On rate of increase (r): patterns of variation in Australian mammals and the implications for wildlife management

Authors


*Present address and correspondence: Applied Ecology Research Group, University of Canberra, Canberra, ACT 2601, Australia. E-mail: hone@aerg.canberra.edu.au.

Summary

1. The annual rate of increase of a wildlife population is the change in abundance or density from one year to the next. This paper compares the expected and observed means and frequency distributions of annual instantaneous rates of increase (r) for unmanipulated field populations of the European rabbit Oryctolagus cuniculus, red fox Vulpes vulpes and house mouse Mus domesticus in Australia.

2. The observed mean r was not significantly different from zero, the expected value, for each species, and the observed frequency distributions were not significantly different from the expected normal distributions.

3. The observed distributions were used to estimate the threshold proportions of pests that need to be killed, harvested or sterilized to stop population growth. To stop maximum population growth, as can occur after large population reductions, the proportion to kill, harvest or sterilize ranged from 0·65 year–1 for foxes to 0·97 year–1 for house mice. It is recommended that the estimates are tested experimentally.

4. The estimated proportions of pests to control were compared with extensive field data detailing attempts to control the pests. Estimated required levels of fox control were often less than observed levels of control. The required levels of rabbit and house mouse control could be higher than observed levels.

Introduction

The annual change in abundance or density of a wildlife population is described as the rate of increase. The annual finite rate of change (λ) is estimated as Nt + 1/Nt where Nt + 1 is abundance in year t + 1 and Nt is abundance in year t. The annual instantaneous rate of change (r) is estimated as ln(Nt + 1/Nt), or average annual instantaneous rate of change as the slope of the regression of lnNt over time, with λ = er (Caughley 1980). The maximum rate of increase under ideal conditions when no resource is in short supply is the intrinsic rate of increase (rm) (Caughley 1980).

Caughley (1980) reduced population management to three problems: conservation, sustained yield and pest control. These can be abbreviated as increasing abundance (r > 0), maintaining abundance (r = 0) and decreasing abundance (r < 0). Pest control for conservation, production or human health usually rests on the assumption that a reduction in pest abundance causes a reduction in pest damage (Braysher 1993; Hone 1994). Evidence for the simultaneous drop in pest abundance and damage is equivocal, especially for vertebrate pests (Hone 1994, 1996). To reduce abundance, population growth must be stopped, and to hold abundance at reduced levels the higher population growth that is generated after pest control must be stopped.

Caughley (1980) stated that ‘most populations, although fluctuating from year to year, have a rate of increase which, averaged over several years, is close to zero’. Caughley & Sinclair (1994) state that the ‘rate of increase of a population of vertebrates usually fluctuates gently for most of the time, around a mean of zero’. The frequency distribution of r is expected to be a normal distribution (Harris 1986). Aspects of the estimation of rate of increase and biases have been explored by many authors (Caughley & Birch 1971; Harris 1986; Eberhardt 1987; Gerrodette 1987; Eberhardt & Simmons 1992; Johnson 1994) with the general conclusion that such analysis is quite robust but precise estimates require considerable data. Attempts have been made to understand the determinants of rate of increase, for example by using life tables and demographic parameters (mortality and fecundity) (Caughley 1976; Johnson 1994; Sibly & Smith 1998) or using the numerical response (Caughley 1976, 1980, 1987), although these are not examined here.

This paper describes the expected and observed patterns of variation, and hence the means and frequency distributions of rate of increase (r), for three species, the European rabbit Oryctolagus cuniculus L., red fox Vulpes vulpes L. and house mouse Mus domesticus Rutty in Australia, and examines the implications of the patterns for wildlife management. The proportions of pests that must be controlled to stop population increases are estimated for each species and compared with observed levels of control achieved in the field.

