.( A.1 )
We employed logistic and nonlinear regression methods to estimate the following parameters: (1) The probability that a female sub-adult reaches maturity at any given time step (see Eq. 2 [in article], and Table A1), (2) The probability to die as a function of rainfall (see Eq. 7 [in article]), and (3) The von Bertalanffy growth equation (see Eq.10 [in article], Table A2, and Fig. A1).
For all other parameters we made educated guesses based on literature and expert advice. The consequences of the uncertainty in all parameter estimations are examined within the sensitivity analysis.
The logistic regression for maturity in females was based on annual rainfall at Currawinya National Park in southwestern Queensland. The Menindee rainfall, R, was converted to an equivalent amount at Currawinya using the formula
|
. |
( A.1 )
|
and 
are
the average and standard deviation of annual rainfall at Menindee (M) and Currawinya
(CW). This conversion assumes that there are regional differences in the responses
of kangaroo populations to rainfall, presumably reflecting differences in soil
and plant communities (Cairns 1989).
Juvenile mortality is generally higher and more variable than adult mortality in red kangaroo populations (Frith and Sharman 1964, Newsome 1977, Shepherd 1987, Pople 1996), and mortality of male red kangaroos is generally higher than that of females, particularly during drought (Robertson 1986, Pople 1996). To reflect these qualitative observations we arbitrarily set the minimum mortality of adult kangaroos to 0.01 and of juveniles to 0.2. In the model, we assume that the minimum mortality of males and females are the same, but males are affected more by droughts than females.
There is considerable evidence that the mortality of kangaroos increases with decreasing rainfall (e.g., Shepherd 1987, Pople 1996), but no data are available to estimate the relationship between red kangaroo mortality and rainfall directly. The relationship between the exponential rate of increase of red kangaroos and rainfall has been described by a number of authors (e.g., Bayliss 1987, Cairns and Grigg 1993). Bayliss (1987) developed a numerical response model relating red kangaroo rate of increase and rainfall at Kinchega National Park:
 |
( A.2 )
|
where r is the annual exponential rate of increase between winter aerial surveys and R is the previous calendar year rainfall. The underlying mechanisms for varying r are changes in survival and birth processes as described in the Euler-Lotka equation (McCallum 2000):
 |
( A.3 )
|
where x is the age of
individuals, 
is the age-specific survivorship, 
is the age-specific fecundity and r is the rate of increase of the
population when it has a stable age distribution.
In our model we use Eq. 7 to describe
the relationship between mortality and rainfall. The parameter 
in Eq. 7 is determined by 
and 
(see
Eq. 8). We initially estimated 
from data in Pople (1996). We used the initial 
and
in Eq. 7
to calculate 
at
a range of rainfall values. We can obtain 
from
because
and 
.
We then calculated the corresponding r using (Eq. A.3). The resulting
curve of r as a function of rainfall was then compared with Eq. A.2.
We iteratively improved our initial estimates of 
,
until the r-values corresponded to those generated by Bayliss' model.
As a result of data limitations we used a stage-structured version of Eq. A.3
with x
{adult,
juvenile}, and made the following simplifying assumptions:
1. The maximum survival rate is
constant for all ages: 
for
adults and 
for
juveniles.
2. The fecundity is independent
of age:
(Tyndale-Biscoe and Renfree 1987).
The final estimated survival curve is shown in Fig. A2. Pople (1996) found no influence of age on fecundity in mature red kangaroos. However, droughts can induce delays in the onset of sexual maturity (Frith and Sharman 1964) and induce anoestrus (Newsome 1964). In our model, anoestrus is included indirectly as decreased juvenile survival in droughts.
At high rainfall, the 
and
schedule
used in Eq. A.3 cannot return the high r-values (i.e.
)
predicted by Bayliss' model. This is almost certainly due to the unstable age
distributions underlying Bayliss' data, resulting from low juvenile and adult
male survival during drought (Bayliss 1985a, Pople
1996).
Table A1. Logistic regression to estimate the influence of weight, age and rainfall on the probability to mature.
Estimate Std. Err. z value (Intercept) 1.13672 5.61169 0.203 0.83948 Weight 0.65345 0.14089 4.638 3.52e-0.6 Age -8.29777 3.40324 -2.438 0.01476 Rain -0.04551 0.01601 -2.843 0.00447 Age:Rain 0.03277 0.01046 3.134 0.00172 Notes: From 1991–2000 annual rainfall, weight and maturity status of 474 female kangaroos was recorded. Individuals were classified as mature if they carried a young or if their teats were enlarged (Pople, unpublished data). Null deviance = 453.32 on 360 degrees of freedom; Residual deviance = 104.92 on 356 degrees of freedom. The age and rainfall coefficients are both negative because the positive effects they both have on the probability a female is mature are incorporated into the interaction term.
Table A2. Nonlinear regression to estimate Bertalanffy growth parameters.
Bul 184.43 2.236 82.4809 152.969 0.8506 179.837 c 0.2613 0.0244 10.713 0.5402 0.0393 13.7322 -3.2515 0.4085 -7.9594 -1.3852 0.2193 -6.3163 Mul 181.94 2.2875 79.5368 153.86 0.6952 221.324 c 0.2554 0.0286 8.9224 0.4651 0.0374 12.4491 -3.2692 0.5344 -6.1179 -1.9794 0.3064 -6.4592 Cw 181.741 1.5762 115.304 152.756 0.5214 292.975 c 0.2352 0.0196 11.9932 0.4105 0.0283 14.5005 -3.7041 0.4032 -9.1863 -2.4267 0.2667 -9.0993 Notes: Data were collected in 1991 at Bulgunnia in NW SA (Bul), Mulyungarie in NE SA (Mul), and Currawinya in Qld (Cw).
is the asymptotic growth, c is the growth rate and
is the shift parameter. The residual standard errors, RSE, and degrees of freedom, df, for the different locations are as follows. Bul: RSE = 4.78 on 147 df for males, or RSE = 4.07 on 134 df for females; Mul: RSE = 5.82 on 92 df for males, or RSE = 4.08 on 149 df for females; Cw: RSE = 6.04 on 158 df for males, or RSE = 4.05 on 178 df for females.
FIG.
A1. Von Bertalanffy growth curves for male and female red kangaroos (see
Eq.10). Parameters for males (solid line):  ,
 , c
= 0.25; parameters for females (dotted line):  ,
 , c
= 0.47. |
| FIG. A2. Survival probability as a function of annual rainfall for adult females (solid line), adult males (dotted line), and juveniles (broken line). |
Literature cited
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Shepherd, N. 1987. Condition and recruitment of kangaroos. Pages 135–158 in G. Caughley, N. Shepherd, and J. Short, editors. Kangaroos: their ecology and management in the sheep rangelands of Australia. Cambridge University Press, Cambridge, UK.
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