DYNAMIC RESERVE SITE SELECTION Christopher Costello UCSB ...
-
Upload
duongtuyen -
Category
Documents
-
view
213 -
download
0
Transcript of DYNAMIC RESERVE SITE SELECTION Christopher Costello UCSB ...
DYNAMIC RESERVE SITE SELECTION
Christopher Costello UCSB
Stephen Polasky
University of Minnesota
Andrew Solow Woods Hole Oceanographic Institution
May 2001
Abstract: The conversion of large areas of natural habitat to human dominated landscapes has been a major cause of the loss of biodiversity. Establishing biological reserves in key habitats is one method to prevent the further loss of biodiversity. In this paper, we analyze a dynamic reserve site selection model where a conservation planner receives a budget each period with which to use to purchase sites (of heterogeneous cost and ecological value) to achieve a goal of maximizing species coverage by the end of the planning period. Sites left outside the reserve system are threatened by development. We formulate this problem as a stochastic dynamic integer programming problem and solve several examples to illustrate how sequential choice compares with static choice in which all selections can be made at one. We also compare optimal solutions with solutions obtained using simple heuristic choice rules.
2
1. INTRODUCTION
The conversion of large areas of natural habitat to human dominated landscapes
has been a major cause of the loss of biodiversity. Only about fifty percent of the forest
area that existed at time of the rise of agriculture remains, of which less than half
remains in large tracts capable of sustaining a full range of biological diversity (World
Resources Institute 1998). Less than half of the wetland in the United States remain from
the time coming of European settlers still remain, and in California, less than 10%
remains (Viliesis 1997). Habitat conversion is likely to continue into the foreseeable
future as the human population continues to grow, perhaps by another three billion or
more over the next fifty years. Given present and projected future trends in land use and
habitat conversion, the question of how best to conserve biological diversity is an urgent
one. Which areas of habitat are the most important to protect in order to conserve
biodiversity and how should conservation agencies target their action so that they
accomplish the most?
An important strategy to conserve biological diversity is to protect habitat through
the establishment of a system of biological reserves. Examples of this approach include
the system of National Wildlife Refuges in the U.S., national parks in many countries,
and private conservation reserves established by such groups as the Nature Conservancy.
Since it is not possible to protect all biologically important sites in a reserve network with
limited resources, it is important that the conservation agency choose wisely among
potential reserve sites.
Starting in the early 1980’s, a fairly substantial literature in conservation biology
has addressed the “reserve site selection problem” (see, for example, Kirkpatrick (1983),
3
Margules et al. (1988), Pressey et al. (1993)). A simple formulation of the reserve site
selection problem is to choose sites to include in a reserve network that conserve the
greatest number of species possible given a budget constraint. A species is considered
conserved if it is present in at least one site included in the reserve network. A species
that is not present in any selected site is not considered conserved. If the presence or
absence of each species at each site is known with certainty, the reserve site selection
problem is what is called a “maximal coverage problem” in operations research (Church
and ReVelle 1974). Branch-and-bound methods can be used to find optimal solutions.
Applications of these methods to find optimal solutions to the reserve site selection
problem include Cocks and Baird (1989), Saetersdal et al. (1993), Church et al. (1996),
Kiester et al. (1996), Csuti et al. (1997), Pressey et al. (1997), Ando et al. (1998), Snyder
et al. (1999), Polasky et al. (2001).
All of these papers are static in the sense that they assume there is a one-time
decision about which sites will be protected and which sites will not. In reality, there are
several important dynamic factors affecting conservation and land use decisions. .
Budgets for conservation agencies, public or private, are not set in a one-time lump sum
amount. Rather, public agencies receive an allocation on a regular budget cycle. These
agencies typically cannot borrow against future budget outlays to make land purchases in
the present. Private conservation groups receive donations through time. These groups
also find it difficult to borrow extensively against the promise of future donations.
