Conclusions

This article has introduced a number of new ideas and methods. These are:

1.

The synthesis of ecosystem management modeling, model estimation, and management option assessment into a publicly-accessible web software system to encourage a data and model-driven dialogue between the public and private sector on the management of ecosystems that are being affected by multiple countries.

2.

Use of an influence diagram to model ecosystem functioning that consists of an interacting combination of qualitative and quantitative variables.

3.

Use of consistency analysis to fit such a model that incorporates both a sample and prior knowledge wherein the sample may be small, incomplete, and/or noisy. As with the cheetah example, such samples are often all that is available for studying wildlife populations and other ecosystem status variables.

In general, an influence diagram is well-suited to environmental management decision making due to its ability to represent both management options (through decision nodes), and uncertain effects of environmental factors (by placing appropriate distributions on the associated chance nodes). If reliable mechanistic or statistical models are available for subunits of the environmental process being managed, these can be embedded in the influence diagram to create a single model that is informed by both quantitative models and the qualitative knowledge of experts.

As the consistency analysis example showed however, a drawback to estimating a mixed influence diagram that does not have an analytical likelihood function is high computation expense. Improvements in algorithms and/or hardware will be needed before all parameters in such an influence diagram can be jointly estimated.

id, the analysis software component of the author's EMS for cheetah management (see Sections 2.2.3 and 1, respectively) was used to perform all of the above computations and produce all associated maps and map overlays. id can also compute descriptive statistics, represent spatio-temporal chance nodes in an influence diagram (see Haas (1995)), and optimally redesign a monitoring network (see Haas (1992)).

Note that much of the cheetah viability influence diagram consists of geographic nodes and geographic information. This inclusion of geography may seem intuitively obvious but the integration of GIS and stochastic viability models, as can be seen above, is not trivial. The need for such integration has been known for some time, see Norton and Possingham (1993). Here, examples of such integration are the percent coverage computations in Tables 2 and 3, and the maps of Figures 3 and 5.

Appendix: Influence Diagram Parameter
Estimation with Consistency Analysis

Consistency Analysis Algorithm

Let $g_S(\mbox{\boldmath$\beta$})$ be a goodness-of-fit statistic that measures the agreement of an influence diagram's joint distribution identified by the values of $\mbox{\boldmath$\beta$}$ and the (possibly) incomplete sample, S. Let $g_H(\mbox{\boldmath$\beta$})$ be the agreement between the influence diagram's hypothesis distribution (identified by the values of $\mbox{\boldmath$\beta$}_H$) and the influence diagram's distribution identified by the values of $\mbox{\boldmath$\beta$}$. Let gsmax be the unconstrained maximum value of $g_S(\mbox{\boldmath$\beta$})$ over all $\mbox{\boldmath$\beta$}$ and let $\mbox{\boldmath$\beta$}_S$ be the parameter values that correspond to this maximum. Let ghmax be the unconstrained maximum value of $g_H(\mbox{\boldmath$\beta$})$ over all $\mbox{\boldmath$\beta$}$. Up to sampling variability in the estimation of $g_H(\mbox{\boldmath$\beta$})$,this value is $g_H(\mbox{\boldmath$\beta$}_H)$. The consistency analysis estimator maximizes $g_{CA}(\mbox{\boldmath$\beta$}) \equiv (1-c_H)g_S(\mbox{\boldmath$\beta$}) / \vert gsmax\vert + c_H g_H(\mbox{\boldmath$\beta$}) / \vert ghmax\vert$where, as discussed above, $c_H\in(0,\:1)$ is the analyst's priority of having the consistent distribution agree with the hypothesis distribution as opposed to agreeing with the empirical distribution. Let $\tilde{\mbox{\boldmath$\beta$}} \equiv {\rm argmax}_{\mbox{\boldmath$\beta$}}(g_{CA}(\mbox{\boldmath$\beta$}))$be the consistency analysis estimate of $\mbox{\boldmath$\beta$}$.If the influence diagram consists of only qualitative chance nodes and the sample is incomplete, a value for $\mbox{\boldmath$\beta$}$ can be found such that the consistent model exactly reproduces the empirical distribution of the observed nodes. This is the case studied in Haas (1991b).

