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Keywords:

  • Oryctolagus cuniculus;
  • plant–herbivore;
  • predator–prey;
  • simulation models;
  • Vulpes vulpes

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

1. Rabbit calicivirus disease (RCD; also known as rabbit haemorrhagic disease) has been introduced recently as a biocontrol agent for rabbits in Australia. The consequences for fox populations that use rabbits as primary prey, for populations of alternative native prey, and for pastures, were examined using a model for rabbit- and fox-prone areas of semi-arid southern Australia.

2. Existing data were used to quantify the interactions of foxes, rabbits and pasture. A generic model for predation on native herbivores was constructed by modifying the density-dependent (Type III) functional response of foxes to rabbits to a depensatory (Type II) response that is appropriate for alternative prey. Similar dependence on pasture biomass was assumed for the dynamics of both rabbits and alternative prey in order to identify clearly the consequences of differing predation. In the absence of quantitative data for Australian conditions, the epidemiology of RCD was simulated empirically to mimic a range of potential patterns of occurrence.

3. For semi-arid Australia the model predicts that as the frequency and intensity of RCD epizootics increases: (i) the mean abundance of rabbits will decline, as will the frequency of eruptions of rabbits; (ii) there may be little increase in mean pasture biomass and a small decrease in periods of very low pasture biomass when competition between herbivores is most intense; (iii) the mean abundance of foxes will decline; (iv) there will be a reduced frequency of occasions when rabbit density is low but fox density is high due to a lag in the response of predator populations; and (v) there is potential for an increase in the mean abundance of alternative prey and in the proportion of time their density exceeds a threshold comparable to that currently required for eruptions of rabbits.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

In 1984 rabbit calicivirus disease (RCD) was reported to have caused high mortality in domestic rabbits Oryctolagus cuniculus (L.) in China (Liu et al. 1984). Over the next decade the virus was detected in wild and domestic rabbits in Europe and central America. Following the arrival of the disease in Spain, wild rabbits declined to 20–80% of their former abundance (B.D. Cooke, unpublished data). These observations provided the impetus for RCD to be considered as a biological control agent in Australia, where rabbits are a major pest species (Williams et al. 1995).

In 1995 a series of outdoor pen trials commenced in a quarantine area on Wardang Island in South Australia (B.D. Cooke, K.A. McColl & N. Amos, unpublished data). The trials used wild rabbits in natural warrens and were designed to measure mortality rates, persistence of the virus in the environment, and the humaneness of RCD. In October 1995, the virus inadvertently transferred to rabbits outside the enclosures and subsequently to mainland Australia. By the end of 1995, RCD was confirmed over much of the north-east of South Australia, in western New South Wales and the extreme south-west of Queensland. Despite the widespread distribution of the virus and apparently substantial mortality rates in some populations of rabbits, the ultimate efficacy of RCD as a biocontrol agent for rabbits is not yet known. Therefore a range of possible epidemiological outcomes needs to be considered in order to assess the direct and indirect impacts of introducing RCD.

There are many potential benefits of controlling rabbits in Australia. Domestic animals and herbivorous marsupials may benefit from lessened competition and there may be enhanced recruitment of perennial native plant species, some of which are highly susceptible to grazing by rabbits (see, for example, references cited in Williams et al. 1995; p. 66). There may be a decline in the abundance of exotic carnivores, such as foxes Vulpes vulpes (L.) and feral cats Felis catus (L.) (Newsome et al. 1997), with a subsequent reduction in predation on alternative prey species. These predators have been implicated directly in the dramatic declines in abundance, in some cases to extinction, of many small-to-medium sized native mammals (see, for example, Kinnear, Onus & Bromilow 1988; Morton 1990; Recher & Lim 1990; Short et al. 1992; Short 1998).

Detrimental environmental effects from controlling rabbits could arise through a variety of causes. There may be increased predation on native wildlife by foxes and feral cats following RCD-induced crashes in rabbit populations. Apart from any short-term impacts of the initial introduction of RCD, the likely longer-term effects of changes in the abundance of predators are unknown. Other consequences of introducing RCD may include a decline in the abundance of native raptors that have become dependent on rabbits as primary food (Newsome et al. 1997), and increases in weed species that are currently suppressed by rabbits (see, for example, Leigh et al. 1987). However, most public concern has focused on the potential of predators to exacerbate further the already substantial loss of vertebrate species in Australia (Woinarski & Braithwaite 1990).

Any changes in the abundance of rabbits will affect other herbivores through direct competition for food. The impact on particular native species will depend on how often pasture biomass declines to levels where their competitive ability is tested. Indirect, or apparent, competition between rabbits and native herbivores can also occur if changes in rabbit density affect the abundance of predators which in turn determines the level of predation on alternative prey.

The primary purpose of this paper is to address the issue of indirect competition, i.e. if the abundance of rabbits is changed by the introduction of RCD, what will be the consequences for predators’ alternative prey? Because more information is available to quantify predation by foxes than other carnivores, the focus of this paper is on the links between foxes and rabbits and foxes’ alternative prey species (Fig. 1). The approach is to construct a model describing the dynamics of the predator and prey populations and the food supply for prey. Differences in herbivore species’ ability to compete for food are not examined but their relative susceptibility to predation and their function as primary or alternative prey are explicit in the model.

image

Figure 1. Schematic model of the interactions of pasture, rabbits, foxes, and alternative prey for areas where rabbits are the primary prey of foxes. Pasture growth (1) is driven by rainfall. The functional response (2) determines consumption of pasture by rabbits, and the numerical response (3) of rabbits depends on pasture biomass. Per capita predation of rabbits by foxes is modelled with a functional response (4), and the numerical response of foxes depends on the density of rabbits (5) and alternative prey (7). Predation on alternative prey is determined by an appropriate functional response (6) but for the purposes of comparison their interactions with pasture biomass, (8) and (9), are assumed to be similar to those of rabbits. The abundance of rabbits can be reduced by epizootics of RCD (10). Data are available to quantify the interactions shown with solid arrows but hypothesized generic relationships are used for the remainder.

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Many of the elements of the model (Fig. 1) can be constructed from existing information for arid and semi-arid Australia. In the absence of detailed epidemiological data, the effect of RCD on rabbits is treated empirically to mimic, in general terms, the possible frequency and intensity of RCD epizootics. There are no data available to quantify the interaction between foxes and alternative native prey species, so the conceptual model described by Pech, Sinclair & Newsome (1995) is adapted to predict the consequences for this category of prey.

Two general forms of the functional response are used (Holling 1959), each corresponding to different susceptibilities of a species to predation at low densities. If a particular prey species is vulnerable to predation at all prey densities, for example if it has no physical refuge from predators, then consumption by predators will take place as soon as prey becomes available and will increase as prey density increases until a maximum intake level is achieved. This can be expressed as a Holling Type II functional response and results in the inverse density-dependent total response of an alternative, or secondary, prey species (Pech, Sinclair & Newsome 1995). However, if a prey species is relatively protected when it is at low density, but predation increases with density above a certain threshold, then a Holling Type III functional response results. This will produce a density-dependent total response characteristic of a primary prey species: the predator and the prey can co-exist with densities fluctuating around a stable equilibrium point (Pech, Sinclair & Newsome 1995).

