Linking modelling, monitoring and management: an integrated approach to controlling overabundant wildlife

Summary

1. Overabundant wildlife can cause economic and ecological damage. Therefore population control typically seeks to maintain species’ abundance within desired control limits. Efficient control requires targets, methods for estimating population size before and after control, and reliable means of forecasting population size. Demographic stochasticity, environmental variability and model uncertainty complicate these tasks. Monitoring provides critical feedback in the control process, yet examples of integrated monitoring and management are scarce.

2. We developed an integrated Bayesian population modelling and monitoring algorithm to assist with dynamic cull control of an overabundant population. We describe components of the control algorithm and their combination to produce a structured, sequential prescription for implementing control of a kangaroo population. We demonstrate its application within a single management year and evaluate its performance over a multi-year horizon under a range of scenarios reflecting uncertainties about population dynamics.

3. Simulation testing of the algorithm demonstrates that it provides a coherent, flexible, efficient and robust basis for managing population control. It is coherent in that connections between management objectives, models and operating rules are explicit and logically integrated. It is flexible in that the management objectives can be freely varied. It is both cost and operationally efficient because: (i) it avoids the need for an expensive, dedicated sampling process to estimate population size prior to culling; (ii) a relatively small number of culls produces reasonable population size estimates and (iii) the estimation by removal process enables direct assessment of whether control has been achieved. Lastly, it is robust because even when there is substantial uncertainty about system state and dynamics, the algorithm performs well at keeping the population under control over the duration of the management horizon.

4. Synthesis and applications. We provide a general and flexible framework for integrated monitoring and culling when the objective is to keep a species’ abundance within control limits. Our framework explicitly deals with uncertainty arising from demographic stochasticity, ecological complexity and lack of knowledge, and provides the foundation for maximizing efficiency and cost-effectiveness of control operations. Our approach could be applied in any instances where control is effected via culling.

Introduction

Overabundant native wildlife can cause economic and ecological damage. As these species generally have protected species status, total eradication from affected areas is neither feasible nor permissible. Instead, population control typically seeks to maintain species’ abundance within desired upper and lower control limits to avoid both overabundance and local extinctions. Population control management has much in common with harvest management (Shea & NCEAS Working Group on Population Management 1998). Control requires target setting of control limits while harvesting requires decision rules for allowable catch or effort, but both require methods of estimating population size before and after harvest/control and for forecasting population size in the next management cycle. These tasks are complicated by demographic stochasticity, environmental variability, partial observability and lack of knowledge. Monitoring is critical to the process, but examples of integrated monitoring and management are surprisingly scarce in wildlife control management. There are many possible reasons: (i) rigorous monitoring of population trends is often very expensive; (ii) ‘affordable’ levels of monitoring tend to have inadequate power for discerning changes of interest; (iii) standard statistical analysis and interpretations of monitoring data often fail to provide outputs that can directly inform management action; and (iv) managers and scientists are often unwilling to specify exactly how monitoring results will be used to alter management or policy (Maxwell & Jennings 2005; Nichols & Williams 2006).

In fisheries harvest management, Management Strategy Evaluation (MSE, also known as the Management Procedure approach) was developed and has been progressively refined since the 1970s to address such issues (Butterworth & Punt 1999; Smith, Sainsbury & Stevens 1999; Sainsbury, Punt & Smith 2000). MSE is conceptually and methodologically equivalent to Walters & Hilborn’s (1976)‘adaptive management’. Key ingredients of MSE include: (i) specifying clear management objectives; (ii) developing quantifiable performance measures for each objective and specifying how measurements will be made, analysed and used; (iii) identifying alternative management strategies or decision options; (iv) evaluating (using quantitative performance measures) the performance of each strategy/option against specified objectives, taking account of uncertainty and (v) communicating the results to decision-makers (Smith, Sainsbury & Stevens 1999; Sainsbury, Punt & Smith 2000). This list points to MSE’s foundations in decision analysis. Particular emphasis is placed on explicating the links between data analysis and decision-making, understanding the impact of uncertainty on achieving management objectives and rigorous evaluation of alternative strategies via simulation testing (Cooke 1999; Sainsbury, Punt & Smith 2000).

