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A model-based decision aid for species protection under uncertainty
Biological Conservation 94 (2000) 23±30
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A model-based decision aid for species protection under uncertainty Martin Drechsler* Department of Ecological Modelling, Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany Received 18 June 1999; received in revised form 2 September 1999; accepted 1 October 1999
Abstract A method of decision making is presented that can be used to compare alternative management actions under ecological uncertainty and to identify which one is likely to have the strongest eect on population viability. The method combines decision analysis with population modelling and uses both information about population patterns (e.g. spatio-temporal) and processes (e.g. reproduction). The process knowledge is used to construct the population model and determine plausible ranges for its parameters. The values of these parameters are likely to have an impact on the rank order of the most eective management actions, and unless their ranges are small, there is uncertainty in the management rank order. This uncertainty is encompassed by considering a number of dierent population parameter combinations, called scenarios. For each of them a sensitivity analysis is performed and a management rank order determined. In the following decision analysis each key scenario contributes with a certain weight that re¯ects its biological plausibility. To determine the weight of a particular scenario, the population dynamics are simulated and the generated patterns are compared with those observed in the real population. The higher the similarity between the two patterns the higher the weight assigned to the scenario. The decision analysis ®nally synthesises the results of sensitivity and pattern analyses and generates a single rank order of the most promising management actions. The method is demonstrated on a case study of the endangered Orange-bellied Parrot Neophema chrysogaster (Australia). # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Decision analysis; Uncertainty; Population model; Conservation management; Extinction
1. Introduction A central issue in conservation biology and species protection is the ranking of alternative management actions (Possingham et al., 1993; Lindenmayer and Possingham, 1996). Such a decision is dicult where the eects of the actions are subject to uncertainty. The eect of creating a habitat network on the viability of a species, for instance, naturally depends on the species' mobility. If this is not precisely known the eect of a habitat network will be uncertain. In such a situation it is desirable to have a tool that helps to synthesise the uncertain information and provide a rational decision. Decision theory which is especially used in areas such as economics and medicine (for an overview, see, e.g. Watson and Buede, 1987; Corner and Kirkwood, 1991) can be such a tool. It encourages the decision maker to be explicit about the relevant fac-
* Tel.: +49-341-235-2039; fax: +49-341-235-3500. E-mail address: [email protected] (M. Drechsler).
tors and use the information in a rational manner. This cannot exclude the possibility of errors (as these are a natural phenomenon in the presence of uncertainty) but can minimise their likelihood. Since the mid eighties decision analysis has also been used in conservation biology (Maguire, 1986, 1991; SouleÂ, 1989; Ralls and Star®eld, 1995; Possingham, 1996, 1997; Burgman et al., 1999; and others). Apart from providing a structure for the decision process, decision analysis can actively help in the decision process only if it is combined with an instrument that can predict the outcomes of the management actions available. Such an instrument can be a simulation model which synthesises all the accessible information. This study demonstrates how a simulation model can be combined with decision analysis to make conservation decisions in the presence of uncertainty. The example is based on the case of the Orange-bellied Parrot Neophema chrysogaster in south-eastern Australia. It breeds in Tasmania and overwinters in the coastal regions of the Australian mainland. Industrial development in these regions has contributed to a population
0006-3207/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0006-3207(99)00168-8
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M. Drechsler / Biological Conservation 94 (2000) 23±30
decline to about 200 birds (Menkhorst et al., 1990). Drechsler et al. (1998) used a population model to simulate various management actions and determined their eects on the viability of the population. The actions were ranked according to these eects. The rank order of the management actions was shown to depend on the values of the model parameters which are subject to uncertainty. Therefore several rank orders had to be considered to encompass this uncertainty. Although Drechsler et al. (1998) were able to deduce some practical rules from this set of rank orders, this process will in general be tedious and may be suboptimal. The decision analysis presented in this paper can greatly simplify the evaluation of rank orders under uncertainty. The result of the analysis is a single rank order of the management actions that can be easily interpreted. 2. Decision analysis under uncertainty For the problem described above the appropriate type of decision analysis is the so-called decision analysis under uncertainty (e.g. Bayes, 1958; Eckhoudt and Gollier, 1995) or quantitative ecological risk assessment (Possingham, 1995). It consists of the action space, the state space and the results function. The action space contains the available actions (e.g. `go swimming' or `go to the cinema'). The state space contains the states that the described system can assume (e.g. `®ne weather' or `bad weather'). The result (e.g. happiness) of a given action depends on the state of the system. This relationship is expressed in the results function. It tells which result Rij is achieved by action Ai if the system is in state Sj. This can be summarised in the so-called results matrix (Table 1). The rows of the results matrix are given by the states Sj (j=1. . .n) and the columns by the actions Ai (i=1. . .m). If the state is known the best action can immediately be obtained from the results matrix. In general this is not the case, but even without the exact knowledge of the state it is possible to derive a decision from the results matrix, namely when the probabilities of the states are known. Let the probability of observing state Sj be denoted by Pj. Then for instance, the expectedvalue criterion (e.g. Eckhoudt and Gollier, 1995) can be applied to identify the action that is most likely to lead Table 1 Example of a results matrix. The matrix elements Rij give the result as a function of the selected management measure Ai and the present state Sj. The quantity Pj is the likelihood of state Sj State
Pj
A1
A2
S1 S2 S3
0.2 0.3 0.5
10 25 30
15 15 20
to the best result. Given the Rij and Pj, the expected result, E(Ai) for action Ai is obtained by taking the average over the results for the various states. The probabilities Pj serve as weight factors, i.e. the result Rix obtained in a likely state Sx will contribute more than the result Riy of an unlikely state Sy: E Ai P1 Ri1 P2 Ri2 . . . Pn Rin
1
In the example of Table 1 we ®nd E(A1)=24.5 and E(A2)=17.5. The expected result achieved by action A1 exceeds that of action A2. Table 1 further shows that the result for action A1 varies and considerably depends on the state. A measure for the variability of the results for a particular action Ai is their standard deviation S(Ai): 1=2 S Ai P1 R2i1 P2 R2i2 . . . Pn R2in ÿ E Ai 2
2
In the example of Table 1 we ®nd S(A1)=7.6 and S(A2)=2.5. The smaller S the less the likelihood of observing a result that strongly deviates from the expected value E. In the example of Table 1 the standard deviation in the results for action A1 is higher than the dierence between the two expected values E(A1) and E(A2), which suggests that the results of A1 and A2 do not dier much with respect to their uncertainty. This is similar to the logic used in a t-test where the dierence between the expected values of two normally distributed random numbers is compared to their pooled standard deviations. In this case we may adopt the dierence E(A1)ÿE(A2) relative to S as a qualitative guide to the importance of dierence. Furthermore, as the variation in the results for action A1 is much higher than that for action A2, action A1 is more likely to lead to a below-average result than A2. This means that there is a trade-o between maximising the expected value and minimising the risk of a poor result. In this case the decision will depend on the risk tolerance of the decision maker, i.e. on whether he or she will tolerate a higher risk in order to achieve a higher expected result or be satis®ed with a smaller expected result in order to reduce the risk of a very poor result. As the expected results in Table 1 are not very dierent, a risk averse decision maker may prefer action A2 whose result is subject to less variation. The aim of the two following sections is to use a population model in order to generate the results matrix and the probabilities Pi, and to derive a rank order of management actions for the population of the Orangebellied Parrot. In contrast to the example above, the `states' of the results matrix will be made of alternative combinations of population model parameters. Uncertainty in the analysis then will be caused by our incomplete knowledge of the correct parameter values rather than by the unknown state of the population. It is