Models

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

Structure of the model

Two types of herbivores are included in the model system that is shown schematically in Fig. 1. One type is rabbits, with density N, which is primary prey for a population of foxes at density, P. The other herbivore, with density M, is alternative prey. The interactions between the populations of predators, prey species and the food supply of the prey can be expressed in a mathematical form as follows (see, for example, May 1981). For changes in pasture biomass:

  • image(eqn 1)

where rV is the rate of change in pasture biomass through growth or senescence, in the absence of grazing. This rate depends on rainfall, R, and the standing biomass, V. Removal of pasture by rabbits occurs with a per capita rate of consumption of gN(V) and by the herbivore with density M at a rate of hM(V).

Based on the model of Pech et al. (1992), predation by foxes on rabbits is, to a reasonable approximation, independent of the abundance of alternative prey. (The validity of this approximation will be discussed later.) Therefore the rate of change of the rabbit population is:

  • image(eqn 2)

where rN(V) is the numerical response of rabbits in the absence of predation. Mortality due to predation depends on the density of foxes, P, and their functional response, gP(N) (i.e. the rate of consumption of rabbits per fox). In eqn 2 the second term on the right hand side, gP(N)P, is the ‘total’ response of foxes and determines the contribution to changes in rabbit density due to predation.

Predation on alternative prey will depend on their availability and whether the food requirements of foxes are satisfied by rabbits. If k is the maximum food intake per fox per unit time, then N rabbits will contribute a proportion gP(N)/k of a fox's maximum consumption rate. This is equivalent to PgP(N)/k foxes being satiated, leaving the remainder, [1 − gP(N)/k]P, to find other food. The alternative prey species will contribute to the additional food requirements of foxes according to the functional response, hP(M). Therefore the rate of change of the alternative prey population is:

  • image(eqn 3)

where rM(V) is the numerical response of the alternative prey species in the absence of predation.

The rate of change of the fox population is:

  • image(eqn 4)

where rP(N,M) is the numerical response of foxes and is dependent on the density of rabbits, N, and alternative prey, M.

The Ivlev form of the numerical response has been used in models for kangaroos Macropus spp. (Bayliss 1987; Caughley 1987; Cairns & Grigg 1993), feral pigs Sus scrofa (L.) (Caley 1993; Choquenot 1994), rabbits (Choquenot 1992) and small-to-medium sized native herbivores (D. Choquenot, S. McLeod & N. Dexter, unpublished data) in Australian semi-arid rangelands. A similar approach will be used for foxes and for their prey. If the abundance of food is F, then the Ivlev form of the rate of increase is:

  • image(eqn 5)

where the parameters for each species are denoted by an appropriate subscript (x = N for rabbits, P for foxes and M for alternative prey). The maximum rate of decrease is ax, cx sets the upper limit to rx when food is abundant, and the maximum rate of increase is rm = cx − ax. The demographic efficiency, dx, determines the shape of eqn 5 and is a measure of the consumers’ ability to use sparse food.

The Holling Type II and Type III forms are used for the functional response of foxes. The Type II functional response for the alternative prey is:

  • image(eqn 6)

where k is the asymptotic upper limit to a predator's consumption and DII determines the slope for small M. A Type III functional response was used by Pech et al. (1992) for fox predation on rabbits and this has the form:

  • image(eqn 7)

where k is the same limit as in eqn 6 and DIII is the density of rabbits at the inflection point.

The model is developed in two stages. The simpler model applies only to the interaction of foxes and rabbits (the solid arrows in Fig. 1) and consists of eqns 1, 2 and 4 with M = 0. All the parameters for predation and herbivory can be estimated from field data but the effects of RCD are simulated empirically to represent a range of possible outcomes. The presence of alternative food for foxes is implicit in this first stage model because a Type III functional response is used for rabbits.

In the second stage, the model is extended to include alternative prey explicitly. Both rabbits and the other prey species consume pasture according to eqn 1. The dynamics of the alternative prey population is modelled with eqn 3, including predation with a functional response in the form of eqn 6. The additional parameters are derived for a generic rabbit-sized herbivore. The rate of increase of the fox population is determined by the total food supply, i.e. the combined densities of the primary and alternative prey.

Sources of data

Data and results are used from previous studies at two sites, Kinchega National Park (latitude 32°29′ E, longitude 142°21′ S) and Yathong (latitude 32°38′ E, longitude 145°34′ S), in western New South Wales. Both sites are semi-arid with highly variable, non-seasonal rainfall patterns, although total annual rainfall at Yathong is higher than at Kinchega. The quarterly rainfall statistics for Kinchega and for Lerida (latitude 31°42′ E, longitude 145°42′ S), which is the nearest meteorological station to Yathong with long-term rainfall data, are summarized in Table 1.

Table 1.  Annual and quarterly rainfall statistics [mean, standard deviation (SD) and coefficient of variation (CV)] for Lerida (100 km north of Yathong) and Kinchega National Park (from Caughley 1987). A cube root transformation was used to simulate the rainfall recorded at Lerida from 1883 to 1993. Shown in parentheses are the parameters for the transformed data and the probability, p, that the distribution of the transformed rainfall does not differ from normal (Kolmogorov–Smirnov test)
 Lerida (mm)ActualTransformedKinchega (mm)
Summer (December–February)
Mean99(4·37)62 
SD69(1·12)59 
CV70% 95% 
p (0·385)  
Autumn (March–May)
Mean87(4·12)57 
SD67(1·19)47 
CV78% 82% 
p (0·541)  
Winter (June–August)
Mean82(4·22)59 
SD39(0·78)34 
CV47% 57% 
p (0·104)  
Spring (September–November)
Mean80(4·12)61 
SD49(0·92)44 
CV61% 72% 
p (0·345)  
Annual
Mean349 236 
SD133 107 
CV38% 45% 

Robertson (1987) estimated the rate of increase of pasture (rV; interaction 1 in Fig. 1) and Short (1985) measured the functional response of rabbits (gN(V); interaction 2) as part of the Kinchega study. A value for the demographic efficiency of rabbits (dN) is available from Choquenot (1992). Suitable data for estimating the other two parameters in the numerical response of rabbits (rN; interaction 3), and for the numerical response of foxes (rP; interaction 5), are available from a predator-removal experiment conducted by Newsome, Parer & Catling (1989) at Yathong. Data from that study were used by Pech et al. (1992) to calculate the functional response, gP(N) (interaction 4). The generic model for the alternative prey species (interactions 6–9) is derived from the analogous interactions for rabbits.

Pasture growth

To model the dynamics of kangaroo populations at Kinchega National Park, Caughley (1987) modified Robertson's (1987) model of pasture growth to compensate for a bias in comparing caged and uncaged plots. Caughley's modified pasture-growth model for the quarterly change in pasture biomass is:

  • image(eqn 8)

For computer simulations, ΔV replaces rV in eqn 1. Pasture biomass is in units of kg ha−1, R is the quarterly total rainfall in mm and V* is an additional random component not accounted for in the regression on R and V. In Caughley's model, V* is drawn from a normal distribution with the mean equal to the estimate from the regression equation and a standard deviation of 52 kg ha−1 (Caughley 1987; p. 162).

Consumption of pasture by rabbits

The functional response of rabbits was measured by Short (1985) using an intensive grazing trial in Kinchega National Park. The daily per capita consumption of pasture by rabbits, adjusted for body weight and expressed as kg animal−1 day−1, is:

  • image(eqn 9)

where V is the pasture biomass in kg ha−1 and w is the mean weight of a rabbit in kg.

The age structure of the rabbit population, and therefore the mean body weight of rabbits, varies throughout the year. For the purposes of this model, w is approximated by the mean body weight of rabbits eaten by foxes, i.e. it is assumed that predation on each age class is proportional to its abundance. For rabbits up to 900 g, weight increases linearly with age according to the relationship (Southern 1940; Parer 1977): (age in days) = 0·188 + 104·06 (weight in kilograms).