Methods

Estimates of annual rates of increase were obtained from a large number of papers in the literature and from unpublished records, in which populations in many locations in Australia were monitored over time (at least 1 year). Hence the raw data were counts. Some data were counts in 1 year and at the same time of the next year. Other data comprised a series of counts over many years. In the latter case, one estimate of rate of increase was obtained for each year, and estimates obtained for each successive year. Estimates were made using data from the same time in each successive year, for example January to January. The results may be biased if a serial correlation exists between successive estimates. The magnitude of any such bias is not known. Care was taken to avoid spurious estimates of population increases known, or suspected, to be caused by immigration or use of unreliable survey methods. All data were for unmanipulated wild populations and were calculated on an annual basis. The minimum value of r was interpreted as an estimate of the maximum rate of decrease parameter (a) (Caughley 1976, 1980, 1987; Bayliss 1987).

The observed mean rates of increase were tested for difference to zero (r = 0) using a paired Student's t-test (Snedecor & Cochran 1967). The goodness-of-fit to a normal distribution of the observed frequency distribution of annual instantaneous rate of increase (r) for each species was tested using chi-square analysis (Snedecor & Cochran 1967). For each normal distribution it was assumed that the mean rate of increase was zero (see the Results) and the variance was the observed variance. The rate of increase (r) is expected to have a normal distribution when estimated from a series of counts of animals (Harris 1986). The skewness of each frequency distribution was tested for differences from that expected for a normal distribution, following Snedecor & Cochran (1967). For each species the observed ratio of the variance to the mean of r was compared with that expected (ratio of 1·0) if the underlying distribution was a Poisson distribution. The analysis was a t-test, following Manly (1992).

Regional variation in the rates of increase (r and λ) was examined separately for eight bioclimatic regions in Australia. The regions were: subtropical (23° to 30°S), subalpine (over 1400 m elevation), arid (less than 250 mm mean annual rainfall), semi-arid (greater than 250 mm but less than 375 mm mean annual rainfall), mediterranean (wet winters, dry summers), mallee (a structural form of Eucalyptus woodland in semi-arid winter rainfall area), temperate (greater than 375 mm mean annual rainfall in southern Australia) and irrigated agricultural land in southern Australia. Data were not available for all species in each bioclimatic region.

The interpretation of the results assumes that these data come from sites and populations representative of the species in Australia. No formal test of the assumption is possible here. Sites may have been selected by researchers because they had plenty of animals for research, rather than being a truly random sample. However, if the sample is substantially biased then the observed results should be different from the expected pattern (for example a mean rate of increase of zero). The results may be sample-size dependent, although the minimum sample size across all regions was 43. Larger samples would be expected to have higher maxima and lower minima, although the relationship should show diminishing returns. Arino & Pimm (1995) argued that observed maximum and minimum population densities increased as sample size increased.

The observed maximum annual instantaneous rates of increase were compared with estimates of the intrinsic rates of increase (rm) of species obtained from equations linking average adult body weight and rm of a broad range of organisms (Blueweiss et al. 1978) and of mammals (Caughley & Krebs 1983; Sinclair 1996). Weights were obtained from Strahan (1995).

Modelling

Management strategies may aim to stop subsequent growth of a population following a control programme. The threshold proportional reduction (p) in the finite rate of increase (λ) needed to make the rate of increase equal 1·0 is:

p = 1−(1/λ)(eqn 1)

Because λ = er, then:

p = 1−e−r(eqn 2)

where r is the annual instantaneous rate of increase. The proportion p is a finite rate to achieve sustained reduction in abundance. Barlow, Kean & Briggs (1997) estimated the instantaneous rate of culling (c) to stop population growth, and stated that the culling rate (c) must balance the rate of increase (r) to achieve the aim. If c = r then equation 2 becomes p = 1 – ec.

Caughley (1980) used equation 2 to estimate the proportion of a population then alive to harvest in a short period of time to hold the rate of increase to zero over the year. The population abundance was assumed to be low after deliberate reduction by, for example, culling. If that proportion is exceeded then the population will decline and could be pushed to extinction. The population need not be harvested but could be reduced by lethal control, such as trapping, or by fertility control. Barlow, Kean & Briggs (1997) concluded that sterilization at a rate equal to the rate of increase would stop population growth but the effect of sterilization would usually be slower to occur than for culling, although the social structure of the pest species could modify the effect.

