Uncertainty in population dynamics and its consequences for the management of the orange-bellied parrot Neophema chrysogaster

Biological Conservation, Vol. 84, No. 3, pp. 269-281, 1998 PII:

S0006-3207(97)00125-0

© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0006-3207/98 $19.00+0.00

ELSEVIER

U N C E R T A I N T Y IN P O P U L A T I O N D Y N A M I C S A N D ITS C O N S E Q U E N C E S F O R THE M A N A G E M E N T OF THE O R A N G E BELLIED P A R R O T Neophema Chrysogaster M a r t i n D r e c h s l e r , a* M a r k A. B u r g m a n a & P e t e r W . M e n k h o r s t b aEnvironmental Science, School of Botany, University of Melbourne Parkville, Victoria 3052, Australia bFlora and Fauna Branch, Department of Natural Resources and Environment 4/250 Victoria Parade, East Melbourne, Victoria 3002, Australia

(Received 13 January 1996; revised version received 26 March 1997; accepted 20 June 1997)

The Orange-bellied Parrot, Neophema chrysogaster, is one of Australia's most threatened species and is listed as 'critical' by the IUCN (1994). The population size is below 200 (Loyn et al., 1986; Menkhorst et al., 1990; Starks et al., 1992; McCarthy, 1995). Birds breed during summer in coastal areas of south-western Tasmania (Brown and Wilson, 1982). Following breeding, they migrate north to the Australian mainland where they over-winter in coastal areas of south-eastern Australia. Causes for the rarity of the Orange-bellied Parrot are uncertain. Though the fecundity of the birds is high, the population size appears to be stable at low numbers. Therefore it is believed (Menkhorst et al., 1990) that high mortality during winter is responsible for the persistent small population size. There are two major objectives in the management of the Orange-bellied Parrot: to increase the (mean) population size and to reduce the risk of population decline. We quantify the latter by the quasiextinction risk which is defined as the probability that the population size falls below a particular threshold within a particular time horizon (Ginzburg et al., 1982). Management options exist for both breeding and wintering sites (Menkhorst et al., 1990; Stephenson, 1991). In the breeding habitat they include fire management, artificial feeding and the provision of nest boxes. The winter habitat management options include control of predators and competitors, management of sheep grazing, and the expansion of suitable habitat. Another management measure, the captive breeding and release of birds, is already underway. In order to manage the population most effectively, it is necessary to know how strongly each of the management options affects the viability of the population. We will investigate these questions by sensitivity analysis of a population model. The model contains elements of a model developed by McCarthy (1995). In the development of a population model for the Orange-bellied Parrot, several types of uncertainty have to be considered. As population data on fecundity and mortality are available for only a few years, there is considerable uncertainty in the values of

Abstract The population dynamics of a rare and dispersed species like the Orange-bellied Parrot Neophema chrysogaster include many uncertainties, especially concerning mortafity. Taking these uncertainties into account we evaluated several options for management of the Orange-bellied Parrot habitat. Options were ranked by their effects on the viability of the population. There was considerable variability in the resulting rank orders. A few general features appeared to be rather stable with respect to all forms of uncertainty considered. It was found that survival of birds during the winter season was more important than their reproductive success in summer and qualitative features of the habitat, such as the composition of vegetation, were more important than quantitative features such as the habitat size. © 1998 Elsevier Science Ltd. All rights reserved

Keywords: population viability analysis, conservation, habitat management, sensitivity analysis, Neophema.

INTRODUCTION Management of an endangered population is improved when the manager knows which factors most decisively affect the viability of the population. Synthesis of the available data into a population model followed by a sensitivity analysis of the model is one tool that may be used to identify these factors (Watt, 1968; Caswell, 1978; Beck, 1983). The sensitivity analysis calculates how strongly the model output (for instance, the growth rate or the extinction risk of the population) depends on the model parameters. One of the most useful features of sensitivity analysis is that model parameters can be ranked in terms of their influences. Such information may be used to guide management decisions affecting populations (Caswell, 1978; Beck, 1983; Possingham et al., 1993). *To whom correspondence should be addressed. Department of Ecological Modelling, Centre for Enviromental Research, Permoserstr. 15, 04318 Leipzig, Germany. 269

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these parameters. Further uncertainty arises from the fact that in a population which experiences densitydependence, mortality is composed of a density-independent component and a density-dependent component, the latter often being described by a carrying capacity (e.g. Krebs, 1985). Estimates of mortality rates from field observations generally contain both components and it is difficult to separate them. This problem is generally ignored and not only leads to uncertainty in the values for the model parameters describing densitydependent and density-independent mortality, but there is also uncertainty about the magnitude of annual variability in those parameters. Even though variability in annual survivorship can be calculated from the field data, this variability may arise from variability in the density-dependent or the density-independent components of mortality. We call this type of uncertainty 'structural uncertainty', because it has to be considered in different population models with different mathematical structures. Structural uncertainty adds another dimension to sensitivity analysis, because the rank order of parameter sensitivities may change, depending on the model structure. It is necessary to conduct a complete set of sensitivity analyses for each alternative model structure. Another source of structural uncertainty is the fact that there is not much known about the behavioral interaction of the birds. Intraspecific competition may be of 'scramble' or 'contest' type (e.g. Krebs, 1985), and even this dichotomy is unresolved for Orange-bellied Parrots. If competition is of scramble type, in a large population the number of survivors may decline with increasing population size. This may be of concern in the discussion of Orange-bellied Parrot breeding habitat management to increase reproduction, because that is likely to increase the size of the wintering population. Usually the two types of intraspecific competition have to be considered in separate models with different structures. In this study we develop a new model for density-dependence in which either contest or scramble competition may be represented by different parameter settings. The uncertainty in the parameter values produces considerable difficulty in the sensitivity analysis of a model. Sensitivity analysis generally assumes that the variability in the model parameters is small so that nonlinearities and parameter interactions can be ignored (Beck, 1983). In many real populations including the Orange-bellied Parrot these uncertainties are so large that non-linearities and interactions between the model parameters have to be considered. The practical consequence is that the magnitudes and rank order of parameter sensitivities will not be constant in the entire parameter space, but will depend on the parameter values--an aspect that is often ignored in the sensitivity analysis of population models. With respect to conservation biology this means that conclusions concerning which factor most strongly affects the viability of a

population will depend on the values of other factors. To cover all possible situations in a model with n variables where each variable can take one of m different values, a systematic scan of m" different parameter combinations would have to be considered. For each of them a sensitivity analysis would have to be carried out. Unless the number of parameters is small, such a procedure is very time consuming and produces a vast amount of data which may itself be difficult to synthesise. In this work we present a new approach that avoids a systematic scan of the parameter space. This approach includes the identifcation of those parameters that are most responsible for the variability in the rank order of parameter sensitivities. Based on the results of this procedure, a set of parameter combinations is constructed that is likely to encompass most of that variability. For each of these parameter combinations the rank order of parameter sensitivities is determined. The results of such analyses are interpreted through summary tables of parameter sensitivities, making sensitivity analysis of the complex model for Orange-bellied Parrots a tractable exercise.