The idea of consistency analysis is to find parameter estimates such that, under the priorities dictated by c_H, the fitted model is as close as possible to both the model defined by the substantive theory and to the model suggested by the sample. Consistency analysis is best used as a sequential learning procedure: $\tilde{\mbox{\boldmath$\beta$}}$from one sample is used as $\mbox{\boldmath$\beta$}_H$ in the consistency analysis conducted with the next sample.

The function g_S(.) measures how close a model with parameter values $\mbox{\boldmath$\beta$}$ agrees with the sample. When some of the model's variables have not been observed, the incomplete sample can only be compared to the PDPF of these remaining or observed chance nodes, called here the marginal PDPF. Finding parameter estimates that result in the unconstrained maximum of $g_S(\mbox{\boldmath$\beta$})$ is called Minimum Distance (MD) parameter estimation and was first studied analytically by Wolfowitz (1957). MD estimation is now known to be robust to model misspecification (see Bickel et al. (1993) and Lindsay (1994)).

Often, the influence diagram's decision nodes effectively define independent influence diagrams given each combination of values on these nodes. Say that there are d unique combinations of these nodes. If the sample is a time series, the model's PDPF should capture the temporal dependence structure. Let ${\bf U}_t^{(i)}$ be the vector of observed chance nodes at time t under decision node combination i. Say that these nodes are observed at m times, $t_1,\ldots,t_m$. Then, a PDPF is needed for the random vector ${\bf U}^{(i)} = ({\bf U}_{t_1}^{(i)'},\ldots,{\bf U}_{t_m}^{(i)'})',$ $i=1,\ldots,d$. Because ${\bf U}^{(i)}$ and ${\bf U}^{(j)}$ are independent for $i\neq j$, the PDPF of ${\bf U}= ({\bf U}^{(1)'},\ldots,{\bf U}^{(d)'})'$is the product of the constituent joint PDPF's: $pf_{{\bf U}\vert\mbox{\boldmath$\beta$}}({\bf u}) = \prod_{i=1}^{d} pf_{{\bf U}^{(i)}\vert\mbox{\boldmath$\beta$}}({\bf u}^{(i)})$.

Using n_MC simulated realizations from the influence diagram model of ${\bf U}^{(i)}$,the PDPF of ${\bf U}$ can be estimated with the local gaussian kernel estimator (Thompson and Tapia 1990, p. 180). Say that S consists of observations ${\bf u}_{obs}$ on the variables ${\bf U}$.For this particular sample, the value of the marginal likelihood function can be estimated with this simulation-based density estimator, i.e., $\hat{L}(\mbox{\boldmath$\beta$}\vert{\bf u}_{obs}) = \hat{pf}_{{\bf U}\vert\mbox{\boldmath$\beta$}}({\bf u}_{obs})$.Further, if c_H = 0 and $g_S(\mbox{\boldmath$\beta$}) = \hat{L}(\mbox{\boldmath$\beta$}\vert{\bf u}_{obs})$,then $\tilde{\mbox{\boldmath$\beta$}}$ is exactly the MML estimate of $\mbox{\boldmath$\beta$}$.

Let pf_e() be the the empirical PDPF of the observed variables. Another option is to define both g_H(.) and g_S(.) as the negative of a Kullback-Leibler Divergence (KLD) measure, i.e.,

$$g_H(\mbox{\boldmath$\beta$}) \equiv -\left\{E_{\mbox{\boldma... ...rt\mbox{\boldmath$\beta$}_H}({\bf u})} \right) \right] \right\}$$ (5)

and

$$g_S(\mbox{\boldmath$\beta$}) \equiv -\left\{E_e\left[ \ln \l... ...bf u})}{ pf_{{\bf U}}^{(e)}({\bf u})} \right) \right] \right\}.$$ (6)