Equations 1 and 2 assume control occurs at random across individuals in the population and that individual variation in reproductive success is low. However, field data across a range of birds and mammals (Clutton-Brock 1988) show that a small proportion of individuals contributes a large proportion of the young to the next generation. The effect of this is that p will underestimate the proportion of animals to kill, harvest or sterilize if those key individuals are missed when control occurs.

Equations 1 and 2 predict that not all animals have to be at risk of some pest control at a particular time for a population to go extinct. Hence a threshold exists, as noted by Bomford & O’Brien (1995). A qualifier for this is needed, however, as a remnant uncontrolled population (not killed or not sterilized) may be too small to persist even in the absence of pest control. Hence, not all reproductive individuals need to be exposed to control if the remnant population is very small, because it has a high probability of going extinct naturally.

To estimate the average value of p for each species over a period of 100 years, the frequency distributions of rate of increase (r) were randomly sampled with replacement and equation 2 solved for p. To estimate p it was assumed that no pests needed to be controlled (culled, harvested or sterilized) when the population was stable (r = 0) or declining (r < 0). Other management strategies are possible but only this one was studied here. Hence, this was an estimate of the level of control needed for a ‘tracking’ strategy; control tracks rate of increase. Such estimation assumes no compensatory changes in the rate of increase in response to control, so estimates should be interpreted as thresholds to be exceeded.

Results

The mean annual instantaneous rate of increase (r) for rabbits was 0·14 year–1, for foxes –0·19 year–1 and for house mice –0·07 year–1 (Table 1). The means for each species were not significantly different (P > 0·05) from zero (Table 1). The respective median annual rates of increase (r) were 0·17 year–1 for rabbits, –0·16 year–1 for foxes and 0·0 year–1 for house mice. The observed maximum annual rates of increase (r) were 2·06 year–1 for rabbits, 1·05 year–1 for foxes and 3·41 year–1 for house mice, and the observed minima were –1·77 year–1 for rabbits, –1·97 year–1 for foxes and –3·91 year–1 for house mice (Table 2). The observed frequency distributions of the annual instantaneous rate of increase (r) were not significantly different (P > 0·05; Table 1) from normal distributions (Fig. 1). Each of the frequency distributions was not significantly (P > 0·05) skewed (Fig. 1a–c) from a normal distribution (Table 1). The frequency distribution for each species was significantly different (P < 0·001) from a Poisson distribution (Table 1) as the variance to mean ratio was substantially greater than 1·0.

Table 1.  Observed means, variances (Var) and standard errors (SE) of annual instantaneous rates of increase (r) of rabbit, red fox and house mouse populations in Australia. The hypothesis tested by the first Student's t-test was that the mean r was not different to zero, and the hypothesis tested by the second t-test was that the observed variance to mean ratio was not different from that of a Poisson distribution. The chi-square analysis was a goodness-of-fit test to a normal frequency distribution. NS = not significant, n = sample size
ParameterRabbitRed foxHouse mouse
Mean r0·14–0·19–0·07
Var (r)0·750·432·74
SE (r)0·090·100·19
n924376
t (r = 0)1·561·900·38
SignificanceNSNSNS
Chi-square5·474·403·63
(normality)
d.f.437
SignificanceNSNSNS
Skewness0·04–0·43–0·27
SignificanceNSNSNS
t (Poisson)27·1215·25232·96
SignificanceP < 0·001P < 0·001P < 0·001
Table 2.  Regional variation of population parameters of rabbit, red fox and house mouse populations in Australia. Parameters are the sample size (n), maximum observed annual instantaneous rate of increase (r), maximum observed annual finite rate of increase (λ) and the maximum rate of decrease (a). The maximum rate of decrease is shown as a positive number (although it corresponds to a decline in abundance) following the convention of Bayliss (1987, table 8.1). The threshold annual proportion (p) of pests to kill, harvest or sterilize to stop maximum population growth, as estimated by equation 2, is also given. A dash indicates no data were available for analysis
Population parameters
Bioclimatic regionSpeciesnrλap
Subtropics
 Rabbit21·263·540·72
 Red fox
 House mouse103·2024·533·910·96
Subalpine
 Rabbit140·772·161·350·54
 Red fox
 House mouse
Arid
 Rabbit502·067·851·770·87
 Red fox241·052·861·970·65
 House mouse
Semi-arid
 Rabbit71·494·450·550·78
 Red fox
 House mouse23·4130·270·97
Mediterranean
 Rabbit191·926·821·470·85
 Red fox150·631·881·720·47
 House mouse462·4011·023·910·91
Mallee
 Rabbit
 Red fox
 House mouse82·4611·713·910·92
Temperate
 Rabbit
 Red fox40·151·160·670·14
 House mouse20·601·830·45
Irrigated land
 Rabbit
 Red fox
 House mouse81·615·002·120·80
Across all regions
 Rabbit922·067·851·770·87
 Red fox431·052·861·970·65
 House mouse763·4130·273·910·97
Figure 1.