METHODS Simulation of management options and introduction of the model parameters We distinguish between four different basic components of the population dynamics. They include reproduction and survival, each of which consists of a density-dependent and a density-independent 'subcomponent.' Density-dependent reproduction or survival are the result of intraspecific competition for limited resources. In the breeding habitat Orange-bellied Parrots are likely to compete for a limited number of suitable nest sites ('breeding capacity,' Kb) (Brown and Wilson, 1982). The density-independent component of reproduction is described by the fecundity, f, which measures the breeding success of each breeding pair independent of the population size. It may depend on the abundance of food. Mortality is believed to be highest in winter (Menkhorst et al., 1990). The number of parrots that survive the winter season is likely to depend on intraspecific competition for limited food resources. The maximum number of birds that can survive under optimal conditions is measured by the 'winter capacity,' Kw. Even in the absence of intraspecific competition there is likely to be mortality in winter, for instance, due to competition with other species or insufficient quality (nutritional value) of food (McDonald, pers. comm.). This densityindependent component of survival is described by the annual survival rates of juveniles and adults, sj and Sa. The management actions that may affect reproduction and survival are shown in Table 1. In the model analysis, a management action will be simulated by

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Table 1. Management measures and likely effects on population dynamics and model parameters

Management measure

Effect

Model parameter affected

Breeding habitat •Artificial feeding •Fire management •Provision of nest boxes

Increase of reproduction and number of suitable nest sites

Density-independent Density-dependent

Fecundity Or) Breeding capacity (Kb)

Winter habitat •Control of predators •Decrease of human disturbance •Management of sheep grazing •Food quality

Increase of chance survival

Density-independent

Annual survival rates (si andsa) for juvenile and adult birds

Increase of maximum number of survivors

Density-dependent

Winter capacity (Kw)

•Expansion of habitat •Control of competitors •Management of sheep grazing

varying the corresponding model parameter and recording the effect on the population. The survival rates, sj and sa, are likely to fluctuate annually in response to environmental variation (see below). Management m a y affect their means (i.e. their long-term average), m/ and ma, or their standard deviations (annual variability), or/and ira. The means of the survival rates are likely to be affected by permanent improvement of winter habitat while their annual variability m a y be reduced by targeting m a n a g e m e n t activities in years where survival conditions are below average. Although the survival rates, s / a n d s~, themselves are partly independent (see below), their means are likely to be related to the same environmental factors. Therefore we make the plausible assumption that a m a n a g e m e n t action that changes mean adult survival, ma, by a particular proportion, will change mean juvenile survival, mj, by the same proportion. The same applies to the standard deviations, tr/ and cry, which are assumed to be always affected in the same way. There will be two other parameters in the model: a parameter, S, that describes the type of intraspecific competition and a parameter, r, that describes the correlation between the survival rates of juveniles and adults. Both are mainly determined by the biology and the behaviour of the birds and would be difficult to modify by habitat management. Their sensitivities on viability of the population are not discussed in this study.

ing season to the beginning of the next. Juvenile survival covers the time from fledging to the beginning of the next breeding season. After this time juveniles become adults and breed. Age-dependence in the survivorship and the reproduction of adults is not considered.

Reproduction The model distinguishes between males and females. Reproduction is determined by the number of breeders and their breeding success. F o r breeding the parrots need hollows in eucalypt trees (Eucalyptus nitida) that are close enough to food sources (Brown and Wilson, 1982). The number of suitable nest sites, Kb ('breeding capacity'), limits the number of breeders. As the birds are believed to be monogamous, the number of breeders is further limited by the number of males or females whichever is smaller. The number of fledglings per breeding pair varies between zero and five, depending on r a n d o m influences such as predation and weather (Brown and Wilson, 1982), and is given by the distribution in Table 2. For each breeding pair the number of fledglings is sampled from this distribution. The sex ratio of fledglings is assumed to be even.

Survival We assume that in a small population there is no intraspecific competition and describe the number of juveniles, N/(t + 1), and adults, Na(t + 1), that survive to the next breeding by

Nx(t + 1) = Bin[Nx(t), s~]

The model

The model simulates the annual life cycle of the parrots. This cycle starts with the reproduction of birds and is followed by the calculation of the number of survivors to the next breeding season. The simulated population is censused at the end of the annual cycle, i.e. immediately prior to the breeding season. Adult survival is measured from the beginning of one breed-

(1)

Table 2. Probability that a breeding pair raises a particular number of fledglings (M. Holdsworth, unpub, data)

Number of fledglings Probability

1

2

3

4

5

0.10 0.05 0.05 0.15 0.20

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In this and the following equations 'x' stands for ' f (juveniles) or 'a' (adults). Nj(t) and N~(t) are the numbers of juveniles and adults in year t, and sj and sa are their respective annual survival rates in the absence of intraspecific competition. Bin[Nx(t),sx] is a binomial variate with mean Sx • Nx(t) and variance Nx(t) .Sx. ( 1 - s~) (Akqakaya, 1991). The use of the binomial distribution accounts for demographic stochasticity. In a large population we assume that birds have to compete for limited (food) resources which decreases the survival probability of each individual. In a first step we consider contest competition (see above) and assume that if the population size, N(t) = Nj(t) + Na(t), exceeds a threshold, Kw, the mortality of each individual is increased correspondingly. If, for instance, the winter capacity is exceeded by 10%, such that N(t) =1.1 gw, we assume that the survival probability of each individual is decreased by a factor of 1/1.1:

Nx(t + 1) = Bin[Nx(t), Sx. Kw/N(t)], (N(t) > Kw)

The right hand side of this inequality can be approximated by sm.Kw where sm is the mean of juvenile and adult survival rates: Sm= (sj + Sa)/2. It means that the number of survivors is roughly limited by Kw multiplied by the average survival rate of an individual. Consequently, the number of survivors can never exceed Kw, and we call Kw the 'winter capacity'. There are various alternative formulations to model scramble competition (e.g. Burgman et al., 1993). We choose one that includes contest competition as a limiting case and allows us to vary continuously between contest and scramble competition. We start from the contest competition model above and assume that at population sizes above a 'scramble threshold', K, the survival probability of each individual (Sx if N(t) < Kw, or Sx. Kw/N(t) if N(t) > Kw) is reduced by an amount

S.

