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Spatial conservation management of the orange-bellied parrot Neophema chrysogaster
Biological Conservation,Vol, PII:
S0006-3207(97)00124-9
84, No. 3, pp. 283-292, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0006-3207/98 $19.00 + 0.00
ELSEVIER
SPATIAL C O N S E R V A T I O N M A N A G E M E N T OF THE O R A N G E - B E L L I E D P A R R O T Neophema chrysogaster Martin Drechsler* Environmental Science, School of Botany, University of Melbourne, Parkville, Victoria 3052, Australia
(Received 26 March 1997; accepted 20 June 1997)
ham (1996) who discuss the effects of various forest management strategies on the viability of a metapopulation. McCarthy et al. (1996) discuss questions like 'How many subpopulations are needed, and which ones are the most important?' These and related questions are concerned with setting priorities for conservation that affect particular subpopulations. Orange-bellied Parrots Neophema chrysogaster breed in south-western Tasmania and overwinter in the southeast of mainland Australia where they are dispersed over about 1000 km of coastline stretching from eastern Victoria to eastern South Australia. The population numbers in South Australia appear to be quite independent from those in Victoria (Menkhorst et al., 1990; Starks et al., 1992). Further, the habitat in South Australia differs considerably from that in Victoria, and the birds overwintering in Victoria and those overwintering in South Australia may be treated as separate 'subpopulations'. It should be noted, however, that the birds probably breed in the same areas in Tasmania and are likely to mix there. Therefore the two wintering subpopulations are unlikely to be distinct subpopulations in a genetical sense. To use financial resources effectively it would be helpful to know whether conservation management should focus mainly on the Victorian or the South Australian subpopulation. Furthermore, the proportion of birds overwintering in Victoria and South Australia may be affected by the release of captive-bred birds (Brown et al., 1994). Knowledge of the optimal distribution of the population can influence decisions on where to release those birds. In this study different management options are compared through their influences on the viability of the population as measured by the quasiextinction risk (Ginzburg et al., 1982) and the mean population size. The analysis is based on a simulation model of the Orange-bellied Parrot, an extension of the model by Drechsler et al. (1998) which assumes that there is only one single wintering population. The aim of these analyses is to explore the consequences of different assumptions about the dynamics of the population and to evaluate interactions between these assumptions and
Abstract
The Orange-bellied Parrot is endangered in south-eastern Australia. In the winter it is dispersed over a long band of coastal habitat ranging from Victoria to South Australia. The present model study compares the benefits of habitat management in the Victorian and the South Australian parts o f the winter habitat. Various management options are compared in a sensitivity analysis. The results show that management in Victoria has a stronger influence on the population than management in South Australia. The population parameters are subject to considerable uncertainty. The sensitivity of the results to the values assigned to the model parameters is explored. The results are valid over a wide range of the plausible parameter space. © 1998 Elsevier Science Ltd. All rights reserved Keywords: population viability analysis, conservation management, spatial dynamics, sensitivity analysis, Neophema.
INTRODUCTION Spatial structures in populations have been an issue in ecology for many years. Important theoretical concepts are, for instance, the theory of island biogeography (MacArthur and Wilson, 1967) and the metapopulation concept (Levins, 1970). These works and the many experimental and theoretical studies of fragmented populations (for instance, Gilpin and Hanski, 1991; and references below) highlight the important mechanisms that may affect the survival of a spatially structured population. In many cases, isolation and destruction of habitat appear to be critical factors. Beside such general knowledge about the requirements of spatially structured populations, studies which take the spatial pattern of human impacts on populations into account are relatively rare. Examples are Lamberson et al. (1994) and Lindenmayer and Possing*Present address: Department of Ecological Modelling, Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany. Fax: + 49 341 235 3500; e-mail: [email protected] 283
M. Drechsler
284
various management alternatives. Uncertainty in the parameter values is a major problem in applying the model. This is overcome by identifying a small basic subset of parameter combinations which encompass most of the variability in the model results.
METHODS
Management options In Drechsler et al. (1998), management options for the reproductive season and the winter season were discussed. It was shown that winter management is more important than summer management, as suggested from earlier work, e.g. Loyn et al. (1986). In this study only the management of winter habitat is considered. Options are: • •
increase mean survival rates, or increase the winter (carrying) capacity (i.e. the maximum number of winter survivors).
As described in Drechsler et al. (1998), the survival rates are likely to be related to the quality of the winter habitat (such as diversity of food plants) while the winter capacity is a measure of habitat quantity (such as habitat size). In the present work both management options are considered for Victoria and South Australia separately. The management goals considered are the increase of the mean population size (by 20%, for instance) and the reduction of the risk of population decline (quasiextinction risk; Ginzburg et al., 1982). Until now, most of the captive-bred birds have been released in Tasmania (Brown et al., 1994). A few birds were released in Victoria in August 1996. Alternatively, birds could be released in South Australia. In order to identify the optimal location it is important to know the optimal proportion of birds that overwinter in South Australia. If, for instance, in the model analysis the present proportion of birds in South Australia turned out to be below the optimal value, then captive birds should be released there. The proportion of birds that overwinter in South Australia is described by a 'migration probability'. In order to evaluate the importance of the South Australian subpopulation for the persistence of the species, the sensitivity (influence) of the mean migration probability on the viability of the population will be determined.
The model The life cycle of the Orange-bellied Parrot With the exception of the environmental variability in the survival rates and the migration of birds, the model has been described in Drechsler et al. (1998). The main aspects are summarised below. The model follows the annual life cycle of the parrots. It starts with the reproduction of birds in summer and is followed by
the winter season where most of the mortality is assumed to occur. After winter, juveniles become adults and breed. Age-dependence in survivorship and reproduction of adults is ignored in the model. Reproduction in the population depends on the maximum number of suitable nest sites (Brown and Wilson, 1982) ('breeding capacity' Kb) and the fecundity distribution which is the proportion of breeding pairs that produce a particular number of juveniles. The number of breeding pairs is limited by the number of nest sites, the number of males and the number of females, whichever is smaller. For each breeding pair the number of fledglings is sampled from the fecundity distribution. The offspring of all breeding pairs are summed to obtain the total numbers of juveniles raised. The sex ratio of fledglings is assumed to be even. After the breeding season, the parrots migrate from their breeding grounds in Tasmania to the mainland. Here the population is assumed to split into two subpopulations, one in Victoria and the other in South Australia. The separation into the two wintering subpopulations is described further below. In each of the two wintering sites, Victoria and South Australia, survival is described by an annual survival rate, and a 'winter' carrying capacity. These parameters may differ between the two wintering subpopulations. If the size, N, of a wintering subpopulation is small and well below the corresponding winter capacity, Kw, then the number of annual survivors in that subpopulation is sampled from a binomial distribution which is determined by the number of birds, N, and the survival rate. The number of survivors is calculated for juveniles and adults separately, because the observed survival rate of juveniles, sj, generally differs from that of adults, Sa. With increasing subpopulation size intraspecific competition is assumed to reduce the chance of survival of each individual. In the Orange-bellied parrot it is not known whether competition is of contest or of scramble type (Krebs, 1985). Both alternatives are considered in the model. In contest competition all resources are divided more or less equally among successful individuals, and individuals that compete unsuccessfully, do not consume any resources. Contest competition is modeled by multiplying the survival rate of each individual by a factor of Kw/N if the population exceeds the 'winter capacity' Kw. The expected number of survivors in the model then is limited approximately by s.Kw, where s = (sj + sa)/2 is the average of juvenile and adult survival rates. In scramble competition some of the resources are consumed by individuals that do not survive. This reduces the chance of survival of each individual to a value that is lower than it would be in contest competition. If the population size, N, exceeds a threshold, x, the survival of each individual is reduced from its 'contest value' above by an amount of
s.(1 -Kw/N).{Kw/(Kw-K)}
(1)
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S p a t i a l conservation m a n a g e m e n t o f the Orange-bellied P a r r o t
Here the 'scramble factor', S, measures how strongly scramble competition reduces the survival of birds and x is set at Kw/2. As discussed in Drechsler et al. (1998) this formulation captures the essential aspects of scramble competition. After the winter season the two subpopulations in Victoria and South Australia migrate back to Tasmania and 'recombine' to form a single breeding population again. E n v i r o n m e n t a l variability
Observations show that the survival of individuals is subject to annual variation ('environmental fluctuations'; for instance, G o o d m a n , 1987; Burgrnan et al., 1993). The model assumes that environmental variation affects the survival rates of juveniles and adults, sj and so, and that these rates are normally distributed with means, m; and ma, and standard deviations, aj and aa. These quantities m a y differ between Victoria and South Australia. Altogether, the model includes four different survival rates, each of which is normally distributed. The means in Victoria are denoted as mvj and mva, and in South Australia they are ms; and msa. The standard deviations are denoted as avj and tTva, and as a~j and a~a in Victoria and South Australia, respectively. Survival in Victoria and South Australia and survival of juveniles and adults will be correlated to some extent. The correlation between adults and juveniles is assumed to be the same in Victoria and in South Australia and is denoted as r. The correlation between survival in Victoria and South Australia (for adults and for juveniles) is given by p. It remains to specify the 'cross correlations', i.e. the correlation between juvenile survival in Victoria and adult survival in South Australia and between adult survival in Victoria and juvenile survival in South Australia. They are likely to be smaller than either of the two 'direct correlations', r and/9, and are modeled as y =
(2)
p.r
F r o m these correlations, r, p, and y, and the standard deviations of the four survival rates the variance-covariance matrix, P, can be constructed (Table 1). F r o m this matrix a correlation matrix A is calculated such that AA t = P where A t is the transpose of A (for instance, Faddeev and Faddeeva, 1963). The matrix A is calculated following the 'square root method' (for instance, Faddeev and Faddeeva, 1963, p. 144; Knuth, 1981, p. Table 1. The variance-covariance matrix, P.-y = p.r
Victoria
Adults Vic. Juveniles Vic. Adults SA Juveniles SA
South Austrailia
Adults
Juveniles
Adults
Juveniles
ava.ava r.avi.O'va p'asa'a~a y.~vi.tTva
r.trva.avi tTvi.t~vi ~/'tTsa'avi p.tTsi.tTvi
p.O'va.tTsa y.avi.asa asa'O'sa r.O'si.tysa
)/.O'va.asi p.O'vi.asi F'O'sa'O'si t~si.tTsi
Table 2. The correlation
matrix A. r'={1-r2} 112 and
p ' = {1-p2}'/2 avo r'avi p'a sa p.r.trsi
0 r ~.avi 0 p.r t.O'si
0 0 p" a sa r.pt.tTs i
0 0 0 p t .r t.O.si
551). The result is given in Table 2. F r o m the matrix A the four survival rates are calculated by (e.g. Burgman et al., 1993) i
Si : mi -k- E aijXj, j=l
= 1 . . . 4.