Conservation agencies therefore must sequence their selections, setting aside some land
today knowing that they will set aside other lands in the future. However, habitat
targeted for future purchase may be converted in the meantime.
4
The goal of this paper is to analyze a dynamic reserve site selection problem. The
objective is to maximize the number of conserved species at the end of the planning
horizon by selecting sites as biological reserves each period given a per period budget
constraint. Each period, all sites that have not been selected have some probability of
conversion to developed land; conversion precludes them from being selected in the
future. Given the threat of conversion and the per period budget constraints, what is the
optimal sequence of reserve site selection? In section 2 we set up the problem as a
stochastic dynamic integer programming problem. In sections 3 and 4 we analyze several
relatively simple examples that illustrate important points about solutions to this problem.
Section 5 contains concluding remarks.
2. A MODEL OF DYNAMIC RESERVE SITE SELECTION
There are J sites, indexed by j = 1, 2,..., J, and I species, indexed by i = 1, 2,..., I
are distributed among the sites. The JxI matrix A has a typical element, Aji which equals 1
if site j contains habitat suitable for species i, and 0 otherwise. The planning horizon is
from 1 to T with time index t, t =1, 2,…, T. At the start of each time period t every site is
either “developed”, “natural”, or “reserved”. Once a site is developed, we assume that no
species survives at that site. The development process is assumed to be irreversible.
Similarly, once a site is selected as a reserve, all species occurring at that site are
protected and assumed to survive. Sites selected as reserves are assumed to remain
protected forever. The third category of land use, “natural”, can either remain natural, be
developed, or be selected as a reserve. The cost of selecting site j at time t as a reserve is
Cjt. If site j, which starts period t as natural, is not selected as a reserve during period t,
5
then it is developed at the end of period t with probability Pjt, and remains natural with
probability 1 - Pjt.
Let Rt be a Jx1 vector where Rjt equals 1 if site j has been selected as a reserve
prior to the beginning of period t, 0 otherwise. Let Xt be a Jx1 vector where Xjt equals 1 if
parcel j is selected as a reserve in period t, 0 otherwise. Therefore, Rt+1 = Rt + Xt. Let Nt
be a Jx1 vector where Njt equals 1 if site j is natural at the beginning of period t, 0
otherwise. Let St be a Jx1 random vector where element Sjt equals 1 if site j is converted
from natural to developed in period t (following the allocation decision in that period), 0
otherwise.
In each period, the planner faces a budget constraint. In period t, the planner is
given funds bt. We assume that the planner may not borrow money. However, funds that
are not spent during period t may be carried forward to period t + 1. Funds carried
forward earn interest at rate δ . Let Bt ≥ 0 represent the amount of money the planner
begins the period with, prior to receiving the budget amount for period t. Then at the
start of period t + 1, the planner will have ( )( )B B b X Ct t t t t+ = + − ′ +1 1 δ .
We formulate the dynamic reserve selection problem as a stochastic dynamic
integer programming problem. There are three period t state variables in this model: Nt,
Rt, and Bt. There is one period t control variable in this model: Xt. The objective of the
planner is to maximize the total number of species conserved at the end of the planning
horizon (i.e., the beginning of period T+1). To do so, the planner chooses Xt ∈ Nt for t =
{0,1,...,T}. The symbol ∈ constrains reserve purchases to those sites that are currently
natural. A species is considered conserved at the end of the planning period if the species
exists in at least one site that is in reserve or natural at the beginning of period T+1.
6
At the beginning of period t, the planner observes Nt, Rt, and Bt. The planner
receives budget outlay bt. The planner then chooses Xt ∈ Nt. Elements of Nt that are not
selected as reserve sites are then subject to possible development. With probability pjt a
natural site j is converted to developed. Any remaining unspent budget is then carried
forward to period t + 1 and earns interest.