Taking these elements from MSE, we designed a control management framework within which management objectives can be achieved (population maintained within control limits), performance can be demonstrated with adequate precision, and monitoring data obtained in the process used to inform future management (sensuJohnson et al. 1997). Our integrated monitoring and management algorithm for sequential population control decisions is underpinned by a Bayesian population model, a Bayesian removal estimation model and a set of operating rules. First, we describe individual components of the control algorithm and explain how they are combined to produce a structured, sequential prescription for implementing population control. We demonstrate its use within a single management year, given parameter estimation uncertainty and then test its performance over a 20 year horizon under a range of scenarios designed to reflect environmental variation and errors in model parameter estimation arising from biased or high variance data. The context for these illustrations is the monitoring and management of western grey kangaroo Macropus fuliginosus Desmarest in Wyperfeld National Park of south-eastern Australia.

Methods and results

Western grey kangaroos in Wyperfeld NP

Wyperfeld NP (3570 km²) in semi-arid northwest Victoria features the ephemeral terminal lake system of the Wimmera River. The diverse vegetation communities in Wyperfeld include Eucalyptus camulduensis and Eucalyptus largiflorens floodplain woodlands, lakebed herbfields, Callitris gracilis and Allocasuarina leuhmanii woodlands, mallee scrub and sand plain heath. European settlement began around the 1840s, bringing changes such as displacement of the indigenous Wotjobaluk people, introduction of sheep and cattle grazing, alien pasture, and rabbits as well as severe reduction in dingo numbers. Grazing was concentrated around the lakebeds and floodplain and continued until the late 1960s (D. Morgan & P. Pegler, unpublished data).

Western grey kangaroos are the only macropodid species in Wyperfeld NP. After stock removal, their numbers appeared to increase substantially. In 1998, Parks Victoria commenced active management of the kangaroo population with the objectives of reducing grazing pressure, enhancing the regeneration of degraded floodplain vegetation and reducing the extremes of ‘boom-bust’ cycling in the kangaroo population (D. Morgan & P. Pegler, unpublished data).

The population model

Kangaroo population monitoring in Wyperfeld began in 1972. It has continued annually at approximately mid-year (prior to the main recruitment period between August–November) in all but three years (1974, 1989 and 1990). Monitoring is based on line transect distance methods with pairs of observers walking ≈50 km of transects through the floodplain mainly, but also including all major vegetation (habitat) types. Density estimates are made for each habitat type, combined using stratification methods to achieve an overall estimate, then multiplied by the study area to obtain a total population estimate (D. Morgan, unpublished data).

Kangaroo population dynamics models have generally concentrated on capturing the effects of intrinsic processes such as density dependence and environmental variables such as rainfall, a random but important driver of plant growth and hence food availability (Caughley, Bayliss & Giles 1984; Bayliss 1985; McCarthy 1996). We use a similar approach and assume that kangaroo density λ_t, varies according to a Ricker-type function with intrinsic growth rate a, effect of density dependence b, effect of rainfall c, and environmental variation ɛ:

• (eqn 1)

where N_t is the population density after the culling period in year t. R_t is the cumulative precipitation over 25 months from October in year t– 3 to October in year t– 1 (the rainfall interval determined to be important to population growth rate of kangaroos in Wyperfeld, D. Morgan, unpublished data). Environmental variation ɛ is represented by normal random deviates with mean 0 and variance σ². To incorporate demographic stochasticity, we assume that the predicted population density N_t+1 is drawn from a Poisson distribution: . The population model was fit to the 26 years of monitoring data, using non-informative priors and WinBUGS (Lunn et al. 2000) code as provided in Appendix S1 (Supporting Information). Model parameter estimates are presented in Table 1.