[1 - K / N ( t ) ] 1 -

K/Kw

(4)

(2)

The effect on the number of survivors can be seen in Fig. 1 (upper line). Neglecting demographic fluctuations (Nx(t + 1) = Sx. Nx(t); see above), the mean number of survivors, N ( t + 1), increases linearly with increasing population size N(t), until it reaches a plateau which depends on the threshold Kw, the survival rates, sj and sa, and the ratio of juveniles and adults. I f we neglect demographic stochasticity the number of survivors is limited by (cf. eqn 2)

N(t + 1) < [sj. Nj(t) + sa • Na(t)]" Kw Nj(t) + Na(t)

(3)

where the 'scramble factor', S, measures the strength of scramble competition (Fig. 1, lower line). In a population of exactly Kw individuals, scramble competition will reduce the number of survivors by Kw.S (eqn (4)). Contest competition is obtained in this model by setting the scramble factor, S, to zero. As can be seen from Fig. 1, our competition model captures the essential qualitative aspects of contest and scramble competition mentioned in the Introduction, but in contrast to m a n y other models (e.g. Burgman et al., 1993), its parameters (K, Kw, and S) have intuitive meanings, and contest and scramble competition are obtained simply by different settings of one parameter, the scramble factor, S.

Environmental variability in survival 0.6

As mentioned in the Introduction, environmental variability m a y affect the density-independent component of survival (age-specific survival rates) or the density-dependent component (winter capacity). In the Orange-bellied Parrot any of the two options (or even a mixture) is possible. To investigate the consequences of this uncertainty on the management of the Orange-bellied Parrot we consider each of the two alternatives in a separate model.

0.4-

N(t+l ) 0.2-

0-

0

0.5

I

1.5

2

2.5

3

N(t)

Fig. 1. Number of survivors, N(t+ 1) as a function of the present number of individuals, N(t) (demographic fluctuations neglected). N(t) and N(t+ 1) are measured in units of the winter capacity, Kw. For simplicity the numbers of juveniles and adults are assumed to be equal. The upper curve represents contest competition, the lower one represents scramble competition. Juvenile and adult survival rates are sj=0.49, s~=0.63, the scramble factor is S=0-1 and the 'scramble threshold' is K = Kw/2; Kw denotes the winter capacity. The shaded area between the two curves represents the additional mortality caused by scramble competition.

Environmental variability in survival rates (S-model) We assume that the survival rates, sj and sa, fluctuate and are normally distributed with means, mj and m,, and standard deviations, ~rj and ~ra. The winter capacity is held constant. As discussed above, the survival probability of an individual is given by its survival rate (sj or sa) which in a large population is multiplied by Kw/N and reduced by S.(1-N/K). Therefore in the r a n d o m sampling of sj and sa, the survival probability of an individual could become negative. As negative probabilities are not plausible, negative values are set to zero.

Uncertainty in population dynamics f o r the management o f the Orange-bellied Parrot

Fluctuations in the survival rates of juveniles and adults will naturally be correlated to some extent (see below). This correlation is described by sj = m/ + vl . ~rj Sa : ma + ~a"

(Pl

"

r -4- I) 2 " ~/1 - r 2

(5)

where vl and v2 are normally distributed r a n d o m numbers with a zero mean and a standard deviation of one (Ripley, 1987; Burgman et al., 1993). The correlation coefficient, r, can take values between + 1 and - 1 . Environmental variability in the carrying capacity (K-model) In this model we assume that there is no environmental variability in the survival rates, s / a n d sa, which are held constant at values mj and m a. Instead we assume that the winter capacity is a normally distributed r a n d o m number with mean and standard deviation, mk and crk. To ensure consistency with the S-model above we assume that the fluctuations in the winter capacity have the same coefficient of variation, ~rk/mk, as the environmental fluctuations in the S-model. This implies (Appendix) that the standard deviation of the winter capacity is given by: v/y2o'2 + ~2 -{- r . y . ~rj . Cra

ak = mk

y . mj + ma

(6)

where y ~ f / 2 and f is the mean number of offspring per breeding pair. Plausible ranges for the model parameters Parameter estimates are based on data provided by members of the Orange-bellied Parrot Recovery Team. The fecundity parameters in Table 2 yield an average number of fledglings per breeding pair o f f = 1.95. As the fecundity parameters shown in Table 2 have been obtained from the monitoring of nest-boxes in the vicinity of artificial feeding sites, they are likely to overestimate the actual breeding success of birds, and the value o f f = 1-95 is used as an upper estimate. A lower estimate is set at f = 1.75 which is close to earlier estimates based on data from natural tree hollows with no artificial food provided (Brown and Wilson, 1982). We vary the fecundity between these two extremes by keeping the relative magnitudes of all values in Table 2 fixed and varying all values by the same proportion. The effect of variability in the relative magnitudes is negligible (see Appendix). Banding-resighting data exist for juveniles and adults for the past 5 years. A m a x i m u m likelihood calculation (Pollock et al., 1990; Lebreton et al., 1992) led to the annual survival rates given in Table 3. The mean survivorships (Table 3) have been obtained from a real population that almost certainly is experiencing densitydependent regulation (density-dependent regulation is

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the most likely explanation for the population size being rather stable in spite of considerable fluctuations in the survivorships). In the models described above, the mean survivorship of an individual in a density-regulated population is effectively determined by a number of parameters including the mean survival rates, sj and sa, the winter capacity, Kw and the scramble factor, S. The winter capacity and the intensity of scramble competition adversely affect the survivorships of individuals. The 'effective' mean survivorship of an individual therefore lies below its respective mean survival rate, mj or ma. Therefore the mean survivorships in Table 3, 0.43 and 0-54 serve as a lower bound on the mean survival rates, m/ and ma. As an upper bound on the mean survival rates we use the upper-95% confidence limits of the mean survivorships, 0-52 and 0.74 (juveniles and adults, respectively; Table 3). The observed standard deviations for juvenile and adult survivorships, 0.09 and 0.13 (Table 3) certainly overestimate the actual environmental variation in the survivorships, because the observed values also include variation arising from demographic stochasticity, density-dependent regulation and measurement error. They are therefore regarded as upper estimates for the parameters crj and era. As lower bounds we use the lower-95% confidence limits of the standard deviations, 0.04 and 0.08 (juveniles and adults, respectively; Table 3). The observed correlation between adult and juvenile survival is nearly zero (Table 3). As upper and lower bounds on the correlation, r, we use the upper and lower 95% confidence limits which are + 1 and - 1 (Table 3). Records indicate that almost all females in the population nest (Fig. 2 in Brown et al., 1994). In a population with an average adult size of at least 90 individuals (Menkhorst et al., 1990, Fig. 3) and an even sex ratio, this means that the number of nest sites, Kb, exceeds 45. An upper limit for Kb was set at 95. In recent years the adult population size has been estimated to be around Table 3. Annual survivorships of juveniles and adults (obtained from banding-resighting data provided by M. Holdsworth of the Orange-bellied Parrot Recovery Team) Juvenile Adult survivorship survivorship 1991 1992 1993 1994 1995 Mean Standard deviation Correlation juvenile/adult survivorships

0.36 -0.53 0.42 0.51 0.66 0.33 0-40 0.43 0-62 0.43 0.52 (0.32,0.54) (0.31,0.74) 0-09 0.13 (0-04, 0.14) (0.08, 0.19) 0-07 (-1-00, + 1.00)

Means, standard deviations of, and correlation between annual survivorships of juveniles and adults. In brackets the 95%-confidence limits for the correlation, obtained by Jackknife resampling (Sokal and Rohlf, 1981).

