Comparing population growth rates using weighted bootstrapping: Guiding the conservation management of Petrogale xanthopus xanthopus (yellow-footed rock-wallaby)

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Comparing population growth rates using weighted bootstrapping: Guiding the conservation management of Petrogale xanthopus xanthopus (yellow-footed rock-wallaby) Mark R. Lethbridgea,*, Peter J. Alexanderb,1,* a Geography, Population and Environmental Management, Flinders University, South Australia, Australia, GPO Box 2100, Adelaide, South Australia 5001, Australia b Department for Environment and Heritage, GPO Box 1047, Adelaide, South Australia 5001, Australia

A R T I C L E I N F O

A B S T R A C T

Article history:

In this study we compare estimates of the long-term mean stochastic population growth

Received 22 March 2007

rates, E[r] for Petrogale xanthopus xanthopus (yellow-footed rock-wallaby), a threatened Aus-

Received in revised form

tralian species. In fluctuating environments such as semi-arid areas, herbivore populations

17 September 2007

respond directly to changes in the biomass of food resources. Biomass is generally corre-

Accepted 23 September 2007

lated with rainfall, so it is often useful to model annual population growth rates of herbi-

Available online 16 May 2008

vores directly with rainfall. Models of this nature are referred to as numerical response models. The factors that are thought to threaten this species include competition from

Keywords:

introduced herbivores and predation from foxes. Annual aerial survey data collected from

Petrogale xanthopus

1997 to 2004 over approximately 600 km of transect line were analyzed in seven zones

Weighted bootstrap

within South Australia. Using the Ivlev numerical response model, the annual population

Population growth

growth rates were found to correlate best with the rain that fell in the seven-month period

Population models

immediately prior to the surveys. Not surprisingly, positive growth rates were found to be

Aerial survey

associated with higher rainfalls in this period, while negative growth rates were associated with lower rainfalls. We also used weighted bootstrapping to calculate confidence intervals around our estimates of long-term mean stochastic population growth rates, E[r]. The findings suggest that the estimates of E[r] are positive in areas where there is fox and herbivore management. However, we found no evidence that this species will decline in the absence of these treatments. Ó 2007 Elsevier Ltd. All rights reserved.

1.

Introduction

Petrogale xanthopus xanthopus (yellow-footed rock-wallaby) is a medium-sized macropod that inhabits the semi-arid rangelands of South Australia and New South Wales. Its conservation status is Vulnerable C2a (Australasian Marsupial and Monotreme Specialist Group, 1996). A second sub-species, P. x. celaris, occupies a smaller area in Queensland. The first

extensive research and regional field survey of P. x. xanthopus in South Australia was conducted in 1980 and 1981 by Copley (1983), with the primary aim of defining its geographical range. His comparison with historical data suggested that the species had declined since early European records. The factors that are thought to have contributed to this decline include competition from introduced herbivores (e.g. goats and rabbits) and predation, primarily from foxes (Lim and Giles,

* Corresponding authors: Tel.: +61 8 8201 5640; fax: +61 8 8201 3521. E-mail addresses: [email protected] (M.R. Lethbridge), [email protected] (P.J. Alexander). 1 Tel.: +61 8 8204 1910. 0006-3207/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.biocon.2007.09.026