Table 1.   Parameter estimates for the Wyperfeld kangaroo population dynamics model based on annual density data from 1972 to 1997 (D. Morgan & Parks Victoria, unpublished data). Four chains; 5000 burn-in and 30 000 samples per chain. Convergence was assessed by checking time-series and autocorrelation plots and the Gelman-Rubin statistic

Parameter	mean	sd	95% credible interval
Intrinsic growth rate, a	0·621	0·231	0·178–1·09
Density dependence, b	−0·025	0·008	−0·042–−0·008
Rainfall, c	0·061	0·055	−0·044–0·173
Environmental variation, σ	0·264	0·074	0.137–0.427

Estimating population size by removal

Removal (or depletion) sampling is used for estimating the abundance of demographically closed animal populations (Schnute 1983). A population is sampled on separate occasions over a short period of time and on each occasion the animals observed are removed. The sequence of catch numbers (x₁, x₂, …, x_n) and corresponding catch-effort constitute the data to which models are fitted to estimate the initial population size. The remaining population size can be subsequently deduced. The main assumptions are that there is no immigration, emigration, birth or natural mortality over the sampling period (so that changes occur only through removal) and that the capture (removal) probabilities (p_i), can be reliably estimated.

Principal methods for analyzing removal data include regression of the catch against the cumulative catch (Leslie & Davis 1939; DeLury 1947; Ricker 1975), a multinomial model with maximum likelihood estimation (Carle & Strub 1978; Schnute 1983; Gould & Pollock 1997) and estimating functions/equations (e.g. Chao & Chang 1999; Wang 1999), using both frequentist and Bayesian approaches (e.g. Wyatt 2002).

The simplest models assume a closed population, equal effort expended on each sampling occasion and identical capture probabilities (p_i) for each individual in the population on each sampling occasion (Moran 1951; Zippin 1956). However, it has long been recognized that removal probabilities may vary over sampling occasions due to behavioural differences and selectivity of sampling technique or equipment. Data from removal sampling often exhibits overdispersion. Assuming a constant capture probability has been shown to lead to under-estimation of true population size and estimation variance (Bohlin & Sundström 1977; Mahon 1980; Schnute 1983; Wang & Loneragan 1996; Mäntyniemi, Romakkaniemi & Arjas 2005). Approaches to modelling unequal catchability include refinements such as constructing parametric functions to govern p_i’s (e.g. Schnute 1983) and treating capture probabilities as random variables following some parametric distribution (e.g. Wang & Loneragan 1996).

Here we adopt a Bayesian model developed by Mäntyniemi, Romakkaniemi & Arjas (2005). It assumes a closed population, equal-effort removal events and randomly distributed individual catchabilities in the population. Because heterogeneous catchability among individuals results in the more ‘catchable’ individuals being removed, the distribution of catchabilities in the remaining population changes after each removal event. This decline in mean catchability at each successive removal is explicitly accounted for in the model. Key model features are briefly described below (see Mäntyniemi, Romakkaniemi & Arjas (2005) for full details of model development).

Individuals in the closed population (k = 1, ..., N) respond independently to removal such that the catchability of an individual p_k, is an independent random draw from a probability density function f(p_k|μ, σ) with population-specific hyperparameters, mean μ, and standard deviation σ. The number of individuals captured in the jth removal event x_j, is binomially distributed given the population size N_j, and the mean catchability q_j of individuals during the jth removal event:

• (eqn 2)

After the jth removal event, N_j+1 = N_j - x_j individuals are left in the population.

As we expect that individuals with the highest catchabilities have been removed, the distribution of catchabilities in the remaining population has changed from the initial distribution and is now proportional to f(p_k|μ, σ)(1 - p_k)^j-1. By assuming that individual catchability p_k is initially distributed according to a Beta (α, β) distribution (a common choice for modelling proportions), we get the probability density function:

• (eqn 3)

where B(α, β) is the beta function . The distribution of catchability before the jth removal is obtained by multiplying equation (3) by (1 - p_k)^j-1. This produces a Beta (α, β + j - 1) density function with mean q_j = α/(α + β + j − 1).

Extra-Poisson variation in the sampling of initial population size N₁ due to spatial clumping is accommodated by assuming that N₁ follows a negative binomial distribution. This is implemented by compounding a Poisson distribution through drawing the ‘rate’ parameter Λ, from a gamma distribution. Λ is obtained as a product of kangaroo density λ_t and the size of the area of interest.