Let V(Nt, Rt, Bt) be the value of the optimal program given the state variables at
the beginning of period t. Then we can write the stochastic dynamic integer
programming equation as follows:
( ) ( )V N R B E V N R Bt t tX N
S t t tt t
t, , max , ,=
∈+ + +1 1 1 (1)
s t X C Bt t t. . ′ ≤ + bt (2)
N N X S S N Xt t t t t t t+ = − − ∈ −1 , (3)
R R Xt t t+ = +1 (4)
( )( )B B b X Ct t t t t+ = + − ′ +1 1 δ (5)
Equation 1 is the stochastic dynamic integer programming equation. ES is the expectation
operator over the vector St, the random vector that shows which unprotected sites are
converted to development in period t. Equation 2 is the period t budget constraint.
Equations 3-5 provide the equations of motion which govern the transitions of the state
variables from one time period to the next, given a reserve allocation in period t, Xt.
This dynamic program is solved by backward induction starting at the end of the
planning period (the beginning of period T+1). The value of the optimal program at the
end of the planning period is:
( ) ( )V N R B e N R A eT T T xI T T Ix+ + + + += +′
1 1 1 1 1 1 1, , min , (6)
7
where emxn is an mxn matrix of ones. In words, equation (6) says that the value of the
program at the end of the planning horizon is equal to the number of species that remain
conserved (i.e. those species that occur in reserves or natural lands) at the beginning of
period T+1. The value of the program is independent of the money left over at the end of
the planning horizon.
Stepping back one period to the beginning of period T and taking advantage of the
fact that in period T we know the value of endowing the future (T+1) with the levels of
each state variable, we can write the stochastic dynamic programming problem (equation
1) as:
( ) ( )V N R B E V N R BT T TX N
S T T TT T
T, , max , ,=
∈+ + +1 1 1 (7)
11 ][ )(,minmax IxTTTxIS eARSNeE rTNTX
′+−=∈
subject to equations (2-5) for t = T. This problem is a static stochastic integer
programming problem that cannot be solved analytically, but standard computation
techniques have been developed that solve problems of fairly high order. The solution
simultaneously gives both the optimal XT (those natural sites in period T to purchase and
place in reserve) and the value function one period hence. The process is then repeated in
period T-1, and can be continued all the way backwards, to period 1.
Note that the complexity of solving this problem is determined by the number of
sites, which determines the number of possible combinations of the state variables Nt and
Rt, and the number of values that the carryover budget, Bt, may assume. Solving a
problem with a larger number of species or a larger number of time periods does not
8
greatly affect the difficulty of solving the stochastic dynamic integer programming
problem.
3. EXAMPLES In this section, we develop a few examples to accomplish three objectives. First, a
analyze a “warm-up exercise” to help build intuition about the dynamic reserve selection
problem. Second, we compare the difference between the dynamic reserve selection
problem and the traditional static approach that assumes that all sites are selected at once.
With a non-zero conversion threat of habitat conversion, a planner can always conserve at
least as many species by making decisions about which sites to select as reserves all at
once rather than sequentially. We measure the expected gain in protected species from
relaxing the constraint that selections must be made sequentially. In the examples, we
find that even for fairly small problems, allowing planners to make a one-time allocation
decision involves a gain of between 5-60% of species. Third, since this program is
difficult to solve (even for relatively small problems), our last objective is to evaluate the
performance of several "heuristic" approaches to the dynamic selection process. We find
that a heuristic (“greedy”) approach, which is much simpler to solve, performs nearly as
well as the optimal dynamic reserve selection algorithm.
In the examples below, we simplify the problem so that the planner chooses a
fixed number of sites to reserve each period. We assume that the budget cannot be
carried forward from period to period.
3.1 A WARM UP EXERCISE
9
To help get a feel for the dynamic site selection problem, we begin with a simple
example involving three sites and two time periods. In each period, one site may be
selected. Table 1 lists the species that occur in each site and the probability of conversion
for each site in each period.