Frequency distributions of the observed annual instantaneous rate of increase (r) for (a) European rabbit, (b) red fox and (c) house mouse populations in Australia. The solid line is the normal frequency distribution estimated assuming a mean of zero and the observed variance. The squares are the observed data.

Estimated proportions of pests to cull, harvest or sterilize to stop maximum population growth were usually higher for house mouse than for rabbit and fox populations (Table 2). The estimates across all regions were highest for the house mouse (0·97 year–1), then rabbit (0·87 year–1), and lowest for the fox (0·65 year–1). Regional variation in the proportion of pests (p) to cull, harvest or sterilize was high (Table 2), although this may have been sample-size dependent as sample sizes varied considerably across regions.

The threshold levels of control needed to stop population growth using a tracking strategy over 100 years were lower, as expected, than for stopping maximum growth. The mean (± SE) value of p per year for the house mouse (0·27 ± 0·04) was higher than for the rabbit (0·24 ± 0·03) and red fox (0·13 ± 0·02).

Discussion

The results described here demonstrate the frequency distributions of the annual instantaneous rate of increase (r) of three mammals in Australia. The normal distributions have not been described previously. The observed results support the statements of Caughley (1980) and Caughley & Sinclair (1994) regarding the mean rate of increase of species being close to zero. The theoretically expected normal frequency distribution of r (Harris 1986) was not significantly different from the observed distributions for all three species.

That the observed means of r are not significantly different from zero is not surprising. Each species has been established in Australia for over 100 years, so should have achieved a long-term equilibrium (although variable not constant) abundance. The data were from unmanipulated populations and hence the results constitute a set of data from experimental controls. If pest control, such as poisoning, trapping or shooting, is considered an experimental treatment, and if pest control is achieving a long-term and broad-scale trend of reduction in abundance, then in treatment areas average rates of increase would be negative. The data for rabbits were collected while myxomatosis did or could have occurred in the populations. This suggests the disease may have initially reduced rabbit abundance to a lower level after release in 1950, but during the period (1966–94) over which the current data were collected abundance was relatively stable. The mean rates of increase for fox and house mouse populations were negative, and larger sample sizes would be needed to determine if the declines were real.

Aspects of the observed patterns of variation in rate of increase can be compared with that reported in the literature and with theoretical models of r. The intrinsic rates of increase estimated from body weight equations were higher than observed maximum values for the house mouse but lower for the rabbit and fox (Table 3). Such differences could be associated with sample size effects, or with the estimates from the body weight equations being means for a body weight estimated by best-fit regressions, whereas the observed data were point estimates not means.