M. Drechsler / Biological Conservation 94 (2000) 23±30
therefore appropriate not to use the term `state' (which is usually something that can change in time) but the term `scenario'. A scenario then represents a particular assumption about the values of the population model parameters. 3. The results function 3.1. The population model and simulation of the management actions To apply the above decision analysis to the ranking of conservation measures a results matrix is needed that relates the results to the actions and the scenarios. The aim of the conservation actions is to increase the viability of the Orange-bellied Parrot population. In the analysis, population viability is measured by the logit, ln[Q/(1ÿQ)], of the quasiextinction risk Q. The quasiextinction risk is the probability that within a certain time frame (e.g. 20 years) the population will fall under a certain threshold (Ginzburg et al., 1982). Four alternative management actions are considered to achieve the aim: to increase the breeding capacity (see below) by the provision of nest boxes, to increase fecundity by arti®cial feeding, to increase the mean annual survival rates of the parrots by winter habitat management, and to increase the winter carrying capacity by the expansion of wintering habitat. The rank order of these management actions is likely to depend on the scenarios. The relationship between the results of the management actions and the scenarios is provided by the population model by Drechsler et al. (1998) which will be outlined in the following. In that model, the annual dynamics of the population were divided in a summer and a winter season. In summer the population breeds in Tasmania south of the Australian mainland. The number of juveniles produced by the population is determined by the number of nest sites (`breeding capacity') and the number of juveniles produced per breeding pair (`fecundity'). After the breeding season, adult and juvenile birds migrate to the Australian mainland and overwinter in coastal regions. In the following spring the surviving birds return to their breeding habitats. By that time the juveniles have grown to adults. Annual mortality is concentrated in the winter months (Menkhorst et al., 1990). A capture±mark± recapture program provides estimates for the annual survival rates of juveniles and adults. From these values long-term means and standard deviations of adult and juvenile survival rates can be calculated, as well as a correlation coecient between adult and juvenile mortality. The model also contains a `winter capacity' which gives the number of birds that can survive a winter under optimal conditions. This winter capacity results from intraspeci®c competition for food and is related to
25
the size of the wintering habitat. Winter mortality is also aected by the type of intraspeci®c competition (scramble or contest competition; e.g. Begon et al., 1990). If scramble competition is present, survival is reduced by a certain amount which is determined by the `scramble factor' (Drechsler et al., 1998) and re¯ects the strength of competitive interactions. All these population parameters are subject to considerable uncertainty (Table 2). The population model was implemented as a computer simulation program and the dynamics were simulated for 20 years. The eect of the four management actions was determined by Drechsler et al. (1998) in a sensitivity analysis (e.g. Caswell, 1978). For this, all population parameters were set to their median values (the `standard scenario; Table 2). Starting from this standard scenario, the four `management parameters', fecundity, mean survival and breeding and winter capacities were varied in turn by a certain proportion and the corresponding changes in the viability [i.e. changes in the logit of the quasiextinction risk, logit(Q)] of the population were determined. For each of the four management parameters a sensitivity coecient, C=logit(Q)/, was calculated that measures its in¯uence on population viability. The rank order of these sensitivity coecients tells which management action is best, second best and so on. 3.2. Formulation of the scenarios and construction of the results matrix Table 2 shows that in some population parameters there is considerable uncertainty. Therefore the rank order of the four management parameters which depends on the population parameters, can be expected to be uncertain, as well. Therefore it is not sucient to consider only the standard scenario. To obtain a clear picture of the in¯uence of the population parameters on
Table 2 Plausible ranges of the population parametersa Parameter Fecundity Breeding capacity Mean annual survival rate of adults Mean annual survival rate of juveniles Standard deviation of adult survival Standard deviation of juvenile survival Correlation between adult and juvenile survival Winter capacity Scramble factor a
Maximum 1.95 95 0.74
Median 1.85 70 0.64
Minimum 1.75 45 0.54
0.52
0.475
0.43
0.13
0.11
0.09
0.09
0.07
0.05
+1 540 0.1
0
ÿ1
400 0.05
270 0.0
Note that the winter capacity is not the size of the real population which is lower. A scramble factor of 0.0 represents contest competition.
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M. Drechsler / Biological Conservation 94 (2000) 23±30
the management rank order, a large number of alternative scenarios has to be considered. Each scenario represents a plausible assumption about the values of the population parameters and for each a management rank order will be determined. For convenience only a relatively small subset of all possible scenarios is considered. For this it is assumed that each population parameter may either take its maximum or its minimum value. If the model has n population parameters, a number of 2n scenarios is considered. In the case of the Orange-bellied Parrot this number could be reduced to eight (Drechsler et al., 1998). These eight scenarios are de®ned by whether mean population growth (fecundity and the mean survival rates) takes its maximum or minimum value, whether the ratio of breeding and winter capacities and the scramble factor is maximal or minimal (i.e. whether the population is regulated mainly in winter or not), and whether the variation of population growth (standard deviation of the survival rates and the correlation between adult and juvenile survival) is maximal or minimal. Altogether, there are only three independent quantities that determine the rank order of management actions. For convenience they will be called population growth, winter regulation and population variation. The population parameter space is only threedimensional now, spanned by the three parameter sets termed `super parameters', population growth, winter regulation and population variation. While the standard scenario (all parameters at median values) occupies the centre of this three-dimensional parameter space, the eight additional scenarios are located at its corners (Fig. 1) For each of the nine scenarios the sensitivity of population viability to the management parameters is calculated, allowing the rank order of the four management actions to be determined (Table 3; cf. Drechsler et al., 1998). Table 3 has the desired structure of a results matrix (cf. Table 1). The sensitivity coecients can be regarded as the results Rij. A large sensitivity coef®cient represents a strong eect of the corresponding measure on population survival, i.e. a good result. 4. Decision making 4.1. Determination of the probabilities of the scenarios All of the nine scenarios introduced above lie within the plausible parameter space spanned by the values in Table 2. However, here the term `plausible' only means that each of the nine scenarios could be the one that best describes the real population, but it does not tell whether a particular scenario is more likely to be valid in the real population than another. What is missing is a criterion that decides how well a particular scenario characterises the real population. For such an evaluation additional information is required.
Fig. 1. Graphical representation of the three-dimensional super parameter space.