Rabbits weigh 30–35 g at birth (Lloyd 1977) and, at the upper end of the scale, the 1250 g rabbits used by Short (1985) for his grazing trial are representative of adult male rabbits in arid Australia whose average weight was estimated by Myers (1970) as 1245 g. Therefore, using the age classes of rabbits found in stomachs of foxes collected throughout the year at Yathong (Catling 1988), the mean body weight of rabbits is approximately 782 g (Table 2).

Table 2.  Mean body weight of rabbits caught by foxes. Proportions of rabbits in each age class were estimated by Catling (1988). The weight range for each age class is based on the estimated weight gained per day for rabbits less than 900 g (Southern 1940; Parer 1977) and the average weight of the adult rabbits used by Short (1985)
Age classWeight range(g)Value used(g)Proportionin fox dietContribution to weight ofaverage eaten rabbit (g)
Small kittens (<50 days)30–4802550·251
Large kittens (50–80 days)480–7706250·2125
Adult rabbits770–125010100·6606
Total   782

Numerical response of rabbits

Choquenot (1992) estimated the annual rate of increase of rabbits as:

  • image(eqn 10)

where V is the prevailing pasture biomass. The demographic efficiency of rabbits, dN = 0·0045, is independent of time in eqn 10. However, both pasture biomass and the abundance of rabbits can fluctuate significantly between seasons, and estimates of the rates aN and cN (see eqn 5) are required on a quarterly, rather than an annual, basis to model this variability.

For the present model, defined by eqn 2, the numerical response, rN, should be calculated for rabbits not subject to fox predation; the effect of predation on rabbits’ rate of increase is included separately in the term −gP(N)P. The numerical response can be estimated from data obtained during the predator-removal experiment conducted by Newsome, Parer & Catling (1989). Unfortunately the pasture biomass estimates they reported are an order of magnitude lower than that recorded by Short (1985) during the graze-down trial at Kinchega, and I. Parer (unpublished data), who collected the data at Yathong, has indicated that, while they accurately reflect the relative abundance of pasture biomass, the absolute values are unreliable. Therefore an alternative approach to direct estimation is required for the numerical response.Bayliss (1987) and Cairns & Grigg (1993) showed that the numerical response of red and western grey kangaroos can be modelled as a response to lagged rainfall as well as directly to pasture biomass. A similar approach can be used for rabbits by using rainfall as the independent variable to estimate the time-dependent parameters, aN and cN, in the numerical response.

The methods for the predator-removal experiment at Yathong are detailed in Newsome, Parer & Catling (1989). Briefly, foxes were regularly removed from a block of 70 km2 from the beginning of 1980 to the middle of 1983. Spotlight counts of rabbits were conducted at approximately 6-week intervals during this period. Similar data were collected during a plague of rabbits in 1979, when estimates of the abundance of foxes and feral cats indicated that, due to a lag in the build-up of predator populations, predation would have had minimal impact on the rate of increase of rabbits. These counts were used to estimate the rate of increase of rabbits per 3 months for each season (Table 3). The 1979 plague of rabbits ended in a drought when the abundance of rabbits declined by >99% over a 4-month period (Newsome, Parer & Catling 1989). This corresponds to an estimated maximum rate of decrease of 4·6 per quarter (i.e. 3 months) over a period of very low rainfall (mean monthly rainfall of 4 mm, from Fig. 4 in Leigh et al. 1989). The parameters for the Ivlev form of the numerical response (eqn 4) for rabbits were estimated by maximum likelihood (Rothamsted Experimental Station, Genstat 5 Committee (1993) Genstat 5 Release 3 Reference Manual. Clarendon, Oxford) using the data in Table 3. Rainfall was lagged by 3 months because rabbits breed in response to pasture growth (references in Williams et al. 1995; page 48), there is a delay in the birth of rabbits due to a gestation period of 30 days (Myers & Poole 1962), and young rabbits do not appear in spotlight counts until they emerge from warrens at about 3 weeks (Mykytowycz & Fullagar 1973). Deleting one outlier for the quarter from March to May (1981) but including the value for the drought-induced population crash as an estimate of the maximum rate of decrease, the parameters are aN = 4·63 ± 0·39, cN = 5·49 ± 0·41 and dN = 0·042 ±0·008, or:

Table 3.  Quarterly rate of increase of rabbits and the seasonal rainfall (from Fig. 4 in Leigh et al. 1989) recorded at Yathong Nature Reserve. The rates of increase were calculated using spotlight counts (a) from an area where foxes were removed (Block A Fig. 1 in Newsome, Parer & Catling 1989), and (b) from a period prior to the removal of foxes but during a plague in 1979 when spotlight counts of rabbits increased from 58·2 km−1 in May to 348·0 km−1 in November. Spotlight counts suggest there was a delay before a corresponding increase occurred in the abundance of predators and that predation would have had little impact on the rate of increase of rabbits during this period in 1979 (Newsome, Parer & Catling 1989)
SeasonBreeding (B) ornon-breeding (NB)Rainfall: quarterlytotal (mm)Rate of increase ofrabbits per 3 months
(a)
December 1980–February 1981NB62 
March–May 1981NB82 0·036
June–August 1981B133 0·14
September–November 1981B52 0·55
December 1981–February 1982NB11 0·38
March–May 1982NB97 0·25
June–August 1982B19 1·20
September–November 1982B18−1·57
December 1982–February 1983NB56−1·86
March–May 1983NB170 0·39
(b)
March–May 1979NB68 
June–August 1979B72 0·51
September–November 1979B  1·27
image

Figure 4. Typical simulated trajectories of rabbit density over 100 years (a) with no RCD and (b) with an 85% reduction in rabbits by RCD in spring on average every 2 years. Identical rainfall sequences were used for this comparison.

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  • image(eqn 11)

where rN is the quarterly rate of increase and R is the total rainfall for the previous quarter (Fig. 2). The fitted relationship (eqn 11) accounts for 95% of the variance of rN. The maximum rate of increase of rabbits is 0·86 per quarter.

image

Figure 2. The numerical response of rabbits in the absence of predation. Quarterly rates of increase were estimated from data reported by Newsome, Parer & Catling (1989). One outlier (Δ) for the quarter from March to May was deleted in fitting the Ivlev model.

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In the model defined by eqn 3, the rate of increase of rabbits should depend on the abundance of their food. There will be no change in the maximum rates of increase and decrease when food availability is used instead of lagged rainfall as the independent variable. These rates, which set the limits for rabbit population dynamics on a quarterly time frame, can be combined with Choquenot's (1992) value for the demographic efficiency based on a response to pasture (eqn 10). Therefore the quarterly rate of increase is:

  • image(eqn 12)

where V is the pasture biomass. In eqn 12, the rate of increase, particularly the contribution from recruitment, should depend on lagged pasture biomass (for the reasons discussed above for rainfall) but the mortality rate may be more responsive to current conditions. The sensitivity of the model to both options, and to changes in the value for the demographic efficiency, is assessed in the Appendix.