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175 individuals with a lower 95% confidence limit of 144 (McCarthy, 1995). Therefore a plausible estimate for the maximum population size prior to breeding should be at least 150, but it is very unlikely to be greater than 300. As shown above, the maximum size of the model population after the winter season is approximately s m . K w where sm= (sj+ Sa)/2 (eqn (3)). An average estimate for Sm is the mean of the upper and the lower plausible limits of mj and m~ (see above): Sm~0"56. The winter capacity, Kw, then lies between 150/0.56 m 270 and 300/0.56 ~ 540. This applies to the Smodel where the carrying capacity is fixed at Kw and the survival rates fluctuate around their means, mj and m~. As described above, in the K-model the survival rates are fixed at mj and m a and the winter capacity is a normally distributed random number with mean, mk. To ensure consistency with the S-model, we assume that this mean has the same bounds as the winter capacity in the S-model, i.e. 270
error. In that case the sensitivity coefficient is given by the relative change in the state variable, A V/V, without division by 8). In this study the state variable is the quasiextinction risk, Q. Quasiextinction risks are better evaluated by using their logit, defined as logitQ = In[Q/ ( I - Q ) ] , because quasiextinction risk curves are Sshaped (Ginzburg et al., 1982; Burgman et al., 1993) and the logit transforms an S-shaped curve into a straight line. The logit function can be introduced into eqn (7) by a slight rearrangement. We consider that [Q(p+8.p)-Q(p)]/Q(p) can be approximated by l n Q ( p + 8 . p ) - l n Q ( p ) . Replace lnQ by logitQ and (with a change of the sign; see below) eqn (7) becomes cp =

logitQ(p + 8. p) - logitQ(p) 8

(8)

With the minus sign added in eqn (8), a positive sensitivity coefficient represents the positive outcome of a management measure: an increase in fecundity, survival rates, breeding capacity or winter capacity reduces the quasiextinction risk. A negative sensitivity coefficient represents an adverse outcome. The sensitivity coefficient, eqn (8), can be applied to evaluate the influences of fecundity, the mean survival rates, and the two carrying capacities. In the case of the environmental fluctuations, it leads to an inconsistency (see Appendix). Instead of varying the standard deviations, aj and or,, by proportions, 8, they have to be varied by amounts of 8.mj and 8.m~, respectively. The sensitivity coefficient for the environmental fluctuations then is calculated by

Sensitivity analysis

To provide a basis for the evaluation of management options (cf. Table 1) we have to calculate the sensitivities of five parameters which are: fecundity (simultaneously varied fecundity parameters of Table 2), mean survival rates of juveniles and adults (varied simultaneously), environmental fluctuations (simultaneously varied standard deviations of juvenile and adult survival rates), breeding capacity and winter capacity. A sensitivity coefficient, Cp, quantifies the influence of a parameter, p, on a particular state variable (model output), V (Caswell, 1978; Beck, 1983; de K r o o n et al., 1986). It is obtained by varying the parameter, p, by a relative proportion, 8, and dividing the resulting relative change, 8 II/V, in the state variable by 8

Cp =

V(p + 3. p) - V(p) 8. V(p)

(7)

(Caswell, 1978; Beck, 1983; de K r o o n et al., 1986) (apart from evaluating management options, a sensitivity coefficient can be used to investigate the potential model

Co = logit V(trj + 8. mj; tra+ 8. ma) - logit V(crj; cry) (9)

The structure of eqn (9) is similar to that of eqn (8), but although the denominator contains the proportion 8 as usual, the standard deviations are now varied by amounts of 8.mj and 8'ma. Note that there is no minus signs in front of the fraction. With this formulation, a positive sensitivity coefficient represents the usual case where a decrease in the environmental fluctuations decreases the quasiextinction risk of the population. For each of the parameter combinations ('scenarios') identified in the next subsection, the sensitivity coefficients were calculated as follows. A time horizon was set at 20 years (and kept fixed in the entire study). In a preliminary analysis the population dynamics were simulated about 100 times to determine a threshold, T, such that in 50 of the 100 simulation runs the population size had fallen below T at least once. This means that the quasiextinction risk corresponding to the threshold, T, was at a median value around 50%

Uncertainty in population dynamics for the management of the Orange-bellied Parrot (Ginzburg et al., 1982). Then the population dynamics were simulated 1000 times over 20 years and the quasiextinction risk of crossing the threshold, T, were determined. After this, the parameters representing the five management actions were varied in turn to obtain five new parameter combinations. Specifically, fecundity was varied by a proportion of ~ = 10%, the mean survival rates simultaneously by 5% each, the standard deviations simultaneously by amounts of 0.05mj and 0.05m~, respectively, and the two carrying capacities by 33% each. For each of the five new parameter combinations the model was run 1000 times. Quasiextinction risks were determined and compared to those obtained for the original parameter combination. Sensitivity coefficients were calculated applying eqn (8) and (9). As a result we obtained a rank order of sensitivity for the five model parameters varied. This procedure was carried out ten times and the results were averaged to improve the precision. Identification of important parameter combinations The sensitivity analysis above produces a rank order of sensitivity for a particular parameter combination (scenario). A different scenario may lead to a different rank order of sensitivity, and to cover the entire parameter space, a large set of different scenarios has to be considered. In each of the nine model parameters, we consider the maximum and minimum values with respect to their plausible ranges, assuming that the results for median parameter values can be obtained by interpolation (Drechsler, submitted). A systematic combination of nine model parameters each with two possible values leads to 29 = 512 different scenarios. The calculation of 512 rank orders would have been too time-consuming and the results too difficult to synthesise. Therefore we concentrated our analysis on a small basic subset of all the possible scenarios which cover most of the variation in the rank order of sensitivity coefficients. The subset of basic scenarios was identified following a method suggested by Drechsler (submitted). We formed 18 'test scenarios' by setting all parameters at their median values except for one which was set either at its maximum or minimum. Table 4 shows the test scenarios and the corresponding sensitivity coefficients obtained from the S-model. Table 4 allows us to assess the influence of the model parameters on the sensitivity coefficients. The comparison of the first two rows in Table 4, for instance, reveals the influence of fecundity on the sensitivity coefficients, the following pair of rows represents the influence of the mean juvenile survival rate, and so on. Rows 7 and 8, for instance, are almost identical which suggests that the standard deviation of juvenile survival has only a weak influence on the rank order o f sensitivity. In Drechsler (submitted) the influences of any two parameters were compared in a qualitative and a quantitative manner, based on Table 4. The results are summarised below. The quantitative test showed that the

