Complementarity in the provision of ecosystem services reduces the cost of mitigating amplified natural disturbance events

Significance

Climate change has been implicated as a root cause of the recent surge in natural disturbance events, leading to a greater focus on disturbance-mitigating benefits of ecosystem management. Quantifying these benefits requires knowledge of the relationship between natural and anthropogenic disturbances, which is lacking at the temporal and spatial scales needed to inform ecosystem-based management. This study investigates a specific relationship between timber harvesting and climate-amplified insect outbreaks. If harvesting is located to mitigate long-distance insect dispersal, there is potential for a win–win outcome, in which both timber production and forest conservation can be increased. This spatially targeted harvesting strategy lowers the cost of providing disturbance-mitigating ecosystem services and produces net gains in forest conservation across various stakeholder groups.

Abstract

Climate change has been implicated as a root cause of the recent surge in natural disturbance events such as storms, wildfires, and insect outbreaks. This climate-based surge has led to a greater focus on disturbance-mitigating benefits of ecosystem management. Quantifying these benefits requires knowledge of the relationship between natural and anthropogenic disturbances, which is lacking at the temporal and spatial scales needed to inform ecosystem-based management. This study investigates a specific relationship between timber harvesting and climate-amplified outbreaks of mountain pine beetle. If harvesting is located to mitigate long-distance insect dispersal, there is potential for a win–win outcome in which both timber production and forest conservation can be increased. This spatially targeted harvesting strategy lowers the cost of providing disturbance-mitigating ecosystem services, because valuable timber products are also produced. Mitigating long-distance dispersal also produces net gains in forest conservation across various stakeholder groups. These results speak to ongoing federal efforts to encourage forest vegetation removal on public forestlands to improve forest health. These efforts will lower the cost of responding to climate-amplified natural disturbance events but only if vegetation removal efforts are spatially located to reduce disturbance risk. Otherwise, efforts to improve forest health may be converting forest conservation services to timber services.

Ecosystem service provisioning is a joint production problem using natural capital (1). Jointly produced services may be complementary or competitive. When they are complementary, the provision of one service accompanies the expansion of another service. When they are competitive, the provision of one can be increased by reducing the provision of another service. These relationships between ecosystem services are being altered by climate-amplified natural disturbance events (CANDEs), such as storms, wildfires, and insect outbreaks (2–5). CANDEs depreciate natural capital in unprecedented ways and elevate the importance of disturbance-mitigating ecosystem services (6). Because CANDEs differ from historical baselines, ecosystem service provisioning should be reevaluated (7, 8) with increased attention on the impact of climate on natural disturbance processes (9–11). Unfortunately, ecosystem service provisioning is often considered in the absence of natural disturbances (8, 12–15). We couple a forest assessment model with a stratified insect dispersal model to investigate management of a natural capital stock (forests) in response to a CANDE (insect outbreak). We show that seemingly competitive ecosystem services (forest conservation and timber production) become complementary over the course of the outbreak. This complementarity has the potential to decrease the future cost of mitigating CANDEs.

Forest management represents one of the few economically viable strategies available to mitigate CANDEs (16) and has been adopted on a large scale (17). Unfortunately, interactions between timber harvesting and natural forest disturbances are poorly understood (18). Large temporal and spatial variations, nonlinear or threshold changes, and inabilities to predict the effect of climate on natural disturbance processes cloud causal relationships (9, 19). The model developed here shows this interaction for the current mountain pine beetle (MPB) outbreak in the western United States and Canada. The current MPB (Dendroctonus ponderosae Hopkins) outbreak began in the late 1990s and is more severe (70–90% mortality of mature lodgepole pines) and expansive (88 million acres impacted) than any previous outbreak. MPB-induced forest mortality is resulting in billions of dollars in economic impact (20), converting vast tracts of forest to a carbon source (21) and occurring in new habitats with unknown ecological consequences (5). Climate change, fire suppression, and changes in timber harvesting have been implicated as causes of the outbreak (22). Here, we show how forest management could be altered to alleviate impacts to ecosystem services.

