Fires and Outbreaks and Pests, Oh My!
Spatially Correlated Risk in Reserve Site Selection
Abstract
Establishing nature reserves protects species from land cover conversion and lost habitat.
Even when a species is contained in a reserve, however, many factors still threaten their
survival. To address the risk after reserve establishment, reserve networks can be created
that allow some redundancy of species coverage and maximize the expected number of
species that survive. In some regions, however, the threats to species within a reserve
may be spatially correlated. As examples, fires, diseases, and infestations all spread from
a starting point and threaten neighboring parcels in addition to the initial location. This
paper develops a reserve site selection optimization framework that compares the reserve
networks from cases in which risks do and do not reflect spatial correlation. Finding
differences in the optimal distribution of reserves across a stylized landscape, the paper
then considers the example of an ecoregion in southeastern Oregon. In addition to
demonstrating the role of spatially correlated risk in creating more dispersed reserve
networks, we also identify differences between the optimal networks to achieve various
objectives such as maximizing the expected number of species versus minimizing the
chance that no species survive.
1. Introduction
Networks of nature reserves offer protection to biodiversity and individual species.
Because we face limits on the amount of area allocated to biodiversity conservation, the
reserve site selection and related literatures examine the issue of selecting parcels to be
conserved across a landscape with species distributed across that landscape. In the
simplest maximal covering problem, the objective is to choose parcels for preservation so
that the greatest number of species is represented, or “covered,” in the reserve. Building
from that foundation, this literature solves the maximal species coverage problem for a
variety of settings including addressing uncertainty about species occurrence through
probabilistic approaches and issues of contiguity of reserves through connectivity and
border length constraints.
The probabilistic reserve site selection models typically deal with uncertainty about
species presence/absence data (Camm, et al. (2002)). In such cases, the objective
becomes maximizing the expected number of species covered by a reserve network and,
because the probability of a species being present on a parcel is less than one, it can be
optimal to conserve two or more plots with the same species to increase the expected
value of the species conserved. Our framework addresses a different kind of uncertainty:
whether or not the species will survive on a parcel within the reserve. Here, we assume
that the presence/absence data are perfect but we model risks to the species that may
cause them to be unprotected despite being located on a parcel within the reserve.
Species within the reserve are protected from lost habitat, assuming that the reserve is
large enough to provide sufficient habitat, but these species can still be killed by natural
causes like disease and by human-based causes like pollution or hunting. Converting the
standard coverage model to one that maximizes the expected number of species in the
network based on the probability of their survival addresses some of the issues
surrounding risks to species even though they are in reserves, and would lead to similar
redundancies in species types across reserves within the network as are found in the
probabilistic presence/absence data case.
In many cases, the probability that a species survives within the reserve network may not
be independent across space. As part of the SLOSS (single large or several small)
debate, many biologists suggest that contiguous or connected reserves increase the
survival probabilities of many species. With that biological information, some research
extends the reserve site selection models to contain constraints on the length of borders or
the amount of connectivity in a reserve system. The “several small” side of the SLOSS
debate has received less attention in the RSS literature, but argues that wildlife corridors
may also create corridors for the dispersal of disease and pests and that spreading
reserves apart may decrease some risks to species within reserve systems. Because
disease, pests, invasive species, and fires all spread spatially from a starting point, they
pose risks to parcels that are connected to each other in a way that permits or encourages
that spread. Although the risk associated with the starting point, such as the location of
the lightning strike that ignites a fire, may be independent across a landscape, once that
threat to species has begun, the risk it poses on the landscape is spatially correlated to the
initial location. In the case of spatially correlated risk, the probability of species survival
on any one parcel is no longer independent of the risks facing neighboring parcels.