(3)
Here s~ = S w , s2=Sv;, s3=Ssa, S4=Ssj, the quantity a~j is the element in the i- th row and the jth column of the matrix A, and X1 to X4 are uncorrelated normally distributed r a n d o m numbers with zero mean and standard deviation one. With the values of the matrix A, eqn (3) becomes Sva = mva + Gva X1 "
Sq = mvj + avj " (rXl + r'X2) sm = m m + asa " (pX1 +/9I)(3)
(4)
Ssj = msj + Osj " (prX1 + p r ' X 2 + p ' r X 3 + p' r'X4)
where r ' = ( l - r 2 ) 1/2 and p ' = (1 _p2)1/2. Migration
As mentioned above, in winter the population splits into two subpopulations. After reaching the mainland, most of the birds stay in Victoria while some of them travel further west to South Australia. Each bird is assumed to migrate to South Australia with a certain migration probability, e. The number of birds that overwinter in South Australia, N s A , then will be binomially distributed NsA = Bin[N, e]
(5)
where N is the population size immediately before the winter season. One m a y call the resulting fluctuation in N s A the 'demographic variability' in migration (c. above). In addition, the migration probability itself may be a r a n d o m number fluctuating in response to environmental variation. It is assumed to be normally distributed with a mean, m e and a standard deviation, ae. Above, migration to South Australia was assumed to be a r a n d o m process and independent of other factors in the dynamics. However, even the birds that overwinter in South Australia are likely to migrate through Victoria. Based on their perception of food supply, e.g. they may depend migration to South Australia on the environmental conditions in Victoria. If
286
M . Drechsler
these conditions are below average, fewer birds m a y stay in Victoria and the migration probability to South Australia would be above average. In the model, the environmental conditions in Victoria are represented by the survival rates of juveniles and adults, svj and sw. For instance, a p o o r year is represented by below-average survival rates. A correlation between high migration and p o o r environmental conditions now means that there is a negative correlation between the survival rates in Victoria and the migration probability. For simplicity only the survival rate of adults is considered. This is not unplausible, because the South Australian subpopulation has a relatively high proportion of adults (Orangebellied Parrot Recovery Team, pers. comm.). As modeled above, both the adult survival rate, s ~ and the migration probability, e, are normally distributed rand o m numbers. Their correlation is now described by the equation (e.g. Burgman et al., 1993) e = me + ae . (qeX1 + ¢ 1 -
q2e . l 0
(6)
where qe is the correlation between migration and adult survival in Victoria and X1 is the r a n d o m number used in eqn (3) above. The quantity Y is another normally distributed r a n d o m n u m b e r that is not correlated to the Xi of eqn (4) and has a zero mean and standard deviation one. With eqn (6), if qe = - - 1 , the migration probability, e, is perfectly negatively correlated with the survival rate, sva, and if qe = 0 the two rates are uncorrelated. Parameter ranges Where applicable, parameter ranges are adopted from Drechsler et al. (1998). There is no information on whether the mean survival rates in South Australia are smaller or larger than in Victoria and the plausible range of survival rates in South Australia is assumed to be the same as that in Victoria. The Victorian subpopulation is much larger than the South Australian subpopulation. Therefore the variability in the survival of the entire population, which was determined by Drechsler et al. (1998), reflects mainly the conditions in Victoria and is used to describe the variability of survival rates in the Victorian subpopulation. In South Australia the birds are dispersed over a much wider range and food appears to be more variable than in Victoria. Therefore variability of survival in South Australia is likely to be higher than that in Victoria (Orange-bellied parrot Recovery Team, pers. comm.). Its upper bound is assumed to be twice that in Victoria. The correlation, p, between survival in Victoria and South Australia cannot be deduced from field data. It is certainly not negative and is assumed to lie between zero and one. The winter capacity which in Drechsler et al. (1998) measures the total number of survivors in the entire wintering population, now has to be distributed between Victoria and South Australia. The winter capacity in South Australia, Ks, is probably smaller than
that in Victoria, Kv, with an upper bound for Ks at 0-8.Kv. A lower bound for the ratio between the two capacities is set at Ks/Kv = 0.2 which is approximately the mean ratio of the observed subpopulation sizes (Table 3). The migration probability, e, is estimated from data collected during the winter population counts over the last 17 years (Table 3) which includes data taken from Starks et al. (1992) and the Orange-bellied Parrot Recovery T e a m (pers. comm.). For each year, the ratio of South Australian birds to the total wintering population size was calculated. This ratio reflects the probability that a bird migrates to South Australia, and so its mean and standard deviation provide estimates for the mean and standard deviation of the migration probability, me and ae (Table 3). Because it is more difficult to survey the South Australian habitat for birds than the Victorian wintering sites, the count data are likely to underestimate the actual proportion of South Australian birds. The median estimate in Table 3, 0.15, is used as a lower bound on the mean migration probability, me, and the upper 95%-confidence limit, 0.21, as an upper bound. The variability in the observed proportion of birds in South Australia is caused by several factors. These include measurement error because of the practical difficulties involved in the population counts in South Australia, the 'demographic variability' in migration (see above), and the actual variability in the migration probability itself. So the latter is likely to be overestimated by the variability in the measured proportion of South Australian birds. The median estimate in Table 3, 0-12, is chosen as an upper bound on the standard deviation of the migration probability, ae, and the lower 95%-confidence limit, 0.08, as a lower bound.
Table 3. Results of the winter population counts, provided by Starks et al. (1992) and the Orange-bellied Parrot Recovery Team
Year
NSA
Ntot
NsA/Ntot
1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 Mean SD
7 7 4 27 22 18 13 8 0 18 28 18 0 0 5 2 2
72 90 82 70 122 85 86 79 76 70 108 67 32 51 57 36 40
0.10 0.08 0.05 0.4 0.18 0-21 0.15 0.10 0 0.26 0.26 0.27 0.00 0.00 0.09 0.06 0.05 0.15 (0.09, 0.21) 0.12 (0.08, 0.16)
Mean and SD of the proportion of birds counted in South Australia. The 95% confidence limits are in parentheses.
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Spatial conservation management o f the Orange-bellied Parrot
The correlation between migration and adult survival in Victoria, qe, is unlikely to be positive (that would mean that birds escape from a good wintering site). It m a y be zero (which means that migration is independent o f survival in Victoria) or negative. U p p e r and lower bounds are 0 and - 1 , although a correlation of - 1 m a y be regarded as rather unlikely. Altogether, the model has 15 independent parameters (Table 4). Sensitivity analysis As described above, in each of the two wintering habitats various m a n a g e m e n t options are considered, aiming at an increase in the mean survival rates, mv~, mvj, m~a and msj, and the two winter capacities, Kv and Ks. In the sensitivity analysis these six parameters are varied by a certain proportion and the resulting changes in the quasiextinction risk of the population are recorded. A m a n a g e m e n t option that affects mean adult survival is likely to affect mean juvenile survival, as well, and in the sensitivity analysis the two mean survival rates are varied simultaneously. Four independent quantities are considered which include the mean survival rates in Victoria and South Australia, denoted as mv and ms, and the two winter capacities, Kv and Ks. For each of these four quantities a sensitivity coefficient is obtained by dividing the change in the quasiextinction risk by the relative change in the varied quantity. The four sensitivity coefficients are denoted as: Cmv , Ckv , Cms
and
£ks"
As described above, each Victorian sensitivity coefficient has to be compared to its South Australian counterpart. This is done by the following sensitivity ratios
C1=
Cmv/Cms
(7)
and (:72 = Ckv/Ck~"
The third management option introduced above is to change the mean migration rate. The effect on the quasiextinction risk is measured by a sensitivity coefficient
(8)
C 3 = Cme
which is calculated in the same way as Cmv and c,,s. F o r simplicity, C1 to C3 will be called the state variables of the analysis. Further details about the sensitivity analysis and the mathematical definition of the sensitivity coefficients can be found in Drechsler et al. (1998). Identification of important parameters and parameter combinations The analysis consists of two parts. The first part determines how the effect of the state variables, CI, C2, and C3, which measure and compare the sensitivities of the management options, depend on the model parameters. In the second part the results of the first part are used to construct a small basic subset of scenarios which encompass most of the variability in the state variables and the variability in the sensitivity of management options. For each of these basic scenarios the five sensitivity coefficients, cmv, Ckv, Cms, Cks and Cme, introduced above, are determined. The first part of the analysis involves test scenarios where all parameters but one are set at median values, and one is either at its upper or lower plausible bound. As the model has 15 parameters, 15 pairs of scenarios have to be considered. In the first scenario of each pair the parameter of concern is set at its upper bound and in the second scenario at its lower bound. For each of the 30 scenarios the three state variables, C1, C2, and C3, are determined. The comparison of two corresponding
Table 4. The model parameters and their plausible bounds Parameter Mean migration probability (me) SD of migration probability (tre) Correl. b/w survival and migration (qe) Mean number of offspring per pair (]) Breeding capacity (max. no. breeders) (Kb) Mean survival rate in Victoria (my) SD of survival rate in Victoria (t~v) Correl. b/w juvenile and adult survival (r) Total winter capacity K = Kv + Ks Scramble factor in Victoria (Sv) Mean survival rate in South Australia (ms) trs/crv(trs = SD of survival in SA) Correl. b/w survival in Victoria and South Australia (/9) Ratio of the two winter capacities (Ks/KO Scramble factor in South Australia (Ss)
Upper
Median
Lower
0.21 0.12 0 1.95 95 (0.74, 0.52) (0.13, 0.08) +1 540 0.1 (0.74, 0.52) 2 1 0.8 0-1
0.18 0.10 -0.5 1-85 70 (0.64, 0.475) (0.11, 0-06) 0 405 0.05 (0.64, 0-475) 1.41 0.5 0.4 0.05
0.15 0.08 -1 1-75 45 (0.54, 0.43) (0-09, 0-04) -1 270 0 (0.54, 0.43) 1 0 0.2 0
The winter capacity and SD of survival rates in South Australia are defined as ratios to their Victoria counterpart. For such a ratio it is reasonable to calculate the median of the plausible range as the geometric mean between upper and lower bound. In parentheses, the first value refers to adults, the second value to juveniles. In the analysis the parameters for adult and juvenile survival are varied simultaneously, i.e. both take either their upper, lower or median value.