Table 3.  Estimates of average adult body weight, intrinsic rate of increase (rm) from body weight equations and maximum observed annual rate of increase (r) for rabbit, red fox and house mouse populations in Australia, and associated sources of rm body weight equations and observed data
Species
ParameterRabbitRed foxHouse mouse
Body weight (kg)1·560·013
Estimated intrinsic rate of increase (rm per year)
Blueweiss et al. (1978)1·360·954·68
Caughley & Krebs (1983)1·300·797·16
Sinclair (1996)1·210·785·40
Maximum observed rate of increase (r per year)2·061·053·41
SourceWood (1980)B. Cooke (unpublished data)Redhead (1982)

Myers (1970) reported that the dynamics of rabbit populations could vary regionally, and the results here support this, even though the estimates of Myers (1970) were later considered biased (Myers et al. 1994). The observed maximum rate of increase of rabbits (2·06 year–1) was less than that reported for an enclosed field population. Mykytowycz & Fullagar (1973) reported that a fenced rabbit population in a 33-ha paddock increased 16-fold in 1 year, which is an observed r of 2·77 year–1. The population bred nearly all year. The estimated intrinsic rate of increase (rm) of foxes in central New South Wales (NSW) was 0·75 year–1 (Pech et al. 1997), which is less than that estimated in the present study (1·05 year–1). The difference could be associated with a skewed sex ratio and, or, an unusual age structure favouring rapid increases, which are reflected in the data presented here. Such unusual events are possible and maximum rates of increase may occur infrequently. Caughley & Sinclair (1994) demonstrated mathematically that a sex ratio skewed towards females could substantially amplify population changes. Chambers, Singleton & Hood (1997) estimated, by computer simulation, the rate of increase in an enclosed house mouse population. Their population parameters were based on observed data. The predicted rate of increase (r) was 7·9 year–1, which is similar to the value of rm predicted by the Caughley & Krebs (1983) body weight equation (Table 3), but higher than the observed maximum annual rate (Tables 2 and 3). McCallum & Singleton (1989) assumed exponential growth of a house mouse population with a rate of increase of 7·3 year–1, which is higher than reported here. Hence it is concluded that the observed rates for house mouse populations probably underestimate the maximum rate of increase possible in the field.

The problems, or aims, of wildlife management suggested by Caughley (1980) can be linked to the frequency distribution of rate of increase. Conservation of endangered species is directional selection acting against negative rates of increase to shift the frequency distribution to the right and increase its height (Fig. 2a–c). The processes that generate population decline are the focus of the declining population paradigm (Caughley 1994). Sustainable harvesting is stabilizing selection acting against extremes of high or low rates of increase. Initially abundance is reduced (Fig. 2d,e) and then the harvest is taken from the population so that abundance stays at the reduced low level (Fig. 2e). The frequency distribution does not shift but extreme values are removed. After harvesting the population returns to its initial stable abundance (Fig. 2f). Pest control is directional selection acting to decrease abundance and hence shift rates of increase to lower or negative levels (Fig. 2g,h). After pest control, the population returns to its initial stable abundance (Fig. 2i). The suggested changes in frequency distributions in Fig. 2 assume that the results of pest control occur at random with respect to r. However, if management occurs strategically then the changes will be different. For example, if conservation acts only on declining populations and pest control only on increasing populations then the frequency distributions will be skewed.

Figure 2.

Temporal changes in the expected frequency distribution of the annual instantaneous rate of increase (r), (a) before, (b) during and (c) after conservation management of endangered species, (d) before, (e) during and (f) after sustainable harvesting, and (g) before, (h) during and (i) after pest control.

The results in the present study suggest that, in general, animals with higher body weights may be easier to control than animals with lower body weights. Larger animals have lower maximum rates of increase (Blueweiss et al. 1978; Caughley & Krebs 1983; Sinclair 1996), and hence require a lower percentage of pests to be exposed to the control procedure to achieve a population reduction. The conclusion, based on modelling, of Lebreton & Clobert (1991), that long-lived species are more sensitive to changes in adult survival and short-lived species more sensitive to changes in fecundity, provides a potentially useful guide for methods of population control (lethal or sterilization) rather than guidance on the required level of control. Barlow (1997) concluded, based on modelling, that sterilization is less effective in reducing density, relative to culling, of a population with a low r (hence high body weight) than a population with a high r (hence low body weight). Application of these results to wildlife management is not automatic. Green & Hirons (1991) were cautious about interpreting results of modelling, for management, as they may not always indicate the best management options.