Functional response of foxes to rabbits

Pech et al. (1992) calculated the per capita consumption of rabbits by foxes for Yathong at a time when rabbits were their primary prey (Catling 1988). They showed that a Type III functional response fitted the dietary data and was consistent with the observed dynamics of an eruptive rabbit population. Re-analysis of the data using the methods proposed by Casas & Hulliger (1994) confirms the difficulty of distinguishing between Type II and Type III responses, especially when there are errors in the measurement of rabbit density corresponding to each estimate of the predation rate. However, the regression test used by Pech et al. (1992) is similar in approach to the locally weighted regression method advocated by Trexler & Travis (1993) and both methods confirm a Type III response. Therefore the response curve fitted by Pech et al. (1992) is used for the current model and the daily consumption of rabbits in grams per fox per day is:

  • image(eqn 13)

where the density, N, is expressed in rabbits per spotlight km. The index of rabbits per spotlight km was converted to an absolute density based on 40% sightability (Robinson & Wheeler 1983; Leigh et al. 1989; I. Parer, unpublished data) within the transects, which were 150 m wide (Leigh et al. 1989). In principle, the estimation of absolute abundance from spotlight counts should take into account factors such as the height of vegetation, terrain and time of year (Williams et al. 1995). However, there are insufficient published data to make these corrections. Therefore the estimated offtake by foxes in gram of rabbit per fox per day is:

  • image(eqn 14)

where N is the number of rabbits ha−1. eqn 14 can be converted to offtake in units of rabbits per day by using the estimated mean body weight of rabbits (Table 2).

Numerical response of foxes

The areas of Australia particularly relevant to modelling the effects of RCD are those where rabbits are abundant and where rabbits are most likely to be the primary prey of foxes. Although foxes can take a wide variety of prey items in these areas (Catling 1988 and references cited therein), as a first approximation the rate of increase of foxes is assumed to be dependent on the density of rabbits. The most direct way to estimate the demographic parameters dP, rm (=cPaP) and cP in eqn 5 is by fitting a numerical response function to measurements of rabbit density and rates of increase for foxes. A 26-year data set is available from a semi-arid area of South Australia (B.D. Cooke, unpublished data) but spotlight counts of foxes are too low and erratic to be a useful measure of fox abundance, a problem typical of records from areas with low fox densities. An alternative approach is to estimate each parameter separately.

The maximum rate of increase can be estimated from the allometric relationship of Caughley & Krebs (1983) or Sinclair (1996). For an average adult body weight for foxes of 5 kg (Coman 1983; J. McIlroy & R. Pizel, unpublished data), rm = 0·84 per annum. The maximum rate of decrease is estimated as aP = −2·26 per annum from the observations at Yathong, where the density of foxes declined over 12·6 months from 1·61 per spotlight km in November 1979 to 0·15 in December 1980. This followed a drought-induced collapse of a rabbit plague in 1980 (Newsome, Parer & Catling 1989). The demographic efficiency of foxes has not been estimated previously, so this was determined by a process of trial and error to match the computer simulations of predator–prey dynamics against two criteria (see the Appendix). These ensure that the model represents general characteristics of fox and rabbit populations that can be linked to reliable, robust field observations. The resulting expression for the numerical response of foxes to rabbits is:

  • image(eqn 15)

where rP is the quarterly rate of increase of foxes and N is the density of rabbits in the previous quarter. The sensitivity of the model to changes in dP is assessed in the Appendix.

Predation rate and numerical response for alternative prey

In theoretical terms, predation on primary and alternative prey can be differentiated using the total response function of the predator (Pech, Sinclair & Newsome 1995). There is a density-dependent total response for primary prey and an inverse density-dependent response for alternative prey.

Pech et al. (1992) used the density of rabbits as the single independent variable to fit a Type III functional response for predation of rabbits by foxes. Apart from a seasonal difference in the maximum offtake of rabbits, probably due to an enhanced supply of carrion in winter, the functional response appeared to be mostly independent of the abundance of alternative food. The clearest exception to this generalization was one occasion when foxes fed almost exclusively on a short-lived eruption of caterpillars (Pech et al. 1992). For the present model, it is assumed that fox predation on rabbits is dependent on rabbit density only. Therefore, ignoring any effects of surplus killing by foxes, predation on species other than rabbits occurs only if foxes are not satiated on rabbits. However, the level of this predation may depend on the availability of alternative prey, as modelled by the functional response, hP(M), in eqn 3.

Fox predation rates on species other than rabbits have not been measured directly, partly because of the difficulty of obtaining suitable data for endangered species. However, recent analyses of rates of increase for re-introduced, or declining, populations of native species suggest that inverse density-dependent predation is the key process threatening these species’ survival (Sinclair et al. 1998). Examples include the eastern barred bandicoot Perameles gunnii (Gray 1838), the quokka Setonix brachyurus (Quoy & Gaimard 1830) and the black-footed rock-wallaby Petrogale lateralis (Gould 1842). For the present model, it is assumed that a Type II functional response is the cause of the inverse density-dependence in predation.

In the absence of quantitative data, and in order to achieve some generality in the predictions, a generic Type II functional response was constructed to compare the effect of predation on rabbits with predation on an alternative species. The alternative species is assumed to be a rabbit-sized herbivore (mean body weight approximately 780 g) with the same physiological constraints on fecundity and survival as rabbits, i.e. the same numerical response as rabbits (eqn 12) in the absence of fox predation but a different vulnerability to foxes at low densities. The value of this approach is that it estimates any differences in the viability of populations of rabbits and alternative prey species that can be attributed solely to predation.

The Type II functional response for predation on the generic alternative prey species has the general form of eqn 6. The asymptote at high prey densities is assumed to be the same as for rabbits (k = 1096 ± 160 g day−1). The slope of the Type II response was set by minimizing the sum of the squared differences between the generic Type II response and the Type III response for rabbits over the range of prey densities from 0 to 5 ha−1. In this way the consumption per predator, averaged over low-to-medium densities of prey, is similar for the two responses. The resulting generic Type II functional response in grams per predator per day is:

  • image(eqn 16)

where M is the density (animals ha−1) of the alternative prey species.

With the inclusion in the model of an alternative prey species to rabbits, the numerical response function for foxes (eqn 15) depends on the combined density of both prey species. This requires an assumption that individuals of both species provide equal resources for foxes but it is consistent with the aim of constructing a comparative model where the difference between primary and alternative prey is the shape of the functional response.

Dynamics of rcd

Until realistic epidemiological models and more local field data become available, the impact of RCD on rabbit populations in Australia can only be modelled empirically, primarily using reports from overseas. Initial outbreaks of RCD in wild rabbits in Spain resulted in mortality rates of up to 95%, although reductions as low as 65% were also reported (B.D. Cooke, unpublished data). Mortality rates from the initial epizootics in Australia in 1995 and 1996 appear to have been in the upper end of this range (B.D. Cooke & A.E. Newsome, unpublished data). A 2-year pattern of RCD epizootics seems to have become established in Spain, with a shift in timing from summer to winter or spring (Cooke 1995). While there has been no apparent change in the virulence of RCD in Spain over the last decade, an avirulent rabbit calicivirus has been reported from Italy (Capucci et al. 1996) and avirulent strains may be present in Britain and Europe (Chasey & Trout 1995). At this stage no data are available that indicate attenuation of RCD or development of resistance in rabbits in Australia, although the example of myxomatosis (Fenner & Ratcliffe 1965) suggests that co-evolution of the host and pathogen is likely to occur.