288
M . Drechsler
scenarios allows us to determine the effect of the model parameters on the state variables. In each pair, the state variables of the second scenario are subtracted from those of the first. The resulting differences, Ai, show how the state variables, Ci, change when one of the 15 model parameters is increased from its lower to its upper bound. A large difference, Ai, means that the corresponding model parameter has a strong influence on the state variable, Ci, and the sign of Ai indicates whether the state variable Ci is increased ( + ) or decreased ( - ) when the parameter is increased from its lower to its upper bound.
RESULTS I n f l u e n c e o f the m o d e l p a r a m e t e r s o n the e f f e c t o f management options
The first analysis involved identification of model parameters that have important influences on the three state variables. The change in a state variable, Ci, which is caused when a model parameter is changed from its lower to its upper bound, is given by the Ai introduced above (Table 5). The comparison of the A i in a particular column, i, of Table 5 allows the identification of those model parameters that have the strongest influence on the state variable, C;. F o r instance, mv has the strongest influence on C1 (A,{mv)=-0.151), and with the definition of Ai above, the negative sign indicates that an increase in m~ will decrease Cl. In each column, i, the bold faced Ai in Table 5 indicate those parameters which have the strongest influence on the state variable, Ci. In particular, the sum of the bold faced numbers in a column is 80% of the sum of the entire column (signs ignored). So in each column, i, the parameters with bold Table 5. Influences of the 15 model parameters on the three state variables
me
cre qe f Kh
A1
A2
A3
0.087 0.016 -0.097 0.004 0.013
0.123 0.126 --0.215 0.043
--0.459 0.051 0.045 0.057
--0-265 --0.334
0.640 - 1-188
0.005
0.052
0.080
0.129
-0.438 --0.211 -0-593 1-502 2.177
my Cry r
-0-151
K
-0.012
0.062
Sv
- 0.014 0.098
- 2.743
ms a,la~
-0.014
p
0.075
Ks/Kv
0.007 0.019
Ss
0.075 -0-086 0-076 - 1-060 0-360
-0.845
-0.608 3.061 -0.961
A plus sign indicates that a state variable is increased by an increase of a model parameter, a minis sign indicates a decrease in that state variable. For each state variable, the strongest parameter influences are bold faced. They make up about 80% of all parameter influences on the respective state variable.
A i explain 80% of the variation in the corresponding state variable, Ci. For instance, the parameters me, qe, m~, r, ms and p explain about 80% of variation in the first state variable, C1. faced
E x t r e m e s c e n a r i o s and s e n s i t i v i t y a n a l y s i s
Tables 6-8 (first two columns) show the parameters with strong influences on the three state variables, C1, C2 and C3. Each of the three tables allows the identification of that parameter combination which maximises the corresponding state variable, Cb C2 and C3. For instance, maximising me or minimising rnv will increase C1 (Table 6). Consequently, C1 will be maximal when me is at its upper and rnv at its lower bound. In this way the two extreme parameter combinations which lead to m a x i m u m and minimum C1 were constructed by setting the model parameters at their appropriate upper or lower bounds (Table 9 first two rows). These two extreme parameter combinations fully encompass the variation in C1 that is caused by the parameters included in Table 6. As pointed out above, the parameters in Table 6 explain about 80% of the total variation in Ci caused by all parameters. Consequently, the two extreme scenarios encompass 80% of the total variability in C1. The same analysis was done using Tables 7 and 8 to construct extreme scenarios that encompass 80% of the variation in Cz and C3, respectively. The result is shown in Table 9. For each of the scenarios shown in Table 9 the sensitivity coefficients, Cmv, Ckv, C,,,s, Cks and Cme, were determined (Table 10). They measure the influence of the five model parameters which represent management actions, on the viability of the population. The main results that can be deduced from Table 10 are (a) The mean survival rate in Victoria always has the strongest influence on the viability of the population, even in Scenario (1) which was designed to maximise the influence of the mean survival rate in South Australia. Generally, the next most important parameter is the mean survival rate in South Australia. (b) In the default parameter combination where all parameters are at their median values, mean survival in South Australia is more important than the winter capacity in Victoria, but this can be different in Scenario (4) where the breeding capacity, the mean survival rate in Victoria and the ratio between the winter capacities in South Australia and Victoria are maximal, and where scramble competition is strong in Victoria but weak in South Australia. (c) The winter capacity in South Australia and the migration probability are comparatively unimportant, even in Scenarios (3) and (5) which were designed to maximise their influence. Instead of reducing the risk of population decline (quasiextinction risk) one may wish to increase the
Spatial conservation management o f the Orange-bellied Parrot
289
Table 6. Influence of the model parameters on the state variables, (71
Making... m e
increases ....
and also leads to...
which results in...
C1
large number of migrants to South Australia migration especially when conditions are poor in Victoria few survivors in Victoria
large South Australian population South Australia acts as a 'safe haven' in critical years small Victorian population
many survivors in South Australia
large S. Australian population
large
qe large and negative my small r large ms large p large
The last two columns show likely effects of the model parameters on the population dynamics. These effects were not explicitly deduced in the model analysis but appear to be plausible. Table 7. Influence of the model parameters on the state variables (?2
Making...
increases..,
and also results in...
Kb small my small Sv small K~/Kv small S~ large
survival rates in Victoria more important than winter capacity* no scramble competition in Victoria stronger intraspecific competition in South Australia and less in Victorian strong scramble competition in South Australia
C2
The last column shows likely effects of the model parameters on the population dynamics. The effect of my, marked by a (*), was found by Drechsler et al. (1998) Table 8. Influence of the model parameters on the state variables (73
Making... Kb large my small Sv large m~ large trs/trv small Ks/ Kv large Ss small
increases..,
and also results in ....
C3
poor average conditions in Victoria strong scramble competition in Victoria good average conditions in Victoria stronger intraspecific competition in Victoria and less in S. Australia no scramble competition in South Australia
The last column shows likely effects of the model parameters on the population dynamics. Table 9. The six extreme parameter combinations which encompass 80% of the variability in the model results
me
tre
qe
f
Kb
1
+
2 3 4 5 6
0 0 0 0
mv
try
r
K
0
-
0
0 0 0 0 0
+ 0 0 0 0
0 0 0 0 0
Sv
ms
trs/trv
p
Ks/Kv
Ss
0
-
0
+
0 + + -
+ + +
0 0 0 0 0
0 0 0 0
0
0
+
0
+
0
0
0 0 0 0 0
0 + + -
0 0 + -
0 0 0 +
0 0 0 0
0 + + -
0 + +
A zero entry means that the respective parameter is at its median value, a plus (minus) sign means that it is at its upper (lower) bound (c. Table 4). m e a n p o p u l a t i o n size. As these objectives m a y be indep e n d e n t to some extent ( B u r g m a n et al., 1993), the m o d e l analysis was repeated using m e a n p o p u l a t i o n size as the m a n a g e m e n t target. T h e results were a l m o s t identical.
DISCUSSION Model assumptions
T h e Orange-bellied P a r r o t is a very rare species a n d despite c o n s i d e r a b l e effort in the field, its ecology is n o t fully u n d e r s t o o d . F o r instance, there is n o t m u c h k n o w n
a b o u t the factors which influence m o r t a l i t y in the population. Density-dependence and environmental variability h a d to be m o d e l e d in a very general way, based o n expert j u d g m e n t . M i g r a t i o n o f birds to South A u s t r a l i a was also m o d e l e d in a very simple m a n n e r . O t h e r aspects such as site fidelity which might play a role, were n o t considered in the study. The c o n c l u s i o n s represent what appears to be i m p o r t a n t in the m a n a g e m e n t o f the Orange-bellied P a r r o t using c u r r e n t knowledge. C a t a s t r o p h e s a n d genetics were n o t considered in this study. As discussed in Drechsler et al. (1998),
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Table 10. Results of the sensitivity analysis performed for the six extreme scenarios defined in Table 5 (Scenarios 1-6) and a Scenario 0 where all parameter have their median values Cmv
Ckv
Cms
Cks
0
31
1
18
2 3 4 5 6
46 24 25 23 26
Cme
2
8
0
1
1
10
0
2
5 0 10 2 0
7 6 6 8 4
1 0 0 0 0
-1 1 2 2 -2
The sensitivity coefficients measure the influences of the mean survival rates in Victoria and South Australia (c,,v and cms), the influences of the winter capacities in Victoria and South Australia (Ckv and Cks), and the influence of the mean migration probability (Cme)on the quasiextinction risk. The numbers have been rounded to a precision that is appropriate with respect to statistical variation in the results. Values greater than 2 may be interpreted as important and values greater than 5 indicate parameters which have a strong influence.