Even without precise knowledge of the size and distribution of the Orange-bellied Parrot population, there are rough estimates on population trend and variation. Regression analysis of the estimated maximum winter population sizes from 1978 to 1989 (Menkhorst et al., 1990; Fig. 3) leads to a long term population change of +0.6 birds per year (95%-con®dence interval: ÿ2.6. . .+3.8 birds per year) which is very close to zero. However, there has been a sharp decline in the number of parrots observed in the annual winter counts from 108 in 1989 to about 40 birds in 1995 and 1996 (Rounsevell, 1996). This would correspond to a population trend (average change in population size) of about ÿ10 birds per year. This decline is likely to be caused by a shift in habitat use by the parrots reducing their sighting probability in recent years. Hence the observed trend of ÿ10 birds per year should not re¯ect the actual trend in the population size. Nevertheless, it may serve as a lower bound on the population trend. Altogether, the population is believed to be stable (Menkhorst et al., 1990; Rounsevell, 1996), i.e. a zero population trend is most likely. Larger population trends are less likely and values below ÿ10 birds per year should have a likelihood of zero. For simplicity, within the range of plausible population trends between 0 and ÿ10 birds per year, a linear relationship between a population trend and its likelihood is assumed (Fig. 2). The coecient of variation of the estimated maximum winter population sizes from 1978 to 1989 (Menkhorst et al., 1990) is around 0.17. The jackknife estimate of the upper bound of the 95%-con®dence interval is close to 0.2. This value may be used as an upper plausible bound on the coecient of variation in the population size. However, the estimates in years 1978, 1979 and 1984±1986 which all dier considerably from the estimates for the other years, are likely to be wrong (Menkhorst et al., 1990). If these years are excluded from the analysis the coecient of variation in the estimated maximum winter population size is about 0.05. Its lower 95%-con®dence limit is close to zero. A value of zero then may be used as a lower plausible bound on the coecient of variation of the population size. In a
M. Drechsler / Biological Conservation 94 (2000) 23±30
27
Table 3 Sensitivity coecients (eect on population viability) of the four management measuresa Scenarios
1 2 3 4 5 6 7 8 9
Actions
Population growth
Winter regulation
Population variation
A1
A2
A3
A4
Median Maximum Maximum Maximum Maximum Minimum Minimum Minimum Minimum
Median Maximum Maximum Minimum Minimum Maximum Maximum Minimum Minimum
Median Maximum Minimum Maximum Minimum Maximum Minimum Maximum Minimum
23 0 3 16 19 20 22 24 27
73 73 71 71 64 74 73 74 71
1 0 0 13 17 0 0 2 2
3 27 26 0 0 6 5 0 0
a
A1={increase fecundity}, A2={increase mean survival rates}, A3={increase breeding capacity} and A4={increase winter capacity} for various scenarios. In each scenario the four sensitivity coecients are normalised to a sum of 100.
Fig. 2. Likelihood of the mean population trend. A value of 1 means that this trend is very likely to be present in the real population. A value of zero means that this trend is most unlikely in the real population.
determined. Figs. 2 and 3 may be used to determine how well the simulated and real dynamics agree. For example, for scenario 1, the simulated population dynamics have a trend of ÿ1 bird per year (Table 4) which is quite likely to be valid in the real population (likelihood of 0.9 in Fig. 2). In this respect, the simulated and real populations agree well. In contrast, the simulated coecient of variation is 0.2 which is much higher than is believed to be valid in the real population (likelihood of 0.0 in Fig. 3). A scenario is accepted as realistic if the simulated and real populations agree both in their trend and coecient of variation. For this the likelihood of a scenario is given by the product of the two weight factors in Table 4. The resulting likelihoods then are normalised (all divided by a common number), so their sum equals one and they can be interpreted as probabilities (Table 4). 4.2. The rank order of management actions The results matrix (Table 3) can be evaluated with the probabilities from Table 4. For the four management actions, the expected-value criterion leads to the following results: E A1 13; E A2 68; E A3 7; E A4 12:
Fig. 3. Likelihood of the coecient of variation of the population size (cf. Fig. 2).
workshop with the Orange-bellied Parrot Recovery Team, a coecient of variation around 5% was regarded as most likely. Larger or smaller coecients of variation are less likely. Assuming a linear relationship between a coecient of variation and its likelihood leads to Fig. 3. With the help of the model, population dynamics may be simulated and population trend and levels of variation
The expected result, i.e. increase in population viability, is maximal when the mean survival rates are increased (action A2). The eects of fecundity and winter capacity (A1 and A4) are medium, the breeding capacity (A3) has a weak in¯uence. The standard deviations of the results are S A1 9; S A2 4; S A3 8; S A4 13: The results are also shown graphically in Fig. 4. The expected result of action A2 is much higher than those of the other three actions. The dierences between E(A2) and the expected results of the other actions are large compared to the standard deviations of the results.
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M. Drechsler / Biological Conservation 94 (2000) 23±30
Table 4 Mean trend (change of mean size per year) and coecient of variation of the modelled population size for the nine scenarios after Table 3a Scenario
1 2 3 4 5 6 7 8 9
Simulated population dynamics Trend (individuals per year)
Weight 1
Population variation
Weight 2
Likelihood (=Weight 1 Weight 2)
Probability (normalised likelihood)
ÿ1 ÿ1 ÿ1 0 0 ÿ15 ÿ8 ÿ15 ÿ8
0.9 0.9 0.9 1 1 0 0.2 0 0.2
0.2 0.24 0.06 0.25 0.09 0.2 0.08 0.25 0.08
0 0 0.93 0 0.73 0 0.8 0 0.8
0 0 0.873 0 0.73 0 0.16 0 0.16
0 0 0.44 0 0.39 0 0.08 0 0.08
a The weight factors were obtained from Figs. 2 and 3. The likelihood of a scenario is the product of the two weights and the probabilities were obtained by normalising the likelihoods so the nine probabilities have a sum of one. They measure how likely each scenario is present in the real popu-
lower than that in the result of action A4. Therefore in action A1 the risk of a poor result is lower than in action A4. Because of its slight superiority in the expected value and its lower risk, A1 outperforms A4. Altogether the rank order of the four actions is A2 > A1 > A4 > A3 where x>y means `x is preferred to y'.
Fig. 4. Expected results (bars) and error (error bars, showing one standard deviation) for the four management actions A1±A4 (from top to bottom: increase fecundity; increase mean survival rates; increase breeding capacity; increase winter capacity). For each action the numbers give the expected value and in parentheses the standard deviation of the result. Nine scenarios (cf. Tables 3 and 4) are considered.
