping, accordingly lamping was the most commonly used method for controlling foxes (57%), followed by snaring (13%).  The annual mean number of foxes killed per month varied greatly (Figure 2.2), though for methods other than lamping there is no record of the level of effort applied to each.  There were two distinct peaks in the annual cull, one in the spring caused by an increase in foxes killed at breeding earths, and one in the summer caused by an increase in foxes killed by lamping. The number of foxes killed by other methods was relatively similar across months. The regional distribution of method use was less clear, due mainly to the small number of estates in some regions meaning any inference was not representative.  Lamping was the most common method in each region (Table 2.2), but estates in different regions put variable resources into culling foxes at breeding earths.  The use of snares appears to differ between regions, but the regions showing the extremes in proportional snaring use are also those with only two estates contributing data (W Midlands and NW England). A more detailed analysis of the lamping data is made possible by the effort data associated with each lamping session, which totalled 15,893 hours of lamping.  Analysis was performed by pooling data across estates for each month.  Differences in the mean number of foxes killed by lamping each month was not independent of the varied level of lamping effort used each month as the two trends very similar, with the exception of the months September-December where the number of foxes killed dropped off sharper than the number of hours spent lamping (Figure 2.3a, c).  The number of foxes killed per month followed a very similar pattern to the number of foxes seen (Figure 2.3b, c), with a mean monthly lamping success (proportion of foxes seen that were successfully shot) of 31% (range 26-40%).  24  Lamping success was slightly higher in the summer months (May-August), but was relatively constant throughout the rest of the year.  Lamping sighting rate (number of foxes seen per hour) peaked in July and August and then showed a steady decline throughout the rest of the year until it began to increase again in June (Figure 2.3d).  The sighting rate on individual estates shows a large amount of variation, but on most estates the general pattern of higher values in late summer is apparent (Figure 2.4).  Regional mean sighting rate shows a decreasing trend from south to north (Figure 2.5).  Mean lamping efficiency (number of foxes killed per hour) followed a similar monthly pattern to sighting rate (Figure 2.3e). Lamping appears to be a habitual practice for most gamekeepers, with 84% either undertaking one or two sessions per week (mean = 1.61 lamping sessions per week).  The timing of lamping sessions followed a seasonal pattern, with sessions starting around 1800 hours during winter months and from about an hour after the fall of darkness in summer months (Figure 2.6).  Lamping was typically completed before midnight, except in midsummer. 2.3.2.3 Demographics of foxes killed Adults and sub-adults (juveniles) made up 80% of the foxes killed.  Contributors determined the sex of only 42% of the cubs killed, but these showed an even sex ratio.  Contributors determined the sex of a greater proportion (90%) of the older foxes which had a slightly higher proportion of males in the sample, though the sex ratio showed seasonal variation (Figure 2.7).  More males were killed during the autumn and winter, with the proportion of females in the cull increasing during the spring. Of the foxes killed, body parts from 12% (n=661) from 30 estates were made available for autopsy.  Within those estates that did contribute jaws or carcasses, the mean percentage of foxes provided was only 27% of the total cull on each estate (range 0.5-100%), with only four estates providing >50% of foxes killed.  The monthly distribution of date of death for foxes made available for autopsy did not match that of the overall cull, with proportionally more foxes provided during the spring, and fewer provided during the late summer and autumn (Figure 2.8).  Analysis of the methods used to kill the foxes provided for 25  autopsy showed that a smaller proportion of foxes killed by lamping and a greater proportion of foxes killed at breeding earths were in the autopsied sample compared to the total cull (Figure 2.9). Of the foxes autopsied, 95% had intact canine teeth, with the remainder too damaged by gunshot to be usable.  As with other carnivores, the pulp cavities of a fox’s teeth fill progressively during the first years of its life, and it is possible to use this feature to distinguish young-of-the-year from older foxes.  