Table 1
Data for Simple Example Site A Site B Site C Species 1, 2, 3, 4 5, 6, 7 1, 2, 5, 6, 8 Probability of Conversion
0.4
0.5
0.2
It is not immediately apparent from inspection of Table 1 which site should be
selected in the first period. Site A contains the most species that are not contained in the
any other site (species 3 and 4). Site B faces the highest threat of conversion. Site C
contains the greatest number of species.
In Table 2, we show the expected number of species conserved after two periods
when one begins by choosing site A, site B or site C in period one. After a site is selected
there is conversion risk for each of the other two unprotected sites, which results in four
possible outcomes for remaining natural areas in period two. If either one or zero natural
areas remain in period two, the period two choice of what to select, if anything, is clear.
When two natural areas remain in period two, a choice must be made between the two
remaining sites over which to select. The site that remains unprotected is then subject to
another conversion risk. For example, if site A is chosen in period 1, there is a 0.4
probability that both sites B and C will not be converted by the start of period 2. If site B
is selected in period 2, there is an 0.8 probability that site C will remain natural at the end
10
of period 2, which means that all 8 species would survive, and a 0.2 probability that site
C would be converted, which means that only seven species would survive. Conditional
on both sites B and C being natural at the start of period two, choosing site B in period
two yields an expected value of 7.8. On the other hand, following similar reasoning, one
can show that choosing site C in period 2 yields an expected value of only 7.5.
Therefore, it is optimal to choose B rather than C in the second period when both are
possibilities. Similar calculations can be done if either site B or site C are chosen in
period one.
Table 2 Expected Number of Species Conserved under Different First Period Choices
Period 1 Selection
Possible Pattern of Remaining Natural Areas in Period Two - Probability of Occurrence
- Expected Value
Expected Number of
Species
A B and C
0.4 7.8*
B 0.1 7
C 0.4 7
None 0.1 4
7.02
B
A and C 0.48 7.8*
A 0.12
7
C 0.32
6
None 0.08
3
6.744
C
A and B 0.3 7.5*
A 0.3 7
B 0.2 6
None 0.2 5
6.55
* Indicates that the expected value reported is for the optimal choice in period two.
As shown in Table 2, the optimal choice is to select site A in period one. Site A
scores well whenever it is combined with at least one other site. There is a fairly low
probability that both sites B and C will be converted in period one when not protected.
Site C is the most likely to remain natural even without being protected so that selecting
11
site C is not a good choice. Since site C will likely remain, there is little value to
choosing site B because there is only one species (species 7) present at site B that is not
present at site C.
3.2 FOUR-SITE EXAMPLES
We now turn to slightly more computationally intensive examples: two period
(T=2) problems with four sites (J=4), where the planner can select one reserve every
period. As noted above, we develop these examples with two primary objectives in
mind:
(1) to determine the expected loss in species when the reserve allocation problem
is constrained to be sequential, rather than all at once, and
(2) to assess the performance of “heuristic” (much less computationally- intensive)
approaches to solving the sequential reserve selection problem.
We examine three very different configurations of species on the four-site
landscape. Configuration C1 has 15 species with very little overlap between sites.
Configuration C2 has significant overlap between two sites, and very little overlap
between the other two sites (and between those sites and the two with large overlap).
Three sites in configuration C3 have significant overlap, while the fourth site is unique.