For this paper, the impact of RCD in Australia was assessed using computer simulations with a range of mortality rates, m, and for each mortality rate the annual probability of occurrence of an epizootic, pe, was varied from 1·0 to 0·2, i.e. from annual epizootics to one every 5 years on average. Therefore, in year t, the proportion of rabbits that is unaffected by RCD is Dt = (1 − pem). In the absence of data on the rate of evolution of the virus or resistance in rabbits, RCD virus is assumed to maintain the same virulence indefinitely. However, the range in mortality rates can be viewed as potential equilibrium endpoints of the co-evolution process. Mortality for each epizootic is imposed by simply applying the appropriate percentage reduction in the abundance of rabbits within a specified season, for example in spring. The precise timing of epizootics is likely to make little difference because the model uses the rainfall pattern for Lerida which, although having a slightly higher summer rainfall compared to other seasons, is characterized by a high variability (Table 1).

Spatial and stochastic effects

eqns 2, 3 and 4 do not include immigration or emigration by foxes, rabbits or alternative prey. No data are available to determine whether the area at Yathong used by Newsome, Parer & Catling (1989) contained sink or source populations for foxes, or to estimate the effects of dispersal. However, the presence of large well-used warrens suggests a long-term resident rabbit population and therefore a reliable food supply for foxes, i.e. the model is based on data for persistent populations of predators and prey. The non-spatial model used in this paper can result in any of the species being driven to extinction during computer simulations. Since there is no evidence that such extinctions occur at Yathong, minimum densities of 0·1 foxes km−2 and 0·17 rabbits ha−1 are included in the model. Both lower limits are an order of magnitude less than the average densities. For computer simulations that include rabbits and an alternative prey species, the fixed minimum prey density is divided equally between the species, i.e. 0·08 animals ha−1 each, so that the base level of food for foxes with two types of prey is the same as for rabbits only. The minimum densities also avoid the difficulty of using deterministic models such as eqns 2, 3 and 4 when the number of animals is very small. Under these conditions a stochastic model, such as that used by McCallum (1995) to model individual predation events, is more appropriate.

The sequence of calculations in each iteration of the model is summarized in Table 4. The initial population densities were set at the minimum values. Although any transient effects due to the initial densities disappeared rapidly, the results of the first 10 years of each computer simulation were deleted from the analyses. Each time step in the model represents one quarter of a year and changes in the density of each species were calculated using the estimated rate of increase for the quarter, for example Nt + 1 = Ntexp (rN). Seasonal rainfall was generated by cubing random samples drawn from the distributions of transformed data for Lerida (Table 1).

Table 4.  Parameter values and sequence of calculations for each iteration of the model. One iteration represents 3 months (=365/4 days)
Sequencefor time tVariable or processAlgorithmRelevanteqn in textParameter values, units
 Initial values   
 Pasture biomass(First 10 years not included in analyses to remove transient effects) V0 = 300 kg ha−1
 Rabbit density  N0 = 0·08 rabbits ha−1 (or N0 = 0·17 ha−1 if M0 = 0)
 Density of alternative prey  M0 = 0·08 animals ha−1 (=N0)
 Fox density Disease  P0 = 0·001 foxes ha−1
1Annual probability of RCD epizootic0 ≤ pe ≤ 1 in spring; pe = 0 for all other seasons  
2Proportion of rabbits that survive RCD epizootic Rainfall (R)Dt = 1 − pem m = 0·65, 0·75, 0·85, 0·95
3Total rainfall per quarter Vegetation (V)Rt = (R*)3 where R* is drawn from a normal distribution for each season mm (see Table 1)
4Change in ungrazed pasture biomass per quarterV(1),t = −55·12−0·0153Vt−1−0·00056V2t−1 + 2·5Rt ΔVt is drawn from a normal distribution with mean = V(1),t and SD = 52 kg ha−1(8)kg ha−1
5Total ungrazed pasture biomassV(2),t = Vt−1 + ΔVt kg ha−1
6Pasture removed by herbivoresCt = (w0·75)v[1 − exp (−V(2),t/f )] (Nt−1 + Mt−1) (365/4)(9)v = 0·068 kg kg−0·75 day−1 f = 138 kg ha−1 w = 0·782 kg
7Pasture biomass at time t Rabbits (N)Vt = V(2),t − Ct(1)kg ha−1
8Predation rate per fox per daygP,t = (k/w)N2t−1/(N2t−1 + D2III)(7), (14)k = 1096 g day−1 w = 782 g DIII = 1·32 rabbits ha−1
9Total predation rate per rabbit per quarterGP,t = (365/4) (gP,tPt−1)/Nt−1  
10Net per capita rate of increaserN,net = N−1 (dN/dt)(2)aN = 4·6 quarter−1
  = − aN + cN[1 − exp (−dNVt−1)] − GP,t(5), (12)cN = 5·5 quarter−1 dN = 0·0045
11Rabbit density at time tNt = DtNt−1 exp (rN,net) rabbits ha−1
 Alternative prey (M)or Nt = Nmin if [DtNt−1 exp (rN,net)] < Nmin Nmin = 0·17 ha−1 if M = 0 Nmin = 0·08 ha−1 if M > 0
12Predation rate per fox per dayhP,t = (k/w)Mt − 1/(Mt−1 + DII)(6), (16)k = 1096 g day−1 w = 782 g DII = 0·99 animals ha−1
13Total predation rate per animal per quarterHP,t = (365/4)[hP,t(1 − gP,t/k)Pt−1]/Mt−1(3) 
14Net per capita rate of increaserM,net = M−1 (dM/dt)(3)aM = 4·6 quarter−1
  = − aM + cM[1 − exp (− dMVt−1)] − HP,t(5), (12)cM = 5·5 quarter−1 dM = 0·0045
15Density of alternative prey at time tMt = Mt−1 exp (rM,net) animals ha−1
 Foxes (P)or Mt = Mmin if [Mt−1 exp (rM,net)] < Mmin Mmin = 0·08 ha−1
16Rate of increaserP = − aP + cP{1 − exp [−dP (Nt−1 + Mt−1)]}(5), (15)aP = 0·56 quarter−1 cP = 0·77 quarter−1 dP = 3·2
17Fox density at time tPt = Pt−1 exp (rP) foxes ha−1
  or Pt = Pmin if [Pt−1 exp (rP)] < Pmin Pmin = 0·1 km−2

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

The effects of RCD were assessed, as a first approximation, using the simple fox–rabbit model without explicit representation of alternative prey, i.e. with M = 0.

Impact on rabbit populations

The results of simulations of the impact of RCD on rabbit populations are summarized in Fig. 3. Impact is characterized in three ways: the mean density of rabbits over 100 years of simulated rainfall, the coefficient of variation (CV) in rabbit density, and the percentage of quarters when the density of rabbits is greater than 15 rabbits per spotlight km (or 2·5 rabbits ha−1). The last index corresponds to the upper limit for the densities where Pech et al. (1992) estimated that foxes can regulate rabbits. This limit is less than the threshold of four rabbits ha−1 used in the Appendix to distinguish periods of high rabbit abundance from background densities. However, the lower value of 2·5 ha−1 is more appropriate for assessing the indirect impact of RCD on the efficacy of predator regulation.

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Figure 3. The impact of RCD on the density of rabbits. Mortality due to RCD was imposed in spring and the annual probability of an epizootic ranged from 1·0 to 0·2. Values with no RCD are included for comparison. The summary statistics were derived from 100 computer simulations, each corresponding to 100 years of rainfall with a mean and standard deviation for Lerida (near Yathong) in western New South Wales. Quarterly estimates were used to calculate (a) the mean density of rabbits and (b) the mean percentage of quarters when the density of rabbits was >2·5 ha−1.