catastrophes should be studied separately. Genetic influences can be important in metapopulations where there are genetically separated subpopulations and a turnover of local extinction and recolonisation (for instance, Gilpin, 1991). However, the population of the Orange-bellied Parrot is very likely to be a single population, because individuals mix in breeding colonies in Tasmania. Genetic threats are related to population size and risk of decline (Kimura and Crow, 1963; Koenig, 1988). Therefore management which increases population size and reduces the risk of decline, will also reduce genetic risks. M a n a g e m e n t implications
Each of the results (Tables 6-8 and Points (a)-(c)) can be directly translated into recommendations for the conservation management of the Orange-bellied Parrot. 1. Table 6 shows that those parameter values that maximise C1, i.e. the importance of habitat quality (mean survival rates) in South Australia, are also likely to maximise the size of the South Australian subpopulation compared with the Victorian subpopulation (last column of Table 6). The effect of the correlations, p and r, on the population sizes are less clear. However, their influence on C1 is comparatively small, so within the scope of this discussion they can be ignored. One can conclude that the relative importance of habitat quality in Victoria and South Australia is positively correlated to the relative sizes of the Victorian and the South Australian subpopulations. 2. The parameter values that were found to maximise C2, i.e. the influence of the winter capacity in South Australia on population viability, also represent a situation where there is strong intraspecific competition in South Australia and weak
intraspecific competition in Victoria (Table 7, last column). The role of the breeding capacity, Kb, is not clear, but its effect on (72 is rather small. One can conclude that the relative importance of the Victorian and the South Australian habitat capacities is positively correlated to the strength of intraspecific competition in the corresponding subpopulations. 3. The parameter values that were found to maximise C3, i.e. the benefit the population would gain from an increase in migration to South Australia, also represent a situation where (average) conditions in South Australia are better than those in Victoria (Table 8). The roles of the breeding capacity, Kb, and the standard deviation of survival in South Australia, trs, are not clear, but their effect on (73 is rather small. Consequently, whether migration to South Australia should be increased or reduced, i.e. whether captive birds should be released in South Australia or in Victoria, depends on whether the conditions in South Australia are better or worse than those in Victoria. This is not surprising, but validates the choice of the extreme scenarios (5 and 6 in Table 9) which were based on this result and lead to result (c) below./li > (a) Survival rates are generally the most important parameters, which confirms previous results. The mean survival rate in the Victorian subpopulation which may be related to the quality of Victorian winter habitat, was found to be more important than that in South Australia. (b) Generally, an increase in survival rates (habitat quality) in the South Australian subpopulation, appears to be more important than an increase in the Victorian winter capacity, i.e. the size of Victorian habitat. However, this is reversed if there is a large post-breeding population (caused by a large number of breeders) and if survival in Victoria is limited mainly by intraspecific competition (high mean survival rate, small ratio between the winter capacities in Victoria and South Australia, and high scramble factor in Victoria) (Scenario 4 in Table 10). In this case the winter capacity in Victoria is of considerable importance and management in South Australia is unimportant. (c) The winter capacity in South Australia is never important and management of habitat quantity in South Australia is not expected to have much influence on the viability of the population. As described in Methods, the relative sizes of the wintering subpopulations (migration probability) can be affected to some extent by different release strategies. The mean migration probability (i.e. the long term average of the relative subpopulation sizes) turned out to have a comparatively weak influence on population viability, although it should be noted that the relative
Spatial conservation management of the Orange-bellied Parrot subpopulation sizes might be changed more easily by m a n a g e m e n t than other parameters, such as the mean survival rates. A p a r t from affecting population viability, the release of birds at different sites on the mainland m a y lead to important insights into the migratory behaviour of the birds and their fidelity to their wintering sites. Thus, release strategies m a y be most beneficial if they are designed with a view to improving understanding of the species' dynamics. Mean survival rate and winter capacity of the Victorian subpopulation were always more important for the viability of the total population than their South Australian counterparts. This results from the fact that the Victorian subpopulation is larger than the South Australian subpopulation. M a n a g e m e n t in Victoria will affect more birds than management in South Australia, which suggests that m a n a g e m e n t should target the largest subpopulation of the wintering population. Although this study assumes that the whole wintering population consists of only two subpopulations, the wintering sites of a large proportion of the population are not known (the counts in Table 3 are substantially smaller than the population estimate of 175 provided by M c C a r t h y (1995). The arguments above suggest that in order to maximise the efficiency of conservation management, the large subpopulations should be targeted, and for this the distribution of the wintering population should be identified more comprehensively.
CONCLUSIONS The result that m a n a g e m e n t should target the largest of the two subpopulations agrees quite well with metapopulation theory. The two wintering subpopulations recombine every summer to a single breeding population, and therefore they are not really subpopulations in the sense of Levins (1970). The system is more analogous to a mainland-island system where there is one large mainland population and one or several small island populations (Harrison, 1991, 1994). Several works, for instance, Harrison, 1991, 1994 and Gyllenberg and Sylvestrov (1994) showed that the fate of the island(s) depends on the fate of the mainland. With this explanation in mind, some of the results are not very surprising. They reflect w h a t - - a t least in hinds i g h t - - a p p e a r s to be plausible. However, it is important to note that these results are not based on a single, 'best guess' parameter combination, but each of them was confirmed within a range between extreme scenarios which encompass 80% of all parameter combinations. Results (1) to (3) describe whether a particular management action becomes more or less effective when model parameters are varied. The analysis of pairs of extreme scenarios provides us with 'confidence intervals' on the results (a) to (c). As each pair encompasses 80%
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of all parameter combinations, one can say that the results (a) to (c) can be expected in about 80% of the entire parameter space. This adds a new aspect to a sensitivity analysis. N o t only was the sensitivity of certain management actions determined, but so was the reliability of the results. ACKNOWLEDGEMENTS I would like to thank M a r k Burgman and Peter Menkhorst for reading the manuscript and providing valuable comments. The advice and population data provided by the Orange-bellied Parrot Recovery Team helped much in the development and analysis of the model. The comments of an anonymous reviewer were very helpful in the revision of the manuscript. This work was supported by the German Research Community D F G . REFERENCES Brown, P. B. and Wilson, R. I. (1982) The Orange-bellied Parrot. In Species at Risk: Research in Australia, eds R. H. Groves and W. D. D. Ride, pp. 107-115. Australian Academy of Science, Canberra. Brown P. B., Holdsworth, M. C. and Rounsevell, D. E. (1994) Captive breeding and release as a means of increasing the Orange-bellied Parrot population in the wild. In Reintroduction Biology of Australian and New Zealand Fauna, ed. M. Serena, pp. 135-141. Surrey Beatty and Sons, Chipping Norton, UK. Burgman, M. A., Ferson, S. and Akakaya, H. R. (1993) Risk Assessment in Conservation Biology. Chapman and Hall, London. Drechsler, M., Burgman, M. A. and Menkhorst, P. W. (1998). Uncertainty in population dynamics and its consequences for the management of the Orange-bellied Parrot Neophema chrysogaster. Biological Conservation 84, 269-281. Faddeev, D. K. and Faddeeva, V. N. (1963) Computational Methods of Linear Algebra. Freeman, San Francisco and London. Gilpin, M. E. (1991) The genetic effective size of a metapopulation. In Metapopulation Dynamics: Empirical and Theoretical Investigations, eds M. E. Gilpin and I. Hanski. Academic Press, London, pp. 165-175. Gilpin, M. E. and Hanski, I. (eds) (1991) Metapopulation Dynamics: Empirical and Theoretical Investigations. Academic Press, London. Ginzburg, L. R., Slobodkin, L. B., Johnson, K. and Bindman, A. G. (1982) Quasiextinction probabilities as a measure of impact on population growth. Risk Analysis 2, 171-181. Goodman, D. (1987) The demography of chance extinction. In Viable Populations for Conservation, ed. M. E. Soul+, pp. 11-34. Cambridge University Press, Cambridge. Gyllenberg, M. and Sylvestrov, D. S. (1994) Quasi-stationary distribution of a stochastic metapopulation model. Theoretical Population Biology 42, 35-61. Harrison, S. (1991) Local extinction in a metapopulation context: an empirical evaluation. In Metapopulation Dynamics: Empirical and Theoretical Investigations, eds M. E. Gilpin and I. Hanski, pp. 73-88. Academic Press, London. Harrison, S. (1994) Metapopulations and conservation. In Large-scale Ecology and Conservation Biology, eds P. J. Edward, R. M. May and N. R. Webb, pp. 111 128. Blackwell Scientific Publication, Oxford.
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Kimura, M. and Crow, J. F. (1963) The measurement of effective population number. Evolution 17, 297-288. Knuth, D. E. (1981) The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 2nd edn. AddisonWesley, Reading, MA. Koenig, W. D. (1988) On determination of viable population size in birds and mammals. Wildlife Society Bulletin 16, 230-234. Krebs, C. J. (1985) Ecology: The Experimental Analysis of Distribution and Abundance. Harper and Row, New York. Lamberson, R. H., Noon, B. R., Voss, C. and McKelvey, K. (1994) Reserve design for territorial species. Conservation Biology 8, 185-195. Levins, R. (1970) Extinction. In Some Mathematical Problems in Biology, ed. M. Gerstenhaber, pp. 77-107. American Mathematical Society, Providence, RI. Lindenmayer, D. B. and Possingham, H. P. (1996) Ranking conservation and timber management options for Leadbeaters Possum in southeastern Australia using population viability analysis. Conservation Biology 10(1), 235-251.
Loyn, R. H., Lane, B. A., Chandler, C. and Carr, G. W. (1986) Ecology of Orange-bellied Parrots Neophema chrysogaster at their main wintering site. Emu 86, 195-206. MacArthur, R. H. and Wilson, E. O. (1967) The Theory of Island Biogeography. Princeton University Press, Princeton, NJ. McCarthy, M. A. (1995) Stochastic population models for wildlife management. Ph.D. thesis, University of Melbourne. McCarthy, M. A., Burgman, M. A. and Ferson, S. (1996) Logistic sensitivity and bounds for extinction risks. Ecological Modelling 86(2-3), 297-303. Menkhorst, P. W., Loyn, R. H. and Brown, P. B. (1990) Management of the Orange-bellied Parrot. In Management and Conservation of Small Populations, eds T. W. Clark and J. H. Seebeck, pp. 239-251. Chicago Zool. Soc., Chicago, IL. Ripley, B. D. (1987) Stochastic Simulation. Wiley, New York. Starks, J., Brown, P., Loyn, R. and Menkhorst, P. (1992) Twelve years of winter counts of the Orange-bellied Parrot Neophema chrysogaster. Australian Bird Watcher 14, 305312.