Table 3 gives the data for these configurations:
Table 3 Species Data for Four Site Examples
Site A Site B Site C Site D
C1 Species 1, 2, 3, 4 1, 5, 6, 7 8, 9, 10, 11, 12 8, 13, 14, 15 C2 Species 1, 2, 3, 4 1, 2, 3, 5 6, 7, 8, 9 10, 11, 12 ,13 C3 Species 1, 2, 3, 4 1, 2, 3, 5 2, 3, 5, 6 7, 8, 9, 10
12
The threat of conversion also has an impact on results. We examine various
combinations of threats for each site. The following table describes the six cases (types
T1-T6):
Table 4 Conversion Probability for Four Site Examples
Site A Site B Site C Site D
T1 Conversion Probability .80 .80 .80 .80 T2 Conversion Probability .20 .20 .20 .20 T3 Conversion Probability .80 .80 .20 .20 T4 Conversion Probability .20 .20 .80 .80 T5 Conversion Probability .80 .80 .80 .20 T6 Conversion Probability .20 .20 .20 .80
For each of the three configurations of species on the landscape, we examine both
the case where all sites have high probabilities of conversion (type T1) and the case
where all sites have low probabilities of conversion (type T2). For configuration C2,
where two sites overlap, and two sites are unique, we examine both the case where the
overlapping sites have high conversion probability and the unique sites have low
conversion probability (type T3) and the case where the overlapping sites have low
probability and the unique sites have high probability (type T4). For configuration C3,
where three sites overlap and one site is unique, we examine the case where the
overlapping sites have high probability and the unique site has low probability (type T5),
and the case where the overlapping sites have low probability and the unique site has
high probability (type T6).
4. RESULTS OF NUMERICAL EXERCISES
We break our results into two subsections – those that pertain to our first objective
(determining the loss in species associated with the constraint that acquisitions need to be
13
sequential), and those that pertain to our second objective (evaluating the performance of
computationally simple heuristic algorithms).
4.1 LOSS FROM SEQUENTIAL ACQUISITION
Making decisions sequentially involves a loss in species compared to the case
where all acquisitions are made at one time. This arises because in the process of
sequential decision-making, some sites are lost to development, thereby precluding their
future purchase as part of a natural reserve system1. However, the reality of most reserve
acquisition situations is that funds are available on a sequential basis, and therefore
planners are constrained to make decisions through time.
We wish to measure the expected gain in species from allowing decisions to made
at a single point in time rather than sequentially. In Table 5 we report the expected
number of species conserved under sequential and all at once selection. In column two of
Table 5, we report the expected number of species either in a reserve or in a natural site
at the end of the second period given that decisions are made optimally in a sequential
fashion. In column three we report the expected number of species preserved if two
parcels can be selected in the first period. We report the expected percentage gain in
species conserved by allowing reserves to be selected in period one in the column 4.
Table 5 Comparison of Expected Values in Static Versus Sequential Choice
Data
Config. Expected # species – sequential allocation
Expected # species – one time allocation
Expected % gain in species
1 As an aside, if “learning” can occur, in the sense that better information about species locations or future budgets or future land prices becomes available through time, it would not necessarily be the case that sequential decision making involved a loss compared to a one-time allocation decision.
14
C1-T1 6.9 10.2 48% C1-T2 12.3 13.9 13% C2-T1 6.0 9.6 60% C2-T2 11.2 12.5 12% C2-T3 9.9 11.4 15% C2-T4 9.1 12.5 37% C3-T1 6.0 8.5 42% C3-T2 9.3 9.8 5% C3-T5 7.5 9.2 23% C3-T6 9.3 9.8 5%
The results shown in Table 5 suggest that even for a small example such as this,
the gains from relaxing this constraint, and allowing all purchases to take place at one
point in time, can be significant. The highest percentage gains occur for configurations
C1-T1, C2-T1, C2-T4, C3-T1, and C3-T5. These configurations have one thing in
common – they all have high probabilities of conversion of sites that are relatively
unique. The expected gain is not as high from relaxing the constraint for configurations
with high probabilities of conversion on sites with high overlap. This can be seen by
examining results from C2-T3 and C3-T6. Although the probability of conversion of the
overlapping sites is high, the optimal strategy is to select the unique sites and take the
chance that at least one of the overlapping sites will survive.