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standard deviations of juvenile and adult survival have a comparatively weak influence on the rank order of sensitivity coefficients. In the qualitative test it was found that mean juvenile and mean adult survival rates, mj and ma, and the fecundity, f, have similar influences on the sensitivity coefficients. This is plausible, because they all can be expected to affect the population dynamics in a qualitatively similar way. An increase in any of them will increase the average growth of the population. This similarity makes the three parameters additive, i.e. a small increase in all of the three, mj, m , and f, has a similar effect on the sensitivity coefficients as does a larger increase in mj (or m~ or f) alone. Consequently, from the 512 scenarios above we have to consider only the 'extreme' scenarios where mj, ma a n d f are all maximal or all minimal. These two scenarios encompass the full variability in the sensitivity coefficients. The results for the 'mixed' scenarios where one o r t w o of mj, m a and f is(are) maximal and the other one(s) is(are) minimal, will lie in between the results for the 'extreme' scenarios. The two carrying capacities, Kb and Kw, have similar but opposite or 'compensatory' influences on the sensitivity coefficients. This means that an increase in one of the two carrying capacities has a similar effect on the sensitivity coefficients as does a decrease in the other. Similarly, an increase in the scramble factor and a

Table 4. Sensitivity coefficients of model parameters in the Smodel, including mean survival rates, m, standard deviations of survival rates, a, fecundity, f, breeding capacity, Kb, and winter capacity, Kw Scenario

Cf Cm

Ca

CKb

CKw

f maximal fminimal mi maximal

15 17 14 19 10 25 15 17 16 16 14 21 16 17 16 15 14 18

26 52 30 16 35 7.3 25 22 24 20 34 3.8 20 24 26 20 26 24

2.2 1.5 3.4 0.93 5.7 0.56 1.8 1.9 1.5 2.0 1.5 1.9 0.09 5.0 3.6 -0.13 0.43 5.4

4.3 3.5 4.4 2-9 4-6 1.3 3.7 4.4 3.3 4.7 2.8 5.4 5.4 0.67 1.4 6.4 5.7 0.06

mi ma

minimal maximal

rna minimal ~ri maximal aa minimal aa maximal aa minimal r maximal r minimal Kb maximal Kb minimal Kw maximal Kw minimal S maximal S minimal

52 34 48 61 45 65 55 55 56 58 48 68 58 53 52 59 55 52

The letters m and ~r stand for simultaneously varied juvenile and adult survival (see Methods: Sensitivity analysis) and f stands for simultaneously varied fecundity parameters (see Methods: Parameter ranges). Each row of five sensitivity coefficients is normalised to a sum of 100. The sensitivity coefficients are given for 18 parameter combinations. In each parameter combination all parameters take median values except for the one indicated which assumes either its plausible maximum or minimum (see text).

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Table 5. Rank orders of sensitivity coefficients representing the relative influences of mean survival rates, m, standard deviations, or, fecundity, f, breeding capacity, Kb, and winter capacity, Kw, on the quasiextinction risk in the S-model

Kb, S minimal

Kb, S maximal

and Kw maximal

and Kw minimal

maximal r minimal r maximal r minimal mj, m~ and f a l l maximal

mj, m,, and f a l l minimal

Cm (43)

Cm (55) Cf (16)

C,~ (38) Cy (10) CKb (8)

Co (15) CKb (15)

C~ (57) Cm (30) CKw (1l) CKa (0)

CKw (0)

CKw (0)

Cf (0)

CKb (0)

Cm (64)

C,,, (68) CU(26)

Cm (65) Cf (18) Co (12) CKw (5) CKb (0)

C,,, (71) Cf (21) Cxw (5) C,~ (1)

Cf (21) Co (12) Cm (2)

CKw (0)

Co (2) CKb (2)

CKw (0)

Cm (60) CKw (22) Co (10) Cf(3)

Table 6. Rank orders of sensitivity coefficients representing the relative influences of model parameters on the quasiextinction risk in the K-model. Details as in Table 5

Kb, S minimal and Kw maximal

Kb, S maximal and Kw minimal

r maximal r minimal r maximal r minimal

mi, ma and f a l l maximal

mi, ma and f all minimal

Cm (69)

Co (60)

Cm (55) CKw (36)

Cf(17) CKb (13) C~(I)

C,n (68) Cf(17) CKb (14) Co (1)

Cm (18) CKw (17) Cf (-5)

Co (13) Cf (-4)

CKw (0)

CKw (0)

CKb (0)

CKb (0)

Cm (71) Cf (25) CKb (2)

Cm (68) Cf (20)

co (l)

Cm (70) Cf (25) CKb (2) co (2)

Cm (73) Cf (22) CKw (5) Co (1)

CKw (0)

CKw (0)

C~ (7)

CKw (5) CKb (0)

CKb (0)

Cxb (0)

The rank orders are given for eight scenarios of the S-model, defined by the following alternatives: mean survival rates and fecundity all maximal or all minimal ({mj=0-54, ma =0.74, f=1-95} or {mj=0.43, ma=0-52 and f=1.75}); breeding capacity and scramble factor maximal and winter capacity minimal or vice versa ({Kb = 95, Kw = 270, S = 0-1 } or {Kb = 45, Kw = 540, S = 0}); correlation between adult and juvenile survival maximal or minimal (r = + 1 or r = -1). Standard deviations of juvenile and adult survival are fixed at crj= 0.07 and tra= 0.11. In each scenario the five sensitivity coefficients are normalised to a sum of 100. decrease in the winter capacity have similar influences on the sensitivity coefficients. This results from the fact that Kb, Kw, and S determine the extent to which the population is limited by reproduction or by survival. The smaller Kb and S and the larger Kw, the more the population will be limited by reproduction and the less by survival. Consequently, the combination of Kb, Kw, and S will affect the decision whether summer or winter habitat m a n a g e m e n t is more important. The full variability in outcomes is encompassed by the two extreme scenarios where Kb and S are maximal and Kw is minimal on the one side, and where Kb and S are minimal and Kw is maximal on the other. Altogether, the 512 scenarios could be reduced to 23= 8 basic scenarios, formed by the systematic consideration of the following three alternatives: 1. mean juvenile and adult survival rates and fecundity all maximal or all minimal, 2. {winter capacity maximal and breeding capacity and scramble factor minimal} or {winter capacity minimal and breeding capacity and scramble factor maximal}, 3. correlation between adult and juvenile survival maximal or minimal. The standard deviations of the survival rates were kept fixed at their median values, because their influence on the sensitivity coefficients appeared to be weak. These basic scenarios encompass most of the

variability in sensitivity coefficients. F o r each of the basic scenarios and each of the two models ('S' and ' K ' ) a rank order of sensitivity coefficients was determined (Tables 5 and 6). RESULTS S-model