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1987; Lim et al., 1987; Copley and Alexander, 1997; Hornsby, 1997). Despite these declines, the geographic range of P. x. xanthopus extends over a broad area of the rangelands of South Australia. Commencing in 1992, the Department for Environment and Heritage South Australia developed a conservation management strategy aimed at the recovery of P. x. xanthopus, supported by a number of organizations and landholders. The strategy endorses integrated pest management using 1080 baiting, spotlight and aerial shooting, mustering and warren ripping (De Preu and Pearce, in litt.). The Department for Environment and Heritage has conducted monitoring of P. x. xanthopus on a regional scale in South Australia for over a decade. Aerial surveys of P. x. xanthopus are preferred to ground-based monitoring techniques because of the high costs associated with field surveys and the relatively inaccessible country in which P. x. xanthopus is found. In this study we compare estimates of the long-term mean stochastic population growth rates, E[r] in areas receiving pest management treatment and untreated areas. In fluctuating environments such as semi-arid areas, herbivore populations respond closely to changes in the biomass of food resources. This is usually correlated with rainfall (Bayliss, 1987; Cairns, 1989; Copley and Alexander, 1997). This relationship has been found to be non-linear. The key food resources of P. x. xanthopus include chenopods and succulent round or flat leafed plants with stellate trichomes: Abutilon leucopetalum, Sida petrophila, Ptilotus obovatus, P. sessiliflorus, Solanum sturtianum, S. petrophilum and S. ellipticum. Forbs, browse and grasses are also consumed (Copley and Robinson, 1983). The accuracy of aerial surveys for estimating abundance or population trends has been the subject of discussion in much of the literature (Caughley, 1974, 1977; Caughley et al., 1976; Hill et al., 1985; Pollock and Kendall, 1987; Hone, 1988; Stoll et al., 1991). The factors that reduce aerial sightability include vegetation density (Caughley, 1974; Caughley et al., 1976), the time of day (Caughley, 1974; Hill et al., 1985; Short and Hone, 1988), observer experience, aircraft height and speed, light, turbulence, wind, shadow intensity (Caughley et al., 1976; Short and Hone, 1988), and habitat type (Short and Bayliss, 1985; Short and Hone, 1988). Aerial sightability is negatively biased and only a factor of the animals are ever seen from the air (Caughley, 1974; Caughley et al., 1976; Southwell, 1989). Lim et al. (1987) attempted to calculate an aerial sightability factor for P. x. xanthopus. This work was carried out in Middle Gorge in the southern Flinders Ranges of South Australia. They estimated the correction factor to range between 4.61 and 5.85. Unfortunately, the Lim et al. (1987) study was conducted over a relatively short transect distance (less than 4 km) and is not likely to be reliable over a broad area of diverse landforms. However, a key influence on the power to detect population trends over time lies in the variability of this factor from year to year (Gibbs, 2000). While Tracey et al. (2005) found variations in aerial sightability for a range of medium-sized mammals between survey periods, these variations were largely seasonal. In our study the population growth rates were derived from raw counts collected at the same time each year. The aerial census conditions were as close as possible to con-

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stant at each site (i.e. weather, aircraft speed, observer capability), thus we have assumed the proportion of animals seen in each survey was constant.

2.

Materials and methods

2.1.

Study area

The study area consists of approximately 600 km of transect line in the rangelands of South Australia. The dominant land use in the rangelands is pastoralism. Occupying a much smaller area are three National Parks and several private Sanctuaries (De Preu and Pearce, in litt.) The study area was divided into seven zones according to different land systems or land management regimes. Fig. 1 shows the location of the study area and the aerial survey routes analyzed in this study. Plumbago Station and Bimbowrie Station are in the Olary Hills and Hawker, Depot Flat, Bunker North Range, Wilkawillina Gorge and the ABC Range are in the Flinders Ranges (De Preu and Pearce, in litt.). Rainfall in the rangelands is variable with annual falls between 250 mm and 300 mm. In elevated areas of the central and southern Flinders Ranges (approximately 750–1000 m above sea level), annual falls of up to 500 mm are experienced. Rainfall is generally unreliable with infrequent high rainfall events (Bureau of Meteorology, 2004). Vegetation is characterized by Eucalyptus camaldulensis drainage lines with and without Melaleuca glomerata. Mallee woodland communities consist of E. socialis, E. dumosa and E. flindersii. Casuarina pauper woodlands are commonly found in small pockets on mid-slope areas. Callitris glaucophylla woodlands are common in the Flinders Ranges. Acacia victoriae shrublands often include Dodonaea spp. and/or Senna spp. Low shrublands may consist of Atriplex vesicaria, Maireana astrotricha, M. sedifolia, M. pyramidata or Chenopodiaceae spp. Rock plants in the steeper rocky slopes include Sida petrophila and Ptilotus obovatus. Triodia spp. is common to this region on open mid-slope to steeply sloping areas (Brandle et al., in litt.). Rabbit, fox and goat control have been in place in Wilkawillina Gorge and the ABC Range since 1993 and the treatment program is ongoing. These zones fall inside the Flinders Ranges National Park. Goat and fox control also commenced on Plumbago Station in 1993 and was maintained throughout the study period. On the neighboring Bimbowrie Station, fox control was the only consistent treatment until the property was purchased to become a National Park in 2004, after which fox and goat control have been maintained. Occasional mustering of goats and limited fox control has taken place to the south of the Bunker North Range zone. The Hawker and Depot Flat zones primarily consist of pastoral properties with no known pest treatment. These zones were originally flown as control sites (De Preu and Pearce, in litt.).