The analysis in the present study shows that in a few exceptional years, vertebrate pest control requires a high proportion (p) of pests to be killed, harvested or sterilized to stop maximum population growth, especially for rabbit and house mouse populations. In contrast, red fox populations may be easier to control, because a lower proportion of pests needs to be killed, harvested or sterilized annually. Experimental testing of the estimates of p is recommended. Furthermore, if control is based on the rate of increase that would have occurred (as in the tracking strategy), not maximum r, then the required level of control is greatly reduced. In about half the years no control is needed as populations decline anyway. The effect of this is to greatly lower the average long-term level of control required if a tracking strategy is used. It is recognized that alternative strategies occur, such as timing pest control to occur when a population is declining, although such a strategy is not evaluated here.

Pest control may reduce pest abundance, but the effects may be reduced by compensatory changes in fecundity and, or, mortality rates (Sinclair 1997). The tracking strategy assumes no compensatory changes occur in, for example, mortality, as density is reduced. The implication of this is that estimates of p reported here should be interpreted as thresholds to be exceeded if sustained reduction in pest abundance is to occur. Nichols et al. (1984) reviewed the evidence for reduced mortality (compensatory mortality) in hunted waterfowl populations, and described a model of additive or compensatory mortality (Fig. 3a). As hunting mortality increased, survivorship declined in a linear manner if mortality was additive. In contrast, with compensatory mortality, survivorship at the population level showed no response until a threshold was reached, and then a linear decline occurred (Fig. 3a). Lethal control applied to pests could act in a similar way to hunting mortality. When fertility control is applied to vertebrate pests the population may or may not respond, such as by higher survival of sterilized individuals or increased fecundity. Individuals can have higher survivorship when reproduction does not occur, as reported for red deer Cervus elaphus (Clutton-Brock, Guinness & Albon 1983). Evidence of compensatory changes in juvenile mortality, and hence survivorship, in response to imposed sterility has been reported (Fig. 3b) in rabbit populations (Williams & Twigg 1996). The model of Nichols et al. (1984) can be extended to include changes in birth rates in response to imposed sterility (Fig. 3c). Survivorship can increase when births are reduced.

Figure 3.

(a) A model of additive and compensatory changes in survivorship of a vertebrate population in response to different levels of imposed mortality (after Nichols et al. 1984). The additive hypothesis predicts that each increase in mortality lowers survivorship. The compensatory hypothesis predicts that below a threshold, d, extra mortality has no effect on survivorship at the population level. (b) Annual survival of young rabbits as a function of the level of imposed sterility (proportion of females sterilized). Data are means, estimated from figure 4e,f in Williams & Twigg (1996), for young rabbits. (c) Extension of the model in (a) to include possible changes of survival in response to a reduction in birth rates following imposed sterility. Solid lines show additive and compensatory effects with no imposed sterility, and dashed lines show effects with imposed sterility and reduced birth rates.

The maximum percentage of rabbits to cull to stop population growth was estimated to be 87% (Table 2). The estimate agrees with the statement by Williams et al. (1995) that a decline in percentage kill (by poisoning) from 95% to 85% will often prevent effective management of rabbits. Observed data on the percentage reduction of rabbit populations after pest control show variable results. For example, the mean percentage kill of rabbit populations after poisoning with sodium monofluoroacetate (compound 1080) in southern Western Australia was 73% in the summer and 50% in the winter during 1971–75. However, during the earlier time period of 1958–62 the mean kills were 88% in the summer and 76% in the winter (Oliver, Wheeler & Gooding 1982). Robinson & Wheeler (1983) estimated that the percentage reductions after 1080 poisoning were greater than 85% in a series of studies. In eastern Australia the percentage reductions ranged from 44% to 99% 1 month after a combination of lethal control methods (Williams & Moore 1995). The field results of such lethal control of rabbits are sometimes higher or lower than the 87% kill estimated in the present study (Table 2) to stop maximum population growth. The estimate (87%) is higher than the maximum level of sterility (80%) imposed in an experiment to test the effects of sterilization on rabbit populations (Williams & Twigg 1996).