Using canine teeth, which have a large size, the percentage occlusion of a canine tooth pulp cavity can be measured by first taking a transverse section the tooth along the line of the jaw and measuring the maximum diameter of the pulp cavity and of the tooth at the base of the enamel, then calculating the percentage of tooth diameter that is solid material, i.e. the percentage pulp cavity occlusion (Goddard & Reynolds 1993).  Tooth cementum development begins in the autumn following birth and the first dark staining annulus (annual growth line) appears during January to March of the year following birth, after which incremental annuli are used to indicate age (Grue & Jensen 1979; Goddard & Reynolds 1993).  Canine teeth <58% occluded have yet to develop the first cementum annuli (Goddard & Reynolds 1993), and by plotting the percentage occlusion against the date of death relative to an assumed mean birth date (April 1; Hewson 1986), it is seen that this percentage corresponds with the time when teeth from the young-of-the-year become of a similar percentage occlusion as adult animals, at about 8 months of age (Figure 2.10).  For foxes killed away from breeding earths this percentage occlusion can therefore be used to distinguish between these age groups, despite the maximum percentage occlusion not being reached until foxes are 15-18 months old (Goddard & Reynolds 1993).  By using this method, it can be seen that 76% of foxes in the autopsied sample were older animals, i.e. not young-of-the-year.  Limited sample sizes prevented further age structure analysis of the FMS dataset as only 87 of the autopsied foxes (13%) were aged to yearly age classes by counting the number of incremental dark-staining lines in the tooth cementum.  As these foxes did not represent a random sample, the survivorship curve derived from the data was unreliable as there were far fewer juvenile foxes than would be expected. 26  By counting the number of placental scars it was possible to estimate the litter size for the 23% of female foxes autopsied that both had intact uteri and were killed between March and July inclusive.  Outside of this period the progressive fading of scars makes it difficult to detect them all (Heydon & Reynolds 2000b).  The mean litter size was 5.62 ± 0.26 cubs per female (1 s.e., n=65).  This sample size was too small to allow a comparison between regions.  The body weights of foxes killed were contributed for 117 foxes from eight estates.  These estates were located throughout Britain so potentially represent the range of body weights observed in Britain.  The mean body weight in this sample was 6.50 kg (range 3.17-9.55 kg).  60% of these foxes were male.  This mean and range is similar to observations from populations in different parts of Britain (Baker & Harris 2008). 2.4 Discussion Uniquely, the FMS provides data on how the fox cull responds to variation in both lamping effort and the ability to locate foxes at breeding earths during the year.  As with postal questionnaire-type surveys for trapping effort of other species (McDonald & Harris 1999), it was not possible to determine whether this was a representative sample of fox managers, but that seems unlikely.  The voluntary nature of the survey is likely to have favoured a proportion of fox managers, i.e. those more likely to keep detailed records of their control efforts, and those who favoured lamping as a control method were especially sought.  However, the contributors came from several different types of occupation representing a wide range of management strategies, from those who controlled foxes year-round to those who only put control effort in at certain times of year, and from those who used several different control methods to those who use only their preferred one.  The self-selecting nature of the survey meant it was not possible to achieve a regionally representative sample, but the 75 estates still gave widespread coverage across Britain in areas that differ greatly in their reasons for controlling foxes.  These diverse reasons included lowland pheasant shooting estates wanting to increase survival of reared birds upon release, to upland grouse moors wanting to increase breeding success, and to farmers wanting to eliminate livestock losses. 27  By examining mean monthly values and taking Britain as a whole, the FMS identified two periods during which the cull increases, one in spring (March-May) and another in late summer (August-September).  Earths are consistently occupied only during the breeding season (Baker & Harris 2008) so an increase in the cull at earths might be expected during the spring when adult foxes can be found near to them.  