2.2.

Survey methodology

Aerial surveys were conducted annually from 1997–2004 in a Bell Jetranger helicopter on clear and still mornings in June between 7:00 and 11 hours. This is when the animals are more consistently sighted because they converge on high

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Fig. 1 – Location of the study area and the aerial survey zones.

points in the terrain to catch the early morning sun. Conversely, in warmer weather they are widely dispersed and take refuge from the heat in caves. The helicopter flies close to ridgelines and cliff faces at speeds of 74–93 km h1. Both clockwise and anticlockwise routes are flown in an attempt to remove sightability bias due to the direction of approach. A computer-based aerial survey system was used for each survey. The system consists of four components: three data entry keypads (for three observers) and a junction box that links the keypad information to a GPS and computer. The keypads are strapped to each observer’s leg. A computer software application called ‘Flystat’ (Lethbridge and Walkom, 1996) records and simultaneously displays all entries for the three observers together with GPS positions. Two experienced observers are located on opposite sides of the aircraft (forward left and aft right) and a trainee observer is located behind the forward observer.

2.3.

Annual population growth rates and rainfall

The annual population growth rate r, is calculated using Eq. (1).   Ntþ1 ð1Þ r ¼ ln Nt Here, Nt+1 is the abundance at time t and Nt+1 is the abundance at time t + 1. As previously mentioned, only a fraction of the animals are ever seen during the aerial survey, thus indices of abundance were used instead of abundance estimates in Eq. (1). These were calculated as the maximum of observer counts (excluding the trainee) of the clockwise and anticlockwise route for each four-kilometer segment of the route. The maximum count of the clockwise and anticlock-

wise route in each segment has been found to be a more stable measure than the mean. Plumbago Station and Bimbowrie Station were not surveyed in 2001. Consequently the population growth rates between 2000 and 2002 could not be derived for these zones. Numerical response models describe the effect of resources on the population growth rate of a species (Cairns and Grigg, 1993; Caughley and Sinclair, 1994). In semi-arid areas of Australia it has been useful to model the population growth rates of herbivores directly with rainfall because it is difficult to collect long-term information about available food resources, compared with the accessibility of rainfall data. For example, Caughley et al. (1984) modeled kangaroo counts with six-month lagged annual rainfalls using a second-order polynomial and obtained reasonably good model fits of R2 = 0.42–0.60 for Macropus rufus, M. giganteus and M. fuliginousus. However, the difficulty with first-, second- and third-order polynomial models is that little biological meaning can be derived from the percentiles. Alternatively, Sharp and Norton (2000) used the Ivlev (1961) numerical response model to correlate rainfall with population growth rates of P. x. xanthopus derived from 11 years of aerial survey data in New South Wales. They also found the population growth rate to be correlated with six-month lagged annual rainfalls (R2 = 0.43). Cairns (1989) tested a range of numerical response models for kangaroo populations and lagged rainfalls. Interestingly, Cairns found regional differences in the length of rainfall lags when comparing kangaroo responses in New South Wales and South Australia. Following Sharp and Norton (2000), in this study the Ivlev (1961) numerical response model Eq. (2) was used to model annual population growth rates in relation to rainfall. This is also known as the Mitscherlich model (Bayliss, 1987).