The spatiotemporal model combines timber harvesting, stratified MPB dispersal, and threshold effects important for explaining MPB outbreaks (23, 24). Local dispersal is governed by non-Fickian diffusion, whereas long-distance dispersal (LDD) is modeled using a logistic regression model that identifies probable locations for outlier outbreaks (Materials and Methods). The model is fit to a study area in northern Colorado to explain the spread and intensity of the current outbreak given harvest activities in the area. Alterations to the temporal and spatial profile of timber harvesting are used to predict the intensity and extent of future MPB outbreaks. Although spatial dynamic models of forest disturbance have received increased attention, most of the previous work abstracts from forest management (25, 26) or includes forest management in a stylized landscape (27, 28). The modeling approach that we developed could be applicable to any forest disturbance that is (i) spatially concentrated, with clear hotspots and dispersal of damage, and (ii) amenable to management by harvesting.

Northern Colorado is an appropriate study area because of the severity of the recent MPB outbreak in this area, the proximity to population centers, such as Denver and Fort Collins, and the variety of forest management objectives within the region. The 11,862-km² study area encompasses Rocky Mountain National Park, three national forests, four wilderness areas, the Arapaho National Recreation Area, Eldora ski area, resort communities, and smaller towns (Fig. 1A and SI Appendix, Data). The study area is placed on a Cartesian spatial grid with 120-m resolution, giving 823,760 total cells. Regions in the area range from no pine trees to dense forests with up to 1,400 trees per hectare (Fig. 1B and SI Appendix, Data). The US Department of Agriculture (USDA) Forest Service (FS) Aerial Detection Survey (ADS) is used to create a spatial database depicting the location and amount of MPB-induced tree mortality (redtops) from 1995 to 2010 (Fig. 1C and SI Appendix, Data). The uptick in MPB activity in 2000 is taken to be the start of the outbreak.

Fig. 1.

Characteristics of the northern Colorado study area. The black line delineates the boundaries of the study area. Where possible, boundaries were selected in areas with few pine trees to minimize MPB dispersal from outside of the study area. (A) USDA FS management areas reflect the diversity of management objectives. (B) The study area ranges from regions with no pine trees to dense forests with up to 1,400 trees per hectare. (C) ADS data of MPB-induced tree mortality in the northern Colorado study area. Nat., natural; Pub and Pvt, public and private.

To explore potential strategies to enhance disturbance-mitigating services, we contrast several alternative harvest scenarios to a benchmark that matches northern Colorado harvest activity from 2000 to 2013. It is constructed by proportionally allocating annual USDA FS measures of volume harvested to cells that (i) are forested, (ii) have an average slope < 20%, (iii) are in the appropriate national forest unit, and (iv) fall within management areas that allow timber harvesting (categories 3–5 in Fig. 1A). Model fit across the grid is summarized by comparing benchmark predictions and ADS data for the total number of redtops ($R^2=0.93$) and the total number of infested cells ($R^2=0.78$) (SI Appendix, Figs. S1 and S2).

Alternative harvest scenarios vary with respect to the timing, intensity, and location of harvest activities (SI Appendix, Table S1). The first scenario considers a harvest moratorium during the outbreak. The second and third scenarios increase the amount of harvesting on all cells in the benchmark scenario by 100%, 500%, and 1,000% after 2000 (reactive management) and from 1995 to 1999 (proactive management). The 500% increase roughly corresponds to harvest activity in timber-intensive areas of the western United States before 1990. In addition to increases in harvest timing and intensity, managers may alter the location of harvests to limit MPB dispersal. A fourth hotspot scenario focuses benchmark harvesting on areas where the predicted probability of becoming an outlier outbreak is larger than 8%, similar to current efforts to thin dense forests to prevent insect outbreaks and wildfires (17). A fifth rapid response scenario uses ADS data to strategically harvest in cells with redtops to prevent local spread, consistent with efforts to control the spread of invasive tree insects and diseases (18). Although political viability and cost-effectiveness vary across scenarios, comparisons identify adjustments in current forest management along the extensive (number of cells harvested), intensive (number of trees harvested), and temporal (harvesting before or during an outbreak) margins.

Results

Ecosystem Service Frontiers.

The dashed lines in Fig. 2 A and B represent ecosystem service frontiers during the outbreak—the maximum level of live trees (Fig. 2A) and undisturbed forest area (Fig. 2B) attainable for a specified level of timber production. Points inside the frontiers are technically inefficient in the sense that it is possible to increase forest conservation and timber production by moving to a strategy on the frontier. Technical efficiency can be improved without reducing the provisioning of either ecosystem service by altering current spatial and temporal patterns of timber harvesting to mimic hotspot and proactive (500% and 1,000%) strategies.

Fig. 2.