This period coincides with the bird nesting period when fox control to protect ground-nesting birds from predation is often critical, so managers traditionally make a greater effort to control foxes at this time.  It is also a time of year when a large impact on the fox population can be made as cubs can be culled before they disperse.  FMS contributors recorded the number of occupied earths on an estate, but not the effort used to achieve the cull at earths (time spent locating earths and waiting for foxes to be active near the entrances) so further examination of how much effort was used to achieve the cull at earths is not possible.  Foxes were controlled at earths in all NGC regions, but though regional variation in the intensity of fox control at earths in spring has been documented elsewhere (Lloyd 1980; Heydon & Reynolds 2000a), the variable number of estates in each region did not permit comparison between them here. The number of occupied earths has been suggested as a measure of fox density assuming that each litter represents a social group, but it is an unreliable one (Sadlier et al. 2004).  To convert earth densities to fox densities, uncertain assumptions must be made about adult sex ratios and the proportions of non-breeding females and itinerant foxes in the population (Heydon, Reynolds & Short 2000).  Added to this is the problem of accurately censusing earths.  Difficulty in distinguishing a currently occupied earth from one recently abandoned can lead to overestimation of earth density, especially in areas where human disturbance causes vixens to move litters frequently, while underestimation can occur in some landscapes as it is difficult to find every available earth, particularly in rocky or densely forested areas.  The number of occupied earths is also closely related to food availability (Hewson 1986) and where this is subject to sudden change, i.e. in areas with cyclic rodent populations, foxes can modify their reproductive effort (Lindström 1988), resulting in a lower than expected number of occupied earths. 28  The other increase in the cull occurs in late summer due to lamping.  The influence of seasonal variation in effort on the lamping cull can be examined further due to the unique level of detail about lamping effort the FMS data contain.  In absence of effort data, an analysis would suggest the lamping cull increases due to fox density being higher in late summer.  With many juvenile foxes that are naive to the threat of man with a spotlight and rifle starting to disperse away from their natal territories, they are easier to shoot.  Most juvenile males disperse compared to a lesser proportion of females (Macdonald & Reynolds 2004), which seems a reasonable explanation for the increasing proportion of males in the cull.  Though this is undoubtedly a factor, the coincident increase in lamping effort during this period that follows the harvest and removal of cover may also make it easier to see foxes whilst lamping.  Lamping efficiency is therefore highest during late summer as there is a larger pool of foxes that are easy to detect.  An increase in lamping effort also occurred during the spring, presumably due to managers attempting to reduce predation during the nesting season, but the fox cull did not respond to this.  This is both because there are fewer foxes left to shoot following a winter of culling and that previous experience of being shot at and surviving is likely to have induced some avoidance behaviour (lamp shyness) in foxes.  These two factors result in lamping efficiency remaining low.  Since the FMS survey, the development of low-cost night-vision optics is likely to have led to improved efficiency at this time of year as spotlights do not need to be used as frequently. Compared to the number of foxes culled within a time interval, the sighting rate of foxes is a more useful index of fox density because it accounts for changes in effort over time.  It is therefore equivalent to the catch per unit effort that has long been used both as an abundance index and to estimate abundance (Seber 1982), but indexes the target population and not the attempt to control it.  The sighting rate of foxes on spotlight counts has often been used as an index of relative abundance (Sadlier et al. 2004) as it has been shown to agree closely with estimated fox density (Heydon, Reynolds & Short 2000).  The mean sighting rate pooled across estates shows a similar seasonal trend to the fox cull, but the coincidental trend in effort with the number of foxes seen means that there is less variation in the sighting rate compared to the fox cull, and the trend is smoother.  