4.2 PERFORMANCE OF HEURISTICS
In section 2 we developed an approach to optimally solving the sequential
allocation model. As pointed out in that section, solving the stochastic dynamic integer
programming problem for a problem with a large number of sites is computationally-
intensive. The dynamic programming approach is far more efficient than a brute-force
forward optimization approach that analyzes all branches of a decision-tree. The
difficulty of the brute-force method rises exponentially with both the time horizon (T)
15
and the number of sites (J). The difficulty of the dynamic programming method rises
exponentially only with J. Still, given the difficulty of finding optimal solutions in
problems with a large number of sites, we wish to assess the performance of two simple
heuristic choice rules.
The first rule (“myopic 1”) treats every period (in our example, periods 1 and 2)
as a static decision problem. The rule is to choose the site that adds the largest number of
expected species versus when no site is selected, considering the fact that unreserved sites
are subject to development at the end of the period. This rule is myopic because it does
not consider the fact that another site may be selected in the future. The second rule
(“myopic 2”) is exactly like the first rule, except that it assumes that all parcels not
selected at the end of the period, are developed. This is the de facto assumption in much
of the existing literature on reserve site selection. These myopic rules are similar to the
“expected greedy algorithm” where at each step the site that adds the greatest expected
number of species to what is already conserved is selected (Polasky et al. 2000).
We compare both the expected number of species saved and the actual site chosen
using each of the three rules, “optimal”, “myopic 1”, and “myopic 2”. Since the choice in
the second period is probabilistic, we only report the choice in the first period using each
rule. We report the results in Table 6.
Table 6 Comparison of Heuristic Rules Versus the Optimal Dynamic Rule
Data
Config. Optimal (# species/site
chosen) Myopic 1 (# species/site chosen)
Myopic 2 (# species/site chosen)
C1-T1 6.9 (C) 6.9 (C) 6.9 (C) C1-T2 12.3 (C) 12.3 (C) 12.3 (C) C2-T1 6.0 (D) 6.0 (D) 5.9 (D) C2-T2 11.2 (D) 11.2 (D) 11.2 (D) C2-T3 9.9 (B) 9.9 (B) 7.7 (D)
16
C2-T4 9.1 (D) 9.1 (D) 9.1 (D) C3-T1 6.0 (D) 6.0 (D) 6.0 (D) C3-T2 9.3 (D) 9.3 (D) 9.3 (D) C3-T5 7.5 (C) 7.5 (C) 6.1 (D) C3-T6 9.3 (D) 9.3 (D) 9.3 (D)
Based on these results, the simple answer to our initial question is that the myopic
1 heuristic performs quite well. In fact, the rule chooses the exact same site selections as
the optimal dynamic program (so the expected number of species conserved under both
rules are the same). The myopic heuristic rule that assumes all unreserved sites are
developed performs fairly well, except when the rule leads to an incorrect acquisition in
the initial period. For example, in cases C2-T3 and C3-T5, this rule selects site D in the
initial period, when the optimal rule selects sites B and C, respectively. Why did this
heuristic select the wrong site? The myopic 2 rule always selects the site that adds the
most species to the existing reserve, without regard for the future. Therefore, in the first
period, when there is no reserve, the rule will select the site with the most species. For
configuration C1, that is site C. For configurations C2 and C3, all sites have the same
number of species, so selecting site D is as good as selecting any other site, according to
this rule.
We are curious about whether these heuristics, or other simple heuristics will
perform well in problems of larger dimension. The more sites that are chosen, and the
longer is the planner’s time-horizon, the worse we expect these heuristics to perform
(relative to the optimal dynamic decision rule).
5. DISCUSSION
In this paper we define a dynamic reserve site selection problem in which sites
must be chosen sequentially because of budget constraints. Sequential site selection is
17
not as beneficial as being able to select sites all at once because there is a threat of
conversion for all sites not protected as reserves. Building a dynamic approach to
conservation is important. A dynamic approach is both realistic and generates potentially
different optimal strategies than a taking a static view.