Table 5 represents the effects of the model parameters on the quasiextinction risk, under the assumption that environmental variability affects the survival rates (Smodel). It can be seen that the survival rates generally have the strongest influence on the quasiextinction risk. The standard deviations of juvenile and adult survival rates have a weak influence if they are negatively correlated, because then the fluctuations cancel each other. If the two survival rates are strongly positively correlated their variability has a considerable influence, especially when the mean population growth (fecundity and mean survival rates) is high. Apart from this one can identify three distinct groups in Table 5.

Group 1 In the lower half of Table 5 where fecundity and mean survival rates are minimal, these parameters have a comparatively strong influence on the quasiextinction risk. Here the sensitivity, Cm, of the mean survival rate is between 64 and 71% and the sensitivity, Cf, of fecundity is between 18 and 26%. The two carrying capacities have only a weak influence (sensitivities, CK6 and CKw, below 5%). The reason is that in these scenarios reproduction cannot compensate for mortality, resulting in a steady population decline. Therefore intraspecific competition is not dominant, and an increase in the two carrying capacities which would reduce intraspecific competition, would have only a weak effect on the outcome.

Group 2 In the upper half of the table the population growth (fecundity and mean survival rates) is maximal. The

Uncertainty in population dynamics for the management of the Orange-bellied Parrot survival rates are still the most important parameters, but now density-dependent effects play a role, too.

277

tuations generally had a slightly smaller effect on the mean population size than on the risk of decline.

Subgroup 2a In the left half of Group 2 where the winter capacity is maximal and the breeding capacity is minimal, the breeding capacity has a considerable influence on the quasiextinction risk (CKb between 8 and 15%). The winter capacity appears to be unimportant.

Subgroup 2b In the right half of Group 2 the breeding capacity is maximal, and the population is apparently limited by the small winter capacity and strong scramble competition. This can be concluded from the considerable effect the winter capacity has on the quasiextinction risk (CKw between 22 and 36%). An increase in fecundity or in the breeding capacity will only increase the number of birds competing in the winter season and therefore has almost no influence (Cfbelow 3%; CKb=O). K-model Table 6 shows the influences of the parameters on the quasiextinction risk, under the assumption that environmental variability results in fluctuations in the winter capacity while the survival rates are fixed (K-model). As one can expect, those fluctuations play no role (sensitivity, Co, below 7%) when the winter capacity is not limiting, i.e. in those scenarios where mean growth (fecundity and mean survival rates) is minimal and/or where the breeding capacity is minimal and the winter capacity maximal (lower and left halves of Table 6; cf. Groups 1 and 2a in Table 5). If population growth and breeding capacity are maximal and the winter capacity is minimal (upper right quarter of Table 6) the sensitivity coefficient, Cf, of fecundity is negative, indicating that an increase in fecundity can have an adverse effect on the population. The reason is that in these scenarios of large population growth, large breeding capacity and small winter capacity the population is strongly regulated by the fluctuating winter capacity. In poor years this is likely to lead to strong competition. If competition is of scramble type the effect on the population can be severe. In such a situation an increase in fecundity would only magnify competition and lead to a smaller number of survivors (cf. Fig. 1). Apart from these observations, the results for the K-model are similar to those of the S-model. Targeting the mean population size Instead of reducing the risk of population decline (quasiextinction risk) one may wish to increase the mean population size. These objectives may be independent to some extent (Burgman et al., 1993) and therefore we repeated the model analysis using mean population size as the management target. The results were almost identical, except that environmental fluc-

DISCUSSION The results show that the rank order of management options depends to some extent on whether the population parameters are at their upper or their lower plausible bounds. One important factor that affects the rank order is the mean population growth (fecundity and mean survival rates). It determines how much influence density-dependence (breeding capacity and winter capacity) has on population viability (cf. Groups 1 and 2 above). An important aspect that distinguishes this model from many other population models is that the population is limited by two carrying capacities. The breeding capacity limits the number of birds that can reproduce and the winter capacity limits the number of survivors. The ratio of these two carrying capacities (breeding and winter capacity) determines the extent to which the population is limited in its reproduction and its survival (cf. Groups 2a and 2b above). Apart from this, a few general features were found in almost all scenarios. . The survival of birds appears to be more important than their reproduction. The influence of fecundity never exceeds the influence of the mean survival rates. We conclude that the survival of individuals is the limiting component in the population dynamics. This conclusion was reached by Menkhorst et al. (1990) and our results support the notion that management should target juvenile and adult survival. . As described in Methods (see Table 1), fecundity and the survival rates represent the density-independent factors of the population dynamics while the two carrying capacities (breeding and winter capacities) represent density-dependent (regulative) factors. In the Results we found that fecundity always has a stronger influence than the breeding capacity and the mean survival rates, mj and ma, always have a stronger influence than the winter capacity. We conclude that density-independent factors are more important in the population than density-dependent factors. Although it is difficult to distinguish between density-dependent and density-independent factors in a real population, and it is difficult to relate them to habitat parameters, one may suggest that density-dependent factors are likely to be related to quantitative features of habitat, such as size and total abundance of food and/or nest sites (see Methods; Table 1). Density-independent factors which act regardless of the population size, are likely to be related to habitat quality, such as human disturbance and quality and spatial distribution of food. Modifica-

278

M . Drechsler et al.

tions of the habitat of the known wintering sites bear unknown risks (Menkhorst et al., 1990). However, in the rehabilitation of adjacent habitat, habitat quality should be a primary target because of its beneficial effect on density-independent growth in the population.

These results are almost independent of whether environmental variability affects the density-independent or the density-dependent component of survival (Smodel and K-model, respectively). They are also quite independent of whether management is to target mean size or quasiextinction risk of the population. Changing the type of competition from contest to scramble had a similar effect on the rank order of management options as a decrease in the winter capacity. The only point that should be mentioned is that if competition is if scramble type and the population is limited by a small and variable winter capacity (K-model), severe competition may lead to a very small number of survivors (cf. Fig. 1). In such a case an increase of reproduction can have an adverse effect on the population.

population in the absence of catastrophes, as this study did. The second analysis would consider the situation immediately after a catastrophe where the population is small and the recovery of the population plays an important role. Management recommendations could be derived for this particular case, too. Conservation managers then could switch between the two management plans, depending whether the population is large and in the 'normal' state or small and in the 'post-catastrophic' state. Spatial structure plays a role in the Orange-bellied Parrot on a local and a regional scale. On the local scale the winter habitat is fragmented into several patches most of which provide food only for certain periods of the winter season. This forces the parrots to move within the winter season over short distances. On the regional scale, the wintering population appears to consist of two subpopulations separated by about 400 km. Aspects of spatial structure in the population of the Orange-bellied Parrot are discussed in Drechsler (1998).