The changes in sighting rate during the year that are seen on most estates would appear to reflect the general 29  biological processes that cause changes in population density, reproduction in the spring causing it to increase in the summer once cubs become independent of the breeding earth and therefore visible while lamping (Baker & Harris 2008), followed by mortality during the autumn and winter that result in a gradual decline.  This seasonal trend was confirmed on a regional-scale by a study that surveyed fox populations in autumn and spring (Heydon, Reynolds & Short 2000).  The regional north-south trend broadly matches known density estimates based upon landscape types that dominate the different regions (Webbon, Baker & Harris 2004), though the sighting rate in East Anglia was higher than expected compared to other regions in England (Heydon, Reynolds & Short 2000). The use of sighting rate as an index of abundance implies assumptions about the nature of the relationship between sighting rate and fox population density.  In addition to being constant over time, this relationship is often assumed to have a linear form. In this form, the slope parameter is known as detectability, or the proportion of the population seen per unit of effort.  The relationship can in fact take on two other forms, a hyperstable form where the rate stays high as density drops, i.e. foxes become easier to see the fewer there are, or a hyperdepleted form where the rate drops much faster than density, i.e. foxes become harder to see the fewer there are (see Fig. 5.2 in Hilborn & Walters 1992).   The latter of these forms is possible, especially as the remaining foxes in the population after a period of lamping effort are likely to be lamp-shy.  Given a lack of data on seasonal farming practices on estates, the assumption of constant sightability during the year is likely to cause problems on estates where grass and crop cover change significantly during the year as it seems reasonable that sightability would be lower when cover is high, i.e. during the spring/summer growing season.  At a larger scale this might suggest that sightability could vary by region or landscape.  Sightability could also be affected by features of fox behaviour.  The sighting rate shows a small increase during December and January that, barring a large immigration event occurring on all estates, is unlikely to be due to a population increase.  One possible explanation is behavioural changes during the mating season that might result in foxes being less wary of lamps and so easier to detect (as they are distracted).  To determine the exact form of this relationship would require long-term field study.  This would require constant monitoring of the sighting rate during different seasons, with the use of tagging or similar 30  methods to estimate density together with culling (by methods other than lamping to avoid lamp-shyness) to reduce the density by a known amount. Detailed analysis of the snaring cull was not possible due to the lack of data on snaring effort (number of snares set per day).  The cull from snaring was relatively constant throughout the year, though a lower proportion of the snaring cull was achieved in February and March.  Snaring is most successful when there is cover in the form of crops or tall grass in which to disguise snares.  Though the conditions for successful snaring vary greatly during the year, it has been found that though the use of snares varies greatly between gamekeepers, the difference in seasonal use of those that do is less than might be expected based upon the difference in conditions, i.e. gamekeepers use a similar number of snares throughout the year (GWCT, unpublished data).  Assuming a similar pattern in FMS contributors’ effort means that snaring efficiency (foxes killed per snare) is lowest in the early spring, which coincides with when cover is at its most sparse.  However, early spring is also when lamping is least efficient and so low culling efficiency at this time of year is likely to be due to a combination of factors that reduce the number of foxes to be caught in snares or seen when lamping, which include: (1) reduced fox population density at this time of year following over-winter mortality, (2) increased territoriality of foxes during this period reducing the likelihood of itinerant foxes being present, and (3) the relative inactivity of nursing vixens prior to and after birth (Lloyd 1980).  This highlights the potential importance of controlling foxes at breeding earths during this time of year. The FMS contains an unprecedented amount of data on rural fox control effort across Britain, with the level of detail on lamping from individual estates making it very useful for modelling local scale population dynamics.  However, despite the quantity of information, the voluntary nature of the survey means that it does have some flaws.  