Therefore A2 (increase of annual survival rates) is clearly the best action. However, action A2 may be dif®cult to achieve in practice. First, annual mortality includes mortality during migration between the breeding and winter habitats, which is much related to weather conditions and dicult to reduce by management. Second, in contrast to the other management parameters, such as carrying capacities, the annual survival rates (cf. Table 2) can in principle not be increased by more than a factor of 2, as they are bound by 1. It is therefore sensible to concentrate on the other three management actions. The dierence between the expected results of actions A1 and A4 with that of action A3 is of the same magnitude as the standard deviations of the results. Therefore actions A1 and A4 should be assigned a higher rank than action A3. The standard deviations of the results for actions A1 and A4 are higher than the dierence between their expected results, E(A1)ÿE(A4) which means that A1 and A4 are more or less equivalent with a slight advantage of A1. The uncertainty (S) in the result of action A1 is
5. Discussion The decision analysis can be used to rank management actions according to their eects on the viability of a population. Uncertainty in population parameters leads to uncertainty in the eects of the management actions. As a consequence, in some instances, particularly where the uncertainty is large, some of the management alternatives cannot be distinguished, i.e. their eects on population viability do not dier signi®cantly with respect to the uncertainty. The decision analysis takes these uncertainties into account and helps to identify which management alternatives can be clearly ranked and which alternatives cannot be distinguished from each other. This helps to . discard inecient management actions . assess the eect of uncertainty on the decision process . decide whether uncertainty has to be reduced before a reasonable decision can be made. Some of the numerical results of the present study are similar to those given in Drechsler et al. (1998) who found that mean survival is the most important population parameter, followed by fecundity, winter capacity and breeding capacity. Particularly, the importance of survival is well known from many other studies. The
M. Drechsler / Biological Conservation 94 (2000) 23±30
new aspect in this approach is that it provides a transparent protocol for assessing and comparing management options under uncertainty. Drechsler et al. (1998) did not show that mean survival has a much stronger eect (compared to the uncertainty in the population) on population viability than the other parameters and that the dierence between the two carrying capacities is less important. They found their results by ad hoc comparison of various rank orders without using an objective method of combining them. Furthermore, they did not consider that some rank orders should be given more weight than others, because they belong to scenarios (i.e. population parameter combinations) that are more likely to be valid in the real population than others. The method presented in this paper is able to eliminate these de®ciencies and provide a systematic and quantitative procedure of evaluating rank orders of management actions under uncertainty. An open question is the proper selection of the scenarios. This may be done randomly or Ð like in this study Ð systematically to evenly cover the parameter space. One may wonder whether nine parameter combinations is sucient to determine the correct rank order. This may be checked by gradually increasing the number of scenarios and looking at possible changes in the rank order. If these are negligible, the resulting rank order is likely to be correct, although a strict mathematical proof cannot be given at this stage. The outlined procedure was carried out for the Orange-bellied Parrot and after increasing the number of scenarios to 27 [where one scenario was located in the centre of the three-dimensional parameter cuboid (Fig. 2) and the others at its corners, the middles of its 12 edges and the centres of its six faces] the expected values and the standard deviations of the results for actions A1 to A4 were fairly constant and not much dierent to those determined above. The issue of ranking action A2 (increase annual survival rates) pointed to the importance of considering management costs in the analysis. Above it was argued that A2 may be dicult to achieve in practice, which in economical terms means that it will be expensive if not priceless. In the following analysis of the other three management actions it was implicitly assumed that they all have the same costs. However, the provision of one additional nestbox to increase the breeding capacity by about 1±2% may be less costly than an increase of the winter capacity by the same percentage. To come to a fair rank order of all management actions, their costs would have to be considered. This could be done in two ways. First, one could choose the management actions such that all have the same costs and then performing the decision analysis as above but based on the changes in population viability rather than sensitivity coecients. Second, one could consider bene®t-cost ratios instead of the sensitivity coecients. As described above, those sensitivity coecients were calculated by
29
dividing the change in population viability (logit of the quasiextinction risk) by the relative change in the management parameter (). To obtain a bene®t-cost ratio one would replace by the corresponding costs. The present decision analysis was greatly simpli®ed by reducing the number of population parameters from nine to three The remaining three were called `super parameters'. The identi®cation of such super parameters has been described by Drechsler (1998). Certainly, in many population models the number of parameters could not be reduced so readily which means that more than nine scenarios (as in the present case) would have to be considered. This would increase computing time but would not restrict the applicability of the decision method. Particularly, it is those complex cases with many scenarios where a formal decision protocol such as the one presented is indispensable. In the determination of the probabilities of the scenarios many quantities may be used. In the Orange-bellied Parrot, there was only knowledge on trend and variation of population size. If available, spatial parameters, such as spatial distribution, density or aggregation, or demographic parameters, such as the sex ratio or the age distribution should also be used. All these parameters describe the (temporal, spatial and/or demographic) pattern of the population. Such patterns are valuable in the validation and tuning of models and model parameter combinations (Grimm et al., 1996). Which patterns are used depends on their availability and their relevance. A more formal way of determining the probabilities of the scenarios is Bayesian analysis (e.g. Gelman et al., 1995). To establish the probability distribution of the model parameters, one would start with simple ®rst guesses on the distribution (`prior distribution'). Using Bayes' Theorem and additional information (from the population patterns) this prior distribution would be updated and improved (`posterior distribution'). This posterior distribution than would contain the probabilities of the scenarios (cf. Table 4). An advantage of Bayesian analysis is that the posterior distribution can gradually be improved as more population data become available. There are various options to extend the decision analysis. Within this work, the aim was to identify a single rank order of management actions. However, if the necessary information is available, conservation managers may depend their actions on the present state of the population. For instance, if the population is small Ð and if this is known to the decision maker Ð he or she may prefer measures that maximise population recovery. In contrast, at times where the population size is large, reducing intraspeci®c competition by providing resources may be more ecient. For further details about statedependent decision analysis, see Possingham (1997). A possible extension is the consideration of multiple goals (e.g. Stewart, 1992) where there may be trade-os
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M. Drechsler / Biological Conservation 94 (2000) 23±30
between dierent goals. The task is to ®nd a solution that provides a best compromise. Especially in conservation biology and species protection, the systematic consideration of multiple goals under uncertainty is a dicult problem, but it can be solved (e.g. Munda, 1995). The extension of the present single goal analysis to a multiple goal analysis may be valuable. A critical point is that the results rely on the population model used. Naturally, such a model cannot consider all factors and processes that may in¯uence the population. Instead, the modeller concentrates on those factors that are expected to have some in¯uence on the outcome of the decision analysis. The decision about which factors to consider in the model introduces another type of uncertainty, called structural uncertainty which usually cannot be eliminated. Interpretation of results should be conditioned by model structure. The decision method is not a strictly determined protocol. In the course of an analysis, decisions have to be made about the relevant factors and how these are to be considered. Such variability is typical in decision analyses, particularly where uncertainty plays a role. Decision analysis therefore should not be ®nal authority. Instead it is a tool used to synthesise expert knowledge and assist in the decision process. The risk of making a wrong decision can never be excluded, but it can be reduced. Acknowledgements I would like to thank Volker Grimm for his helpful comments on the manuscript. The comments of Mark Burgman and Hugh Possingham were very encouraging and useful in the revision of the paper. References Bayes, T., 1958. An essay towards solving a problem in the doctrine of chances. Biometrika 46, 293±298 (reprint of Bayes' 1763 manuscript). Begon, M., Harper, J.L., Townsend, C.R., 1990. Ecology: Individuals, Populations and Communities. Blackwell, Cambridge, MA. Burgman, M.A., Keith, F.J., Rohlf, C.R., Todd, C.R., 1999. Probabilistic classi®cation rules for setting conservation priorities. Biological Conservation 89 (2), 227±231. 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. Corner, J.L., Kirkwood, C.W., 1991. Decision analysis applications in the operations research literature, 1970±1989. Operations Research 39, 206±219. Drechsler, M., 1998. Sensitivity analysis of complex models. Biological Conservation 86, 401±412.