In this paper, we develop a simple static model of the expected maximum species
coverage problem where each parcel in the landscape, and therefore the species on that
parcel, faces risk of habitat destruction even if the parcel is in the reserve. We develop a
stylized landscape and 3 cases of species distribution on that landscape. We then solve
for the optimal parcels to preserve, given a budget constraint, in the case of independent
risks, the case of spatially-correlated risks, and the case with both types of risk. We
describe the differences in the resulting reserve networks. We also examine objective
functions that reflect some desire to avoid bad events, such as minimizing the chance that
no (of 2 possible) species survive in the reserve, and compare the resulting reserve
networks to the completely risk-neutral maximum expected species network. We then
apply the optimization to an eco-region in Oregon that has several endangered species
and significant threat of habitat destruction from large fires.
2. Spatially Correlated Risk in a Stylized Landscape
In this section we develop a modeling framework that considers spatially independent
and spatially-correlated risk in the choice of parcels to conserve in a reserve network.
Our stylized landscape consists of a 5-by-5 grid of parcels with species present, with
certainty, on some parcels. In ongoing work with explore more species distributions but
for this paper draft we present results from three cases: a single species; two species; and
two species with “hotspots.”
2A. Modeling Approach
In species covering models where species presence/absence data is known with certainty,
the reserve site selection problem is non-stochastic and the objective is simply to
maximize the number of species represented in the reserve. Although we assume perfect
presence/absence data here, we consider the probability that some hazard, such as fire,
disease, or pests, threatens species in the reserve and the problem becomes stochastic and
species survival is probabilistic. Revelle et. al (2002) describe, in general terms, a
probabilistic maximal covering problem (or expected covering problem) where there is
uncertainty in the objective. Although many types of hazards are spatially correlated, for
ease of exposition in what follows, we will develop the examples with a risk of fire.
We begin with a landscape with n parcels indexed j (J = {1,2,…,n}) and m species
indexed i (I = {1,2,…,m}). On each parcel j, the presence (or absence) of species i is
known with certainty. There is some positive probability of the hazard, wildfire in our
case, occurring on parcel j and eliminating species i on that parcel. The probability of
wildfire on parcel j, and whether or not species present on parcel j survive, depends on
how many burn patterns include parcel j and the probability of each burn pattern. There
are k (K={1,2,…,l) burn patterns each with probability bk. Our objective is to design a
reserve that maximizes the expected number of species present in the reserve after a
period of time in which fire may occur, as described by equations [1] and [2].
𝑀𝑎𝑥 ∑𝑛𝑖=0 𝑥𝑖 𝑝𝑖
[1]
Subject to
𝑝𝑖 = ∑𝑘∈𝐵𝑖 𝑏𝑘
[2]
Where
𝑝𝑖 = probability i total species in the reserve survive fire
𝑥𝑖 = total number of species surviving in the reseve
𝐵𝑖 = set of burn patterns where i species survive in the reserve
In all expected covering problems, expected species presence in the same parcel and
across parcels is assumed to be independent (Camm et al. (2002), Haight et al. (2000),
Polasky et al. (2000), Arthur et al. (2004)). In our spatially-correlated risk case, however,
we relax the assumption of independence across parcels. Our contribution to the existing
reserve site selection literature is our effort to include the risk of fire spread in the model.
If there is a fire on parcel j, not only will species on parcel j be affected, but there is also
the possibility that the fire will spread to adjacent parcels, thereby reducing expected
species presence on all surrounding parcels.
With fire spread, in addition to considering all n possible ignition points, we also consider
the probability of fire spreading from a single adjacent parcel k onto the individual parcel
j. In a general model, the probability of species survival is now linked across space. For
our framework, we assume that ignition probabilities are independent across space but
there is some probability that the fire spreads beyond the parcel of ignition. Here, when
the fire spreads beyond the ignited parcel, it spreads to each of the neighboring 8
contiguous parcels but not beyond those closest neighbors. (A more general model could
include another probability of the fire spreading beyond that ring.) We also assume that
fire, whether it ignites on the parcel or spreads to that parcel, eliminates all species on the
burned parcel. We assume there are two types of fires: large and small. Small fires
ignite on an individual parcel and burn only the parcel where the ignition occurred.