Tradeoffs between timber production and forest conservation are determined by the relationship between natural and anthropogenic disturbances. (A and B) Dashed lines represent ecosystem service frontiers—the maximum level of live trees and preserved forest area for a specified levels of timber harvesting. (C and D) Harvest strategies are grouped according to their impact on intensity and spatial extent of the outbreak compared with no harvesting.

Identifying the highest valued combination of forest conservation and timber (allocative efficiency) depends on social preferences for the ecosystem services and the shape of the frontiers. Movement along the frontiers captures the physical tradeoff between timber production and forest conservation. At high levels of timber production, the frontier is downward sloping and convex, suggesting that small increases in one service come at the expense of large decreases in the other. However, below current levels of timber production (<7 million trees), the slope of the frontier is positive, indicating the possibility of an increase in both ecosystem services, regardless of whether forest conservation is measured in terms of trees conserved or undisturbed forest area. This win–win outcome combined with a convex frontier at high levels of timber production make hotspot the preferred harvest strategy for a wide range of forest values (SI Appendix, Comparing Harvest Strategy Performance). Even in areas where live trees are significantly more valuable than harvested timber, a positive amount of timber harvesting can be justified during the outbreak. This motivation for timber harvesting is because of disturbance-mitigating benefits from harvest activities spatially located to mitigate long-distance MPB dispersal.

To explain how timber production and forest conservation can be complements, Fig. 2 C and D shows how timber production affects the intensity (cumulative redtops) and spatial extent (cumulative cells with redtops) of the outbreak. At endemic MPB levels, one tree harvested means roughly one less live tree in the forest. During an outbreak, this rate of exchange depends on the harvest strategy. Strategies are grouped according to impact on intensity and spatial extent relative to the harvesting moratorium. Moderate increases in harvesting (proactive, 100%; reactive, 100% and 500%) and harvesting to limit local MPB dispersal (rapid response) increase both the intensity and extent of the outbreak. Significant increases in harvesting above benchmark levels (proactive, 500% and 1,000%; reactive, 1,000%) reduce outbreak intensity but increase the spatial extent, which results in a net reduction in the stock of live trees (Fig. 2A). Only the hotspot strategy reduces outbreak intensity and extent enough to result in a net gain in live trees. If harvests are spatially located according to this strategy, harvesting one tree during an outbreak actually conserves more than one tree.

Given improvements in ecosystem service provision by adjusting harvesting at the extensive margin (spatial reallocation according to hotspot), could adjustments in harvesting at the intensive margin (number of trees harvested on hotspot cells) achieve additional improvements? Fig. 3 shows how each point on the frontier shifts up by spatially reallocating the reactive harvest strategy according to the hotspot strategy. This shift follows the disturbance-mitigating benefits from harvesting to limit LDD. For both measures of forest conservation considered, these ecosystem services remain complementary, with harvest levels higher than the benchmark level. An increase in timber harvesting could have been justified during the outbreak for disturbance-mitigation services if harvest activities had been spatially allocated to mitigate long-distance MPB dispersal.

Fig. 3.

Spatial reallocation of timber harvesting to mitigate long-distance MPB dispersal produces gains in both timber production and forest conservation during an outbreak. Dashed lines represent ecosystem service frontiers when the spatial location of timber harvesting is determined by management units and site conditions. Solid lines represent the same frontier when timber harvesting is reallocated to hotspot cells that exhibit certain risk characteristics, such as high density, low elevation, and a large amount of forest edge.

Ecosystem Service Winners and Losers.

The results in Figs. 2 and 3 implicitly assume that relative values for forest conservation and timber production are homogenous across the study area. These values can be expected to vary spatially, evidenced by the variety of USDA FS management categories. Also, the effect of timber harvesting on USDA FS parcels will spill over to private landowners and the National Park Service because of MPB dispersal. To determine how a switch to a hotspot strategy may spatially allocate ecosystem service winners and losers, we map the location of cells that experience a gain or a loss in live trees and timber production after a transition to a hotspot strategy (Fig. 4).

Fig. 4.

Winners and losers produced by moving to a hotspot harvest strategy during an MPB outbreak. Green cells represent areas that experience a net gain in (A) timber harvesting and (B) number of live trees when moving to a hotspot harvest strategy. Red cells represent areas that experience a net loss in (A) timber harvesting and (B) number of live trees when moving to a hotspot harvest strategy. All stakeholder groups and management categories experience forest conservation gains at the intensive (change in live trees) and extensive (change in forest area) margins.