S0006-3207(97)00124-9
84, No. 3, pp. 283-292, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0006-3207/98 $19.00 + 0.00
ELSEVIER
SPATIAL C O N S E R V A T I O N M A N A G E M E N T OF THE O R A N G E - B E L L I E D P A R R O T Neophema chrysogaster Martin Drechsler* Environmental Science, School of Botany, University of Melbourne, Parkville, Victoria 3052, Australia
(Received 26 March 1997; accepted 20 June 1997)
ham (1996) who discuss the effects of various forest management strategies on the viability of a metapopulation. McCarthy et al. (1996) discuss questions like 'How many subpopulations are needed, and which ones are the most important?' These and related questions are concerned with setting priorities for conservation that affect particular subpopulations. Orange-bellied Parrots Neophema chrysogaster breed in south-western Tasmania and overwinter in the southeast of mainland Australia where they are dispersed over about 1000 km of coastline stretching from eastern Victoria to eastern South Australia. The population numbers in South Australia appear to be quite independent from those in Victoria (Menkhorst et al., 1990; Starks et al., 1992). Further, the habitat in South Australia differs considerably from that in Victoria, and the birds overwintering in Victoria and those overwintering in South Australia may be treated as separate 'subpopulations'. It should be noted, however, that the birds probably breed in the same areas in Tasmania and are likely to mix there. Therefore the two wintering subpopulations are unlikely to be distinct subpopulations in a genetical sense. To use financial resources effectively it would be helpful to know whether conservation management should focus mainly on the Victorian or the South Australian subpopulation. Furthermore, the proportion of birds overwintering in Victoria and South Australia may be affected by the release of captive-bred birds (Brown et al., 1994). Knowledge of the optimal distribution of the population can influence decisions on where to release those birds. In this study different management options are compared through their influences on the viability of the population as measured by the quasiextinction risk (Ginzburg et al., 1982) and the mean population size. The analysis is based on a simulation model of the Orange-bellied Parrot, an extension of the model by Drechsler et al. (1998) which assumes that there is only one single wintering population. The aim of these analyses is to explore the consequences of different assumptions about the dynamics of the population and to evaluate interactions between these assumptions and
Abstract
The Orange-bellied Parrot is endangered in south-eastern Australia. In the winter it is dispersed over a long band of coastal habitat ranging from Victoria to South Australia. The present model study compares the benefits of habitat management in the Victorian and the South Australian parts o f the winter habitat. Various management options are compared in a sensitivity analysis. The results show that management in Victoria has a stronger influence on the population than management in South Australia. The population parameters are subject to considerable uncertainty. The sensitivity of the results to the values assigned to the model parameters is explored. The results are valid over a wide range of the plausible parameter space. © 1998 Elsevier Science Ltd. All rights reserved Keywords: population viability analysis, conservation management, spatial dynamics, sensitivity analysis, Neophema.
INTRODUCTION Spatial structures in populations have been an issue in ecology for many years. Important theoretical concepts are, for instance, the theory of island biogeography (MacArthur and Wilson, 1967) and the metapopulation concept (Levins, 1970). These works and the many experimental and theoretical studies of fragmented populations (for instance, Gilpin and Hanski, 1991; and references below) highlight the important mechanisms that may affect the survival of a spatially structured population. In many cases, isolation and destruction of habitat appear to be critical factors. Beside such general knowledge about the requirements of spatially structured populations, studies which take the spatial pattern of human impacts on populations into account are relatively rare. Examples are Lamberson et al. (1994) and Lindenmayer and Possing*Present address: Department of Ecological Modelling, Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany. Fax: + 49 341 235 3500; e-mail: [email protected] 283
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various management alternatives. Uncertainty in the parameter values is a major problem in applying the model. This is overcome by identifying a small basic subset of parameter combinations which encompass most of the variability in the model results.
METHODS
Management options In Drechsler et al. (1998), management options for the reproductive season and the winter season were discussed. It was shown that winter management is more important than summer management, as suggested from earlier work, e.g. Loyn et al. (1986). In this study only the management of winter habitat is considered. Options are: • •
increase mean survival rates, or increase the winter (carrying) capacity (i.e. the maximum number of winter survivors).
As described in Drechsler et al. (1998), the survival rates are likely to be related to the quality of the winter habitat (such as diversity of food plants) while the winter capacity is a measure of habitat quantity (such as habitat size). In the present work both management options are considered for Victoria and South Australia separately. The management goals considered are the increase of the mean population size (by 20%, for instance) and the reduction of the risk of population decline (quasiextinction risk; Ginzburg et al., 1982). Until now, most of the captive-bred birds have been released in Tasmania (Brown et al., 1994). A few birds were released in Victoria in August 1996. Alternatively, birds could be released in South Australia. In order to identify the optimal location it is important to know the optimal proportion of birds that overwinter in South Australia. If, for instance, in the model analysis the present proportion of birds in South Australia turned out to be below the optimal value, then captive birds should be released there. The proportion of birds that overwinter in South Australia is described by a 'migration probability'. In order to evaluate the importance of the South Australian subpopulation for the persistence of the species, the sensitivity (influence) of the mean migration probability on the viability of the population will be determined.
The model The life cycle of the Orange-bellied Parrot With the exception of the environmental variability in the survival rates and the migration of birds, the model has been described in Drechsler et al. (1998). The main aspects are summarised below. The model follows the annual life cycle of the parrots. It starts with the reproduction of birds in summer and is followed by
the winter season where most of the mortality is assumed to occur. After winter, juveniles become adults and breed. Age-dependence in survivorship and reproduction of adults is ignored in the model. Reproduction in the population depends on the maximum number of suitable nest sites (Brown and Wilson, 1982) ('breeding capacity' Kb) and the fecundity distribution which is the proportion of breeding pairs that produce a particular number of juveniles. The number of breeding pairs is limited by the number of nest sites, the number of males and the number of females, whichever is smaller. For each breeding pair the number of fledglings is sampled from the fecundity distribution. The offspring of all breeding pairs are summed to obtain the total numbers of juveniles raised. The sex ratio of fledglings is assumed to be even. After the breeding season, the parrots migrate from their breeding grounds in Tasmania to the mainland. Here the population is assumed to split into two subpopulations, one in Victoria and the other in South Australia. The separation into the two wintering subpopulations is described further below. In each of the two wintering sites, Victoria and South Australia, survival is described by an annual survival rate, and a 'winter' carrying capacity. These parameters may differ between the two wintering subpopulations. If the size, N, of a wintering subpopulation is small and well below the corresponding winter capacity, Kw, then the number of annual survivors in that subpopulation is sampled from a binomial distribution which is determined by the number of birds, N, and the survival rate. The number of survivors is calculated for juveniles and adults separately, because the observed survival rate of juveniles, sj, generally differs from that of adults, Sa. With increasing subpopulation size intraspecific competition is assumed to reduce the chance of survival of each individual. In the Orange-bellied parrot it is not known whether competition is of contest or of scramble type (Krebs, 1985). Both alternatives are considered in the model. In contest competition all resources are divided more or less equally among successful individuals, and individuals that compete unsuccessfully, do not consume any resources. Contest competition is modeled by multiplying the survival rate of each individual by a factor of Kw/N if the population exceeds the 'winter capacity' Kw. The expected number of survivors in the model then is limited approximately by s.Kw, where s = (sj + sa)/2 is the average of juvenile and adult survival rates. In scramble competition some of the resources are consumed by individuals that do not survive. This reduces the chance of survival of each individual to a value that is lower than it would be in contest competition. If the population size, N, exceeds a threshold, x, the survival of each individual is reduced from its 'contest value' above by an amount of
s.(1 -Kw/N).{Kw/(Kw-K)}
(1)
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S p a t i a l conservation m a n a g e m e n t o f the Orange-bellied P a r r o t
Here the 'scramble factor', S, measures how strongly scramble competition reduces the survival of birds and x is set at Kw/2. As discussed in Drechsler et al. (1998) this formulation captures the essential aspects of scramble competition. After the winter season the two subpopulations in Victoria and South Australia migrate back to Tasmania and 'recombine' to form a single breeding population again. E n v i r o n m e n t a l variability
Observations show that the survival of individuals is subject to annual variation ('environmental fluctuations'; for instance, G o o d m a n , 1987; Burgrnan et al., 1993). The model assumes that environmental variation affects the survival rates of juveniles and adults, sj and so, and that these rates are normally distributed with means, m; and ma, and standard deviations, aj and aa. These quantities m a y differ between Victoria and South Australia. Altogether, the model includes four different survival rates, each of which is normally distributed. The means in Victoria are denoted as mvj and mva, and in South Australia they are ms; and msa. The standard deviations are denoted as avj and tTva, and as a~j and a~a in Victoria and South Australia, respectively. Survival in Victoria and South Australia and survival of juveniles and adults will be correlated to some extent. The correlation between adults and juveniles is assumed to be the same in Victoria and in South Australia and is denoted as r. The correlation between survival in Victoria and South Australia (for adults and for juveniles) is given by p. It remains to specify the 'cross correlations', i.e. the correlation between juvenile survival in Victoria and adult survival in South Australia and between adult survival in Victoria and juvenile survival in South Australia. They are likely to be smaller than either of the two 'direct correlations', r and/9, and are modeled as y =
(2)
p.r
F r o m these correlations, r, p, and y, and the standard deviations of the four survival rates the variance-covariance matrix, P, can be constructed (Table 1). F r o m this matrix a correlation matrix A is calculated such that AA t = P where A t is the transpose of A (for instance, Faddeev and Faddeeva, 1963). The matrix A is calculated following the 'square root method' (for instance, Faddeev and Faddeeva, 1963, p. 144; Knuth, 1981, p. Table 1. The variance-covariance matrix, P.-y = p.r
Victoria
Adults Vic. Juveniles Vic. Adults SA Juveniles SA
South Austrailia
Adults
Juveniles
Adults
Juveniles
ava.ava r.avi.O'va p'asa'a~a y.~vi.tTva
r.trva.avi tTvi.t~vi ~/'tTsa'avi p.tTsi.tTvi
p.O'va.tTsa y.avi.asa asa'O'sa r.O'si.tysa
)/.O'va.asi p.O'vi.asi F'O'sa'O'si t~si.tTsi
Table 2. The correlation
matrix A. r'={1-r2} 112 and
p ' = {1-p2}'/2 avo r'avi p'a sa p.r.trsi
0 r ~.avi 0 p.r t.O'si
0 0 p" a sa r.pt.tTs i
0 0 0 p t .r t.O.si
551). The result is given in Table 2. F r o m the matrix A the four survival rates are calculated by (e.g. Burgman et al., 1993) i
Si : mi -k- E aijXj, j=l
= 1 . . . 4.