Large fires ignite on a single parcel and spread to surrounding parcels. Large fires can
also ignite outside the focal landscape and spread into the focal landscape.
To explore the impact of spatially correlated risk on optimal reserve site selection, we
first develop a single-period, single-species model on a stylized five-by-five grid
landscape (figure 1) where an initial species distribution and the probability fire are
given. (Ongoing work develops a range of initial species distributions.) To begin, based
on observations, we assume that a smaller number of fires spread to become large fires
than remain on the ignition parcel, which translates for our base case into a higher
probability of a small fire than of a large fire. Further, we assume that there is a
constraint on the maximum number of parcels included in the reserve.
Solving this optimal reserve site selection model requires a numerical approach to fully
depict the resulting landscapes. We develop a mathematical program to solve the
optimization using complete enumeration over our stylized landscape. For the case of
one species on the landscape, the basic algorithm is as follows:
Step 1: Enumerate the species distribution on the landscape for the optimization.
Step 2: Calculate all possible reserve designs with the max parcel constraint.
Step 3: Calculate all possible burn patterns and the probability of each, using the
probability of ignition and of spread.
Step 4: For each possible reserve design, calculate the probability of zero and one
species being present in the reserve following fire.
Step 5: For each reserve design, calculate the expected number of species.
Step 6: Select the reserve design with the maximum number of expected species.
For step 3, the probability of zero species being present in the reserve is calculated as the
probability that fire burns every parcel in the reserve where species one was initially
present. (As in many reserve site selection models, we assume that species that are not in
the reserve do not survive or are not considered in the optimization.) For reserve designs
where the species was not initially present, the probability of the species being present in
the reserve is zero. The probability of one species being present in the reserve is
calculated as the probability that fire does not burn every parcel in the reserve containing
that one species.
Figure 1: Five-by-five landscape
Next we examine two cases where there are two species present on the landscape. The
algorithm for these cases is similar to the single species case except that step 3 now
requires that we calculate the probability of zero, one, and two species being present in
the reserve.
2B. Results
Results: Single-species landscape
On the single-species landscape, illustrated in figure 2, our chosen species distribution
has species 1 present on five of the twenty-five reserve parcels and the problem is to
choose a two-parcel reserve to maximize the expected number of species (see figure for
parcels). In this case, there are 3 reserve designs that maximize the expected number of
species post-fire: (7,20), (11,20), and (12,20). Each parcel in the reserve contains species
1 and the parcels are far enough apart so that a large fire that burns one of the parcels
cannot reach the second parcel in the reserve. For these 3 reserve designs, the probability
of species 1 surviving on at least one parcel is equal to one and the expected number of
species is also equal to one.
Figure 2: Single-species landscape
The expected number of species is at a maximum when one parcel in the reserve is
beyond the limits of burning simultaneously with other parcels in the reserve. Here,
spatially correlated risk makes reserve connectivity undesirable. This result will hold as
long as there are two parcels that contain species 1 that are separated by more than one
parcel, regardless of species distribution.
Results: Two species landscape
Next we examine a five-by-five grid landscape with two species present on a total of 11
parcels (figure 3). In this setting, when there is a positive probability of a large fire, the
reserve design that maximizes the expected number of species includes any reserve with
one parcel containing species 1 and one parcel containing species 2, regardless of the
distance between the two parcels. For example, reserve designs containing adjacent
parcels (8,9), parcels separated by one (7,17), and parcels separated by more than one
(1,20) all result in the same number of expected species, 1.76.