CONCLUSION

Model assumptions The major assumptions in this work concern the choice of the parameter combinations (scenarios) and the models themselves. We have considered only eight different scenarios which, however, were identified to encompass most of the plausible parameter space and the variability in the rank order of parameter sensitivities. The various factors that contribute to the mortality of individuals were described by two different models. These include contest and scramble competition and different types of environmental variation, and though we hope that our choice was representative, these two models can, of course, not cover everything that may affect the mortality of individuals in the real population. Another problem in the model analysis is that actual management costs could not be included. Though the breeding capacity, for instance, has a much smaller influence than the mean survival rates, it is probably much easier to increase than the survival rates which may depend on many environmental factors not yet understood. Three aspects of population dynamics were ignored completely: genetics, catastrophes and spatial population structure. The management recommendations given in this study focus on the maximisation of the mean population size and the minimisation of the risk of decline. We believe that these objectives will also reduce genetic risks which are often related to population size (e.g. Kimura and Crow, 1963; Koenig, 1988). Catastrophes are believed to be rare in the population of the Orange-bellied Parrot. Therefore it is more reasonable to divide the problem of catastrophes into two sub problems. The first analysis considers a 'normal'

The population dynamics of a rare and dispersed species like the Orange-bellied Parrot contain many uncertainties. In the modelling process this is reflected in uncertainty in the model structure and the parameter values. In this work we have investigated the impact of these uncertainties and their influences on alternative management options. An exhaustive systematic analysis of all these aspects would produce a vast amount of data which would be difficult to synthesise. We have therefore reduced the number of plausible parameter combinations to a small number of basic scenarios. These encompass most of the plausible parameter space and the variability in the rank order of management options. For each of these scenarios the rank order of the most promising management options was determined. The results for the different scenarios could be summarised into a few groups which do not contain all of the details but reveal the most important qualitative aspects. This procedure not only provides the conservation manager with different, scenario-dependent rank orders of management options, but it can also help focus research onto those factors and uncertainties that affect the rank order in order to optimise management. In addition, the approach identifies which features are stable with respect to all forms of uncertainty considered. In the Orange-bellied Parrot we found that the survival of the birds seems to be more important than their reproductive success. Density-independent components of mortality which are related to qualitative features of habitat, seem to be more important than density-dependent components of mortality which are related to habitat size.

Uncertainty in population dynamics for the management of the Orange-bellied Parrot ACKNOWLEDGEMENTS We would like to thank the Orange-bellied Parrot Recovery Team for their advice and the provision of population data. Michael McCarthy helped with his expertise on population modelling and with comments on an earlier version of this manuscript. This paper was much improved by the comments of two anonymous reviewers. M. Drechsler is supported by the German Research Community D F G .

REFERENCES Akqakaya, H. R. (1991) A method for simulating demographic stochasticity. Ecological Modelling 54, 133-136. Beck, M. B. (1983) Sensitivity analysis, calibration and validation. In Mathematical Modelling of Water Quality: Streams, Lakes and Reservoirs, ed. G. T. Orlob, pp. 425447. International Series on Applied Systems Analysis, Vol. 12, Wiley, Chichester, UK. 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., Hoidsworth, 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. Caswell, H. (1978) A general formula for the sensitivity of population growth rate to changes in the life history parameters. Theoretical Population Biology 14, 215-230. Drechsler, M. (1998) Spatial management of the Orangebellied Parrot Neophema chrysogaster. Biological Conservation 84, 283 292. Drechsler, M. Sensitivity analysis of complex models. Submitted to Biological Conservation. de Kroon, H., Plaisier, A., van Groenendael, J. and Caswell, H. (1986) Elasticity, the relative contribution of demographic parameters to population growth rate. Ecology 67, 1427-1431. Gardiner, C. W. (1985) Handbook of Stochastic Methods. Springer, London and New York.

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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. IUCN (1994) IUCN Red List Categories. Species Survival Commission. Gland, Switzerland. Kimura, M. and Crow, J. F. (1963) The measurement of effective population number. Evolution 17, 297-288. Koenig, W. D. (1988) On determination of viable population size in birds and mammals. Wildlife Society Bulletin 16, 230-204. Krebs, C. J. (1985) Ecology: The Experimental Analysis of Distribution and Abundance. Harper and Row, New York. Lebreton, J.-D., Burnham, K. P., Clobert, J. and Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs 62, 67-118. 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. McCarthy M. A. (1995) Stochastic population models for wildlife management. Ph.D. thesis, University of Melbourne. 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 Zoological Society, Chicago, IL. Papoulis, A. (1984) Probability, Random Variables and Stochastic Processes. MacGraw Hill, New York. Pollock, K. H., Nichols, J. D., Brownie, C. and Hines, J. E. (1990) Statistical inference for capture-recapture experiments. Wildlife Monographs 107, 1-107. Possingham, H. P., Lindenmayer, D. B. and Norton, T. W. (1993) A framework for the improved management of threatened species based on population viability analysis (PVA). Pacific Conservation Biology 1, 39-45. Ripley, B. D. (1987) Stochastic Simulation. Wiley, New York. Sokal, R. R. and Rohlf, F. J. (1981) Biometry, 2nd edn. Freeman, 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. Stephenson, L. H. (1991) The Orange-bellied Parrot Recovery Plan: Management Phase. Prepared for Australian National Parks and Wildlife Service. Watt, K. E. F. (1968) Ecology and Resource Management. McGraw Hill, New York.