These include the lack of regional representation in the data and the variation in length of contribution, which appears to decrease south-north.  The time and discipline required by managers to record data every day was not insignificant, which most likely explains why the number of estates contributing complete record book sets from January 1996 to August 2000 was relatively small.  The additional effort required to collect fox carcasses was clearly too great for many 31  contributors, meaning that the autopsied sample of foxes reflected the managers ease of carcass collection rather than a random sample.  Foxes killed by lamping are more difficult to recover as they are often shot from several hundred feet away, as shown by the low number of foxes killed by lamping during autumn and winter in the autopsied sample.  This meant that younger foxes were under-represented in the sample and so the age structure determined by the FMS was unreliable.  The collection rate might have been improved by using some form of incentive scheme to encourage collection of killed foxes, but given the large number of foxes culled this would have been prohibitively expensive. Prior to the FMS, the data available on rural fox populations in Britain was not of sufficient spatial or temporal resolution to allow modelling of the within-year population dynamics on estates.  The lack of data on culling effort further limited the potential modelling options.  At this time, gamekeepers would commonly evaluate their fox control efforts simply by the numbers of foxes killed in one year, i.e. fox bag data such as that forming part of the NGC, and comparing this bag to their fox bag in previous years and with their peers.  They would also consider the size of game bags that were achieved.  In the year before the FMS started (1995), the mean number of foxes killed on an estate was just over 3 foxes km-2 yr-1 (range 0-36 foxes km-2 yr-1, Figure 2.11).  The fox bags on 50% of the estates were fewer than 2 foxes km2 yr-1.  Gamekeepers might therefore argue, as a rule of thumb, that if they had removed more than this number of foxes, their fox control was relatively effective.  Since 1995, a number of predator removal studies have demonstrated that fox control can reduce fox density and achieved positive effects on game and other ground-nesting bird species by removal of large fox bags of over 5 fox km-2 yr-1 (Stoate & Leake 2002; Fletcher et al. 2010; Potts 2012), although others have achieved success with smaller bags of closer to 2 fox km-2 yr-1 (Tapper, Potts & Brockless 1996).  These studies serve to reinforce the rule of thumb that a bag of more than 2 foxes km-2 yr-1 was an indicator of effective fox control on their estate. On average, gamekeepers might therefore appear to be successfully controlling fox density using the rule-of-thumb.  Larger bags, in this understanding, would result from repeated replacement of culled foxes through immigration over the course of the year.  32  However, given a fox density of 1 fox km-2 (i.e., within the range of regional and landscape scale estimates; Heydon, Reynolds & Short 2000; Webbon, Baker & Harris 2004) and annual productivity of three cubs per fox, a bag of 3 foxes km-2 yr-1 is comparable only with annual productivity, without invoking immigration from outside the estate.  An annual cull of this magnitude is therefore on the scale of a sustainable harvest rather than the overwhelming mortality likely to control the population.  In the absence of an understanding about the effect of control on fox density on each restricted area estate within the year, the conclusion would be that control on most estates was actually failing to effectively reduce fox density. Of course, local differences in fox density, timing of the cull, and rate of replacement make this an incorrect conclusion.  If foxes are prevented from breeding locally, and are not rapidly replaced by immigration, culling at this level might be effective in reducing fox density.  This example further highlights the problems in using bag data to indicate success when there is no measure of culling effort or any indicator of fox density.  If there are few foxes to remove then the bag can only be small, even with high levels of control effort; the local fox population may nevertheless be strongly suppressed and the cull achieving its objectives.  It is therefore necessary to understand how fox density changes in response to culling effort.  The difficulty of estimating fox density on short time-steps using field methods means that this is only possible using population dynamics modelling.  The importance of the FMS in bridging the gap in data between intensive local studies and long-term survey schemes to provide fox sighting rate data for modelling fox populations at local scales therefore cannot be understated.  