One part of the result here is that, at these probability values, the expected value of
species is always highest when each of the species is contained in the reserve; the
marginal value of covering a species twice is not as high as that of covering both species
once. Going further, this pattern can be explained by examining the probability that both
species survive and the probability that only one species survives for each of the three
reserve designs, as seen in Table 1for cases where there are large and small fires, only
large fires, and only small fires. For each of these cases, the probability of large fires is
equal across parcels as is the probability of a small fire.
As shown in Table 1, there is a greater probability of both species surviving when the
reserve parcels are adjacent because, compared to other spatial reserve designs, there are
fewer burn patterns that include one of the two adjacent parcels. When reserve parcels
are separated by more than one parcel, the number of burn patterns that include only one
of the reserve parcels is greater than when the two parcels are adjacent.
Figure 3: Two-species landscape
However, the reserve design that maximizes the probability that at least one species
survives has reserve parcels separated by more than one parcel. Separating reserve
parcels by more than one parcel also minimizes the probability that zero species survive.
The two parts of the expected value equation, both species surviving and a single species
surviving, work in opposite directions and make the optimal pattern of reserves
indifferent to the spacing as long as each species is covered by a parcel. This result is not
sensitive to the relative probabilities of large and small fires, as long as the probability of
a large fire is positive, as shown in Table 1.
When the probability of a large fire is zero, the spatial arrangement of reserve parcels no
longer matters and all three reserve designs perform equally well, not only in terms of
expected species, but also in terms of maximizing the probability that at least one species
survives, also shown in Table 1. The optimal reserve design in this case includes one
parcel with species 1 and one parcel with species 2, regardless of location.
Table 1: Probability one, two, and zero species survive for three reserve designs
Reserve parcels Reserve parcels
Adjacent
separated by one
parcel
Probability at
When probability of
0.93
0.96
least one species
large fire = 0.01 and
survives
probability of small
fire = 0.02
Probability both
species survive
Probability only
one species
survives
Probability zero
species survive
Reserve parcels
separated by more
than one parcel
1.00
When probability of
small fire = 0
0.86
0.9387
1.00
When probability of
large fire = 0
1.00
1.00
1.00
When probability of
large fire = 0.01 and
probability of small
fire = 0.02
0.83
0.80
0.77
When probability of
small fire = 0
0.7551
0.6939
0.6326
When probability of
large fire = 0
0.92
0.92
0.92
When probability of
large fire = 0.01 and
probability of small
fire = 0.02
0.10
0.16
0.22
When probability of
small fire = 0
0.1224
0.2448
0.3674
When probability of
large fire = 0
0.08
0.08
0.08
When probability of
large fire = 0.01 and
probability of small
fire = 0.02
0.06
0.03
0.00
When probability of
small fire = 0
0.1224
0.0612
0.00
When probability of
large fire = 0
0.00
0.00
0.00
Results: Two species with biodiversity hotspots
Next we examine a landscape with biodiversity hotspots, where some parcels contain
both species 1 and 2 (figure 4). When the two biodiversity hotspots are adjacent, and the
probability of a large fire is 0.01 and the probability of a small fire is 0.02, the reserve
design that maximizes the expected number of species is any 2-parcel combination that
includes one of the biodiversity hotspots and a second parcel with either species 1 or 2
that is at a distance of more than one parcel from the reserved hotspot. These 2-parcel
combinations include: (8,23), (8,25), (7,20), (7,23), (7,25), and (5,7). The expected
number of species for all of these reserve designs is 1.87. Once one hotspot is in the
reserve, conserving the adjacent hotspot does not improve the expected number of
species (1.86 for both hotspots for these parameter values) as much as conserving a more
distant parcel with only one species because of the spatially correlated risk facing the
adjacent parcels.
Figure 4: Two species landscape with biodiversity hotspots
However, increasing the probability of a small fire and decreasing the probability of a
large fire makes the selection of the adjacent hotspot parcels optimal. When the
probability of a small fire increases from 0.02 to 0.02432 and the probability of a large
fire decreases to 0.01 to 0.008, including adjacent parcels 7 and 8 in the reserve
maximizes the expected number of species. These results demonstrate the role of the
spatially correlated risk: that pattern of risk matters more at higher probabilities of fire
spread.