(3)
Here s~ = S w , s2=Sv;, s3=Ssa, S4=Ssj, the quantity a~j is the element in the i- th row and the jth column of the matrix A, and X1 to X4 are uncorrelated normally distributed r a n d o m numbers with zero mean and standard deviation one. With the values of the matrix A, eqn (3) becomes Sva = mva + Gva X1 "
Sq = mvj + avj " (rXl + r'X2) sm = m m + asa " (pX1 +/9I)(3)
(4)
Ssj = msj + Osj " (prX1 + p r ' X 2 + p ' r X 3 + p' r'X4)
where r ' = ( l - r 2 ) 1/2 and p ' = (1 _p2)1/2. Migration
As mentioned above, in winter the population splits into two subpopulations. After reaching the mainland, most of the birds stay in Victoria while some of them travel further west to South Australia. Each bird is assumed to migrate to South Australia with a certain migration probability, e. The number of birds that overwinter in South Australia, N s A , then will be binomially distributed NsA = Bin[N, e]
(5)
where N is the population size immediately before the winter season. One m a y call the resulting fluctuation in N s A the 'demographic variability' in migration (c. above). In addition, the migration probability itself may be a r a n d o m number fluctuating in response to environmental variation. It is assumed to be normally distributed with a mean, m e and a standard deviation, ae. Above, migration to South Australia was assumed to be a r a n d o m process and independent of other factors in the dynamics. However, even the birds that overwinter in South Australia are likely to migrate through Victoria. Based on their perception of food supply, e.g. they may depend migration to South Australia on the environmental conditions in Victoria. If
286
M . Drechsler
these conditions are below average, fewer birds m a y stay in Victoria and the migration probability to South Australia would be above average. In the model, the environmental conditions in Victoria are represented by the survival rates of juveniles and adults, svj and sw. For instance, a p o o r year is represented by below-average survival rates. A correlation between high migration and p o o r environmental conditions now means that there is a negative correlation between the survival rates in Victoria and the migration probability. For simplicity only the survival rate of adults is considered. This is not unplausible, because the South Australian subpopulation has a relatively high proportion of adults (Orangebellied Parrot Recovery Team, pers. comm.). As modeled above, both the adult survival rate, s ~ and the migration probability, e, are normally distributed rand o m numbers. Their correlation is now described by the equation (e.g. Burgman et al., 1993) e = me + ae . (qeX1 + ¢ 1 -
q2e . l 0
(6)
where qe is the correlation between migration and adult survival in Victoria and X1 is the r a n d o m number used in eqn (3) above. The quantity Y is another normally distributed r a n d o m n u m b e r that is not correlated to the Xi of eqn (4) and has a zero mean and standard deviation one. With eqn (6), if qe = - - 1 , the migration probability, e, is perfectly negatively correlated with the survival rate, sva, and if qe = 0 the two rates are uncorrelated. Parameter ranges Where applicable, parameter ranges are adopted from Drechsler et al. (1998). There is no information on whether the mean survival rates in South Australia are smaller or larger than in Victoria and the plausible range of survival rates in South Australia is assumed to be the same as that in Victoria. The Victorian subpopulation is much larger than the South Australian subpopulation. Therefore the variability in the survival of the entire population, which was determined by Drechsler et al. (1998), reflects mainly the conditions in Victoria and is used to describe the variability of survival rates in the Victorian subpopulation. In South Australia the birds are dispersed over a much wider range and food appears to be more variable than in Victoria. Therefore variability of survival in South Australia is likely to be higher than that in Victoria (Orange-bellied parrot Recovery Team, pers. comm.). Its upper bound is assumed to be twice that in Victoria. The correlation, p, between survival in Victoria and South Australia cannot be deduced from field data. It is certainly not negative and is assumed to lie between zero and one. The winter capacity which in Drechsler et al. (1998) measures the total number of survivors in the entire wintering population, now has to be distributed between Victoria and South Australia. The winter capacity in South Australia, Ks, is probably smaller than
that in Victoria, Kv, with an upper bound for Ks at 0-8.Kv. A lower bound for the ratio between the two capacities is set at Ks/Kv = 0.2 which is approximately the mean ratio of the observed subpopulation sizes (Table 3). The migration probability, e, is estimated from data collected during the winter population counts over the last 17 years (Table 3) which includes data taken from Starks et al. (1992) and the Orange-bellied Parrot Recovery T e a m (pers. comm.). For each year, the ratio of South Australian birds to the total wintering population size was calculated. This ratio reflects the probability that a bird migrates to South Australia, and so its mean and standard deviation provide estimates for the mean and standard deviation of the migration probability, me and ae (Table 3). Because it is more difficult to survey the South Australian habitat for birds than the Victorian wintering sites, the count data are likely to underestimate the actual proportion of South Australian birds. The median estimate in Table 3, 0.15, is used as a lower bound on the mean migration probability, me, and the upper 95%-confidence limit, 0.21, as an upper bound. The variability in the observed proportion of birds in South Australia is caused by several factors. These include measurement error because of the practical difficulties involved in the population counts in South Australia, the 'demographic variability' in migration (see above), and the actual variability in the migration probability itself. So the latter is likely to be overestimated by the variability in the measured proportion of South Australian birds. The median estimate in Table 3, 0-12, is chosen as an upper bound on the standard deviation of the migration probability, ae, and the lower 95%-confidence limit, 0.08, as a lower bound.
Table 3. Results of the winter population counts, provided by Starks et al. (1992) and the Orange-bellied Parrot Recovery Team
Year
NSA
Ntot
NsA/Ntot
1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 Mean SD
7 7 4 27 22 18 13 8 0 18 28 18 0 0 5 2 2
72 90 82 70 122 85 86 79 76 70 108 67 32 51 57 36 40
0.10 0.08 0.05 0.4 0.18 0-21 0.15 0.10 0 0.26 0.26 0.27 0.00 0.00 0.09 0.06 0.05 0.15 (0.09, 0.21) 0.12 (0.08, 0.16)
Mean and SD of the proportion of birds counted in South Australia. The 95% confidence limits are in parentheses.
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The correlation between migration and adult survival in Victoria, qe, is unlikely to be positive (that would mean that birds escape from a good wintering site). It m a y be zero (which means that migration is independent o f survival in Victoria) or negative. U p p e r and lower bounds are 0 and - 1 , although a correlation of - 1 m a y be regarded as rather unlikely. Altogether, the model has 15 independent parameters (Table 4). Sensitivity analysis As described above, in each of the two wintering habitats various m a n a g e m e n t options are considered, aiming at an increase in the mean survival rates, mv~, mvj, m~a and msj, and the two winter capacities, Kv and Ks. In the sensitivity analysis these six parameters are varied by a certain proportion and the resulting changes in the quasiextinction risk of the population are recorded. A m a n a g e m e n t option that affects mean adult survival is likely to affect mean juvenile survival, as well, and in the sensitivity analysis the two mean survival rates are varied simultaneously. Four independent quantities are considered which include the mean survival rates in Victoria and South Australia, denoted as mv and ms, and the two winter capacities, Kv and Ks. For each of these four quantities a sensitivity coefficient is obtained by dividing the change in the quasiextinction risk by the relative change in the varied quantity. The four sensitivity coefficients are denoted as: Cmv , Ckv , Cms
and
£ks"
As described above, each Victorian sensitivity coefficient has to be compared to its South Australian counterpart. This is done by the following sensitivity ratios
C1=
Cmv/Cms
(7)
and (:72 = Ckv/Ck~"
The third management option introduced above is to change the mean migration rate. The effect on the quasiextinction risk is measured by a sensitivity coefficient
(8)
C 3 = Cme
which is calculated in the same way as Cmv and c,,s. F o r simplicity, C1 to C3 will be called the state variables of the analysis. Further details about the sensitivity analysis and the mathematical definition of the sensitivity coefficients can be found in Drechsler et al. (1998). Identification of important parameters and parameter combinations The analysis consists of two parts. The first part determines how the effect of the state variables, CI, C2, and C3, which measure and compare the sensitivities of the management options, depend on the model parameters. In the second part the results of the first part are used to construct a small basic subset of scenarios which encompass most of the variability in the state variables and the variability in the sensitivity of management options. For each of these basic scenarios the five sensitivity coefficients, cmv, Ckv, Cms, Cks and Cme, introduced above, are determined. The first part of the analysis involves test scenarios where all parameters but one are set at median values, and one is either at its upper or lower plausible bound. As the model has 15 parameters, 15 pairs of scenarios have to be considered. In the first scenario of each pair the parameter of concern is set at its upper bound and in the second scenario at its lower bound. For each of the 30 scenarios the three state variables, C1, C2, and C3, are determined. The comparison of two corresponding
Table 4. The model parameters and their plausible bounds Parameter Mean migration probability (me) SD of migration probability (tre) Correl. b/w survival and migration (qe) Mean number of offspring per pair (]) Breeding capacity (max. no. breeders) (Kb) Mean survival rate in Victoria (my) SD of survival rate in Victoria (t~v) Correl. b/w juvenile and adult survival (r) Total winter capacity K = Kv + Ks Scramble factor in Victoria (Sv) Mean survival rate in South Australia (ms) trs/crv(trs = SD of survival in SA) Correl. b/w survival in Victoria and South Australia (/9) Ratio of the two winter capacities (Ks/KO Scramble factor in South Australia (Ss)
Upper
Median
Lower
0.21 0.12 0 1.95 95 (0.74, 0.52) (0.13, 0.08) +1 540 0.1 (0.74, 0.52) 2 1 0.8 0-1
0.18 0.10 -0.5 1-85 70 (0.64, 0.475) (0.11, 0-06) 0 405 0.05 (0.64, 0-475) 1.41 0.5 0.4 0.05
0.15 0.08 -1 1-75 45 (0.54, 0.43) (0-09, 0-04) -1 270 0 (0.54, 0.43) 1 0 0.2 0
The winter capacity and SD of survival rates in South Australia are defined as ratios to their Victoria counterpart. For such a ratio it is reasonable to calculate the median of the plausible range as the geometric mean between upper and lower bound. In parentheses, the first value refers to adults, the second value to juveniles. In the analysis the parameters for adult and juvenile survival are varied simultaneously, i.e. both take either their upper, lower or median value.