33  2.5 Tables Table 2.1.  The National Gamebag Census regional classification (Tapper 1992), detailing regional area and the proportion of each of these regions that Fox Monitoring Scheme estates covered. Name (code) Counties Area (km2) FMS estates FMS area [km2 (%)] SW England (1) Avon, Cornwall, Devon, Dorset, Somerset 17,720 7 41.3 (0.23) SE England (2) Berkshire, E Sussex, Hampshire, Kent, Middlesex, Surrey, W Sussex, Wiltshire 19,680 16 64.9 (0.33) E England (3) Cambridgeshire, Essex, Lincolnshire, Norfolk, Suffolk 22,170 26 197 (0.89) C England (4) Bedfordshire, Buckinghamshire, Hertfordshire, Leicestershire, Northamptonshire, Nottinghamshire, Oxfordshire, Warwickshire 16,360 5 35.2 (0.22) W Midlands (5) Gloucestershire, Hereford & Worcester, Shropshire, Staffordshire, W Midlands 13,690 2 21.9 (0.16) Wales (6) All 20,760 3 36.9 (0.18) NW England (7) Cumbria, Cheshire, Greater Manchester, Lancashire, Merseyside 16,810 2 13.0 (0.08) NE England (8) Cleveland, Durham, Humberside, N Yorkshire, Northumberland, S Yorkshire, Tyne & Wear, W Yorkshire 23,960 7 52.6 (0.22) E Scotland (9) Borders, Central, Fife, Grampian, Lothian, Tayside 26,740 3 58.5 (0.22) W Scotland* (10) Dumfries & Galloway, Highland, Strathclyde 41,300 4 13.8 (0.03) *area excludes the Orkney Islands, Shetland Islands and Western Isles of Scotland as foxes are absent from these areas (Harris et al. 1995)     34  Table 2.2.  Percentages of the total cull across FMS estates in each of the NGC regions made by different methods.  (N.B. some rows do not sum exactly to 100% due to rounding error) Region Lamping (%) Snaring (%) Earth – adults (%) Earth – cubs (%) Drives and traps (%) Other (%) Number of estates SW England 54.5 10.5 1.2 7.9 2.0 23.8 7 SE England 73.8 6.6 0.5 2.3 6.4 10.4 16 E England 61.0 10.2 2.0 10.9 5.0 11.0 26 E Midlands 60.1 13.3 0.3 5.7 3.8 16.8 5 W Midlands 57.9 1.1 2.1 12.6 10.5 15.8 2 Wales 59.8 10.8 3.9 12.7 1.0 11.8 3 NW England 41.4 31.2 9.6 7.0 8.3 2.5 2 NE England 61.9 8.5 2.6 7.5 1.6 17.9 7 E Scotland 36.5 23.9 5.3 22.8 3.3 8.1 3 W Scotland 76.2 8.9 1.0 4.0 0.0 9.9 4  35  2.6 Figures  Figure 2.1. Map of Great Britain showing the location (filled black circles) of the 75 rural estates that contributed data to the Fox Monitoring Scheme between January 1996 and August 2000.  National Gamebag Census regions (dashed lines) are shown (1 = SW England; 2 = SE England; 3 = E England; 4 = C England; 5 = W Midlands; 6 = Wales; 7 = NW England; 8 = NE England; 9 = E Scotland; 10 = W Scotland.  Urban areas (grey 1-km squares) were identified based upon the dominant land classification in each 1-km square from the Countryside Survey 2000 dataset using the Countryside Information System (CEH 2005). 36   Figure 2.2.  Monthly variation in mean annual number of foxes killed (pooled across estates) showing the composition of the cull by different control methods. 37   Figure 2.3.  Monthly variation in mean annual lamping data pooled across estates, showing a) the number of hours spent lamping; b) the number of foxes seen; c) the number of foxes killed; d) the number of foxes seen per hour (sighting rate); e) the number of foxes killed per hour (efficiency). 38   Figure 2.4.  Monthly sighting rate of foxes on FMS estates between January 1996 and August 2000.  Estates are labelled by NGC region code (see Figure 2.1), data are all shown on the same scale for comparison.  Blank months indicate either no contribution or zero lamping effort. 39   Figure 2.5.  FMS estates (filled black circles) showing the regional mean annual sighting rates (foxes seen per hour of lamping) across estates.  Greyscale shows sighting rate from low (light) to high (dark).  White indicates no data as foxes absent from these islands. 40   Figure 2.6.  Timing of lamping hours used relative to the hours of darkness in each month. 41   Figure 2.7.  Sex ratio of the 4547 adult and juvenile foxes culled across 75 estates between January 1996 and August 2000. 42   Figure 2.8.  Comparison of the monthly distribution of the total fox cull and the sample made available for autopsy by contributors.   Figure 2.9.  Comparison of the proportional use of different methods used to kill foxes between the total cull and the sample made available for autopsy by FMS contributors. 43    Figure 2.10.  Percentage occlusion of fox canine teeth pulp cavities as a function of the date of death in days from an assumed mean birth date of 1 April.  Dashed line indicates the percentage occlusion below which canine teeth are yet to develop the first cementum annuli.  44   Figure 2.11.  Data from the National Gamebag Census on the numbers of foxes killed on shooti