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r ¼ a þ cð1  edV Þ

ð2Þ

The response r is the annual population growth rate of the species, a is the maximum rate of decrease of a population in the absence of resources, c is the rate at which a is ameliorated by increasing the amount of resources and d is a measure of the demographic efficiency of the population (Bayliss, 1987). Demographic efficiency may be thought of as the ability of a population to increase when resources are in short supply (Caughley and Sinclair, 1994). The variable V represents either resource abundance or, as in this case, rainfall. Importantly, the model asymptotes to a maximum rate of increase rm, where rm = ca. The annual growth rates were tested for their fit to wide range of candidate lags and monthly rainfall periods. Monthly rainfall data were obtained from the Bureau of Meteorology ‘Climate Data Australia’ database (2004). These data were recorded at rainfall gauges nearest to the centre of each of the survey zones. Eq. (2) is non-linear, hence the maximum likelihood distributions of parameters a, c and d are non-linear. This is called parameter-effects non-linearity. The extent of non-linearity in the model and data combination is known as ‘intrinsic non-linearity’ and when this is high, an acceptable convergence to a single set of parameter estimates may be difficult. However, provided intrinsic non-linearity in the model is low, model non-linearity can be reduced by re-parameterization techniques (Ratkowsky, 1983). Here parameters a and c were transformed to the exponents ea and ec respectively. Because parameter d is known to be between 0 and 1, in order to keep it positive Ratkowsky (1983, p81) suggested the transformation e(0d). Including transformed parameters, Eq. (2) became Eq. (3). r ¼ ea þ eb ð1  eðe

ð0kÞ VÞ

Þ

ð3Þ

Here a = ea, c = eb and d = e(0k). Optimal estimates of the ^ and ^k, were then deterparameters a, b and k, denoted ^a, b mined iteratively using a maximum likelihood approach. Finally, estimates of the original parameters denoted ^a, ^c and ^ and ^k, using the relad^ were derived from the estimates ^a, b ^ ^a ð0^ kÞ b ^ . tionships ^ a ¼ e , ^c ¼ e and d ¼ e

2.4.

Weighted bootstrap

Bootstrapping can be used to calculate confidence limits for population parameters without making assumptions about the distributional properties of the data (Efron and Tibshirani, 1993; Manly, 1997; Indurkhya et al., 2001). The method uses the sample frequency to estimate the underlying probability distribution of the whole population (Manly, 1997). This involves randomly drawing a number of subsets from the sample population with replacement. The re-sampling process is then continued a large number of times and each bootstrap subset is separately modeled. The distributional properties and confidence intervals are then derived from the resulting set of model parameters. In our application, however, each sample has an unequal probability of occurring because the frequency of rainfall is not uniformly random. We therefore used an approach called weighted parametric bootstrapping (Tsai, 2002). This

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acknowledges that some samples are more likely to occur than others and it reduces the likelihood of outliers, particular in small sample sizes, biasing the parameter distributions. Using historic rainfall data, the samples were drawn according to the probability that the associated rainfall event will occur. The properties of the resulting parameter distributions obtained by bootstrapping are usually unknown. To deal with this, Efron (1979) introduced a method where the percentiles at either end of the distribution are used to obtain the confidence interval. However, a biased parameter distribution may result, where the median of the distribution does not agree with the original parameter estimate (Manly, 1997). For this reason, Efron (1981) proposed a bias-corrected and bias-corrected accelerated percentile method of calculating confidence intervals. Indurkhya et al. (2001) found this method performed well with outliers. Meyer et al. (1986) also trialed jackknifing and bootstrapping to estimate the sample distribution of rates of increase based upon both simulated and real demographic data. They used bias corrections to reduce bias in bootstrapping, leading to almost identical results between the two methods. Following Chaudhary and Stearns (1996), the bias-corrected percentile technique was used in this study. For j bootstrap replicates ^hj , the median bias ^z adjustment for ^h is given by Eq. (4). ^z ¼ U1 ½Q

ð4Þ

Here U is the standard normal distribution function and Q is the proportion of ^hj bootstrap replicates that are less than the parameter estimate (Briggs et al., 1999). The bias-corrected confidence intervals were then obtained from the distribution at the percentiles using Eq. (5) and Eq. (6). LowerPercentile ¼ ð2^z  za=2 Þ

ð5Þ

UpperPercentile ¼ ð2^z þ za=2 Þ

ð6Þ

Here za=2 ¼ U1 ða=2Þ is the cumulative distribution of the standard normal variate. If the median of the bootstrap parameter distribution is equal to the parameter estimate, then Q = 0.5 and ^z = 0, hence no percentile adjustment will take place (Briggs et al., 1999). A bias-corrected percentile algorithm by Buckland (1985) was incorporated in the bootstrap algorithm described earlier. Supporting subroutines by Beasley and Springer (1977) and Hill (1973) were also used.