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M . Drechsler et al. APPENDIX

Calculation of the standard deviation of the winter capacity in the K-model To ensure consistency between the S-model and the Kmodel we want the coefficient of variation of the winter capacity in the K-model to equal the coefficient of variation of the survival rates in the S-model. As there are two different survival rates, sj and s~, which are correlated to some extent, we cannot use their individual coefficients of variation (crjmj and ~ra/ma) to measure the total variability in the two survival rates. Instead we consider

var[N(t + 1)] = Na(t) 2. (y2tr2 + 2 y . r. crj. % + ~ ) (A6) where y = Nj(t)/Na(t), the ratio of juveniles and adults before the winter season. This ratio can be approximated by y , ~ f / 2 where f is the expected number of offspring per breeding pair (see Methods). With eqn (A5) and (A6) the coefficient of variation in the fluctuating number of survivors in the S-model is CV[N(t + 1)] ~ ~/var[N(t + 1)] m[N(t + 1)]

N(t + l) = Nj(t ÷ 1) + Na(t + l) ----sj . Nj(t) + Sa " Na(t) (A1)

v/y2.~r~ + 2. y . r . ~ r j . Cra +cr2a y . mj + ma

which is eqn (A1) without demographic fluctuations and intraspecific regulation. Therefore the total variability in the two survival rates, sj and Sa, is directly reflected in the variability of N ( t + 1) and is measured by the coefficient of variation of N ( t + 1). As described in Methods, in the S-model the two survival rates, sj and sa, are normally distributed r a n d o m numbers with means and standard deviations, mj and ma, and crj and %, and correlation r. We are interested in the coefficient of variation in the fluctuating number of survivors, N ( t + 1), which is the ratio of its standard deviation and its mean. The mean number of survivors in a population with Nj(t) juveniles and Na(t) adults is given by m[N(t + 1)] = mj . Nj(t) ÷ m~ . N~(t)

(A2)

The variance of a quantity is the mean of the squared difference between that quantity and its mean: var[y] = m [ ( y - m y ) 2] (Sokal and Rohlf, 1981). The variance o f N ( t + 1) then is varIN(t + 1)] = var[sj • Nj(t) + s a • Na(t)] = m[{(sj - mj). Nj(t) ÷ (s a

-

-

ma)" Na(t)} 2] (A3)

We use that the mean of the product of two r a n d o m numbers, m[x.y], is given by m[x . y] = r . Crx . try ÷ mx . my

(A4)

where mx and my and Crx and Cry are the means and standard deviations of x and y, respectively, and r is the correlation between x and y. With some algebra we find that the mean and the variance of the number of survivors are m[N(t + 1)] = Na(t)" (y" mj + ma)

(A5)

As mentioned above, we want this coefficient of variation to equal the coefficient of variation of the winter capacity in the K-model, i.e. C V [ N ( t + 1)] = CV[Kw] = crk/mk. Variability in the shape of the distribution of fecundity parameters The reproduction in the population is described by the number of breeding pairs, Nb, and the probability that Nb breeding pairs produce Nj juveniles. The mean number of juveniles, m(Nj) and the variance, var(Nj) are given by the number of breeders, Nb, multiplied by the mean, f, and the variance, af2, of the distribution of Table 2, respectively (Papoulis, 1984; Gardiner, 1985). Higher moments of the distribution of Nj such as skewness and curtosis, are negligible due to the Central Limit Theorem (Papoulis, 1984; Gardiner, 1985; we calculated the distribution of Nj for a breeding population of 10 or more pairs and found no significant deviation from a normal distribution). T h e r e f o r e f a n d cry2 are the only two fecundity parameters that really affect the population dynamics. They are calculated from Table 2 as f = 1.95 and cry2 = 4.35 (standard deviation of ~rf= 2.09). We tested five alternative distributions of fecundity parameters, all of them with the same mean o f f = 1-95 (Table 7). They include two bimodal, a unimodal and two even distributions. Table 7. Five examples, (a) to (e), of distributions of fecundity parameters (probabilities of raising n fledglings) Example

n= 1

Number of fledglings n= 2 n= 3 n= 4 n= 5

n= 6

of

a b c d e

0.325 0.195 0 0.13 0.093

0 0 0 0.13 0.093

0 0 0 0 0.093

2.16 2.10 2.00 1.83 2.16

0 0 0 0.13 0.093

0 0.195 0.488 0.13 0.093

0-325 0-195 0 0-13 0-093

The resulting mean fecundity is f = 1.95 in all distributions. For each distribution the standard deviation, ~rf,is given in the right column.

Uncertainty in population dynamics for the management o f the Orange-bellied Parrot

281

Introduction of the sensitivity coefficient, eqn 9 s ~ ~•

#t

•

t#

p

•

•

iI

° tg'

..," tt

•

°%°%,

,, •

'% •

• •

0.3

I

0.4

~

%..

,,,~ I

0.5

016

017

018

0.9

Sa

Fig. AI. Normally distributed adult survival rates, sa. The solid line represents an example with mean, ma=0-6 and standard deviation, aa=0.12. From this, the dotted line is obtained by increasing the mean by 10% to ma=0.66. The standard deviations remain constant at aa=0.12. The thin dashed line is obtained from the solid line by reducing the standard deviation by 10% to a~= 0.108. The mean survival rate, ma, remains fixed at m~=0.6. The bold dashed line is obtained from the solid line by decreasing the standard deviation by 10% of the mean survival rate, ma, to a value of a~ = 0.06. The mean survival rate, ma, remains fixed. The vertical line goes through 0.54 which equals m~-a~ in the dotted and the bold dashed curves. The table shows that the standard deviations of all distributions are very close to the standard deviation of the original distribution of a f = 2.09. We conclude that for a given mean fecundity, f, the variability in the relative magnitudes of the fecundity parameters has little influence on the distribution of the number of juveniles produced in the population. We have to consider only the variability in the mean, f. This is done by varying all fecundity parameters by the same proportion, keeping their relative magnitudes fixed.

To demonstrate the inconsistency that occurs when eqn (8) is applied to determine the sensitivity of aj and aa, consider an example of normally distributed adult survival rates with mean, ma and standard deviation, aa (Fig. A1, solid line). To obtain the dotted line, the mean survival rate, ma, was increased by a proportion of 8 = 10% which increased all survival rates by an amount of 8"ma. To obtain the thin dashed line we decreased the standard deviation, a~, by the same relative proportion of a = 10%. This action decreased the frequency of very low and very high survivorships and simulates a reduction of the management effort in 'good' years to spend the saved resources in 'bad' years. However, the quantitative difference between the solid and the thin dashed curve is much less than between the solid and the dotted curve. This means that the proportional change of the standard deviation aa has a much weaker overall effect on the survival rates, sa, than a change of the same proportion in m~. It seems more reasonable not to vary ma and a~ by the same relative proportion, but to vary ma by a relative proportion, 8, as usual, and ~ by an amount of 6.m~ (Fig. A1, bold dashed line). With this, both parameters, ma and a~, are varied by an amount of 3.ma. In Fig. A1 the impacts of both actions appear qualitatively different, but their overall strengths are comparable. In particular, either of these two options will increase the magnitude of m a - ~ a from 0.48 to 0.52. This means that both options reduce the likelihood of poor survival rates, s~, to a value of 15-9%. We prefer this approach, because it weights both actions, increase of mean survival rates and decrease of standard deviations, in a similar way. To accommodate this kind of parameter variation we use the sensitivity coefficient based on eqn (9).