Spatial variation in timber production values will likely be driven by differences in harvest costs between the red and green areas in Fig. 4A. The cost of applying a hotspot strategy could be higher in areas without ready road access or lower in areas where loggers can exploit returns to scale. Although these cost differences are not quantified, hotspot harvesting may be less costly, because it moves harvest activity to historically more timber-intensive areas in the western part of the study area. In terms of forest conservation, a hotspot harvest strategy conserves 17 million trees (over 5% of the live trees in the study area) and increases forest area by 397 km² (an area approximately the size of Denver). These gains may reduce the value of nontimber ecosystem services because of spatial variation in forest conservation benefits between the green and red areas in Fig. 4B. However, as long as the spatially adjusted difference in cost is less than the spatially adjusted value of increased ecosystem services, the hotspot strategy remains preferable.

In addition, the stakeholders (USDA FS, National Park Service, and private landowners) all experience net gains in forest conservation with a move to a hotspot harvest strategy. Thus, even if forest conservation values vary over space, a hotspot harvest strategy will be supported by all stakeholder groups in the study area. However, less than one-half of the gains in forest conservation accrue to the agency with timber harvest authority—the USDA FS. This spillover suggests that there is a potential for stakeholder conflict over the amount of harvesting under a hotspot strategy, even if all stakeholders benefit from a spatial reallocation.

Discussion

With predictions of more severe forest disturbances in the future, it is important to quantify the contribution of public forest management in an attempt to mitigate unintended climatic amplifications of natural forest disturbances. Our findings confirm previous research suggesting that partial harvests in advance of an outbreak can lower the amount of MPB-induced mortality (29–31). This conventional wisdom is also consistent with ongoing efforts to encourage forest vegetation removal to improve the health and resiliency of public forestland. For instance, the USDA FS received over $80 million to remove vegetation in response to recent fires, insect outbreaks, ice storms, and hurricanes as part of the American Recovery and Reinvestment Act of 2009. As part of the 2014 Farm Bill, Congress permanently authorized stewardship contracting on public lands, which provides commercially valuable timber in exchange for the removal of noncommercial forest vegetation that would improve forest health. Although these policies may improve technical efficiency by moving closer to the ecosystem service frontier, our findings suggest that they will not yield the highest valued combination of forest conservation and timber unless vegetation removal efforts are spatially located to reduce disturbance risk. For the MPB outbreak case that we consider, vegetation removal efforts must be focused in areas with a high risk of becoming outlier outbreaks.

Our findings also show that improvements in ecosystem service provisioning can be achieved by adapting to a CANDE after it begins, even when active management is not feasible on large portions of an at-risk forest. This result is of practical importance for forest managers, because predicting the start of an insect outbreak can be difficult; also, the majority of public forestland in the United States is incompatible with active management because of either physical characteristics (e.g., steep slopes) or land designations (e.g., wilderness areas). However, there are two important points to consider for future research. First, a more thorough understanding of the spatiotemporal relationship between forest management and MPB dispersal will be critical for limiting damages from future outbreaks. The dispersal model correctly predicts over 28% of the outlier outbreaks created through LDD. Efforts to increase this percentage will likely yield additional conservation benefits from hotspot harvesting. Second, whereas our analysis tests various assumptions concerning the value of provisioning timber and forest conservation, it does not consider the response of the local timber market to the large increases in timber production considered in many of the harvesting scenarios. For example, large increases in timber production during a CANDE will flood local timber markets, resulting in higher costs, because harvested timber must be transported farther to reach mills with available capacity. This surge in timber supply will likely make harvest strategies that do not increase timber harvesting in response to a CANDE (e.g., hotspot harvesting) even more attractive.

Materials and Methods

The coupled forest assessment–insect dispersal model is spatially dynamic, with space represented by a Cartesian grid and time set in annual increments to match the MPB lifecycle $(t=1,\dots ,T)$. Cells are square, with cell size determined by its length ∆x (i.e., ∆x = 1 km amounts to each cell being 100 ha). We assume cell sizes of 120 × 120 m (∆x = 0.12). Descriptions of the variables are presented in SI Appendix, Table S2.

Forest Dynamics.