Estimating population size by iterative removal

We demonstrate the removal estimation process with a simple simulation. First, using the population model and an arbitrary rainfall value of 0.7 m we predict kangaroo density λ_t, and (assuming an area of 100 km²) compute an informative prior for Λ. Using the following arbitrary parameters: N₁ = 1900 and μ = 0.1, we simulate data for 10 removals (details in Appendix S1, Supporting information). The simulated catch sequence is 177, 159, 144, 152, 121, 101, 84, 112, 72 and 80.

To illustrate the iterative removal estimation procedure, we begin the analysis with data from the first three culls (and the informative prior for Λ), then progressively add data points from further culls while updating the prior for Λ at each iteration. MCMC sampling in the WinBUGS package was used to obtain posterior parameter estimates.

As shown by the means and widths of the 95% credible intervals for N₁, estimates of initial (and remaining) population size increase in both accuracy and precision with an increasing number of removals, though there are diminishing gains in precision with each successive cull (Fig. 1a,b). By the fifth removal, there is substantial narrowing of the 95% credible interval for N₁ and the remaining population size. By the sixth removal, these 95% credible intervals have halved in width compared to the start of the iterative procedure (Fig. 1).

Figure 1.  (a) Mean (Δ), median (•) and 95% credible intervals (−) for initial population size N₁ with increasing number of removals. Dotted line marks the true value of N₁. (b) True (▪) and estimated mean (Δ), median (•) and 95% credible intervals (−) for remaining population size with increasing number of removals. (See Appendix S1, (Supporting Information) for WinBUGS code).

Simulating cull management

The implicit objective of the kangaroo management policy is to allow the population to fluctuate according to environmental conditions and natural demographic processes while ensuring it does not exceed a threshold that would degrade vegetation condition, compromise vegetation regeneration and cause the population to become susceptible to mass die-offs during periods of environmental stress (P. Pegler, pers. comm.). The lower limit L, may represent a population size sufficient to ensure a level of regenerative capacity that avoids quasi- or local extinction while the upper limit U, may represent a population size below the limit of putative vegetation damage.

Pulling together the population dynamics model, management constraints and the removal estimation model, we designed a prescription for implementing population control within the timeframe of a single management year (Fig. 2). The main steps are as follows:

Figure 2.  (a) Population control algorithm for the kangaroo population in Wyperfeld NP, Australia. The population dynamics model takes data from the previous year’s population estimate and combines it with rainfall data for the current year to generate a population size prediction for the start of the management year (zone A). If the lower bound of the 95% credible interval of the prediction is greater than L and the upper bound of the 95% credible interval extends above U, the first cull treatment is implemented (zone B). This first cull treatment consists of three cull-effort units. The number of animals removed constitutes three removal estimation data points with which to update the information on population size and consequently decide if further culling is warranted (zone C). All subsequent cull events consist of just one cull-effort unit. The process is repeated until the lower 95% credible bound for the population size estimate intersects L or until the upper 95% credible bound falls below U (zone C). When this occurs, culling ceases for the management year. The population size estimate based on the final cull is then supplied back to the population dynamics model to be combined with rainfall data for the following year and used to generate a population size prediction for the following year (zone D). (b). Graphical representation of the rules described in (a). Horizontal dashed lines indicate the L and U control limits. Vertical intervals represent the 95% credible intervals for population estimates – the left hand cluster shows examples of circumstances under which culling may proceed; the right hand cluster show examples of situations where culling is prohibited.

• 1
   Estimate population size in the current management year (zone A, Fig. 2a).
• 2
   Decide if culling is warranted. This is based on the conditions shown in zone B, Fig. 2a and expressed schematically in Fig. 2b.
• 3
   Carry out the first cull treatment. Note that the first cull treatment consists of three cull-effort units. Using this data, compute the first removal estimate of the remaining population size and decide if culling should continue. (Subsequent cull events consist of just one cull-effort unit). Then iterate between culling and removal estimation until the remaining population is deemed to be within control limits (zone C, Fig. 2a,b).
• 4
   Obtain an end-of-cull population estimate that can be combined with environmental data (for the ensuing management year) to provide an estimate of the population size at the start of the following year (zone D, Fig. 2a).