Cressie and Read (1984) give a class of objective functions for MD parameter estimation: for $\lambda \in (-2, \: 1)$ and $\delta ({\bf u}_i) = (d({\bf u}_i) - pf_{\mbox{\boldmath$\beta$}}({\bf u}_i)) / pf_{\mbox{\boldmath$\beta$}}({\bf u}_i)$,find $\mbox{\boldmath$\beta$}$ such that $\sum_{i=1}^{n}pf_{\mbox{\boldmath$\beta$}}({\bf u}_i)((1+\delta({\bf u}_i))^{\lambda + 1} - 1) / (\lambda((\lambda + 1))$is minimized. Maximum likelihood, minimum Hellinger distance, and minimum Kullback-Leibler Divergence correspond to $\lambda = 0, \: -1/2$, and 1, respectively (see Lindsay (1994)). Note that ML is an MD estimator. If n=1, $d({\bf u})=1$when ${\bf u}= {\bf u}_{obs}$ and 0 otherwise.

In the cheetah example above, negative Hellinger distance is used to measure agreement with the sample of size one, i.e., $g_S(\mbox{\boldmath$\beta$}) = -2(1 - \sqrt{pf({\bf u}_{obs})})^2$.Also for this example, due to limited computational resources, $g_H(\mbox{\boldmath$\beta$})$measures only the agreement in the first moment between the consistent and hypothesis distributions. This is accomplished by defining $g_H(\mbox{\boldmath$\beta$})$ to be the negative Euclidean length of the difference between two sample mean vectors. The first is computed over a Monte Carlo sample of 20 realizations from ${\bf U}_{\mbox{\boldmath$\beta$}}$and the second from a size-20 sample drawn from ${\bf U}_{\mbox{\boldmath$\beta$}_H}$.These definitions of $g_S(\mbox{\boldmath$\beta$})$ and $g_H(\mbox{\boldmath$\beta$})$makes consistency analysis a penalized MD estimator.

Interpretation and Assignment of c_H

Consistency analysis uses a point assignment of the parameters to represent the substantive science theory, and c_H to represent the analyst's priority weights of having the consistent distribution agree with the hypothesis and empirical distributions. Because these two objectives are being simultaneously optimized, consistency analysis does not force the analyst to ``put his/her eggs all in one basket,'' i.e., $\tilde{\mbox{\boldmath$\beta$}}$ is not solely a function of prior knowledge nor solely a function of the sample.

The following heuristics can be used to guide the selection of a value for c_H:

1.

Setting c_H to 1.0 is equivalent to the opinion that a sample is unnecessary and hence is not gathered. Consequently, no consistency analysis is performed.

2.

If $\mbox{\boldmath$\beta$}_H$ values are taken from peer-reviewed articles reporting on experiments very similar to the one being modeled, and the sample is partly the result of interpretation by the collector, c_H should be high, say between .5 and .9.

3.

If $\mbox{\boldmath$\beta$}_H$ values are based on analyst ``hunches'' and some element of science-based reasoning, and the sample is known to be partly based on interpretation, c_H should be moderate, say between .3 and .7. An example would be when a review of the literature reveals disagreement over the impact of a driver variable on an ecosystem health indicator.

4.

If $\mbox{\boldmath$\beta$}_H$ is specified with little scientific justification, and the sample is known to be highly reliable and solely the result of calibrated instruments, c_H should be small, say between 0.0 and .4.

5.

If $\mbox{\boldmath$\beta$}_H$ is completely unreliable relative to the sample, c_H should be set to zero.

Consistency Analysis Procedures

Before an influence diagram should be trusted to aid ecosystem management decisions, its reliability should be assessed. An influence diagram whose forecasts have excessively high variance (low predictive validity) should not be used to formulate policy. To this end, the RMSPE of policy-relevant output variables is always calculated to provide an assessment of the model's prediction skill.