Drechsler, M., Burgman, M.A., 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. Eckhoudt, L., Gollier, C., 1995. Risk-evaluation Management and Sharing. Harvester and Wheatsheaf. New York, London. Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B., 1995. Bayesian Data Analysis. Chapman and Hall, London. Ginzburg, L.R., Slobodkin, L.B., Johnson, K., Bindman, A.G., 1982. Quasiextinction probabilities as a measure of impact on population growth. Risk Analysis 2, 171±181. Grimm, V., Frank, K., Jeltsch, F., Brandl, R., UchmanÂski, J., Wissel, C., 1996. Pattern-oriented modelling in population ecology. Science of the Total Environment 183, 151±166. Lindenmayer, D.B., Possingham, H.P., 1996. Ranking conservation and timber management options for Leadbeater's Possum in southeastern Australia using population viability analysis. Conservation Biology 10 (1), 235±251. Maguire, L.A., 1986. Using decision analysis to manage endangered species populations. Journal of Environmental Management 22, 345±360. Maguire, L.A., 1991. Using decision analysis to manage endangered species. In: Decker, D.J., Krasny, M.E., Go, G.R., Smith, C.R., Gross, D.W. (Eds.), Challenges in the Conservation of Biological Resources: A Practitioner's Guide. Westview Press, Inc, Boulder, CO, pp. 139±152. Menkhorst, P.W., Loyn, R.H., Brown, P.B., 1990. Management of the Orange-bellied parrot. In: Clark, T.W., Seebeck, J.H. (Eds.), Management and Conservation of Small Populations, Chicago Zoology Society, pp. 239±251. Munda, G., 1995. Multicriteria Evaluation in a Fuzzy Environment. Physica, Heidelberg. Possingham, H.P., 1995. The practical application of population viability analysis for conservation planning. In: Bradstock, R.A., Auld, T.D., Keith, D.A., Kingsford, R.T., Lunney, D., Sivertsen, D.P. (Eds.), Conserving Biodiversity: Threats and Solutions. Surrey Beatty & Sons, Sydney, pp. 292±299. Possingham, H.P., 1996. Decision theory and biodiversity management: how to manage a metapopulation. In: Floyd, R.B., Sheppard, A.W., Wellings, P. (Eds.), Frontiers of Population Ecology. CSIRO Publishing, Canberra, Australia, pp. 391±398. Possingham, H.P., 1997. State-dependent decision analysis for conservation biology. In: Pickett, S.T.A., Ostfeld, R.S., Shachak, M., Likens, G.E. (Eds.), The ecological basis of conservation: Heterogeneity, ecosystems and biodiversity. Chapman and Hall, New York, pp. 298±304. Possingham, H.P., Lindenmayer, D.B., Norton, T.W., 1993. A framework for improved threatened species management using Population Viability Analysis. Paci®c Conservation Biology 1, 39±45. Ralls, K., Star®eld, A.M., 1995. Choosing a management strategy: two structured decision-making methods for evaluating the predictions of stochastic simulation models. Conservation Biology 9, 175±181. Rounsevell, D., 1996. Orange-bellied Parrot Recovery Plan 1997± 2001. Parks and Wildlife Service, Hobart. SouleÂ, M.E., 1989. Risk analysis for the Concho water snake. Endangered Species Update 6, 19±25. Stewart, T.J., 1992. A critical survey on the status of multiple criteria decision making Ð theory and practice. OMEGA International Journal of Managegement Science 20 (5/6), 569±586. Watson, S.R., Buede, D.M., 1987. Decision Synthesis. The Principles and Practice of Decision Analysis. Cambridge University Press, Cambridge.
www.elsevier.com/locate/biocon
A model-based decision aid for species protection under uncertainty Martin Drechsler* Department of Ecological Modelling, Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany Received 18 June 1999; received in revised form 2 September 1999; accepted 1 October 1999
Abstract A method of decision making is presented that can be used to compare alternative management actions under ecological uncertainty and to identify which one is likely to have the strongest eect on population viability. The method combines decision analysis with population modelling and uses both information about population patterns (e.g. spatio-temporal) and processes (e.g. reproduction). The process knowledge is used to construct the population model and determine plausible ranges for its parameters. The values of these parameters are likely to have an impact on the rank order of the most eective management actions, and unless their ranges are small, there is uncertainty in the management rank order. This uncertainty is encompassed by considering a number of dierent population parameter combinations, called scenarios. For each of them a sensitivity analysis is performed and a management rank order determined. In the following decision analysis each key scenario contributes with a certain weight that re¯ects its biological plausibility. To determine the weight of a particular scenario, the population dynamics are simulated and the generated patterns are compared with those observed in the real population. The higher the similarity between the two patterns the higher the weight assigned to the scenario. The decision analysis ®nally synthesises the results of sensitivity and pattern analyses and generates a single rank order of the most promising management actions. The method is demonstrated on a case study of the endangered Orange-bellied Parrot Neophema chrysogaster (Australia). # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Decision analysis; Uncertainty; Population model; Conservation management; Extinction
1. Introduction A central issue in conservation biology and species protection is the ranking of alternative management actions (Possingham et al., 1993; Lindenmayer and Possingham, 1996). Such a decision is dicult where the eects of the actions are subject to uncertainty. The eect of creating a habitat network on the viability of a species, for instance, naturally depends on the species' mobility. If this is not precisely known the eect of a habitat network will be uncertain. In such a situation it is desirable to have a tool that helps to synthesise the uncertain information and provide a rational decision. Decision theory which is especially used in areas such as economics and medicine (for an overview, see, e.g. Watson and Buede, 1987; Corner and Kirkwood, 1991) can be such a tool. It encourages the decision maker to be explicit about the relevant fac-
* Tel.: +49-341-235-2039; fax: +49-341-235-3500. E-mail address: [email protected] (M. Drechsler).
tors and use the information in a rational manner. This cannot exclude the possibility of errors (as these are a natural phenomenon in the presence of uncertainty) but can minimise their likelihood. Since the mid eighties decision analysis has also been used in conservation biology (Maguire, 1986, 1991; SouleÂ, 1989; Ralls and Star®eld, 1995; Possingham, 1996, 1997; Burgman et al., 1999; and others). Apart from providing a structure for the decision process, decision analysis can actively help in the decision process only if it is combined with an instrument that can predict the outcomes of the management actions available. Such an instrument can be a simulation model which synthesises all the accessible information. This study demonstrates how a simulation model can be combined with decision analysis to make conservation decisions in the presence of uncertainty. The example is based on the case of the Orange-bellied Parrot Neophema chrysogaster in south-eastern Australia. It breeds in Tasmania and overwinters in the coastal regions of the Australian mainland. Industrial development in these regions has contributed to a population
0006-3207/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0006-3207(99)00168-8
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M. Drechsler / Biological Conservation 94 (2000) 23±30
decline to about 200 birds (Menkhorst et al., 1990). Drechsler et al. (1998) used a population model to simulate various management actions and determined their eects on the viability of the population. The actions were ranked according to these eects. The rank order of the management actions was shown to depend on the values of the model parameters which are subject to uncertainty. Therefore several rank orders had to be considered to encompass this uncertainty. Although Drechsler et al. (1998) were able to deduce some practical rules from this set of rank orders, this process will in general be tedious and may be suboptimal. The decision analysis presented in this paper can greatly simplify the evaluation of rank orders under uncertainty. The result of the analysis is a single rank order of the management actions that can be easily interpreted. 2. Decision analysis under uncertainty For the problem described above the appropriate type of decision analysis is the so-called decision analysis under uncertainty (e.g. Bayes, 1958; Eckhoudt and Gollier, 1995) or quantitative ecological risk assessment (Possingham, 1995). It consists of the action space, the state space and the results function. The action space contains the available actions (e.g. `go swimming' or `go to the cinema'). The state space contains the states that the described system can assume (e.g. `®ne weather' or `bad weather'). The result (e.g. happiness) of a given action depends on the state of the system. This relationship is expressed in the results function. It tells which result Rij is achieved by action Ai if the system is in state Sj. This can be summarised in the so-called results matrix (Table 1). The rows of the results matrix are given by the states Sj (j=1. . .n) and the columns by the actions Ai (i=1. . .m). If the state is known the best action can immediately be obtained from the results matrix. In general this is not the case, but even without the exact knowledge of the state it is possible to derive a decision from the results matrix, namely when the probabilities of the states are known. Let the probability of observing state Sj be denoted by Pj. Then for instance, the expectedvalue criterion (e.g. Eckhoudt and Gollier, 1995) can be applied to identify the action that is most likely to lead Table 1 Example of a results matrix. The matrix elements Rij give the result as a function of the selected management measure Ai and the present state Sj. The quantity Pj is the likelihood of state Sj State
Pj
A1
A2
S1 S2 S3
0.2 0.3 0.5
10 25 30
15 15 20
to the best result. Given the Rij and Pj, the expected result, E(Ai) for action Ai is obtained by taking the average over the results for the various states. The probabilities Pj serve as weight factors, i.e. the result Rix obtained in a likely state Sx will contribute more than the result Riy of an unlikely state Sy: E Ai P1 Ri1 P2 Ri2 . . . Pn Rin
