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Effect of Uncertainty in Mapped
Biodiversity Data on Optimal
Conservation Decisions
Michael J. Conroyl and Jennifer E. Crocker2
Abstract--Spatially referenced biological diversity data are increasingly
being used to assist with conservation decisions. Examples include the
identification, selection, and design of ecological reserves; the types,
amount, and distribution of forest cutting; and the placement of corridors
to connect reserves. These have in common: (1) an implicit long-term
objective; (2) dynamic systems ; (3) alternative possible decisions ; (4)
alternative uncertain outcomes; (5) uncertainty about both the current
system state (sampling errors), and in predictions about future system
states ("model uncertainty"). Decision-theoretic methods and predictive
models can be used to find decisions that result in a long term optimum
(e.g., a balance between species conservation and timber revenue), subject
to updating through current surveys, research, and other information. We
illustrate the approach for a hypothetical reserve design problem involving
four species with different habitat affinities in a landscape with two habitat
types, with both demographic and statistical uncertainty. In particular, we
examine how uncertainty in predictive models (e.g., models relating
animal distribution and abundance to habitat features) can result in
suboptimal decisions, and how research and monitoring can be used
adaptively to reduce uncertainty and improve decision making.
INTRODUCTION
Increasingly, managers are faced with making decisions about how to manage
systems in which there are multiple, competing goals or objectives, and the system
under management is both complex and incompletely understood. The failure of
public decision makers to make a "good" decision can have severe consequences,
ranging from loss of agency credibility and good will, to unnecessary losses of
revenue or opportunity, to adverse impacts on the resource, including the loss of
species.
Concern over loss of biological diversity has led to efforts in the
' Assistant Leader, Georgia Cooperative Fish and Wildlrfe Research Unit, Athens, GA.
Graduate research assistant, Institute of Ecologv, Universify of Georgia, Athens, GA
685
cartographic analyses of species distributions and landscape characteristics (e.g.,
Burley 1988; Scott et al. 1993) In these efforts, animal species richness is either
observed directly or inferred from vegetation (habitat) maps and models relating
abundance or presencelabsence to vegetation cover attributes and physical features
(Scott et al. 1993). The mapped data are then used land-use decisions directed at
the conservation of biological diversity, such as reserve design (Scott et al. 1993).
Biodiversity mapping programs such as Gap Analysis frequently do not formally
incorporate uncertainty about 1) mapping errors in habitat and animal species
presence distributions, and 2) species-habitat associations. They implicitly assume
that predictions can be made about the future state of biodiversity, given alterations
in current habitat conditions (e.g., through management). Here we consider how
statistical unreliability in spatially referenced data may affect the utility of mapped
species richness patterns for conservation decisions such as reserve selection and
design. We first briefly consider alternative approaches to obtaining optimal
management decisions under the conditions in which the system is 1) stochastic, 2)
dynamic, 3) imperfectly understood, and 4) incompletely observable. We illustrate
the general problem by means of an artificial landscape containing 2 habitat types
and 4 species (Conroy and Noon in press, Crocker and Conroy 1995).
Before proceeding, we note that we are using one specific method (stochastic
dynamic programming) for determining optimal policies. There are a variety of
alternative analytical approaches to this problem, capably reviewed by Williams
(1989). These include non-dynamic methods such as linear programming, simulation
methods, and combinations of simulation and optimization. We recognize that no
single approach is likely to perform adequately under all circumstances, but we
strongly feel that any approach should adequately reflect both the dynamic nature of
ecological systems and decision making, and the role of uncertainty in both
observations and model structure. We also agree with Walters (1986) and Nichols
et al. (1995) that a formally adaptive approach to natural resource management will
in the long run prove most efficacious.
METHODS
Elements of the Problem
We consider a hypothetical 100 hectare landscape with three cover types, which
we have temed forest, open, and developed. Initially, the landscape is 75% forested,
25% open, and 0% developed. This landscape is initially occupied by four
hypothetical species, two forest habitat specialists and two open habitat specialists.
The management objective expresses a tradeoff between the conservation goal of
maintaining the species in the landscape and the economic goal of allowing
development. At five year intervals, the manager must decide what proportion of
each habitat type to conserve, and a proportion of all unconserved land is
permanently lost to development. The manager has a range of hypotheses regarding
population response to habitat management. The population is monitored during the
management period, both to guide management decisions and to reduce uncertainties
about population dynamics. In the development below we freely borrow fiom
Nichols et. al. (1995), whom we thank for their lucid description of the problem.