The consistency component of consistency analysis always entails the computation of (1) parameter estimates that are as consistent as possible with both prior theory and the sample, and (2) the RMSPE. If computational resources permit, the Coefficient of Variation (CV) is also computed for each parameter estimate using the delete-1 jackknife. The $CV(\tilde{\beta})$ is defined to be $S_{\tilde{\beta}} / \tilde{\beta}$where

$$S_{\tilde{\beta}} = \sqrt{\frac{n-1}{n}\sum_{i=1}^{n}\left( ... ...a}_{-i} - \frac{1}{n}\sum_{k=1}^{n}\tilde{\beta}_{-k}\right)^2}$$ (7)

and $\tilde{\beta}_{-i}$ is the consistency analysis estimate of $\beta$when datum i is temporarily removed from the sample (Shao and Tu 1995, p. 333). $CV(\tilde{\beta}$) values larger than (say) 70% indicate so much uncertainty as to render the estimated value of $\beta$ unreliable.

The analysis component of consistency analysis involves the following activities:

1.

Examination of the RMSPE to determine the reliability of predictions computed from the fitted influence diagram.

2.

If computed, examination of parameter estimate CV's to assess parameter estimate reliability.

3.

Assessment of the differences between hypothesis parameter values and the associated consistent values. Large absolute differences indicate areas of prior theory that do not agree with observation. Such theory should be critically reviewed and possibly modified.

4.

Examination of dependency links that appear weak - indicated by consistent parameter values that change little as the parent values of a chance node are varied. Such a situation suggests a chance node is actually independent of its parents.

Computational Implementation of Consistency Analysis

Consistency analysis is implemented by the author as follows. The logic sampling (Henrion 1988) algorithm is used to simulate realizations of the influence diagram. The ``complex'' constrained optimization algorithm of Box (1965) is used to maximize $g_{CA}(\mbox{\boldmath$\beta$})$ subject to the constraints of valid parameter values in $\mbox{\boldmath$\beta$}$ and valid conditional distributions defined by $\mbox{\boldmath$\beta$}$.The same sequence of random numbers is used for each evaluation of $g_{CA}(\mbox{\boldmath$\beta$})$ to ensure smoothness of this objective function.

Comparison of Consistency Analysis With Other Methods

Consistency analysis can be viewed as a penalized goodness-of-fit parameter estimation method. See Easterling (1976) for a discussion of parameter estimation via maximization of a goodness-of-fit statistic. Viewed this way, the penalty function is the deviation from the hypothesis distribution.

Consistency analysis has similarities with Minimum Discrimination Information (MDI) parameter estimation (Gokhale and Kullback 1978). The function relating model parameters to probabilities of the model's joint events (called external constraints by Gokhale and Kullback) is given by the recursive factorization of the influence diagram's joint probability distribution. Consistency analysis differs from MDI in that no functional relationships between model event probabilities and the empirical PDPF (called internal constraints by Gokhale and Kullback) need be satisfied. Also, in MDI the discrimination information statistic measures the distance between the estimated model and the maximum entropy model (constant PDPF over all joint events). The idea of MDI is to minimize this distance subject to satisfying the external and internal equality constraints. In consistency analysis, the maximum entropy distribution is replaced by the hypothesis distribution supplied by the analyst. Further, by manipulating the value of c_H, the analyst can change the relative emphasis consistency analysis gives to finding parameter estimates that agree with this hypothesis distribution relative to agreeing with the empirical distribution. Hence, in consistency analysis, no apriori relationships between the estimated model's distribution and the empirical distribution need be satisfied.

In a multi-parameter estimation problem, classical bayesian methods require the analyst to specify an apriori joint distribution for the parameters (for convenience, independence is often assumed). For example, Dickey et al. (1987) require the analyst to specify a joint Dirichlet distribution for the parameters of a contingency table. Hence, not only is the expected value of each parameter needed, but also all higher moments and cross-moments. If the sample is small and/or incomplete, this very strong distributional assumption can have more influence than the sample on bayesian parameter estimates. In this case, justifying this prior, joint distribution of the parameters with arguments from relevant scientific knowledge is critical to establishing the credibility of the resulting parameter estimates. Several areas of scientific inquiry however, lack sufficient theoretical knowledge to completely specify such a distribution. Instead, available theory can often only support the assignment of point values (``best guesses'') to the parameters with no reliable judgements about the the prior uncertainty of such values. One such area is the use of large, complex environmental process models to support environmental management decision making. In such situations, a bayesian approach requires more from the analyst than the analyst can justify.