Taking a dynamic approach opens up many interesting questions, only some of
which are analyzed in this paper. The examples of the previous section used a simple
budget constraint that did not incorporate heterogeneous land costs or the possibility of
carrying budgets forward through time. The model described in section 2 included the
ability to handle both of these complications. Adding in realism about the budget
constraint adds to the computational difficulty of solving the model. Formally, doing
adds an additional state variable (Bt). The degree of additional computational difficulty
depends upon the number of different values Bt may take. Constraining Bt to a small
number of potential values would only complicate the analysis slightly. On the other
hand, allowing Bt to take on any dollar amount between 0 and the maximum budget
would greatly complicate finding a solution. We feel that the analysis of budget
constraints in dollar terms, rather than in terms of the number of sites, is an important
next step in the analysis.
In this paper, we assumed that the probability of conversion and the land price for
any site at any time was independent of what other sites had been previously reserved or
developed. It is highly likely, however, that the probability of conversion for a particular
site depends upon what has happened on nearby sites. For example, if a reserve is
established on a neighboring site, the desirability of development for a given site might
increase, which may increase the probability of development and the price that would be
18
demanded if the sites was chosen as a reserve site. Development also depends on the
presence of infrastructure (sewer lines, roads, etc…), which introduces spatial correlation
in the conversion probabilities and land prices. In principle, the stochastic dynamic
programming approach can incorporate conversion probability functions that depend
upon the pattern of development and reserve selection, pjt = pjt(Rt, Nt), as well as land
costs, Cjt = Cjt(Rt, Nt). In this case, a planner choosing a particular reserve site would take
account not only of the direct effect of choosing that site as a reserve but also the effect
that doing so would have on altering probabilities of conversion and land prices on other
potential reserve sites. Making land prices and probabilities endogenous in this way
would increase the value taking a dynamic strategic approach. With endogenous land
prices and probabilities, it might make sense for the planner to wait to purchase a large
block all at once rather than trying to purchase small blocks sequentially only to see
either the price of the remaining blocks bid up or sites developed before they are able to
be reserved.
Considering whether species are likely to persist in reserves and natural areas is a
vitally important consideration in conservation planning, and one that is often not
adequately handled. Different species have different home range requirements. Some
sites are perfectly suitable for supporting populations of some species but may be
inadequate for supporting viable populations of other species. Further, the probability of
survival for a species may depend on the entire pattern of reserves and natural areas
rather than on whether a particular site is developed or not. Stochastic environmental or
demographic events may cause a local population of a species to die out. However, if the
population of the species inhabiting a site is part of a larger meta-population, then a site
19
may be recolonized from other sites as long as there is sufficient ability of populations to
move between sites. In addition, there are predator-prey or competitive relationships
among species that make the probabilities of survival among species non- independent.
While introducing dependence in survival probabilities across sites or species
complicates the analytics of the model, the more difficult challenge at present is our lack
of ecological understanding that would allow modeling non- independent probabilities.
Incorporating species survival probabilities into reserve site selection and other large-
scale conservation planning exercises is an important topic on which more research is
needed.
A further source of uncertainty is that there is often incomplete species range
information. In our notation, this would make the elements of the matrix A stochastic
rather known with certainty. Polasky et al. (2000), Camm et al. (2001), and Haight et al.
(2001) analyze a static reserve site selection problem with incomplete information about
species occurrences.
Finally, we have not considered other ecological benefits of conservation beyond
the benefits of species conservation, such as the continued provision of ecosystem
services (water purification, flood mitigation, nutrient cycling, climate regulation…).
With consideration of the benefits ecological services, the “cost” of conserving land
would be amended to be the land price minus the value of ecosystem services provided
by the site. Each site can be viewed as giving some contribution to the value of
ecosystem services. However, ecosystem functions typically depend the pattern of
development over a region. Changing land use on any particular site will then have
effects that spread more broadly through the ecosystem, which would change the
20
contribution of other sites not only of the site on which the land use changed. Just like
the consideration of endogenous land prices, ecosystem service values would typically
depend on the entire pattern of development.