288
M . Drechsler
scenarios allows us to determine the effect of the model parameters on the state variables. In each pair, the state variables of the second scenario are subtracted from those of the first. The resulting differences, Ai, show how the state variables, Ci, change when one of the 15 model parameters is increased from its lower to its upper bound. A large difference, Ai, means that the corresponding model parameter has a strong influence on the state variable, Ci, and the sign of Ai indicates whether the state variable Ci is increased ( + ) or decreased ( - ) when the parameter is increased from its lower to its upper bound.
RESULTS I n f l u e n c e o f the m o d e l p a r a m e t e r s o n the e f f e c t o f management options
The first analysis involved identification of model parameters that have important influences on the three state variables. The change in a state variable, Ci, which is caused when a model parameter is changed from its lower to its upper bound, is given by the Ai introduced above (Table 5). The comparison of the A i in a particular column, i, of Table 5 allows the identification of those model parameters that have the strongest influence on the state variable, C;. F o r instance, mv has the strongest influence on C1 (A,{mv)=-0.151), and with the definition of Ai above, the negative sign indicates that an increase in m~ will decrease Cl. In each column, i, the bold faced Ai in Table 5 indicate those parameters which have the strongest influence on the state variable, Ci. In particular, the sum of the bold faced numbers in a column is 80% of the sum of the entire column (signs ignored). So in each column, i, the parameters with bold Table 5. Influences of the 15 model parameters on the three state variables
me
cre qe f Kh
A1
A2
A3
0.087 0.016 -0.097 0.004 0.013
0.123 0.126 --0.215 0.043
--0.459 0.051 0.045 0.057
--0-265 --0.334
0.640 - 1-188
0.005
0.052
0.080
0.129
-0.438 --0.211 -0-593 1-502 2.177
my Cry r
-0-151
K
-0.012
0.062
Sv
- 0.014 0.098
- 2.743
ms a,la~
-0.014
p
0.075
Ks/Kv
0.007 0.019
Ss
0.075 -0-086 0-076 - 1-060 0-360
-0.845
-0.608 3.061 -0.961
A plus sign indicates that a state variable is increased by an increase of a model parameter, a minis sign indicates a decrease in that state variable. For each state variable, the strongest parameter influences are bold faced. They make up about 80% of all parameter influences on the respective state variable.
A i explain 80% of the variation in the corresponding state variable, Ci. For instance, the parameters me, qe, m~, r, ms and p explain about 80% of variation in the first state variable, C1. faced
E x t r e m e s c e n a r i o s and s e n s i t i v i t y a n a l y s i s
Tables 6-8 (first two columns) show the parameters with strong influences on the three state variables, C1, C2 and C3. Each of the three tables allows the identification of that parameter combination which maximises the corresponding state variable, Cb C2 and C3. For instance, maximising me or minimising rnv will increase C1 (Table 6). Consequently, C1 will be maximal when me is at its upper and rnv at its lower bound. In this way the two extreme parameter combinations which lead to m a x i m u m and minimum C1 were constructed by setting the model parameters at their appropriate upper or lower bounds (Table 9 first two rows). These two extreme parameter combinations fully encompass the variation in C1 that is caused by the parameters included in Table 6. As pointed out above, the parameters in Table 6 explain about 80% of the total variation in Ci caused by all parameters. Consequently, the two extreme scenarios encompass 80% of the total variability in C1. The same analysis was done using Tables 7 and 8 to construct extreme scenarios that encompass 80% of the variation in Cz and C3, respectively. The result is shown in Table 9. For each of the scenarios shown in Table 9 the sensitivity coefficients, Cmv, Ckv, C,,,s, Cks and Cme, were determined (Table 10). They measure the influence of the five model parameters which represent management actions, on the viability of the population. The main results that can be deduced from Table 10 are (a) The mean survival rate in Victoria always has the strongest influence on the viability of the population, even in Scenario (1) which was designed to maximise the influence of the mean survival rate in South Australia. Generally, the next most important parameter is the mean survival rate in South Australia. (b) In the default parameter combination where all parameters are at their median values, mean survival in South Australia is more important than the winter capacity in Victoria, but this can be different in Scenario (4) where the breeding capacity, the mean survival rate in Victoria and the ratio between the winter capacities in South Australia and Victoria are maximal, and where scramble competition is strong in Victoria but weak in South Australia. (c) The winter capacity in South Australia and the migration probability are comparatively unimportant, even in Scenarios (3) and (5) which were designed to maximise their influence. Instead of reducing the risk of population decline (quasiextinction risk) one may wish to increase the
Spatial conservation management o f the Orange-bellied Parrot
289
Table 6. Influence of the model parameters on the state variables, (71
Making... m e
increases ....
and also leads to...
which results in...
C1
large number of migrants to South Australia migration especially when conditions are poor in Victoria few survivors in Victoria
large South Australian population South Australia acts as a 'safe haven' in critical years small Victorian population
many survivors in South Australia
large S. Australian population
large
qe large and negative my small r large ms large p large
The last two columns show likely effects of the model parameters on the population dynamics. These effects were not explicitly deduced in the model analysis but appear to be plausible. Table 7. Influence of the model parameters on the state variables (?2
Making...
increases..,
and also results in...
Kb small my small Sv small K~/Kv small S~ large
survival rates in Victoria more important than winter capacity* no scramble competition in Victoria stronger intraspecific competition in South Australia and less in Victorian strong scramble competition in South Australia
C2
The last column shows likely effects of the model parameters on the population dynamics. The effect of my, marked by a (*), was found by Drechsler et al. (1998) Table 8. Influence of the model parameters on the state variables (73
Making... Kb large my small Sv large m~ large trs/trv small Ks/ Kv large Ss small
increases..,
and also results in ....
C3
poor average conditions in Victoria strong scramble competition in Victoria good average conditions in Victoria stronger intraspecific competition in Victoria and less in S. Australia no scramble competition in South Australia
The last column shows likely effects of the model parameters on the population dynamics. Table 9. The six extreme parameter combinations which encompass 80% of the variability in the model results
me
tre
qe
f
Kb
1
+
2 3 4 5 6
0 0 0 0
mv
try
r
K
0
-
0
0 0 0 0 0
+ 0 0 0 0
0 0 0 0 0
Sv
ms
trs/trv
p
Ks/Kv
Ss
0
-
0
+
0 + + -
+ + +
0 0 0 0 0
0 0 0 0
0
0
+
0
+
0
0
0 0 0 0 0
0 + + -
0 0 + -
0 0 0 +
0 0 0 0
0 + + -
0 + +
A zero entry means that the respective parameter is at its median value, a plus (minus) sign means that it is at its upper (lower) bound (c. Table 4). m e a n p o p u l a t i o n size. As these objectives m a y be indep e n d e n t to some extent ( B u r g m a n et al., 1993), the m o d e l analysis was repeated using m e a n p o p u l a t i o n size as the m a n a g e m e n t target. T h e results were a l m o s t identical.
DISCUSSION Model assumptions
T h e Orange-bellied P a r r o t is a very rare species a n d despite c o n s i d e r a b l e effort in the field, its ecology is n o t fully u n d e r s t o o d . F o r instance, there is n o t m u c h k n o w n
a b o u t the factors which influence m o r t a l i t y in the population. Density-dependence and environmental variability h a d to be m o d e l e d in a very general way, based o n expert j u d g m e n t . M i g r a t i o n o f birds to South A u s t r a l i a was also m o d e l e d in a very simple m a n n e r . O t h e r aspects such as site fidelity which might play a role, were n o t considered in the study. The c o n c l u s i o n s represent what appears to be i m p o r t a n t in the m a n a g e m e n t o f the Orange-bellied P a r r o t using c u r r e n t knowledge. C a t a s t r o p h e s a n d genetics were n o t considered in this study. As discussed in Drechsler et al. (1998),
M. Drechsler
290
Table 10. Results of the sensitivity analysis performed for the six extreme scenarios defined in Table 5 (Scenarios 1-6) and a Scenario 0 where all parameter have their median values Cmv
Ckv
Cms
Cks
0
31
1
18
2 3 4 5 6
46 24 25 23 26
Cme
2
8
0
1
1
10
0
2
5 0 10 2 0
7 6 6 8 4
1 0 0 0 0
-1 1 2 2 -2
The sensitivity coefficients measure the influences of the mean survival rates in Victoria and South Australia (c,,v and cms), the influences of the winter capacities in Victoria and South Australia (Ckv and Cks), and the influence of the mean migration probability (Cme)on the quasiextinction risk. The numbers have been rounded to a precision that is appropriate with respect to statistical variation in the results. Values greater than 2 may be interpreted as important and values greater than 5 indicate parameters which have a strong influence.