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There is a steady decline in rabbit density as the frequency of epizootics increases and as the mortality rate in each epizootic increases (Fig. 3a). In Fig. 3(a), and in similar figures subsequently, the range of intercepts for no RCD is an indication of the variability in mean values estimated from the computer simulations. Perhaps more important is the change in the CV in the density of rabbits, which reflects better the boom or bust dynamics of rabbit populations. In the absence of RCD CV is approximately 280%. This variability is unaffected by occasional epizootics then decreases with increasing impact of RCD, for example to approximately 65% for annual epizootics causing 85% mortality. The ability of RCD to suppress major eruptions of rabbits is also apparent from the percentage of quarters with rabbit density >2·5 ha−1 (Fig. 3b). As expected, the greatest benefits are achieved with frequent epizootics that impose high levels of mortality. However, the intensity of epizootics makes little difference when they occur less often than about once every 3 years.

Figure 4 shows typical results of computer simulations for the trajectory of the rabbit density over 100 years with no RCD and with RCD killing 85% of rabbits every 2 years on average. Without the disease, there are sporadic eruptions reaching the density observed at Yathong in 1979 (Newsome, Parer & Catling 1989). Similar outbreaks are rare but still possible with a 2-year pattern of RCD if a period without disease coincides with high rainfall and low abundance of predators (Fig. 4b).

Indirect effects on pasture biomass

The computer simulations used to generate Fig. 3 were also used to calculate two indices for comparing the predicted changes in pasture for different frequencies of RCD epizootics. These are the mean pasture biomass and the percentage of quarters when pasture biomass falls below 250 kg ha−1. Mean pasture biomass increases as the frequency of epizootics increases (Fig. 5a) but with a surprisingly small overall improvement of approximately 2% if the value for no RCD is compared with that for annual epizootics. However, RCD may be more important in maintaining pastures, particularly those with perennial plants, in the long term if it reduces the number of periods when pasture biomass is sufficiently low to exacerbate competition between herbivores. Williams (1991) observed that sheep production is likely to be reduced by competing herbivores only when pasture biomass is less than 250 kg ha−1. This is because the functional responses of sheep, kangaroos and rabbits, as measured by Short (1985), indicate that maximum per capita consumption is always achieved above this threshold. The model suggests there may be a small decline in the proportion of time with low pasture biomass (Fig. 5b).

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Figure 5. Changes in pasture biomass resulting from the impact of RCD on the density of rabbits. Quarterly estimates derived from the computer simulations used for Fig. 3, were used to calculate (a) the mean pasture biomass in kg ha−1 and (b) the mean percentage of quarters when the pasture biomass was <250 kg ha−1.

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Figure 6 shows typical results of computer simulations for the trajectory of the pasture biomass over 25 years with no RCD compared to RCD imposing 85% mortality on average every 2 years. For example, with no RCD, the very low pasture biomass in year 10 is the modelled outcome of a build-up in rabbit numbers over the previous 2 years followed by a drought and minimal pasture growth. In contrast, there were several years of poor rainfall and low rabbit numbers prior to year 23. Because below-average rainfall, not rabbits, produced the low pasture biomass in year 23 there is little difference between the trajectories with and without RCD. Over the 25-year segment of the computer simulation, mean pasture biomass was essentially unaffected by RCD, but with the disease present a small reduction in periods of low biomass (<250 kg ha−1) is apparent.

image

Figure 6. Typical trajectories of pasture biomass over a 25-year segment of a 100-year computer simulation, with no RCD (——), and with an 85% reduction in rabbits by RCD in spring on average every 2 years (— — —). Identical rainfall sequences were used for this comparison. The approximate threshold for competition between species of herbivores is 250 kg ha−1 (. . .).

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Indirect effects on foxes

The impact on foxes of the loss of their primary prey due to RCD can be estimated with the simulations used for Figs 3 and 5. Figure 7 shows that the abundance of foxes declines as the supply of rabbits decreases with increasing impact of RCD. With no RCD the mean density of foxes is approximately 0·7 km−2 and the CV in fox density is approximately 110%. Occasional RCD epizootics increase the CV in fox density, which suggests the potential for sporadic increased predation pressure on alternative prey species. For example, the CV is approximately 130% for epizootics causing 85% mortality on average every 2·5 years (annual probability of an epizootic equals 0·4). Fox density, and the variation in fox density, decline to very low levels with frequent, high-intensity epizootics. For example, for annual epizootics that impose 85% mortality the CV is approximately 5%, but this estimate and the corresponding low fox density (Fig. 7) are to some extent an artefact of the fixed minimum density of 0·1 foxes km−2.

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Figure 7. Changes in the density of foxes resulting from the impact of RCD on the density of rabbits. The mean density of foxes was calculated using quarterly estimates derived from the computer simulations used for Fig. 3. Note that 0·1 foxes km−2 is the fixed minimum density of foxes in the model (see text).

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Indirect effects on alternative prey

The impact on alternative prey will be greatest when there is a high density of foxes and they are not satiated with rabbits. From eqn 14 for the foxes’ functional response, the latter condition occurs when the density of rabbits is <15 per spotlight km or 2·5 ha−1. For the purposes of comparing the outcomes of different frequencies of RCD epizootics, high fox densities were selected as those greater than the modelled average of 0·7 km−2 in the absence of RCD. The computer simulations suggest that if extended periods of low predation pressure are critical for alternative prey, they should benefit substantially from RCD epizootics (Fig. 8).

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Figure 8. Percentage of quarters with low rabbit density (<15 per spotlight km or 2·5 ha−1) and high fox density (greater than the modelled average of 0·7 km−2 in the absence of RCD), based on quarterly estimates derived from the computer simulations used for Fig. 3.

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The impacts of RCD on rabbits, foxes and alternative prey species of foxes were estimated using the extended second-stage model, which includes explicitly both types of prey. Examples of computer simulations without RCD and with biennial epizootics are shown in Fig. 9. With no RCD (Fig. 9a), the response of the fox population to total prey availability is apparent, as is the ability of foxes effectively to suppress alternative prey for extended periods of up to 10 or 20 years. By comparison, eruptions of the alternative prey are more frequent when RCD controls rabbits on average once every 2 years (Fig. 9b). As the frequency of epizootics increases, the mean density of rabbits (Fig. 10a) is generally lower and declines a little faster for the model with two prey species than with rabbits only (Fig. 3a) because of competition for pasture. However, due to a compensatory increase in the abundance of the alternative prey species there is no net change in pasture biomass (Fig. 10d). Reduction in rabbit density by RCD results in a reduction in the mean density of foxes (Fig. 10b) but, compared to Fig. 7, the effect is mitigated by the contribution of the alternative prey to the numerical response of foxes. The mean density of the alternative prey nearly doubles when RCD occurs annually compared to no RCD (Fig. 10c) but to some extent the density in the absence of RCD is artificially maintained by a fixed minimum in the model. Perhaps of more importance for an arid-adapted species is an ability to recolonize areas after poor conditions have contracted its distribution to refugia (Morton 1990). Figure 10(e,f) compares the abilities of rabbits and an alternative prey species to reach high densities under different regimes of RCD. The threshold for defining ‘high’ density for both prey species was set at 2·5 animals ha−1. This corresponds to the density where rabbits can escape predator regulation and, under good conditions, can increase to plague numbers (Pech et al. 1992). As RCD more effectively suppresses rabbits and forces a decline in predator numbers, the frequency of periods with high densities of the alternative prey increases from approximately 5% with no RCD to approximately 10% with annual epizootics (Fig. 10f). There is a corresponding decline in outbreaks of rabbits (Fig. 10e), with a stronger response to infrequent epizootics than in the model with rabbits as the only prey species (Fig. 3b).