In order to isolate the impact of the spatially correlated risk, current analysis of this
framework sets the total risk to the landscape equal and varies the pattern of risk from
spatially correlated to independent. Preliminary results demonstrate the role of fire
spread in determining spatially separated patterns of sites within a reserve network.
2C. Objective Functions
In the probabilistic maximum coverage models, as here, the typical objective function is
to maximize the expected number of species in the reserve. In the case of conserving
species in the face of risk, such as fire, conservation actors may consider other objective
functions. For example, a conservation land manager may conserve land to minimize the
chance of extreme negative events such as losing all species to the natural hazard. In our
model, that objective function equates to choosing the two parcels to conserve that
minimize the probability of zero species surviving the fire period. For our benchmark
case with a probability of both small and spreading large fires, the reserve network that
minimizes the chance of losing both species is a reserve network with parcels that are
spread far enough apart that the hazard cannot spread from any one reserve parcel to the
other (Table 2), while the expected value objective is indifferent between that and other
distributions of the reserve sites. In contrast, if the manager’s objective function is to
maximize the chance that both species survive, rather than the expected number of
species, under some parameter values including those of the benchmark, the optimal
reserve contains adjacent parcels whenever there are large fires possible on the landscape.
These preliminary results are the subject of ongoing research to further explore the
impact of different objective functions on reserve networks, especially when the objective
function reflects some desire to avoid negative outcomes.
3. Oregon landscape model and results
On the stylized landscape it is possible to numerically solve the reserve site selection
problem with spatially correlated risk, but solving a similar real-world problem is not
possible when the number of species and parcels is large. Heuristics are often used to
solve large problems that cannot be solved by traditional mathematical programming
techniques. In particular, in the field of natural resource management, heuristics have
been applied to many spatial forest planning and harvest scheduling problems (Murray
and Church (1999) and Bettenger et al. (2002) among others).
We use a simulated annealing algorithm to find solutions to the reserve site selection
problem in southeastern Oregon. In forest planning problems, simulated annealing
algorithms have been found to perform well relative to other heuristics, such as genetic
and tabu search algorithms (Boston and Bettinger (1999); Liu et al. (2006)). Because
simulated annealing is a heuristic, it does not guarantee the optimal solution, or even the
same solution, every time. However, by thoroughly searching the solution space we can
be certain of a “good” solution. Our simulated annealing algorithm is as follows:
Step 1: Initialize the solution.
Step 2: Apply fire to landscape and allow to burn through repeated randomized
experiments. Calculate the average number of species remaining in threehexagon reserve after fire.
Step 3: Randomly select a hexagon to input into the solution.
Step 4: Randomly select a hexagon to leave solution.
Step 5: Apply fire to landscape and allow to burn through repeated randomized
experiments. Calculate the average number of species remaining in threehexagon reserve after fire.
Step 6: If solution is better than best so far, save to best and current. If not,
calculate Boltzman constant – there is some positive probability of accepting a
non-improving solution.
Step 7: Increase counter and return to Step 3.
The mathematical programming model is applied to species presence data for terrestrial
vertebrate species in southeastern Oregon, the Northern Basin and Range ecoregion.
There are 80 parcels (outlined in black in figure 5) from which the three-parcel reserve
can be chosen and a total of 310 species present on the southeastern Oregon landscape.
The Oregon data set is part of a larger state-wide data set that has been widely used in the
reserve site selection literature (e.g., Hamaide and others 2009, Csuti and others 1997;
Haight and others 2000; Polasky and others 2001; Arthur and others 2004; Hamaide and
others 2006).