Consistency analysis is one way to incorporate prior knowledge into parameter estimation without inheriting some of the criticisms of the application of bayesian methods to the model assessment problem. In addition to the above concern of excessive demands on the analyst's prior knowledge, other criticisms of the bayesian approach to parameter estimation include (a) what constitutes a noninformative prior, (b) interpretability of an improper prior, and (c) reliable elicitation of priors. See Dennis (1996) for further discussion on the use of bayesian inference in ecological modeling.

From a bayesian perspective, MML is based on the belief that, before observing the sample, all parameter value combinations have the same chance of generating the sample (see Berger (1985, pp. 27, 132)). Further, MMLE's of parameters that define unobserved chance nodes are the result of the likelihood of the data on the observed chance nodes being made as large as possible, i.e., since no information on the unobserved chance nodes is available, parameter estimates for the unobserved chance nodes are based only on the model's structure and the observed chance nodes. In the absence of any substantive theory for the process, these characteristics of MMLE's are reasonable. On the other hand, with prior substantive theory (including previous empirical studies), an MMLE completely ignores any prior or qualitative knowledge about the process and allows absurd combinations of parameter values to contribute the same weight to parameter estimation as combinations having substantive science justification. Through the use of $g_H(\mbox{\boldmath$\beta$})$,consistency analysis downweights such absurd combinations of parameter values without any need to place a prior joint distribution on the parameters.

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1cNode Symbol 1cDistribution 1c(Type)

Time t fixed (decision)

Region q fixed (decision)

Manag. Option m fixed (decision)

Reserve Fraction R_t geography-derived (chance)

Unprotected Land Use U geography-derived (chance)

Climate C geography-derived (chance)

Hunting Pressure H_t simple discrete (chance)

Herbivore Count B_t dB(t) = B(t)(k₀ - B(t))dt

$$\mbox{} + B(t)\sigma dW_t^{(B)}$$ (chance)

$$K_t = \mbox{nearest integer} (\beta_{K_t}^{(0)} + \beta_{K_t}^{(1)}B_t)$$ Carrying Capacity K_t (chance)

$$df_t = -.5(\alpha_f + \beta_f^2(2f_t - 1))(1 - (2f_t - 1)^2)dt$$ Birth rate f_t

$$\mbox{} + .5\beta_f(1 - (2f_t - 1)^2)dW_t^{(f)}$$ (chance)

$$dr_t = -.5(\alpha_r + \beta_r^2(2r_t - 1))(1 - (2r_t - 1)^2)dt$$ Death rate r_t

$$\mbox{} + .5\beta_r(1 - (2r_t - 1)^2)dW_t^{(r)}$$ (chance)

$$dN_t = \left[ f_t(1 - P^{cN_t})N_t - r_tN_t - (f_t - r_t)\frac{N_t^2}{K_t} \right]dt$$ Cheetah Count N_t

$$\mbox{} + \beta_N dW_t^{(N)}$$ (chance)

$$D_t = 0I_{\{d < \xi\}}(d) + (d-\xi)/(\rho - \xi)I_{\{d \in (\xi,\: \rho)\}}(d)$$ Fraction Detected D_t

$$\mbox{} + 1I_{d \gt \rho\}}(d)$$ (chance)

$$L = 10^4I_{\{D_t<.15\}}(D_t) + 1I_{\{.15 < D_t < .5\}}(D_t)$$ Loss L

$$\mbox{} + 5I_{\{.5 < D_t < .8\}}(D_t) + 10I_{\{.8 < D_t\}}(D_t)$$ (value)

 

q Hypothesis Values of Hypothesis Values of

$$\beta^{(e)} \beta^{(l)} \beta_{varid} \beta_{arid} \beta_{sarid}$$

Marsabit .99, .93 .70, .28, .01

Eastern .99, .96 .13, .82, .04

Samburu .99, .97 .23, .53, .12

Tsavo .99, .48 .01, .94, .04

Masailand .99, .94 .01, .48, .26

Laikipia .99, .99 .01, .08, .47

Nakuru .99, .95 .01, .05, .40

Western .99, .99 .01, .02, .01

Central .99, .94 .01, .55, .14

Turkana .99, .99 .32, .50, .08

Coastal .99, .99 .01, .18, .44

 