REFERENCES Ando, A., J.D. Camm, S. Polasky and A. R. Solow. 1998. Species Distribution, Land Values and Efficient Conservation. Science. 279 2126-2128. Camm, J.D., S.K. Norman, S. Polasky and A.R. Solow. 2001. Nature Reserve Site Selection to Maximize Expected Species Covered. Operations Research. Forthcoming. Church, R.L. and C.S. Revelle. 1974. The Maximal Covering Location Problem. Papers of the Regional Science Association. 32 101-118. Church, R.L., D.M. Stoms and F.W. Davis. 1996. Reserve Site Selection as a Maximal Coverage Location Problem. Biological Conservation. 76 105-112. Cocks, K.D. and I.A. Baird. 1989. Using Mathematical Programming to Address the Multiple Reserve Site Selection Problem: An Example from the Eyre Peninsula, South Australia. Biological Conservation. 49 113-130. Csuti, B., S. Polasky, P.H. Williams, R.L. Pressey, J.D. Camm, M. Kershaw, A.R. Keister, B. Downs, R. Hamilton, M. Huso and K. Sahr. 1997. A Comparison of Reserve Selection Algorithms Using Data on Terrestrial Vertebrates in Oregon. Biological Conservation. 80 83-97. Haight, R.G., C.S. Revelle, and S.A. Snyder. 2000. An Integer Optimization Approach to a Probabilistic Reserve Site Selection Problem. Operations Research. 48 (5) 697-708. A.R. Kiester, J.M. Scott, B. Csuti, R.F. Noss, B Butterfield, K. Sahr, and D. White. 1996. Conservation Prioritization Using GAP Data. Conservation Biology 10(5): 1332-1342. Kirkpatrick, J.B. 1983. An Iterative Method for Establishing Priorities for the Selection of Nature Reserves: An Example from Tasmania. Biological Conservation. 25 127-134. Margules, C.R., A.O. Nicholls and R.L. Pressey. 1988. Selecting Networks of Reserves to Maximize Biological Diversity. Biological Conservation. 43 63-76. Polasky, S., J.D. Camm, A.R. Solow, B. Csuti, D. White and R. Ding. 2000. Choosing Reserve Networks with Incomplete Species Information. Biological Conservation. 94 1-10.
21
Polasky, S., J.D. Camm and B. Garber-Yonts. 2001. Selecting Biological Reserves Cost Effectively. Land Economics 77(1): 68-78. Pressey, R.L., C.J. Humphries, C.R. Margules, R.I. Vane-Wright and P.H. Williams. 1993. Beyond Opportunism: Key Principles for Systematic Reserve Selection. Trends in Ecology and Evolution. 8 124-128. Pressey,R.L., H.P. Possingham and J.R. Day. 1997. Effectiveness of Alternative Heuristic Algorithms for Identifying Minimum Requirements for Conservation Reserves. Biological Conservation. 80 207-219. Saetersdal, M., J.M. Line and H.J. Birks. 1993. How to Maximize Biological Diversity in Nature Reserve Selection: Vascular Plants and Breeding Birds in Deciduous Woodlands, Western Norway. Biological Conservation. 66 131-138. Snyder, S.A., L.E. Tyrell and R.G. Haight. 1999. An Optimization Approach to Selecting Research Natural Areas in National Forests. Forest Science 45 (3) 458-469. Viliesis, Ann. 1997. Discovering the Unknown Landscape: A History of America’s Wetlands. Washington, DC and Covelo, CA: Island Press. World Resources Institute, United Nations Environmental Programme, United Nations Development Programme, The World Bank. 1998. 1998-1999 World Resources: A Guide to the Global Environment. New York: Oxford University Press.