A method of rabbit control by immunocontraception has been proposed and is being developed in Australia (Tyndale-Biscoe 1994). Myxoma virus may be a vector of a gene for an antigen that causes rabbits to become immunosterile. The percentage of a rabbit population that survived exposure to myxoma virus is an indicator of the percentage of rabbits that could be sterilized by immunocontraception. Such rabbits should have antibodies to myxoma virus. Parer & Fullagar (1986) reported that the percentage of rabbits in subtropical Queensland with antibodies to myxoma virus (seropositive) varied from 28% to 92% over several years. Kerr et al. (1998) reported that rates of seropositivity in rabbits in south-western Western Australia varied between years from 30% to 93%. As for the poisoning data, the myxoma data suggest that the maximum rate of increase would not always be stopped.

Field data suggest that bait uptake by red foxes in Australia may be similar to that required for substantial reductions in the rate of increase of fox abundance if a control agent was orally delivered. Marlow (1992) reported that 88% of 43 foxes in an arid zone study site in NSW ate baits, and Marks et al. (1996) reported that 94% of 50 foxes in a temperate urban area in Victoria showed evidence of having eaten at least one bait. However, Fleming (1997) reported that only 58% of 19 red foxes in a temperate rural site in NSW consumed baits. These data on bait uptake by foxes generally reached the estimated levels (P = 0·65) required for fox control. Data extracted from the literature (J. Hone, unpublished data) show that the estimated percentage of foxes in non-urban areas in Australia killed by poisoning with compound 1080 has an unweighted mean of 81% and ranges from 66% to 100% (n = 8). The unweighted mean (81%) is higher than the percentage estimated here (65%; Table 2) to stop maximum fox population growth. Saunders, White & Harris (1997) have suggested that the pattern of bait uptake by urban foxes may be increased by more careful placement of baits in areas used intensively by foxes, such as woodland and rough ground.

Data on kills of house mice are similarly available for comparison with estimated required levels of control. The percentage kill of house mouse numbers within 1 week of poisoning with strychnine ranged from 69% to 99% (Mutze 1989), and was 100% in another study in South Australia (Mutze 1993). House mice populations were reduced by between 43% and 62% within 22 days by poisoning with bromadiolone in a part of central NSW (Twigg, Singleton & Kay 1991), and by up to 99% by poisoning with brodifacoum in a part of the Victorian mallee (Brown & Singleton 1998). Many of these estimates are less than the percentage of mice (97%) required to be killed or sterilized to stop maximum population growth (Table 2). A potential vector of immunocontraception of house mouse is murine cytomegalovirus (MCMV) (G. R. Singleton, personal communication). The percentage of house mice seropositive to MCMV in parts of Australia averaged 91–98%, with a range of 62–100% (Singleton et al. 1993). These percentages approach the levels required for stopping population growth of house mouse populations.

In summary, field data on instantaneous rates of increase (r) of unmanipulated populations of the European rabbit, red fox and house mouse in Australia were collated and analysed. The observed means were not significantly different from the expected value of zero, and the frequency distributions were not significantly different from the expected normal distributions for each species. The results were used to predict the levels of pest control required to stop population growth for each species. The required levels were compared with observed levels of control achieved during pest control. The required level of fox control was often less than observed levels, but the required levels of control of rabbit and house mouse populations could be higher than observed levels.

Acknowledgements

I thank Mani Berghout, Lisa Chambers, Brian Cooke, Greg Hood, Alan Newsome, John McIlroy, Roger Pech, Glen Saunders and Grant Singleton for access to data, Roger Pech and Tony Sinclair for discussions, and Peter Bayliss, Steve McLeod, Glen Saunders, Jeff Short, Kent Williams and two referees for useful comments on draft manuscripts. I thank the CSIRO Division of Wildlife and Ecology, the Vertebrate Biocontrol CRC, the University of Canberra, the Australian Academy of Science and the Centre for Epidemiology of Infectious Diseases at Oxford University for support and facilities.

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