If culling is warranted in any given management year, the first cull treatment is carried out and catchability is estimated anew for that year. Catchability (and population size) is then iteratively updated after each successive cull within the year. In this way, our estimation process ‘tracks’ catchability, allowing it to vary within and between years, while making no assumptions about the underlying drivers of variation.

We devised a simulation study to investigate how this population control algorithm might perform in a virtual world where, in addition to demographic and environmental stochasticity, there are substantial estimation errors in key model parameters. The three parameters we focused on were starting population density, and effects of density dependence and rainfall on population growth rate. Each of these parameters can be correctly estimated, under-estimated or over-estimated. The imperfect belief estimates were obtained by halving (for under-estimation) or doubling (for over-estimation) the starting population size and considering the lowest and highest biologically plausible values for the strength of density dependence, and the contribution of rainfall in mediating population growth. Combining the various states of knowledge about starting population density and effects of density dependence and rainfall yielded 27 unique scenarios for testing. They include a single scenario for which all three model parameters are correctly estimated, six scenarios in which two of the three model parameters are correctly estimated, 12 scenarios in which only one of the three model parameters is correctly estimated and eight scenarios in which none of the three model parameters are correctly estimated. The latter two scenario types represent compounding parameter uncertainty. Scenario parameterization is provided in Appendix S2 (Supporting Information) along with the R-code used to run the simulations.

We set a management horizon of 20 years. The hypothetical goal is to keep the population density within 5 (L limit) and 20 (U limit) kangaroos per km² in each management year throughout the 20-year period. We postulate that given the population dynamics model estimates in Table 1 and a rainfall regime comparable to the past 26 years, it is likely the population will spend substantial periods above the desired U limit in the absence of culling. As implied by the population control algorithm in Figure 2, the hypothetical manager demands at least 95% confidence that the population will not be culled to densities below L. Subject to this constraint, the manager aspires to achieve with 95% confidence, population densities below the U limit after cull management each year.

The performance of the algorithm in test scenarios is measured in terms of the frequency with which the ‘true’ population density is kept within control limits. Within the simulation, the ‘true’ population fluctuates according to the best estimates of population dynamics parameters for the Wyperfeld kangaroo population (provided in Table 1), based on the 26 years of monitoring data. We defined ‘control’ as being achieved if at the end of a management year the ‘true’ population was within desired limits. A secondary measure of the performance of the algorithm was the frequency with which the 95% Bayesian credible intervals for population density at the end of each management year captured the ‘true’ density.

The simulation proceeded as follows:

• 1
   Sample a set (trio) of model parameter states from the 27 possibilities.
• 2
   Apply the population control algorithm for 20 consecutive years, utilizing parameter estimates for the selected set of model parameter states.
• 3
   In parallel with step 2, population growth proceeds according to the ‘true’ model parameters (including stochasticity), though the actual population size is impacted by the culling activities undertaken using the control algorithm based on the ‘uncertain’ parameter estimates (step 2).
• 4
   Using the data from steps 2 and 3, tally the number of times out of 20 years that: (i) the population density was maintained within L and U control limits; (ii) the 95% Bayesian credible intervals for population density estimate captured the ‘true’ density.
• 5
   Repeat steps 2–4, 10 times for each of the 27 scenarios.

The population dynamics model includes demographic and environmental stochasticity even when ‘true’ underlying parameter estimates are used. Hence, the 10 replications of each ‘uncertainty scenario’ allowed us to test the control algorithm with multiple plausible population realizations. Implementation of a greater number of replications was precluded by the time taken to run each simulation. The consistency of the results indicate that 10 replications of each scenario was sufficient to characterize the variability in the outcomes of the population control simulations and to understand implications of imperfect knowledge about the population starting state and population growth processes.

Performance of the population control algorithm

As anticipated, total absence of culling results in the kangaroo density spending substantial periods above the U limit over the 20-year period (Fig. 3a). Based on our two evaluation metrics, the algorithm worked almost perfectly in keeping the true (post-cull) population density within control limits when the underlying population growth process parameters and starting population size are correctly estimated, even with substantial stochasticity in population growth (Fig. 3b).