2.5.

Long-term mean stochastic population growth rates

It is not possible to compare mean growth rates or use ANOVA tests to compare growth rates in treated and untreated populations, because the response of each population to rainfall varies depending on the rainfall pattern and land system. In other words, there are other covariates affecting these growth rates. When the rainfall frequency distribution is known, a better approach is to estimate the long-term mean stochastic growth rate of a population, denoted E[r] (Davis et al., 2003). The analytical solution described by Davis et al. uses a moment generating function for the rainfall following a gamma distribution. This is given by Eq. (7). Here the sample mean and variance of the rainfall gamma distribution are denoted as l and r2

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0

1  l2 dr2 r2 A @ E½r ¼ a þ c 1  1 þ l

ð7Þ

Ogutu et al. (2006) tested the precision of herd size and abundance estimates for strip and line transect aerial surveys using bootstrapping. They analyzed the variation in precision by re-sampling with different bootstrap sample sizes. Similarly, we varied the number of bootstrap re-samples n, representing the number of repeat surveys, to estimate a range of different confidence intervals. Field et al. (2007) calculated the trajectory of statistical power for woodland bird surveys in South Australia. They used five years of data to calculate how the statistical power would change in the following five years, to determine whether continuing the monitoring was a sensible investment. Likewise, we increased n beyond the current number of surveys to explore the confidence intervals that might be obtained if the surveys were continued. We used bootstrapping to re-sample the raw data and simulate the likely gains in continuing the aerial surveys over a longer period. This assumes that the data already sampled adequately represents the true distribution.

3.

Results

Table 1 shows the annual population growth rates. We tested a range of candidate monthly rainfall intervals and lags for their fit to these growth rates. The vertical axes in Fig. 2 show the R2 fits of Eq. (2) for each zone. The horizontal axes show the rainfall intervals and lags (in months) that were tested. Although variable in some zones, the rainfall in November to May tends to best fit the population growth rates. This is particularly evident in Fig. 2b–d and f. For consistency, we decided to adopt the seven-month rainfall period immediately prior to the aerial surveys (i.e. no lag). Table 2 shows the final parameters (±SE) and R2 values for each zone using this rainfall interval. Not surprisingly, the standard errors for parameters a and b are inflated because the two parameters are highly correlated. Fig. 3 shows the final fitted Ivlev models using this rainfall interval and the maximum counts from the two flights each year in June. The solid vertical line represents the mean rainfall encountered over the entire period that rainfall had been recorded, while the dashed vertical line represents the median rainfall. These lines can be compared with the rainfall for which r = 0. In all cases the rainfall intercepts where r = 0 are below the median or mean rainfall calculations.

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However, the relationship between the rainfall intercept and the median rainfall still fails to consider the long-term rainfall patterns (Davis et al., 2003). The mean stochastic population growth rate E[r] is a better measure of long-term population trends because it integrates all values of r across the known rainfall distribution. In this study, the rainfall was found to have a gamma distribution. Following Davis et al. (2003), Fig. 4 shows estimates of E[r] with the lower 5 and upper 95 bias-adjusted percentiles. Here, the population in the Hawker zone is more likely to decline in the long-term. However, the upper 95 bias-adjusted percentile suggests this may still not be significantly different from zero. The 5 and 95 percentiles for Depot Flat include E[r] = 0, suggesting the population may be static, but could feasibly be increasing or decreasing. All other zones indicate stronger positive population growth rates except the ABC Range, where the results are uncertain because the confidence interval includes E[r] = 0. The likely gains in continuing the aerial surveys over a longer period are shown in Fig. 5. Except for Bimbowrie Station and Plumbago Station, the results are based upon seven rates of population growth derived over eight years. This is because two years of counts are required to derive one population growth rate estimate. Bimbowrie Station and Plumbago Station were not surveyed in 2001 and consequently only have five measures of population growth rate derived over seven years. Provided the current Ivlev (1961) model is a good approximation of the population’s response, Fig. 6 suggests that continuing the Hawker zone another year to obtain eight measures of the population growth rate may narrow the distribution to be confident that the animals are declining (a = 0.10). However, even after conducting 11 years of surveys in the Depot Flat zone, there may still be no way of knowing if that population is increasing, static or decreasing.