catastrophes should be studied separately. Genetic influences can be important in metapopulations where there are genetically separated subpopulations and a turnover of local extinction and recolonisation (for instance, Gilpin, 1991). However, the population of the Orange-bellied Parrot is very likely to be a single population, because individuals mix in breeding colonies in Tasmania. Genetic threats are related to population size and risk of decline (Kimura and Crow, 1963; Koenig, 1988). Therefore management which increases population size and reduces the risk of decline, will also reduce genetic risks. M a n a g e m e n t implications
Each of the results (Tables 6-8 and Points (a)-(c)) can be directly translated into recommendations for the conservation management of the Orange-bellied Parrot. 1. Table 6 shows that those parameter values that maximise C1, i.e. the importance of habitat quality (mean survival rates) in South Australia, are also likely to maximise the size of the South Australian subpopulation compared with the Victorian subpopulation (last column of Table 6). The effect of the correlations, p and r, on the population sizes are less clear. However, their influence on C1 is comparatively small, so within the scope of this discussion they can be ignored. One can conclude that the relative importance of habitat quality in Victoria and South Australia is positively correlated to the relative sizes of the Victorian and the South Australian subpopulations. 2. The parameter values that were found to maximise C2, i.e. the influence of the winter capacity in South Australia on population viability, also represent a situation where there is strong intraspecific competition in South Australia and weak
intraspecific competition in Victoria (Table 7, last column). The role of the breeding capacity, Kb, is not clear, but its effect on (72 is rather small. One can conclude that the relative importance of the Victorian and the South Australian habitat capacities is positively correlated to the strength of intraspecific competition in the corresponding subpopulations. 3. The parameter values that were found to maximise C3, i.e. the benefit the population would gain from an increase in migration to South Australia, also represent a situation where (average) conditions in South Australia are better than those in Victoria (Table 8). The roles of the breeding capacity, Kb, and the standard deviation of survival in South Australia, trs, are not clear, but their effect on (73 is rather small. Consequently, whether migration to South Australia should be increased or reduced, i.e. whether captive birds should be released in South Australia or in Victoria, depends on whether the conditions in South Australia are better or worse than those in Victoria. This is not surprising, but validates the choice of the extreme scenarios (5 and 6 in Table 9) which were based on this result and lead to result (c) below./li > (a) Survival rates are generally the most important parameters, which confirms previous results. The mean survival rate in the Victorian subpopulation which may be related to the quality of Victorian winter habitat, was found to be more important than that in South Australia. (b) Generally, an increase in survival rates (habitat quality) in the South Australian subpopulation, appears to be more important than an increase in the Victorian winter capacity, i.e. the size of Victorian habitat. However, this is reversed if there is a large post-breeding population (caused by a large number of breeders) and if survival in Victoria is limited mainly by intraspecific competition (high mean survival rate, small ratio between the winter capacities in Victoria and South Australia, and high scramble factor in Victoria) (Scenario 4 in Table 10). In this case the winter capacity in Victoria is of considerable importance and management in South Australia is unimportant. (c) The winter capacity in South Australia is never important and management of habitat quantity in South Australia is not expected to have much influence on the viability of the population. As described in Methods, the relative sizes of the wintering subpopulations (migration probability) can be affected to some extent by different release strategies. The mean migration probability (i.e. the long term average of the relative subpopulation sizes) turned out to have a comparatively weak influence on population viability, although it should be noted that the relative
Spatial conservation management of the Orange-bellied Parrot subpopulation sizes might be changed more easily by m a n a g e m e n t than other parameters, such as the mean survival rates. A p a r t from affecting population viability, the release of birds at different sites on the mainland m a y lead to important insights into the migratory behaviour of the birds and their fidelity to their wintering sites. Thus, release strategies m a y be most beneficial if they are designed with a view to improving understanding of the species' dynamics. Mean survival rate and winter capacity of the Victorian subpopulation were always more important for the viability of the total population than their South Australian counterparts. This results from the fact that the Victorian subpopulation is larger than the South Australian subpopulation. M a n a g e m e n t in Victoria will affect more birds than management in South Australia, which suggests that m a n a g e m e n t should target the largest subpopulation of the wintering population. Although this study assumes that the whole wintering population consists of only two subpopulations, the wintering sites of a large proportion of the population are not known (the counts in Table 3 are substantially smaller than the population estimate of 175 provided by M c C a r t h y (1995). The arguments above suggest that in order to maximise the efficiency of conservation management, the large subpopulations should be targeted, and for this the distribution of the wintering population should be identified more comprehensively.
CONCLUSIONS The result that m a n a g e m e n t should target the largest of the two subpopulations agrees quite well with metapopulation theory. The two wintering subpopulations recombine every summer to a single breeding population, and therefore they are not really subpopulations in the sense of Levins (1970). The system is more analogous to a mainland-island system where there is one large mainland population and one or several small island populations (Harrison, 1991, 1994). Several works, for instance, Harrison, 1991, 1994 and Gyllenberg and Sylvestrov (1994) showed that the fate of the island(s) depends on the fate of the mainland. With this explanation in mind, some of the results are not very surprising. They reflect w h a t - - a t least in hinds i g h t - - a p p e a r s to be plausible. However, it is important to note that these results are not based on a single, 'best guess' parameter combination, but each of them was confirmed within a range between extreme scenarios which encompass 80% of all parameter combinations. Results (1) to (3) describe whether a particular management action becomes more or less effective when model parameters are varied. The analysis of pairs of extreme scenarios provides us with 'confidence intervals' on the results (a) to (c). As each pair encompasses 80%
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of all parameter combinations, one can say that the results (a) to (c) can be expected in about 80% of the entire parameter space. This adds a new aspect to a sensitivity analysis. N o t only was the sensitivity of certain management actions determined, but so was the reliability of the results. ACKNOWLEDGEMENTS I would like to thank M a r k Burgman and Peter Menkhorst for reading the manuscript and providing valuable comments. The advice and population data provided by the Orange-bellied Parrot Recovery Team helped much in the development and analysis of the model. The comments of an anonymous reviewer were very helpful in the revision of the manuscript. This work was supported by the German Research Community D F G . REFERENCES Brown, P. B. and Wilson, R. I. (1982) The Orange-bellied Parrot. In Species at Risk: Research in Australia, eds R. H. Groves and W. D. D. Ride, pp. 107-115. Australian Academy of Science, Canberra. Brown P. B., Holdsworth, M. C. and Rounsevell, D. E. (1994) Captive breeding and release as a means of increasing the Orange-bellied Parrot population in the wild. In Reintroduction Biology of Australian and New Zealand Fauna, ed. M. Serena, pp. 135-141. Surrey Beatty and Sons, Chipping Norton, UK. Burgman, M. A., Ferson, S. and Akakaya, H. R. (1993) Risk Assessment in Conservation Biology. Chapman and Hall, London. Drechsler, M., Burgman, M. A. and Menkhorst, P. W. (1998). Uncertainty in population dynamics and its consequences for the management of the Orange-bellied Parrot Neophema chrysogaster. Biological Conservation 84, 269-281. Faddeev, D. K. and Faddeeva, V. N. (1963) Computational Methods of Linear Algebra. Freeman, San Francisco and London. Gilpin, M. E. (1991) The genetic effective size of a metapopulation. In Metapopulation Dynamics: Empirical and Theoretical Investigations, eds M. E. Gilpin and I. Hanski. Academic Press, London, pp. 165-175. Gilpin, M. E. and Hanski, I. (eds) (1991) Metapopulation Dynamics: Empirical and Theoretical Investigations. Academic Press, London. Ginzburg, L. R., Slobodkin, L. B., Johnson, K. and Bindman, A. G. (1982) Quasiextinction probabilities as a measure of impact on population growth. Risk Analysis 2, 171-181. Goodman, D. (1987) The demography of chance extinction. In Viable Populations for Conservation, ed. M. E. Soul+, pp. 11-34. Cambridge University Press, Cambridge. Gyllenberg, M. and Sylvestrov, D. S. (1994) Quasi-stationary distribution of a stochastic metapopulation model. Theoretical Population Biology 42, 35-61. Harrison, S. (1991) Local extinction in a metapopulation context: an empirical evaluation. In Metapopulation Dynamics: Empirical and Theoretical Investigations, eds M. E. Gilpin and I. Hanski, pp. 73-88. Academic Press, London. Harrison, S. (1994) Metapopulations and conservation. In Large-scale Ecology and Conservation Biology, eds P. J. Edward, R. M. May and N. R. Webb, pp. 111 128. Blackwell Scientific Publication, Oxford.
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Loyn, R. H., Lane, B. A., Chandler, C. and Carr, G. W. (1986) Ecology of Orange-bellied Parrots Neophema chrysogaster at their main wintering site. Emu 86, 195-206. MacArthur, R. H. and Wilson, E. O. (1967) The Theory of Island Biogeography. Princeton University Press, Princeton, NJ. McCarthy, M. A. (1995) Stochastic population models for wildlife management. Ph.D. thesis, University of Melbourne. McCarthy, M. A., Burgman, M. A. and Ferson, S. (1996) Logistic sensitivity and bounds for extinction risks. Ecological Modelling 86(2-3), 297-303. Menkhorst, P. W., Loyn, R. H. and Brown, P. B. (1990) Management of the Orange-bellied Parrot. In Management and Conservation of Small Populations, eds T. W. Clark and J. H. Seebeck, pp. 239-251. Chicago Zool. Soc., Chicago, IL. Ripley, B. D. (1987) Stochastic Simulation. Wiley, New York. Starks, J., Brown, P., Loyn, R. and Menkhorst, P. (1992) Twelve years of winter counts of the Orange-bellied Parrot Neophema chrysogaster. Australian Bird Watcher 14, 305312.