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Figure 9. Typical simulated trajectories of the density of rabbits, foxes and foxes’ alternative prey over 100 years, (a) with no RCD and (b) with an 85% reduction in rabbits by RCD in spring on average every 2 years.

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image

Figure 10. Changes in the abundance of rabbits, foxes and foxes’ alternative prey, and in pasture biomass, resulting from the impact of RCD on rabbits. Computer simulations were used to calculate, on a quarterly basis, the mean density of (a) rabbits, (b) foxes and (c) a generic alternative prey species, the mean pasture biomass (d), and the mean percentage of quarters when the density of (e) rabbits or (f) the alternative prey species were >2·5 ha–1. Each estimate was derived from 1000 100-year computer simulations.

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Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

The model for predators, herbivores and pasture (Fig. 1) was constructed primarily to examine the consequences for predators’ alternative prey of a reduction in primary prey by a new biocontrol agent. Key processes in the model were quantified using available data for foxes, rabbits and native pasture in semi-arid southern Australia, and the epidemiology of RCD was simulated empirically to mimic a range of potential patterns of occurrence. The effect of predation on species that are alternative prey for foxes was modelled using a Type II functional response. Support for this model has come from a recent analysis of demographic data from declining species and from re-introductions of native species in mainland Australia (Sinclair et al. 1998). Similar dependence on pasture biomass was assumed for the dynamics of both primary and alternative prey in order to identify clearly the consequences of differing predation.

A series of predictions can be made for rabbit- and fox-prone areas of semi-arid Australia. Using the core model with foxes and rabbits but without explicit dynamics for alternative prey, the following changes would be expected as the frequency and intensity of RCD epizootics increases: (i) the mean abundance of rabbits will decline, as will the frequency of eruptions of rabbits; (ii) there may be little increase in mean pasture biomass and a small decrease in periods of very low pasture biomass when competition between herbivores is likely to be most intense; (iii) the mean abundance of foxes will decline; and (iv) there will be a reduced frequency of occasions when rabbit density is low but fox density is high due to a lag in the response of predator populations. When the model is extended to include both primary and alternative prey, an additional consequence of increasing the impact of RCD is that (v) there is potential for an increase in the mean abundance of alternative prey and in the proportion of time their density exceeds a threshold comparable to that currently required for eruptions of rabbits. However, in the extended model, increased pasture consumption by the alternative prey species compensates for RCD-induced reductions in the impact of rabbits. These predictions are presented graphically in Figs 3, 5, 7, 8 and 10.

Rabbits

The rate of increase of rabbits, in the absence of predation by foxes, was modelled using an Ivlev function with rainfall or pasture biomass as the independent variable (eqns 11 and 12). One of the major simplifications in using this approach is that the responses through recruitment and mortality are merged. However, it is quite possible that the mortality rate changes more rapidly than recruitment in response to rainfall or food supply, and will be reflected immediately in counts of rabbits. For example, very low rainfall coincided with the post-plague crash in the abundance of rabbits at the end of 1979 at Yathong (Newsome, Parer & Catling 1989). To some extent it might be possible to modify the model by dividing the data into two seasons, the breeding season when recruitment would be more important in determining numbers of rabbits, and the non-breeding season when mortality would tend to dominate, and to use different lags for each. In addition, the maximum rate of increase is likely to be different during winter and spring, when most rabbits are breeding (Gilbert et al. 1987), compared to the remainder of the year. The data (omitting the outlying value for the quarter from March to May 1982) were divided into breeding and non-breeding periods (Table 3) and, with the maximum rate of decrease assumed constant for all seasons, the parameters dN and cN in eqn 5 were estimated separately for each period. However, the estimates differ very little from the values in eqn 11 primarily because there are insufficient data from Yathong during the non-breeding period of the year.

Foxes

The rate of increase of fox populations (eqn 15) is determined by the supply of rabbits in the simpler, first-stage model and by rabbits and alternative prey in the extended model. The models produce somewhat different predictions for the consequences of RCD for foxes. The potential for a much larger reduction in fox density in the single-prey compared to the two-prey model (Figs 7 and 10b) is consistent with the conclusion of Anderson & May (1986) that an introduced pathogen affecting a prey species will have greater impact for a specialist compared to a generalist predator. In addition to simplifying the fox's prey into two classes (rabbits as primary prey and other herbivores as similar-sized alternative prey), the model ignores the role of RCD in supplying carrion, which can be a substantial component of the fox's diet, especially in winter (Catling 1988). However, during the initial 1995 epizootic in the Flinders Ranges area of South Australia, most rabbits infected with RCD appeared to die in warrens rather than on the surface (A.E. Newsome, unpublished data) and the carcasses would have been relatively inaccessible to foxes. As a further simplification, the model does not include seasonal effects on fox demography. For example, in the model of Pech et al. (1997) environmental conditions during the critical late summer/autumn period were assumed to have an immediate impact on survivorship of foxes as well as a delayed influence by affecting recruitment for the following year.

Alternative prey

The first-stage model for foxes and rabbits was modified to illustrate the effect of changing fox density on populations of small-to-medium sized native species that are alternative prey for foxes. Predation on alternative prey was represented by a Type II functional response which, for purposes of comparison, was matched as closely as possible to the Type III response fitted by Pech et al. (1992) for rabbits. It would be preferable to have data that test the hypothesized functional response, particularly for species whose viability is threatened by predation. In addition, the relative abundance of prey species may affect predation rates in a way not included in the model. But this information is not available at present nor is it likely to be in the foreseeable future, partly because of the difficulty of studying rare prey species. Therefore a generic model was constructed to illustrate principles rather than details for a particular native species. The main value of this approach is that it can be used to make general predictions despite limited information and it can form a conceptual model for management and future research.

For computer simulations, the numerical response of rabbits in the absence of predation was also used for generic, small-to-medium sized native herbivores that are alternative prey for foxes. In principle, the model could be improved by modifying this function to match better the demographic characteristics of representative native species. For example, in a recent model for arid-zone species (D. Choquenot, S. McLeod & N. Dexter, unpublished data) it is suggested that physiological constraints, such as litter size, on the maximum rate of increase caused some small macropods to have a more asymmetric numerical response than other similar-sized mammals such as rabbits. The authors hypothesized that following European settlement of Australia the temporal variability in food supply for these mammals increased and the outcome, modelled as an interaction of this variability with the numerical response, was the decline and extinction of some species. This assessment of the rate of increase of Australian native species is supported by the analysis of Sinclair (1997), which shows that marsupial species tend to have a lower instantaneous birth rate per individual per year than eutherian species of equivalent body weight. However, in the absence of detailed data for the numerical responses of native species, the simplest method for drawing general comparisons with the rabbit is to change only the predation component of the rate of increase.

Rcd

The effects of RCD were included in the model in an empirical manner by imposing short periods of additional mortality on the rabbit population. This could be improved by developing an epidemiological model that includes the dynamics of the host and key vectors (see, for example, Anderson & May 1979). Until the methods of transmission of RCD in the Australian environment are understood, or at least the effects of RCD on free-range rabbit populations have been measured over several years, it is not feasible to construct detailed mechanistic models. The model used here does not include any attenuation of the effects of RCD that might occur through co-evolution of genetic resistance in rabbits and virulence of the virus. This process became highly significant after the introduction of myxoma virus to Australia in the 1950s (Fenner & Myers 1978) but as yet no clear evidence has emerged for RCD. The effect of changing the seasonality of RCD epizootics was tested with the current model, but with the non-seasonal pattern of rainfall for Yathong, which is typical of many mid-latitude semi-arid and arid areas, there was no detectable influence of season on the predictions.