4.

Discussion

A range of studies have concluded that both predation from foxes and competition from introduced herbivores have led to the decline of P. x. xanthopus in Australia (Copley, 1983; Lim and Giles, 1987; Lim et al., 1987; Copley and Alexander, 1997; Sharp and Norton, 2000). Fox predation was also found to be a significant factor in the decline of P. lateralis populations in Western Australia, where Kinnear et al. (1988) monitored five P. lateralis populations for six years after

Table 1 – Annual population growth rates calculated over the study period for each zone. Data provided by the Department for Environment and Heritage South Australia Year

Hawker

Depot Flat

Plumbago Station

Bimbowrie Station

Bunker North Range

Wilkawillina Gorge

ABC Range

1998 1999 2000 2001 2002 2003 2004

0.173 0.399 0.418 0.758 0.925 0.095 0.049

0.462 0.713 0.359 0.083 0.509 0.080 0.620

0.185 0.697 0.492 No data No data 0.382 0.075

0.036 0.000 0.981 No data No data 0.163 0.856

0.875 0.588 0.847 0.693 0.127 0.483 0.629

0.000 0.109 0.565 0.179 0.500 0.359 0.835

0.000 0.000 1.196 0.282 0.017 0.594 0.611

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Fig. 2 – Three-dimensional surface of the Ivlev model (1961) fit with a range of monthly rainfall intervals and lags for 1997– 2004 in each zone: (a) Hawker, (b) Depot Flat, (c) Plumbago Station, (d) Bimbowrie Station, (e) Bunker North Range, (f) Wilkawillina Gorge and (g) ABC Range.

implementing fox control at two of the sites. The results included population increases at the two treated sites of between 138% and 223%. Two of the untreated sites experienced declines; 14% and 85%, and one untreated site recorded an increase of 29%.

Positive population responses from P. x. xanthopus can be linked to fox and goat control, with the exception of the ABC Range zone. However, there is no evidence to conclude that P. x. xanthopus populations will decline in the absence of these treatments.

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Table 2 – Final parameters (±SE) and R2 values from the Ivlev (1961) numerical response model for each zone using a seven months rainfall interval not lagged. A SE could not be determined for parameters marked * Zone

a

c

d

R2

Hawker Depot Flat Plumbago Station Bimbowrie Station Bunker North Range Wilkawillina Gorge ABC Range

5.35 ± 2.99 2.40 ± 2.97 0.76 ± 1.81 1.57 ± 0.35 1.58 ± 3.63 1.29 ± 2.55 0.55 ± 2.17

5.58 ± 2.97 2.96 ± 2.51 1.28 ± 1.68 3.30 ± 0.50 14.50 * 31.40 * 1.42 ± 1.77

0.0250 ± 0.007 0.0125 ± 0.007 0.0302 ± 0.013 0.0081 ± 0.004 0.0010 ± 0.048 0.0003 ± 0.043 0.0082 ± 0.047

0.41 0.78 0.55 0.98 0.38 0.43 0.22

4.1.