1
In the example of Table 1 we ®nd E(A1)=24.5 and E(A2)=17.5. The expected result achieved by action A1 exceeds that of action A2. Table 1 further shows that the result for action A1 varies and considerably depends on the state. A measure for the variability of the results for a particular action Ai is their standard deviation S(Ai): 1=2 S Ai P1 R2i1 P2 R2i2 . . . Pn R2in ÿ E Ai 2
2
In the example of Table 1 we ®nd S(A1)=7.6 and S(A2)=2.5. The smaller S the less the likelihood of observing a result that strongly deviates from the expected value E. In the example of Table 1 the standard deviation in the results for action A1 is higher than the dierence between the two expected values E(A1) and E(A2), which suggests that the results of A1 and A2 do not dier much with respect to their uncertainty. This is similar to the logic used in a t-test where the dierence between the expected values of two normally distributed random numbers is compared to their pooled standard deviations. In this case we may adopt the dierence E(A1)ÿE(A2) relative to S as a qualitative guide to the importance of dierence. Furthermore, as the variation in the results for action A1 is much higher than that for action A2, action A1 is more likely to lead to a below-average result than A2. This means that there is a trade-o between maximising the expected value and minimising the risk of a poor result. In this case the decision will depend on the risk tolerance of the decision maker, i.e. on whether he or she will tolerate a higher risk in order to achieve a higher expected result or be satis®ed with a smaller expected result in order to reduce the risk of a very poor result. As the expected results in Table 1 are not very dierent, a risk averse decision maker may prefer action A2 whose result is subject to less variation. The aim of the two following sections is to use a population model in order to generate the results matrix and the probabilities Pi, and to derive a rank order of management actions for the population of the Orangebellied Parrot. In contrast to the example above, the `states' of the results matrix will be made of alternative combinations of population model parameters. Uncertainty in the analysis then will be caused by our incomplete knowledge of the correct parameter values rather than by the unknown state of the population. It is
M. Drechsler / Biological Conservation 94 (2000) 23±30
therefore appropriate not to use the term `state' (which is usually something that can change in time) but the term `scenario'. A scenario then represents a particular assumption about the values of the population model parameters. 3. The results function 3.1. The population model and simulation of the management actions To apply the above decision analysis to the ranking of conservation measures a results matrix is needed that relates the results to the actions and the scenarios. The aim of the conservation actions is to increase the viability of the Orange-bellied Parrot population. In the analysis, population viability is measured by the logit, ln[Q/(1ÿQ)], of the quasiextinction risk Q. The quasiextinction risk is the probability that within a certain time frame (e.g. 20 years) the population will fall under a certain threshold (Ginzburg et al., 1982). Four alternative management actions are considered to achieve the aim: to increase the breeding capacity (see below) by the provision of nest boxes, to increase fecundity by arti®cial feeding, to increase the mean annual survival rates of the parrots by winter habitat management, and to increase the winter carrying capacity by the expansion of wintering habitat. The rank order of these management actions is likely to depend on the scenarios. The relationship between the results of the management actions and the scenarios is provided by the population model by Drechsler et al. (1998) which will be outlined in the following. In that model, the annual dynamics of the population were divided in a summer and a winter season. In summer the population breeds in Tasmania south of the Australian mainland. The number of juveniles produced by the population is determined by the number of nest sites (`breeding capacity') and the number of juveniles produced per breeding pair (`fecundity'). After the breeding season, adult and juvenile birds migrate to the Australian mainland and overwinter in coastal regions. In the following spring the surviving birds return to their breeding habitats. By that time the juveniles have grown to adults. Annual mortality is concentrated in the winter months (Menkhorst et al., 1990). A capture±mark± recapture program provides estimates for the annual survival rates of juveniles and adults. From these values long-term means and standard deviations of adult and juvenile survival rates can be calculated, as well as a correlation coecient between adult and juvenile mortality. The model also contains a `winter capacity' which gives the number of birds that can survive a winter under optimal conditions. This winter capacity results from intraspeci®c competition for food and is related to
25
the size of the wintering habitat. Winter mortality is also aected by the type of intraspeci®c competition (scramble or contest competition; e.g. Begon et al., 1990). If scramble competition is present, survival is reduced by a certain amount which is determined by the `scramble factor' (Drechsler et al., 1998) and re¯ects the strength of competitive interactions. All these population parameters are subject to considerable uncertainty (Table 2). The population model was implemented as a computer simulation program and the dynamics were simulated for 20 years. The eect of the four management actions was determined by Drechsler et al. (1998) in a sensitivity analysis (e.g. Caswell, 1978). For this, all population parameters were set to their median values (the `standard scenario; Table 2). Starting from this standard scenario, the four `management parameters', fecundity, mean survival and breeding and winter capacities were varied in turn by a certain proportion and the corresponding changes in the viability [i.e. changes in the logit of the quasiextinction risk, logit(Q)] of the population were determined. For each of the four management parameters a sensitivity coecient, C=logit(Q)/, was calculated that measures its in¯uence on population viability. The rank order of these sensitivity coecients tells which management action is best, second best and so on. 3.2. Formulation of the scenarios and construction of the results matrix Table 2 shows that in some population parameters there is considerable uncertainty. Therefore the rank order of the four management parameters which depends on the population parameters, can be expected to be uncertain, as well. Therefore it is not sucient to consider only the standard scenario. To obtain a clear picture of the in¯uence of the population parameters on
Table 2 Plausible ranges of the population parametersa Parameter Fecundity Breeding capacity Mean annual survival rate of adults Mean annual survival rate of juveniles Standard deviation of adult survival Standard deviation of juvenile survival Correlation between adult and juvenile survival Winter capacity Scramble factor a
Maximum 1.95 95 0.74
Median 1.85 70 0.64
Minimum 1.75 45 0.54
0.52
0.475
0.43
0.13
0.11
0.09
0.09
0.07
0.05
+1 540 0.1
0
ÿ1
400 0.05
270 0.0
Note that the winter capacity is not the size of the real population which is lower. A scramble factor of 0.0 represents contest competition.
26
M. Drechsler / Biological Conservation 94 (2000) 23±30
the management rank order, a large number of alternative scenarios has to be considered. Each scenario represents a plausible assumption about the values of the population parameters and for each a management rank order will be determined. For convenience only a relatively small subset of all possible scenarios is considered. For this it is assumed that each population parameter may either take its maximum or its minimum value. If the model has n population parameters, a number of 2n scenarios is considered. In the case of the Orange-bellied Parrot this number could be reduced to eight (Drechsler et al., 1998). These eight scenarios are de®ned by whether mean population growth (fecundity and the mean survival rates) takes its maximum or minimum value, whether the ratio of breeding and winter capacities and the scramble factor is maximal or minimal (i.e. whether the population is regulated mainly in winter or not), and whether the variation of population growth (standard deviation of the survival rates and the correlation between adult and juvenile survival) is maximal or minimal. Altogether, there are only three independent quantities that determine the rank order of management actions. For convenience they will be called population growth, winter regulation and population variation. The population parameter space is only threedimensional now, spanned by the three parameter sets termed `super parameters', population growth, winter regulation and population variation. While the standard scenario (all parameters at median values) occupies the centre of this three-dimensional parameter space, the eight additional scenarios are located at its corners (Fig. 1) For each of the nine scenarios the sensitivity of population viability to the management parameters is calculated, allowing the rank order of the four management actions to be determined (Table 3; cf. Drechsler et al., 1998). Table 3 has the desired structure of a results matrix (cf. Table 1). The sensitivity coecients can be regarded as the results Rij. A large sensitivity coef®cient represents a strong eect of the corresponding measure on population survival, i.e. a good result. 4. Decision making 4.1. Determination of the probabilities of the scenarios All of the nine scenarios introduced above lie within the plausible parameter space spanned by the values in Table 2. However, here the term `plausible' only means that each of the nine scenarios could be the one that best describes the real population, but it does not tell whether a particular scenario is more likely to be valid in the real population than another. What is missing is a criterion that decides how well a particular scenario characterises the real population. For such an evaluation additional information is required.