Management options
In this simplistic, non-spatial problem, the manager must decide how much of the
available wildlife forest and open habitat to conserve for the next five year period.
The action chosen in period t is denoted 4. A policy A specifies a sequence of
management actions { a , a,, ..., a,} as a function of the observed population and
habitat conditions.
Models
The models specify the dynamics of the state variables through time. The set of
population models specifies a range of alternative hypotheses about the population
responses to management actions. The population dynamics and environmental
dynamics are combined in a single state dynamics equation,
where x, is a vector of state variables including both population status and habitat
status, z, is a white noise process, and F,(x,,a,,zJis the net change in the state
variables under model i.
We considered three models describing population response to management
actions. These models differ in the degree to which population dynamics is related
to the amount, composition, and spatial arrangement of habitat. Model 1 was a
simple non-dynamic habitat relation model, N=dH, where N=species abundance, d=a
constant density, H=the amount of habitat (forest or open, depending on the species).
Model 2 was an equilibrium source-sink model (Pulliam 1988). For each species,
where N=species abundance, H=amount of source habitat, d=maximum density in
the source habitat, k,= population growth rate in the source habitat, and A, =
population growth rate in the sink habitat.
Model 3 was a dynamic source-sink model (Pulliam 1988). For each species,
dynamics were described by:
N(2)t+1=
{
if a (l)N(l), < dH
if a (I)N(l), 2 dH
N(2), + 1(1)N(i), - dH
where N(l)=population in source, k(l)=population growth rate in source, d=
maximum density in source, H= amount of source habitat, N(2)=population in sink,
a(2) = population growth rate in sink. Values of the species parameters are given in
Table 1.
Table 1.--Model parameters for population dynamics.
Habitat affinity
Species 1
Forest
Species 2
Forest
Species 3
Open
Species 4
Open
AI
A2
-
-
d
-
4
More complicated, spatially-explicit models can and will be constructed, but for
the present Models 2 and 3 allow dispersal from favorable to unfavorable habitats,
and thus are 'spatially informed', whereas for Model 1, in which movement between
habitats does not occur, it is only the amount of favorable habitat that determines
abundance for each species.
Landscape dynamics followed a simple state transition model. Unconserved land
was permanently lost to development at a rate of 0.049 every 5 year period. Of the
remaining habitat, forest converted to open at a 5 year rate of 0.26, and open to
forest at a 5 year rate of 0.95.
Components of uncertainty
There are four components of uncertainty which are important in managing
biological systems. Environmental stochasticity is uncontrollable variation in the
factors influencing population dynamics. We modelled extinction as a stochastic
process, with probability distributions conditioned on population size. Structural
uncertainty reflects imperfect knowledge about the relationship between management
actions and population status. We explored the effect of structural uncertainty by
using a weighted average of the three model predictions in the optimization
procedure, and varying the weights assigned to each model. Partial observability
refers to the imprecision in a monitoring program. Such imprecision leads to
uncertainty about the appropriate action to take, and also can affect the ability to
discriminate among alternative models. We plan to model the effect of partial
observability by developing an optimal policy based on complete observability, and
then using that policy in a simulation model with random variation around the
population state variables. Partial controllability refers to the inability to implement
management decisions without error. We have not considered partial controllability
in this exercise, but plan to do so in future work.
Monitoring
Since decisions are partially based on population status, a monitoring program is
needed. The monitoring program can also serve to discriminate among the
competing models. The data recorded at time t are denoted y,, and are related to the
population according to
with the random variable E independent of z. The set of all data collected up to time
t is denoted Y,.
Model weights
The likelihood at time t that model i is the best model relating population to
habitat is denoted p,(t). The likelihoods are expected to change through time as data
are accumulated and compared with model predictions.
Objective function
Our objective function V is a weighted sum of a species conservation objective
and a resource use objective. In practice, the weights would be determined using
multi-attribute utility theory (Keeney and Raiffa 1976). Our objective is to maximize
where V(AI Y, pJ is the expected value of the objective for policy A given the current
state of knowledge, W,=0.1875 and W, = 0.25 are weights for the two objectives,
S(A) is the number of species at the end of 100 years, and C is the proportion of the
available habitat conserved over all time steps. The weights were chosen so that the
objective function falls between 0 and 1.