Rainfall period and lag

Sharp and Norton (2000) found that 12 months of rainfall lagged by six months correlated with population growth rates in New South Wales. Conversely, our South Australian study found the seven months of rain that fell between November and May to be correlated with annual population growth rates. Cairns (1989) and Cairns and Grigg (1993) observed similar regional differences in the length of rainfall lags when comparing kangaroo responses in New South Wales and South Australia. They found kangaroo population growth rates in the South Australian pastoral zone (including the rangelands) to be correlated with summer to autumn rainfalls, six months lagged by three months, compared with Caughley et al. (1984) and Bayliss (1985, 1987), who found rainfall lags of six to 12 months to better correlate in New South Wales. This concurs with Noble (1977) conclusion that the biomass in South Australian pastoral zones tends to coincide with higher summer rainfalls rather than winter rainfalls. While P. x. xanthopus is thought to breed all year round, a high incidence of parturition generally occurs between December and March every year (Poole et al., 1985). This coincides with the driest period of the year in this region and the competition for food resources between herbivores is high. As such it is likely that the annual population growth rates correlate strongly with rainfall in the seven months immediately prior to the aerial survey because good rainfall over the summer period resulted in sufficient food resources for the juvenile animals to survive and be observed in the following June survey. In semi-arid areas the relationship between the mean or median rainfall and the rainfall intercept (r = 0) has less ecological meaning because a population may decline faster in response to more frequent low rainfall events than it recovers in response to the less frequent higher rainfalls (Davis et al., 2003). This will depend upon the frequency distribution of rainfall events and the slope of the Ivlev curve, particularly in the region where r < 0. For this reason we prefer to use the long-term mean stochastic growth rates E[r]. Plumbago Station and Bimbowrie Station have estimates of E[r] = +0.36 and +0.32 respectively, both indicating the population is likely to increase with the current land management. The former is a working sheep station and has been subject to goat and fox control since 1992. Bimbowrie Station has also had consistent fox control since 1992, and extensive goat control commenced in 2004. Apart from the ABC Range

zone, all of the treated areas in the Flinders Ranges show E[r] to be significantly different from zero (a = 0.10). There is insufficient evidence to suggest that the populations in the Hawker zone will change in the long-term. Both the Hawker and Depot Flat zones are in a region where there is a mixture of different management practices; limited localized fox control and opportunistic goat control. However, even if E[r] remains at zero, many local populations along these ranges have been observed to be small and isolated, and may suffer from inbreeding depression or stochastic environmental events (Boyce, 1992; Lacy, 1993). Increasing the number of bootstrap re-samples n to see how the confidence intervals of E[r] might change if the aerial surveys were continued for up to 11 years yielded some useful insights for the management program. Uncertainty in the estimates for the ABC Range is large and more than 11 years of surveys would be required before evidence of the population’s response is clear. Conversely, at all other sites the diminishing returns in the confidence interval for additional surveys after approximately nine years would not justify their continuation. After eight to nine years the Hawker zone may reveal an estimate of E[r] significantly different from zero. This may not be the case for Depot Flat and the population may remain at current numbers well into the future, unless other effects such as isolation and inbreeding depression take hold. The uncertainty associated with estimating population trends results from the combined effects of process variation and observation errors (Hilborn and Mangel, 1997). However, in this study we did not have sufficient replicate samples at each site to decipher the relative contribution of these two errors. As both authors have participated in the aerial surveys, we are in a position to make some general observations about the likely source of some of this uncertainty. The narrow confidence intervals associated with Bimbowrie Station are likely to be partly due to the relatively open habitat and good visibility associated with this zone. Conversely, the ABC Range zone includes data collected at the nearby Heysen Range and Brachina Gorge in the Flinders Ranges National Park. This zone’s wide confidence intervals are thought to be associated with the steep rocky terrain and habitat complexity providing relatively low visibility. If anything, these simulations suggest there may be no value in surveying zones like the ABC Range to ascertain population trends to guide management. As previously mentioned, we divided the geographical range of the species into these zones according to changes in land management practices or broader changes in the land

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Fig. 3 – Comparison of Ivlev plots for each site 1997–2004, showing a seven-month rainfall interval with zero lagged months and the maximum count approach: (a) Hawker, (b) Depot Flat, (c) Plumbago Station, (d) Bimbowrie Station, (e) Bunker North Range, (f) Wilkawillina Gorge and (g) ABC Range. The dashed vertical bar represents the median rainfall over the sample period. The solid vertical bar represents the median rainfall over the full period of rainfall recording.

systems. However, our estimates of E[r] represent the overall trends of the entire population in each zone, when in fact some local populations within a zone may be declining and others increasing. This suggests that it may also be important

to monitor changes in the distribution of local populations. While it might be tempting to subdivide the zones into smaller areas in order to estimate localized trends, given that only a fraction of the animals are ever seen, the sampling effects of

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Fig. 4 – Long-term mean stochastic growth rates in treated and untreated zones. Error bars represent 90% confidence intervals.