The forest in each cell $(i,j)$ is divided into two categories: juvenile trees $J^{(i,j)}$ and adult trees $A^{(i,j)}$. The density (stems per kilometer²) of trees in each cell varies. Juvenile trees (diameter at breast height ≤ 8 in) are considered immune to MPB attack. The juvenile stock on each cell at time $t$ is divided into M age classes and given by $J_{\mathrm{age},t}^{(i,j)}$ with $\mathrm{age}=1,\dots ,M$. The initial number of stems in each juvenile age class is inversely related to the density of adult trees on the cell:

$$J_{\mathrm{age},0}^{(i,j)}=\frac{\underset{(i,j)}{\max }\left[A_0^{(i,j)}\right]-A_0^{(i,j)}}{bM},$$ [1]

where $A_0^{(i,j)}$ is the initial density of adult trees in each cell, and b is a positive scalar. This specification implies more juvenile trees in cells with few adults, a homogeneous age structure for juvenile trees across the grid, and seedlings that are always available to occupy gaps left by removed adults. Each year, $d=2\%$ of each age class dies, and the remaining stock matures to the next age class:

$$J_{\mathrm{age}+1,t+1}^{(i,j)}=\left(1-d\right)J_{\mathrm{age},t}^{\left(i,j\right)},$$ [2]

for $\mathrm{age}=1,\mathrm{...},M-1$. Each year, the oldest juveniles, $\mathrm{age}=M$, move into the adult category.

Adult trees (diameter at breast height > 8 in) are susceptible to MPB-induced mortality and harvesting. The law of motion for the beginning-of-period density in the adult category is given by

$$A_{t+1}^{(i,j)}=A_t^{(i,j)}+J_{M,t}^{(i,j)}-R_t^{\left(i,j\right)}-h_t^{\left(i,j\right)},$$ [3]

where $h_t^{(i,j)}$ is the commercial harvest of adult trees, and $R_t^{(i,j)}$ is the number of trees that is successfully attacked by MPB in year $t$ and will die in the following year. For simplicity, the model does not consider germination of seeds that may contribute to the juvenile age class. If the model is simulated for $T>M$ periods, this simplification implies an eventual lack of new growth into the adult age class, because all juveniles have already moved into the adult class. In these instances, the initial juvenile age class will need to be reseeded.

MPB-Induced Mortality.

For MPB populations to reach epidemic levels, two requirements must be satisfied. First, there must be a sustained period of favorable weather over several years (32). Weather conditions during the dispersal period and larval development can influence MPB populations directly through survival and/or indirectly through impacts to host tree resistance and timing of new beetle emergence (33). Second, there must be an abundance of susceptible host trees (32, 34). An abundance of susceptible host trees combined with homogenous forest structure aid in moving from endemic to epidemic MPB populations (35). Thus, as areas that are climatically favorable for MPB increase because of climate change, forest structure will be the primary factor influencing the probability and severity of MPB outbreak (36).

Assuming that it takes 1 y for a successfully attacked tree to turn red, the number of new successfully attacked trees in cell $(i,j)$ in year $t$ is

$$R_t^{\left(i,j\right)}=\min \left(z{E}_{t-1}^{\left(i,j\right)},A_t^{\left(i,j\right)}\right),$$ [4]

where $E_{t-1}^{(i,j)}=\max \left(Q_{t-1}^{(i,j)}-D,0\right)$ is the number of effective MPBs that exceeds a critical attack threshold, $D$, in cell $(i,j)$, and $z$ is the number of new redtops per effective MPB. The total number of beetles at $t-1$ arriving in cell $(i,j)$ from all other cells is

$$Q_{t-1}^{\left(i,j\right)}={\left(\mathrm{\Delta }x\right)}^2\left[{\displaystyle \sum _{\left(l,k\right)}{P}_{t-1}^{\left(i,j\right)\leftarrow \left(l,k\right)}+P_{t-1}^{\left(i,j\right)\leftarrow \text{long}}}\right],$$ [5]

where $P_{t-1}^{\left(i,j\right)\leftarrow \left(l,k\right)}$ captures local dispersal from cell $(l,k)$ to cell $\left(i,j\right)$, and $P_{t-1}^{\left(i,j\right)\leftarrow \text{long}}$ captures the density of beetles that spread to cell $\left(i,j\right)$ through LDD.

Local Flight Dispersal.

During local flight dispersal (LFD), beetles fly above the forest understory but below the tree crowns. Most locally dispersed beetles locate suitable host trees within 2 d of emergence by flying downwind until an attractive odor plume is encountered (37). The time that it takes to locate a suitable host tree determines the distance traveled during local dispersal, but typically, beetles will travel no more than 100 m through LFD (38, 39).