General conclusions

Morton (1990) reviewed the processes that have led to the decline and extinction of many vertebrate species in arid Australia. Of major importance is a species’ ability to expand its range from refugia following drought. He hypothesized that small-to-medium sized species in the weight range of 0·1 kg to approximately 3 kg are particularly vulnerable. This range lies within the continental assessment of a ‘critical weight range’ (CWR) of 35 g to 5·5 kg for vulnerable species by Burbidge & McKenzie (1989). Viable populations of CWR species cannot persist in many small refugia, whereas populations of smaller-sized species can. On the other hand, CWR mammals are less mobile than larger species and under good conditions cannot recolonize areas as rapidly. According to Morton (1990), habitat change and loss of refugia brought about by introduced herbivores exacerbated the vulnerability of CWR native species. However, rabbits fit neatly into the CWR category but appear to persist remarkably well. Differences in the shape of the numerical response function for rabbits compared to native species could account for this (D. Choquenot, S. McLeod & N. Dexter, unpublished data). In this paper, different susceptibility to predation by foxes is considered as an alternative, or perhaps complementary, hypothesis.

Morton (1990) ranked the role of introduced predators as secondary in the demise of native species. This view is supported, for example, by Tunbridge (1991) whose evidence indicates that the loss of many native species in the Flinders Ranges coincided with the expansion of pastoralism between 1850 and 1900 but pre-dated the arrival of foxes in that area. However, an analysis of bounty payments between 1883 and 1920 suggests that predation by foxes, not competition from rabbits or from livestock in a rapidly expanding rural industry, caused the extinction of three species of bettongs, Bettongia lesueur (Quoy & Gaimard 1824), B. penicillata (Gray 1837) and B. gaimardi (Desmarest 1822), and the decline of the rufous bettong Aepyprymnus rufescens (Gray 1837) and the long-footed potoroo Potorus longipes (Seebeck & Johnston 1980) in New South Wales (Short 1998). Whatever the original causes of species’ declines, the model suggests that their continued existence or recovery can be strongly influenced by exotic predators such as foxes. By reducing the primary prey of foxes, RCD has the potential to improve the conservation prospects of the remnant populations of native mammals in rabbit-prone areas of arid and semi-arid Australia.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

Thanks are due to A.E. Newsome, J. Hone, S. McLeod and N. Nicholls for critical comments on a draft of this manuscript. The work was funded partly by Environment Australia.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix
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Received 9 September 1997; revision received 11 May 1998

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

No data are available to estimate the demographic efficiency of foxes (dP) directly. In addition, the data on rates of increase of rabbits at Yathong (Table 3) could not be used to estimate the demographic efficiency of rabbits (dN) when pasture biomass is the independent variable. Therefore two criteria that summarize key attributes of fox and rabbit populations were used with computer simulations to select the most appropriate value for dP. They were also used as benchmarks to assess the sensitivity of the model to changes in dP and in the components of the term, (dN.V), in the numerical response of rabbits. The criteria are:

1. The average fox density should be realistic for semi-arid Australia. Estimates are of the order of 0·6–0·9 km−2 (Saunders et al. 1995).

2. The frequency of rabbit plagues should be similar to that observed for semi-arid Australia. Two periods of exceptionally high rabbit abundance, much greater than the background rabbit density of <4 ha−1, were observed during a 26-year study at two sites in the southern Flinders Ranges of South Australia (B.D. Cooke, unpublished data). This corresponds to a plague in approximately 8% of years and is in agreement with the historical overview by Newsome et al. (1997) that estimated an average of one rabbit plague every decade for semi-arid eastern Australia.

The simplified model based on the interaction of foxes and rabbits (the solid arrows in Fig. 1 or eqns 1, 2 and 4 with M = 0) was used to estimate dP. Initially, dN was set at the value from Choquenot (1992) and two options, no time-lag and a time-lag of 3 months, were considered for the numerical response of rabbits to pasture biomass. The results of the computer simulations (Fig. 11a,b) show that the mean density of foxes increases with dP but the proportion of years with rabbit density >4·0 ha−1 declines with dP. Although the model is sensitive to changes in dP, the trade-off between the frequency of rabbit plagues and the mean density of foxes results in a reasonably well-defined range of acceptable values. This outcome is essentially unaffected by the response time of the rabbit population to its food supply. If the numerical response of rabbits to pasture biomass has a lag of one quarter, a plague frequency of 8–10% corresponds to 3·1 < dP < 3·3 and an average fox density in the range 0·65–0·7 km−2 (Fig. 11a). Alternatively, with no lag in the numerical response of rabbits, the corresponding estimates for foxes are 3·2 < dP < 3·6 and mean densities of 0·75–0·85 km−2 (Fig. 11b).

imageimage

Figure 11. Sensitivity of the simplified fox–rabbit model (i.e. M = 0) to estimates of the demographic efficiencies of the rabbit and fox populations, and to a time lag of 3 months, or no time lag, in the numerical response of rabbits to pasture biomass. Each point in the graphs is the mean value from 500 simulations iterated with 3-month time steps for 100 years. Transient dynamics during an initial 10-year period in each simulation were not included in the analyses. A threshold density of four rabbits ha−1 defines a plague year. (a) Mean fox density and percentage of years with rabbit plagues for dN = 0·0045 and a time lag of 3 months in the numerical response of rabbits. (b) As for (a) but with no time lag. (c) Mean fox density and percentage of years with rabbit plagues for dP = 3·2 and a time lag of 3 months in the numerical response of rabbits. (d) As for (c) but with dP = 3·4 and no time lag.

The effect of varying the value for dN was assessed with no lag, and with a lag of one quarter, in the response of rabbit populations to pasture biomass (Fig. 11c,d). In each case the corresponding value for dP was set at the mid-range of the estimates from Fig. 11(a,b), i.e. dP = 3·2 and dP = 3·4, respectively. At low values for dN (<0·004), rabbit populations are unable to increase sufficiently rapidly to erupt. Rabbit plagues occur in 8–10% of years when dN is approximately 0·0045 (Fig. 11c,d), or when 0·0055 < dN < 0·0062 in Figs 11(c) and 0·0051 < dN < 0·0058 in Fig. 11(d). In both cases, the predator–prey dynamics are different in the higher range of dN compared to dN at approximately 0·0045. With dN at approximately 0·0045, mean fox densities are within the expected range for semi-arid Australia. Higher values of dN generate a supply rate of rabbits sufficient to maintain unrealistically high mean densities of foxes, and an additional outcome is that rabbits are regulated to consistently very low densities between plague years. This is qualitatively different to the long-term data on rabbit abundance from semi-arid South Australia (B.D. Cooke, unpublished data) and the observations at Yathong (Newsome, Parer & Catling 1989) where rabbits often reach densities up to 2–3 ha–1 between eruptions.

The sensitivity analyses summarized in Fig. 11 confirm that dN = 0·0045 is a good estimate for the demographic efficiency of rabbits. There appears to be little difference when comparing the two options for a time lag in the numerical response of rabbits to pasture biomass. However, a lag of one quarter is consistent with the estimated relationship based on rainfall (eqn 11). For computer simulations, this is equivalent to using (dN.V) = 0·0045Vt−1 as the exponent in eqn 12. Therefore, from Fig. 11(a), the best estimate for the demographic efficiency of foxes is dP = 3·2.