Fig. 5 – Simulation of the confidence interval (from the 5 percentile to 95 percentile) versus years of survey effort.

monitoring smaller populations using this approach may increase the uncertainty at this scale, even if the number of repeat surveys each year were increased. The Bunker North Range zone is immediately adjacent Wilkawillina Gorge zone and receives limited goat and fox control in its southern-most extent abutting Wilkawillina Gorge. It is perplexing that this zone has an unusually high estimate of E[r]. Sharp and Norton (2000) suggested that for this species the intrinsic rate of increase could be as high as 0.79. However, using gestation and pouch life data from Poole et al. (1985), a more conservative estimate of 0.51 was derived. Interestingly, E[r] = 0.63 in the Bunker North Range indicates that the average long-term population growth rate is somewhere near the intrinsic rate of increase, and that at times

the rate of increase might be higher than the intrinsic rate. This may be due to demographic stochasticity, i.e. variations in age structure, fecundity and survival. For example, Cairns and Grigg (1993) suggested that unstable age distributions and lower population densities might have been responsible for some of the unusual predictions in their model from one year to the next. Another conceivable explanation is there has been movement from the Wilkawillina Gorge zone into the North Bunker Range zone near the boundary.

4.2.

Model fit

Some of the Ivlev (1961) models fit the data poorly and despite using bootstrapping to re-sample the estimates of E[r], we

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Acknowledgements Thanks to the Department for Environment and Heritage, South Australia (Science and Conservation Directorate) for the provision of population data. Operation Bounceback, a project jointly funded by the Natural Heritage Trust and the Department for Environment and Heritage, South Australia, provided funding for this research. The Department for Environment and Heritage, South Australia also supplied topographical data. Many individuals and organizations have contributed to the management controls and monitoring efforts required to undertake this study. We thank you all.

Fig. 6 – Simulation of the actual 5 percentile to 95 percentile versus survey effort for Hawker (dashed line) and Depot Flat (solid line) zones only.

have still treated some of the models with caution. For example, Bunker North Range, Wilkawillina Gorge and ABC Range had relatively low R2 values of 0.38, 0.43 and 0.17 respectively. This may be due to the relatively low number of animals seen in these zones. An average rainfall period was also selected in an attempt to have some consistency when comparing the sites. However, in Fig. 2a the peak R2 fit for the Hawker zone actually fell closer to five months of rainfall lagged by three months. Poor model fit may also be associated with observer fatigue, particularly in the Hawker and Depot Flat zones, which are flown on the same morning.

4.3.

The value of aerial surveys for this species

Scale is important in the science of conservation biology and researchers should always have a clear understanding of scale-dependent processes. This emphasizes the importance of choosing data collection strategies that capture information at meaningful spatial and temporal scales (Bissonette and Broekhuizen, 1995; Merriam, 1995). While population growth rate models are useful at a general level of abstraction, they only provide a summary of the numerous underlying demographic processes in a population (McCallum, 2000). Aerial surveys are useful for highlighting the populations that may require more detailed investigation, but are insufficient as the sole monitoring system for guiding species management. In fact, broad-scale aerial surveys may fail to answer more detailed management questions such as the intensity and frequency of pest control effort required. These questions require site-specific studies to be undertaken. Management interventions for species recovery are generally part of a multi-faceted biodiversity conservation effort. While the results of our study tend to support the notion that sustained integrated management of both grazing and predation pressure will have a positive effect on P. x. xanthopus populations, there is no evidence to date that suggests these populations are necessarily declining in the absence of fox and herbivore treatments. The caveat to this observation is that if these populations are gradually declining in the untreated areas, the current aerial survey methodology is not yet capable of deciphering these trends.

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