Figure 3.  (a) Population density trajectories in the absence of any cull management over the 20 year management horizon. Open circles indicate modelled yearly population density estimates. The top two plots show example trajectories of individual simulation runs while the bottom plot shows results for population densities aggregated across 10 replicates of this scenario. Trajectories reflect the model parameterization described in Table 1 and rainfall patterns similar to that experienced in the past 26 years. Plots differ due to stochasticity in population growth. (b) Cull-controlled population density trajectories under perfect knowledge of system dynamics (notwithstanding stochasticity). The top two plots show example trajectories of individual simulation runs while the bottom plot shows all ‘post-control’ densities for 10 replicates of this scenario. Open circles represent population density prior to control in each year. Solid dots represent the true population density after control (if implemented in that year). Solid lines represent the best estimate of post-control population density with upper, lower 95% credible intervals. Dotted lines indicate L and U control limits in all plots.

In the absence of any cull management, less than 10% of 270 simulations were ‘in control’ for at least half of the management horizon (Fig. 4). In contrast, of the 270 simulations that tested the various combinations of under- and over-estimation of starting state and population growth parameters, 232 (86%) were kept in control by the control algorithm for at least 15 of the 20 years. Only three (1.1%) were out of control for more than half the management period (Fig. 4). The Bayesian 95% credible intervals captured the true population density in 81% of the 5400 (270 simulations × 20 years) management years.

Figure 4.  Distribution of the number of years for which population density was ‘in control’ over the 20 year management period, under complete absence of cull management (dashed histogram) and use of the control algorithm (shaded histogram). Results are based on 270 simulations (3 starting state estimates × 3 estimates of density dependence × 3 estimates of rainfall effects on population growth × 10 replications of each combination).

Under- or over-estimation of starting population density had little impact on the ability to keep the population within control limits (Fig. 5). When population dynamics parameters were well estimated, the population estimate generally converged on the true population value within 1 or 2 years irrespective of whether the starting population density estimate was half or twice the truth.

Figure 5.  Example population trajectories and Bayesian best estimates and 95% credible intervals representing belief about yearly population densities throughout the management horizon under various types of error including under- and over-estimates of starting population size (left-most column), effect of density dependence (centre column), and influence of rainfall on the population growth rate. Open circles represent the true population density prior to culling; solid circles represent the true population density after culling. Solid lines are the best estimates of post-control population density and Bayesian 95% credible intervals (after culling is complete, if implemented). Note that these plots are randomly selected from 270 (27 uncertainty scenarios × 10 replicates of each) possible plots that could be presented. Other error scenarios are not represented here for the sake of compactness, including the worst case scenarios of all parameters being simultaneously under- and over-estimated.

In comparison, imperfect estimation of population growth process parameters (density dependence and the rainfall influence on population growth rate) had a greater influence on the ability to control the population. Poor control arose when the effect of density dependence was over-estimated by 100%. Control was generally returned if the estimate of the strength of density dependence was within 50% of the truth (Fig. 5).

The impact of imperfect knowledge of rainfall influence on population growth rate was less important than imperfect knowledge of density dependence. A total of 50% under- and 100% over-estimation of the strength of the rainfall effect on population growth rate had minimal impact on population estimation and control outcomes (Fig. 5), especially if other parameters were correctly estimated. This is partly because rainfall exerts less influence on population growth than density dependence (within the range of precipitation experienced in the past 26 years; Table 1).

As a whole, the simulation results indicated that even in extreme cases where starting population density and both population growth process parameters were simultaneously under- or over-estimated, the population was maintained within control by the algorithm approximately the same number of times as in the other scenarios. The mean number of years within control for: (i) under-estimation of all parameters was 17.2 (sd = 1.03); (ii) over-estimation of all parameters was 15.5 (sd = 2.50); (iii) under-estimation of density dependence and over-estimation of other model parameters was 17.9 (sd = 2.99); and (iv) over-estimation of density dependence and under-estimation of other parameters was 17.3 (sd = 1.49). Results for all scenarios are summarized in Appendix S2 (Supporting Information).