Solution method
All optimizations and simulations were done using Stochastic Dynamic
Programming (Lubow 1993), which uses backward iteration to find optimal policies
for discrete Markov processes.
Analysis
Our analysis focused on the effect of structural uncertainty on the optimal policy
and the expected value of the objective function. Seven combinations of weights pi
were used (Table 2), reflecting a range of likelihoods ascribed to the three models.
The first three cases reflected structural certainty. In the next three cases there is
greater confidence in one model than the other two, and in the third there is no prior
knowledge about the appropriate model.
In the extreme case in which the data y, indicate with certainty which model is
correct, we obtain perfect information (Lindley 1985) about the system structure.
Given prior weights pi given to the models, the expected value of the objective
function with perfect information is:
The difference between this value and the maximum value of the objective function
calculated using the weighted (by thep,) average of the three models is the expected
value of perfect information (EVPI) (Lindley 1985). The EVPI was calculated for
the four combinations of weights which reflected structural uncertainty.
RESULTS
Table 2.--The effect of varying model weights on optimal decisions and objective V
PI
P2
P3
V
EVPI
forest area
conserved
open area
conserved
In general, the population predictions for Model 2 were higher than those for
Model 1. This was not surprising, since in both models density was proportional to
the amount of preferred habitat, with larger constants of proportionality in Model
2. For all combinations of state variables, the expected utility for Model 2 was equal
to or greater than the expected utility for Model 1. Similarly, expected utilities for
Model 3 were generally equal to or greater than those for Model 2. As habitat was
lost, Model 2 predicted immediate population equilibrium with the new conditions,
while Model 3 predicted a time lag.
The optimal decisions also varied among the models. In general, the optimal
amount to conserve was larger for the model which predicted lower population size.
Varying the likelihoods p, assigned to each model affected both the optimal
decision and the value of the objective function for the optimal decision (Table 2).
DISCUSSION
The degree of confidence placed in each model affected both the optimal
management decision and the expected value of the objective function. Basing
management on an incorrect model without considering structural uncertainty can
lead to suboptimal decisions.
The EVPI refers to the expected increase in the objective function from a
hypothetical study designed to eliminate structural uncertainty. For example, a gain
in objective function units of 0.033 might translate into an increase in the expected
number of species conserved of 0.033/W1= 0.176, or an increase in the proportion
of land open for development of 0.033/W2 = 0.132. In a realistic management
scenario, data will not discriminate among the models exactly. In that case, the
expected gain from gathering the data depends on the model likelihoods (Lindley
1985). The expected value of partial information can be used in an analysis in the
cost-benefit tradeoff of various levels of monitoring intensity.
In this exercise, we chose the model weights arbitrarily. An adaptive approach to
management would involve treating the model weights as state variables whose
dynamics are based on Bayesian updating. In passive adaptive management (Walters
and Hilborn 1978), information feedback into the decision making process is not
considered in evaluating the objective function. Optimization, management, and
monitoring are done iteratively, with model weights recalculated after each iteration.
In active adaptive management, the optimization specifically considers the benefits
of future learning. Active adaptive management presents considerable computational
difficulties, because the information state Y,increases in dimension through time. A
dynamic programming approach incorporating future learning must rely on a set of
summary statistics which evolve in a Markovian fashion.
This model exercise represents a preliminary step in developing adaptive
approaches to conservation planning. Our immediate objectives include exploring
the effect of partial observability on decision-making in our model landscape, and
performing similar analyses using population models with spatial structure. This
will require some refinements in our solution methods due to the dimensionality
limitations of dynamic programming. We anticipate taking an iterative simulationoptimization (SO) approach to dealing with more complex systems. In SO, methods
such as dynamic programming or linear programming is used to develop optimal
policies for simplified or static systems. More detailed simulation models are then
used to track the system dynamics over time using the optimal policy. If SO proves
to be an effective strategy for dynamic optimization in complex systems, we will
have the analytical tools available for addressing optimization that is truly adaptive,
in the sense that the information state is included in the optimization procedure.
ACKNOWLEDGMENTS
We thank Ken Williams, Jim Nichols, Fred Johnson, and Bill Kendall for
invaluable consultation and input.
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