Fig. 1. Graphical representation of the three-dimensional super parameter space.
Even without precise knowledge of the size and distribution of the Orange-bellied Parrot population, there are rough estimates on population trend and variation. Regression analysis of the estimated maximum winter population sizes from 1978 to 1989 (Menkhorst et al., 1990; Fig. 3) leads to a long term population change of +0.6 birds per year (95%-con®dence interval: ÿ2.6. . .+3.8 birds per year) which is very close to zero. However, there has been a sharp decline in the number of parrots observed in the annual winter counts from 108 in 1989 to about 40 birds in 1995 and 1996 (Rounsevell, 1996). This would correspond to a population trend (average change in population size) of about ÿ10 birds per year. This decline is likely to be caused by a shift in habitat use by the parrots reducing their sighting probability in recent years. Hence the observed trend of ÿ10 birds per year should not re¯ect the actual trend in the population size. Nevertheless, it may serve as a lower bound on the population trend. Altogether, the population is believed to be stable (Menkhorst et al., 1990; Rounsevell, 1996), i.e. a zero population trend is most likely. Larger population trends are less likely and values below ÿ10 birds per year should have a likelihood of zero. For simplicity, within the range of plausible population trends between 0 and ÿ10 birds per year, a linear relationship between a population trend and its likelihood is assumed (Fig. 2). The coecient of variation of the estimated maximum winter population sizes from 1978 to 1989 (Menkhorst et al., 1990) is around 0.17. The jackknife estimate of the upper bound of the 95%-con®dence interval is close to 0.2. This value may be used as an upper plausible bound on the coecient of variation in the population size. However, the estimates in years 1978, 1979 and 1984±1986 which all dier considerably from the estimates for the other years, are likely to be wrong (Menkhorst et al., 1990). If these years are excluded from the analysis the coecient of variation in the estimated maximum winter population size is about 0.05. Its lower 95%-con®dence limit is close to zero. A value of zero then may be used as a lower plausible bound on the coecient of variation of the population size. In a
M. Drechsler / Biological Conservation 94 (2000) 23±30
27
Table 3 Sensitivity coecients (eect on population viability) of the four management measuresa Scenarios
1 2 3 4 5 6 7 8 9
Actions
Population growth
Winter regulation
Population variation
A1
A2
A3
A4
Median Maximum Maximum Maximum Maximum Minimum Minimum Minimum Minimum
Median Maximum Maximum Minimum Minimum Maximum Maximum Minimum Minimum
Median Maximum Minimum Maximum Minimum Maximum Minimum Maximum Minimum
23 0 3 16 19 20 22 24 27
73 73 71 71 64 74 73 74 71
1 0 0 13 17 0 0 2 2
3 27 26 0 0 6 5 0 0
a
A1={increase fecundity}, A2={increase mean survival rates}, A3={increase breeding capacity} and A4={increase winter capacity} for various scenarios. In each scenario the four sensitivity coecients are normalised to a sum of 100.
Fig. 2. Likelihood of the mean population trend. A value of 1 means that this trend is very likely to be present in the real population. A value of zero means that this trend is most unlikely in the real population.
determined. Figs. 2 and 3 may be used to determine how well the simulated and real dynamics agree. For example, for scenario 1, the simulated population dynamics have a trend of ÿ1 bird per year (Table 4) which is quite likely to be valid in the real population (likelihood of 0.9 in Fig. 2). In this respect, the simulated and real populations agree well. In contrast, the simulated coecient of variation is 0.2 which is much higher than is believed to be valid in the real population (likelihood of 0.0 in Fig. 3). A scenario is accepted as realistic if the simulated and real populations agree both in their trend and coecient of variation. For this the likelihood of a scenario is given by the product of the two weight factors in Table 4. The resulting likelihoods then are normalised (all divided by a common number), so their sum equals one and they can be interpreted as probabilities (Table 4). 4.2. The rank order of management actions The results matrix (Table 3) can be evaluated with the probabilities from Table 4. For the four management actions, the expected-value criterion leads to the following results: E A1 13; E A2 68; E A3 7; E A4 12:
Fig. 3. Likelihood of the coecient of variation of the population size (cf. Fig. 2).
workshop with the Orange-bellied Parrot Recovery Team, a coecient of variation around 5% was regarded as most likely. Larger or smaller coecients of variation are less likely. Assuming a linear relationship between a coecient of variation and its likelihood leads to Fig. 3. With the help of the model, population dynamics may be simulated and population trend and levels of variation
The expected result, i.e. increase in population viability, is maximal when the mean survival rates are increased (action A2). The eects of fecundity and winter capacity (A1 and A4) are medium, the breeding capacity (A3) has a weak in¯uence. The standard deviations of the results are S A1 9; S A2 4; S A3 8; S A4 13: The results are also shown graphically in Fig. 4. The expected result of action A2 is much higher than those of the other three actions. The dierences between E(A2) and the expected results of the other actions are large compared to the standard deviations of the results.
28
M. Drechsler / Biological Conservation 94 (2000) 23±30
Table 4 Mean trend (change of mean size per year) and coecient of variation of the modelled population size for the nine scenarios after Table 3a Scenario
1 2 3 4 5 6 7 8 9
Simulated population dynamics Trend (individuals per year)
Weight 1
Population variation
Weight 2
Likelihood (=Weight 1 Weight 2)
Probability (normalised likelihood)
ÿ1 ÿ1 ÿ1 0 0 ÿ15 ÿ8 ÿ15 ÿ8
0.9 0.9 0.9 1 1 0 0.2 0 0.2
0.2 0.24 0.06 0.25 0.09 0.2 0.08 0.25 0.08
0 0 0.93 0 0.73 0 0.8 0 0.8
0 0 0.873 0 0.73 0 0.16 0 0.16
0 0 0.44 0 0.39 0 0.08 0 0.08
a The weight factors were obtained from Figs. 2 and 3. The likelihood of a scenario is the product of the two weights and the probabilities were obtained by normalising the likelihoods so the nine probabilities have a sum of one. They measure how likely each scenario is present in the real popu-
lower than that in the result of action A4. Therefore in action A1 the risk of a poor result is lower than in action A4. Because of its slight superiority in the expected value and its lower risk, A1 outperforms A4. Altogether the rank order of the four actions is A2 > A1 > A4 > A3 where x>y means `x is preferred to y'.
Fig. 4. Expected results (bars) and error (error bars, showing one standard deviation) for the four management actions A1±A4 (from top to bottom: increase fecundity; increase mean survival rates; increase breeding capacity; increase winter capacity). For each action the numbers give the expected value and in parentheses the standard deviation of the result. Nine scenarios (cf. Tables 3 and 4) are considered.