Most ‘out-of-control’ events consisted of the population density breaching the preferred U limit. However, some simulations resulted in densities below the L limit. This generally occurred when density dependence was under-estimated, so that the population model produced predictions that were mistakenly greater than was in fact the case. This in turn, could have led to greater culling than was warranted. In 2% of the total simulations (6 of 270), the population was driven to extinction. Five of these were cases in which the density dependence effect was under-estimated.

Discussion

The first simulation study showed that the Bayesian removal estimation model, with its treatment of unequal catchabilities, produced reasonable estimates of the true starting and post-cull remaining population size in a single management year. Using a relatively small number of culls, the iterative process recalibrated our imperfect knowledge of population size after each cull and quickly converged on good estimates of the true population size even when there was substantial uncertainty in a priori belief about population size (Fig. 1).

Simulation testing of control algorithm performance for multi-year management found that when starting population density and underlying population growth process parameters are correctly estimated, the algorithm worked almost perfectly in keeping the true population density within control limits (Fig. 3b). This demonstrates the logical consistency of the integrated population model, iterative removal estimation process and operating rules.

The uncertainty sensitivity analysis indicates that the population control algorithm was generally robust to incorrect assumptions about starting population density and population growth process estimates (Figs 4 and 5). The population density stayed within control limits under most of the scenarios tested because of the robustness of the removal estimation process and the ‘self-correcting’ nature of our control algorithm. If the population model predicts a high population when it is in fact quite low, a cull event may be invoked but the numbers culled will be small compared to what was expected and the posterior population estimate shrinks rapidly toward the true value. Conversely, if the population model predicts a low population density when the population is in fact quite high, then culling is not implemented that management year. If environmental conditions are favourable, this leads to a predicted population increase in the following year and the population is usually then subject to cull treatment, at which time the number of animals culled leads to an upward adjustment of the population estimate toward the true value.

‘True’ population density (post-cull management) was captured by the Bayesian 95% credible intervals of the best post-cull density estimate in 81% of the 5400 management years. This discrepancy in coverage arises because of the systematic errors in starting population density and the effects of density dependence and rainfall in the various test scenarios (effectively in 26 of 27 scenarios). Despite this, the resultant coverage is reasonably close to ‘nominal coverage’. However, the fact that coverage is rather less than 95% has implications for the application of the culling rules in Fig. 2b. It suggests that greater caution is required in situations where the lower 95% credible bound is close to L– as depicted for example, by the vertical interval third from the left in Fig. 2b. If the true value lies close to the lower 95% credible bound and we have imperfect coverage, we could potentially drive the population to extinction in the first cull treatment of the following year (which consists of three cull-effort units).

We can safeguard against such an eventuality by conducting supplementary surveys to reduce uncertainty in population estimates prior to making a cull management decision. This additional information is also important if the true population is at low levels. In such situations, cull management is likely to cease after a just a few cull-effort units resulting in fewer data points for removal estimation and wide 95% credible intervals on the remaining population estimate. An example of this phenomenon can be seen in the last few management years of the population trajectory for the test scenario in which the density dependent effect is under-estimated (Fig. 5). These situations which trigger supplementary survey investment closely resemble those identified by Hauser, Pople & Possingham (2006) in their investigation of how frequently, managed populations should be monitored. Adopting such measures should help insure against a situation of poor estimation of process model parameters which can potentially lead to unintended population decline.

When these considerations are appropriately accounted for, the algorithm presented here provides a coherent, flexible, efficient and robust basis for managing population control in single and multi-year situations. It is coherent in that the connections between management objectives, models and operating rules are explicit and logically integrated. It is flexible in that the management objectives (L and U limits) and the desired level of confidence in achieving them can be easily and freely varied. It is both cost- and operationally efficient because: (i) it avoids the need for an expensive, dedicated survey to estimate population size prior to cull management (except in the conditions described above); (ii) a relatively small number of culls produces reasonable estimates of population size and (iii) the removal estimation process itself enables attainment of population control to be directly assessed. Lastly, it is robust in the sense that even when there is substantial uncertainty about system state and dynamics, the algorithm performs well at keeping the population under good control over the duration of the management horizon.