Chance Node and m q Hypothesis Values Parameter Names

U do nothing Marsabit .75, .23, .01

$$\beta_{nomcam} \beta_{nomcatt}$$ or Eastern .01, .97, .01

$$\beta_{ranch}$$ increase Samburu .01, .90, .01

anti-poaching Tsavo .01, .97, .01

Masailand .01, .77, .01

Laikipia .01, .34, .34

Nakuru .01, .44, .01

Western .01, .28, .01

Central .01, .70, .01

Turkana .01, .97, .01

Coastal .01, .87, .01

expand ranching Marsabit .80, .10, .05

Eastern .10, .80, .05

Samburu .10, .60, .20

Tsavo .10, .80, .05

Masailand .10, .60, .20

Laikipia .01, .05, .90

Nakuru .01, .05, .90

Western .05, .10, .10

Central .05, .10, .10

Turkana .80, .10, .05

Coastal .80, .10, .05

 

m R_t U Hypothesis Value

do nothing 0-.5 nomadic-camel .2

nomadic-cattle .2

ranching .2

farming .2

.5-.99 nomadic-camel .3

nomadic-cattle .3

ranching .3

farming .3

increase 0-.5 nomadic-camel .95

anti-poaching nomadic-cattle .95

ranching .95

farming .95

.5-.99 nomadic-camel .95

nomadic-cattle .95

ranching .95

farming .95

expand 0-.5 nomadic-camel .05

ranching nomadic-cattle .05

ranching .05

farming .05

.5-.99 nomadic-camel .2

nomadic-cattle .2

ranching .2

farming .2

 

C m k₀

very arid do nothing 3000

very arid anti-poaching 8000

very arid expand ranching 2000

arid do nothing 3000

arid anti-poach 8000

arid expand ranching 2000

semi-arid do nothing 3000

semi-arid anti-poach 8000

semi-arid expand ranching 2000

non-arid do nothing 3000

non-arid anti-poach 8000

non-arid expand ranching 2000

 

Influence Diagram Region, District Area 6cYear

Area (km²) (km²) 1887 1962 1975 1977 1986 1990

Marsabit, 73952 Marsabit 1 1 1 1 1 1

Eastern, 162139 Garissa 43931 1 1 1 1 1 1

Mandera 26470 1 1 1 1 1 1

Tana_River 35237 1 1 1 1 1 1

Wajir 56501 1 1 1 1 1 1

Samburu, 62117 Isiolo 25605 1 1 1 1 1 1

Samburu 20809 1 1 1 1 1 1

West_Pokot 5076 1 1 1 1 1

Baringo 10627 1 1 1 1 1

Tsavo, 20821 Tsavo_NP 1 1 1 1 1 1

Masailand, 39476 Kajiado 20963 1 1 1 1 1 1

Narok 18513 1 1 1 1 1 1

Laikipia, 9718 Laikipia 1 1 1 1 1 1

Nakuru, 7024 Nakuru 1 1 1

Western, 37358 Bungoma 3074 1 1 1

Busia 1629 1 1

Elgeyo-Marakwet 2722 1 1 1 1 1

Kakamega 3520 1 1

Kericho 4890 1 1 1 1

Kisii 2196 1 1 1 1

Kisumu 2093 1 1 1 1

Nandi 2745 1

Siaya 2523 1 1 1

South_Nyanza 5714 1 1

Trans_Nzoia 2468 1 1 1

Uasin_Gishu 3784 1 1

Central, 63142 Embu 2714 1 1 1

Kiambu 2448 1 1 1 1

Kirinyaga 1437 1 1 1

Kitui 23020 1 1 1 1 1 1

Machakos 13629 1 1 1 1 1

Meru 9922 1 1 1 1 1

Muranga 2476 1

Nairobi 684 1 1 1 1 1 1

Nyandarua 3528 1 1 1 1 1 1

Nyeri 3284 1 1 1 1

Turkana, 60824 Turkana 1 1 1 1 1

Coastal, 33807 Kilifi 12414 1 1 1 1 1

Kwale 8257 1 1 1 1 1 1

Lamu 6506 1 1 1 1 1

Mombasa 210 1 1 1 1 1

Taita 6420 1 1 1 1 1 1

 