Therefore A2 (increase of annual survival rates) is clearly the best action. However, action A2 may be dif®cult to achieve in practice. First, annual mortality includes mortality during migration between the breeding and winter habitats, which is much related to weather conditions and dicult to reduce by management. Second, in contrast to the other management parameters, such as carrying capacities, the annual survival rates (cf. Table 2) can in principle not be increased by more than a factor of 2, as they are bound by 1. It is therefore sensible to concentrate on the other three management actions. The dierence between the expected results of actions A1 and A4 with that of action A3 is of the same magnitude as the standard deviations of the results. Therefore actions A1 and A4 should be assigned a higher rank than action A3. The standard deviations of the results for actions A1 and A4 are higher than the dierence between their expected results, E(A1)ÿE(A4) which means that A1 and A4 are more or less equivalent with a slight advantage of A1. The uncertainty (S) in the result of action A1 is
5. Discussion The decision analysis can be used to rank management actions according to their eects on the viability of a population. Uncertainty in population parameters leads to uncertainty in the eects of the management actions. As a consequence, in some instances, particularly where the uncertainty is large, some of the management alternatives cannot be distinguished, i.e. their eects on population viability do not dier signi®cantly with respect to the uncertainty. The decision analysis takes these uncertainties into account and helps to identify which management alternatives can be clearly ranked and which alternatives cannot be distinguished from each other. This helps to . discard inecient management actions . assess the eect of uncertainty on the decision process . decide whether uncertainty has to be reduced before a reasonable decision can be made. Some of the numerical results of the present study are similar to those given in Drechsler et al. (1998) who found that mean survival is the most important population parameter, followed by fecundity, winter capacity and breeding capacity. Particularly, the importance of survival is well known from many other studies. The
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new aspect in this approach is that it provides a transparent protocol for assessing and comparing management options under uncertainty. Drechsler et al. (1998) did not show that mean survival has a much stronger eect (compared to the uncertainty in the population) on population viability than the other parameters and that the dierence between the two carrying capacities is less important. They found their results by ad hoc comparison of various rank orders without using an objective method of combining them. Furthermore, they did not consider that some rank orders should be given more weight than others, because they belong to scenarios (i.e. population parameter combinations) that are more likely to be valid in the real population than others. The method presented in this paper is able to eliminate these de®ciencies and provide a systematic and quantitative procedure of evaluating rank orders of management actions under uncertainty. An open question is the proper selection of the scenarios. This may be done randomly or Ð like in this study Ð systematically to evenly cover the parameter space. One may wonder whether nine parameter combinations is sucient to determine the correct rank order. This may be checked by gradually increasing the number of scenarios and looking at possible changes in the rank order. If these are negligible, the resulting rank order is likely to be correct, although a strict mathematical proof cannot be given at this stage. The outlined procedure was carried out for the Orange-bellied Parrot and after increasing the number of scenarios to 27 [where one scenario was located in the centre of the three-dimensional parameter cuboid (Fig. 2) and the others at its corners, the middles of its 12 edges and the centres of its six faces] the expected values and the standard deviations of the results for actions A1 to A4 were fairly constant and not much dierent to those determined above. The issue of ranking action A2 (increase annual survival rates) pointed to the importance of considering management costs in the analysis. Above it was argued that A2 may be dicult to achieve in practice, which in economical terms means that it will be expensive if not priceless. In the following analysis of the other three management actions it was implicitly assumed that they all have the same costs. However, the provision of one additional nestbox to increase the breeding capacity by about 1±2% may be less costly than an increase of the winter capacity by the same percentage. To come to a fair rank order of all management actions, their costs would have to be considered. This could be done in two ways. First, one could choose the management actions such that all have the same costs and then performing the decision analysis as above but based on the changes in population viability rather than sensitivity coecients. Second, one could consider bene®t-cost ratios instead of the sensitivity coecients. As described above, those sensitivity coecients were calculated by
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dividing the change in population viability (logit of the quasiextinction risk) by the relative change in the management parameter (). To obtain a bene®t-cost ratio one would replace by the corresponding costs. The present decision analysis was greatly simpli®ed by reducing the number of population parameters from nine to three The remaining three were called `super parameters'. The identi®cation of such super parameters has been described by Drechsler (1998). Certainly, in many population models the number of parameters could not be reduced so readily which means that more than nine scenarios (as in the present case) would have to be considered. This would increase computing time but would not restrict the applicability of the decision method. Particularly, it is those complex cases with many scenarios where a formal decision protocol such as the one presented is indispensable. In the determination of the probabilities of the scenarios many quantities may be used. In the Orange-bellied Parrot, there was only knowledge on trend and variation of population size. If available, spatial parameters, such as spatial distribution, density or aggregation, or demographic parameters, such as the sex ratio or the age distribution should also be used. All these parameters describe the (temporal, spatial and/or demographic) pattern of the population. Such patterns are valuable in the validation and tuning of models and model parameter combinations (Grimm et al., 1996). Which patterns are used depends on their availability and their relevance. A more formal way of determining the probabilities of the scenarios is Bayesian analysis (e.g. Gelman et al., 1995). To establish the probability distribution of the model parameters, one would start with simple ®rst guesses on the distribution (`prior distribution'). Using Bayes' Theorem and additional information (from the population patterns) this prior distribution would be updated and improved (`posterior distribution'). This posterior distribution than would contain the probabilities of the scenarios (cf. Table 4). An advantage of Bayesian analysis is that the posterior distribution can gradually be improved as more population data become available. There are various options to extend the decision analysis. Within this work, the aim was to identify a single rank order of management actions. However, if the necessary information is available, conservation managers may depend their actions on the present state of the population. For instance, if the population is small Ð and if this is known to the decision maker Ð he or she may prefer measures that maximise population recovery. In contrast, at times where the population size is large, reducing intraspeci®c competition by providing resources may be more ecient. For further details about statedependent decision analysis, see Possingham (1997). A possible extension is the consideration of multiple goals (e.g. Stewart, 1992) where there may be trade-os
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between dierent goals. The task is to ®nd a solution that provides a best compromise. Especially in conservation biology and species protection, the systematic consideration of multiple goals under uncertainty is a dicult problem, but it can be solved (e.g. Munda, 1995). The extension of the present single goal analysis to a multiple goal analysis may be valuable. A critical point is that the results rely on the population model used. Naturally, such a model cannot consider all factors and processes that may in¯uence the population. Instead, the modeller concentrates on those factors that are expected to have some in¯uence on the outcome of the decision analysis. The decision about which factors to consider in the model introduces another type of uncertainty, called structural uncertainty which usually cannot be eliminated. Interpretation of results should be conditioned by model structure. The decision method is not a strictly determined protocol. In the course of an analysis, decisions have to be made about the relevant factors and how these are to be considered. Such variability is typical in decision analyses, particularly where uncertainty plays a role. Decision analysis therefore should not be ®nal authority. Instead it is a tool used to synthesise expert knowledge and assist in the decision process. The risk of making a wrong decision can never be excluded, but it can be reduced. Acknowledgements I would like to thank Volker Grimm for his helpful comments on the manuscript. The comments of Mark Burgman and Hugh Possingham were very encouraging and useful in the revision of the paper. References Bayes, T., 1958. An essay towards solving a problem in the doctrine of chances. Biometrika 46, 293±298 (reprint of Bayes' 1763 manuscript). Begon, M., Harper, J.L., Townsend, C.R., 1990. Ecology: Individuals, Populations and Communities. Blackwell, Cambridge, MA. Burgman, M.A., Keith, F.J., Rohlf, C.R., Todd, C.R., 1999. Probabilistic classi®cation rules for setting conservation priorities. Biological Conservation 89 (2), 227±231. 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. Corner, J.L., Kirkwood, C.W., 1991. Decision analysis applications in the operations research literature, 1970±1989. Operations Research 39, 206±219. Drechsler, M., 1998. Sensitivity analysis of complex models. Biological Conservation 86, 401±412.
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