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.

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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),

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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.

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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,

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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.

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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)

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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

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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

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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

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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

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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

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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

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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.

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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)

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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)

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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

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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

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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

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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

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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.

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