Although mainly applied to fisheries harvest management, we have shown that an MSE-like approach is highly relevant and can be profitably applied to control management. Given that culling is a widely used form of control for managing overabundant wildlife and pest populations [see e.g. Proulx (1997) on pocket gophers Thomomys talpoides; Blackwell et al. (2003) on red-winged blackbirds Agelaius phoeniceus], our population control algorithm offers efficiency gains (and cost savings) for managers who currently spend substantial amounts of money every year on monitoring and cull management. This is however, subject to two very important caveats: firstly, the removal estimation approach is only useful in the context of a coherent modelling and estimation framework such as the one described here; secondly, a rigorous calibration phase is required to ensure that the closed population assumption and other assumptions regarding population growth and density dependence are reasonable. This would involve a short-term investment in some ‘gold-standard’ independent monitoring such as ground, aerial or mark-recapture surveys.

In this paper, we have not attempted to develop a complete adaptive management framework for managing ‘control’ populations. We have instead focused on one aspect of adaptive management – the use of monitoring data, combined with a system dynamics model to understand system state, and identify appropriate management at each time-step. Our framework can be extended to include Bayesian updating of model parameters such as the strength of density dependence and this would allow iterative improvements in knowledge about system dynamics. This process could be undertaken as a Bayesian exercise in refining the parameters of a single model (see McCarthy & Possingham 2007), or by sequential updating of beliefs about the relative plausibility of competing system models (see Johnson et al. 1997; Johnson, Kendall & Dubovsky 2002). Relatively minor modifications to the algorithm (Fig. 2) would allow learning about population dynamics in each iteration of the management cycle. This was not undertaken here in order to maintain simplicity and focus on the removal estimation process, though it represents a logical improvement to our current algorithm that would provide some insurance against poor initial population model calibration. Allowing knowledge of process model parameters to improve over time would also facilitate detection of changes to population dynamics that may arise due to shifts in environmental or climatic processes.

In Wyperfeld NP in south-eastern Australia, controlling kangaroo numbers is a means objective ultimately in service of a fundamental objective related to vegetation management. A detailed analysis of the performance of kangaroo culling as a strategy for achieving a particular fundamental objective (such as enhancing vegetation regeneration) was beyond the scope of this paper. However, we recognize that evaluating the investment in kangaroo population control requires an explicit link to the fundamental objectives. The evaluation question should be ‘what does our culling regime contribute towards enhancing regeneration of key floodplain woodland species?’, rather than ‘are we keeping kangaroos between control limits?’. While the latter question needs to be answered on a regular basis to understand a key system variable (kangaroo densities), the basic question about how culling assists managers in achieving overall Park management objectives must be addressed before culling could be considered part of a coherent adaptive management strategy. Building a credible knowledge base in a true adaptive management fashion requires strategies for testing hypotheses about the relationship between means and fundamental objectives. This is particularly relevant to kangaroo management in Wyperfeld because there is evidence (see Tiver & Andrew 1997) which contradicts the notion that heavy grazing pressure by kangaroos impacts on tree and shrub regeneration.

We have shown that a MSE-like approach to developing a control framework that integrates modelling, monitoring and management activities can improve efficiencies and transparency in decision-making, whilst accommodating practical concerns such as the need for flexibility. Simulation testing of control rules and the impact of various types of uncertainty on achieving management objectives provides a useful basis for examining robustness in decision-making. This framework provides a potentially powerful approach for managing populations in a practical, coherent and robust manner and we hope that this work takes us towards a more complete adaptive management strategy for understanding and controlling overabundant populations.

Acknowledgements

This work was funded by Parks Victoria’s Research Partners Program. YEC was supported by ARC grant LP0667891. BW was supported by an ARC Fellowship and the Applied Environmental Decision Analysis CERF (Australian Government Department of Environment, Water, Heritage and the Arts). We thank John Wright, Phil Pegler, Lorraine Ludewigs, David Morgan, Michael McCarthy, Yakov Ben-Haim and Mark Burgman for assistance and discussions. Cindy Hauser, the editor and two anonymous reviewers provided critical comments that helped improve the manuscript.