Influence Diagram 6cSurvey Year

Region 1977 1978 1980 1981 1982 1983 1985

Marsabit 125142 120018 - 91538 - - 68656

Eastern 98231 135743 19578 - - 23891 32801

Samburu 89282 43219 809 8055 1482 - 31379

Tsavo - - - - - - -

Masailand 262017 473869 197731 38082 40881 252054 -

Laikipia 58315 57068 - 20703 15382 - 21544

Nakuru - - - - - - -

Western - - - - - - -

Central 4730 6050 6036 - - 2650 -

Turkana 40832 - - 22484 27711 - -

 

$$\beta_H \tilde{\beta} \tilde{\beta}$$ Chance Node Parent's

and Conditioning

Parameters Values

B_t very arid, 3000., 1. 1000., 3. 1000., 10. do nothing

$$\sigma$$ very arid, 8000., 1. 8000., 10. 8000., 10. anti-poaching

very arid, 2000., 1. 2000., 10. 2000., 10. expand ranching

arid, 3000., 1. 1000., 10. 1000., 10. do nothing

arid, 8000., 1. 8000., 10. 8000., 10. anti-poaching

arid, 2000., 1. 2000., 10. 2000., 10. expand ranching

semi-arid, 3000., 1. 1900., 10. 5500., 10. do nothing

semi-arid, 8000., 1. 8000., 10. 8000., 10. anti-poaching

semi-arid, 2000., 1. 2000., 10. 2000., 10. expand ranching

non-arid, 3000., 1. 5900., 5. 5500., 10. do nothing

non-arid, 8000., 1. 8000., 10. 8000., 10. anti-poaching

non-arid, 2000., 1. 2000., 10. 2000., 10. expand ranching

K_t B_t 0.0, .10 .000, .107 .000, .189

$$\beta_0 \beta_1$$

f_t t, 0-.5 .2, .05, .281, .094 .3, .080

$$\alpha_f$$ .001 .001 .001

$$\beta_f$$ t, .5-.99 .2, -.05, .242, -.022 .200, -.050

.001 .001 .001

r_t t, infrequent .05, .1, .053, .032 .036, .094

$$\alpha_r$$ .001 .002 .001

$$\beta_r$$ t, frequent .05, -.1, .078, -.200 .059, -.200

.001 .002 .002

N_t t, f_t, r_t, K_t 1000., .600, .050, 1053., .600, .014, 1053, .703, .014

N₀, P, c, .001 .003 .002

$$\beta_N$$

 

Management Option (m) Expected Loss (standard error)

do nothing 6801.6 (4685.9)

increase anti-poaching enforcement 5.0 (0.0)

expand ranching 10000.0 (0.0)

Figure Legends

=.3in Figure 1. A directed, acyclic graph.

=.3in Figure 2. Cheetah viability influence diagram.

=.3in Figure 3. First three letters of influence diagram regions (upper-left), see Table 6. Kenya's climate (upper-right). Kenya's land use areas (lower-left). Kenya's protected areas (lower-right). Each rasterization is over a 50 by 50 grid.

=.3in Figure 4. Observed and one-step-ahead predictions of herbivore count (B_t) and detection fraction (D_t) versus time. A predicted value is the average of 100 realizations of the variable at each of the observation times. All parameter estimates computed with a hypothesis relative importance weight (c_H) of 0.5 (left column), and 0.0 (right column). Squares: observed, diamonds: predicted.

=.3in Figure 5. Mean of detection fraction (D_t) by region for the year 2000 under the management option: do nothing. Consistent (c_H=.5) parameter values were used.