To begin, we assume that all fires are small. In this case, the probability of a small fire
on each of the 80 parcels is 0.04, making the probability of a small fire occurring
somewhere on the landscape equal to one. The reserve design that gives the maximum
observed number of expected species is depicted in figure 5, where all parcels in the
study area are outlined in black and reserve parcels are marked with a star. Because the
three reserve parcels are separated by more than one parcel, the introduction of large
spreading fires does not change the reserve design that gives the maximum number of
expected species. And we see that when all fires are large, igniting on a single hexagon
and spreading to the 5 adjacent hexagons, the reserve design with the greatest number of
expected species is unchanged from that described in figure 5.
Figure 5: Oregon landscape
While the reserve design with the greatest number of expected species may not be
sensitive to the introduction of large fires, the influence of spatially correlated risk on the
number of expected species can be seen by comparing reserve designs when all fires are
small to when all fires are large. For example, suppose two reserve designs are under
consideration for conservation: all parcels in one of three-parcel reserve are separated by
more than one parcel whereas the parcels in the second three-parcel reserve are not.
These two possible reserves are illustrated in figures 6a and 6b. When all fires are small,
and there is no spatially correlated risk, the reserve design in figure 6a – less distance
between parcels – gives a greater number of expected species than the reserve design in
figure 6b. When all fires are large, the reserve design in figure 6b is preferred. Given the
pattern of conservation in figure 6b, it is not possible for a single large fire to burn two
reserve parcels. Although the reserve in figure 6a includes more species, in the presence
of spatially correlated risk, the benefit from reduced hazard risk makes the dispersed
reserve pattern the preferred choice.
Figure 6a: Oregon landscape
Figure 6b: Oregon landscape
4. Discussion and concluding remarks
In this paper, we develop a reserve site selection framework that addresses the issue of
risks to species within the reserve and that recognizes that those risks can be correlated
across space. We focus on an example of fires that spread from an ignition point but
outbreaks of diseases, pest infestations, and invasive species also produce risks that are
correlated across space. Some of the reserve site selection literature incorporates spatial
considerations such as the desire to have connectivity between reserve sites but spatially
correlated risk works in the opposite direction. Which consideration dominates will
reflect the particular setting but large fires, pests, and invasive species are increasingly
threatening species conservation in the western U.S.
In addition to further parameterizations and analysis of both our general framework and
the Oregon case, this model creates a platform for consideration of a variety of related
issues. For example, we intend to use this foundation to explore issues of risk
heterogeneity across a landscape. Similarly, adding a dynamic component with the
location of risk changing over time could provide information about siting reserves in the
context of climate change. Also, expanding the framework to consider the probability of
species survival as a function of connectivity of reserves, and thereby incorporating
concepts from the island biogeography literature, would allow us to test whether
connectivity or spatially correlated risk dominates reserve site selection in a particular
setting.
Our central result here is that spatially correlated risk creates incentives to spread reserve
sites apart from each other but the objective function and competing forces determine
whether that incentive dominates in the reserve pattern. Locating reserve sites at a
distance from each other minimizes the probability that no species will survive because
no single large, spreading fire can destroy species in two distant sites. Working in the
opposite direction, however, locating reserve sites near each other maximizes the
probability that all species survive when the probability of a large fire in any particular
location is relatively small. These forces interact with the objective function to determine
the optimal reserve pattern. We find cases in which maximizing the expected number of
species leads to indifference between agglomerated versus disaggregated reserves while a
more risk-sensitive objective that minimizes the chance of losing half or all species
produces a spatially disaggregated reserve network. We also find that when species are
distributed in a manner that creates “hotspots” with a high number of species, spatially
correlated fire risk can change the pattern of conservation with the hotspots becoming
less desirable due to their susceptibility to risk. The popular decision framework that
simply prioritizes conservation of hotspots does not consider risks and can lead to a
socially undesirable reserve network. Similarly, the current reserve site selection
literature’s emphasis on maximizing the expected number of species conserved may not
adequately address concerns and goals of some conservation managers in the face of risk.
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