The density of beetles (number per kilometer²), which connects LFD from cell $(l,k)$ with the central location $x^{(l,k)}$ to cell $\left(i,j\right)$ with the central location $x^{(i,j)}$, is $P_{t-1}^{\left(i,j\right)\leftarrow (l,k)}$. The in-season (that is, during the few summer weeks during which MPBs fly) dispersal of MPB is assumed to follow the ecological diffusion model (40):

$$\frac{\partial B\left(x,s\right)}{\partial s}={\nabla }^2\left[{\mu }_{t-1}^{\left(l,k\right)}B\left(x,s\right)\right],$$ [6]

where ${\mu }_{t-1}^{\left(l,k\right)}$ represents MPB motility in cell $\left(l,k\right)$ at time $t-1$, and $B\left(x,s\right)$ is the density of (locally) dispersing beetles at $x$ on day $s$ of dispersal (23). Motility in this model is inversely proportional to residence time (41) in a patch. For MPB, large motility would be expected in regions with few trees (where beetles have no reason to linger), whereas motility should decline sharply in patches that are densely stocked with pines (because beetles spend more time interacting with potential hosts). Unlike a normal diffusion model, which eventually spreads all dispersers infinitesimally thin, the ecological diffusion model distributes dispersers into habitats in proportion to the residence time. Following the information in ref. 23, we assume that motility (kilometers² per day) in cell $\left(l,k\right)$ at time $t‒1$ is an exponentially declining function of adult host density in cell $\left(l,k\right)$:

$${\mu }_{t-1}^{\left(l,k\right)}={\mu }_0 \exp \left[-\left({\mu }_1+\log \left({\mu }_0\right)\right)\frac{A_{t-1}^{\left(l,k\right)}}{\mathrm{1,000}}\right],$$ [7]

where ${\mu }_0$ is the maximum beetle dispersal per year in an area with no pine trees, and ${\mu }_1$ is a constant describing the rapid decline of motility in the presence of hosts. To predict the number of beetles dispersing from one cell to another cell, we solve Eq. 6 with the initial condition

$$B\left(x^{\left(l,k\right)},s=0\right)=F\theta R_{t-1}^{\left(l,k\right)},$$ [8]

where F is the number of emerging MPBs per host, and θ is the proportion of emerging MPBs that engage in LFD. Then, the density of beetles arriving at cell $(i,j)$ from cell $(l,k)$ can be written as

$$P_{t-1}^{\left(i,j\right)\leftarrow \left(l,k\right)}=B\left(x^{(i,j)},s=1\right).$$ [9]

Here, individual beetles are assumed to die if they have not found a host within 1 d of leaving their brood tree $\left(s\in \left[\mathrm{0,1}\right]\right)$, and $s=1$ is the end time for dispersal (1 d).

Simulating the model described above requires numerically solving the ecological diffusion equation every simulation year distributed over all cells in space. To implement the dispersal simulations efficiently, we used homogenization to transform Eq. 6 into a related partial differential equation on much larger scales, effectively reducing the number of cells without sacrificing accuracy (40), and then, we solved the equation numerically using the alternating direction implicit (ADI) method (42). The ADI approach is a standard 2D solution procedure for diffusion equations that replaces the usual implicit discretization (which requires inversion of a large, sparse matrix system) with layered tridiagonal systems in alternating directions. Because the ADI and homogenization procedures may not exactly conserve the number of beetles each year, we calculated the total number of emerging beetles initially, normalized solutions, and multiplied by the starting number of beetles to conserve beetles during the solution procedure.

LDD.

There is evidence of LDD at the scale of tens (43, 44) and hundreds (35) of kilometers because of convective winds that transport beetles above the forest canopy. This LDD can create outlying spots of infestation beyond the main zone of the outbreak. These outlier spots represent new initial conditions for each time step of the LFD model and ultimately, may coalesce with the main outbreak or other outlier outbreaks. LDD through pathways, such as wind and human intervention, is an inherently random event, but after MPBs arrive at a distant location, there may be features of the landscape that increase the likelihood that a particular cell is successfully attacked. Previous studies indicate that successful bark beetle attack can be predicted using elevation, direct solar radiation, forest density, beetle pressure, and amount of forest edge (25, 45–48). Our methodology uses these natural features of the landscape to predict where outlier outbreaks of MPB are likely to establish.

Not all new redtop trees in a given year arise from LDD, and therefore, we begin by defining the area that qualifies for LDD. To identify LDD areas, we create a weighted measure of the cumulative number of redtops within a 5-km radius of each cell in our study area ($CRT$). The weights decline linearly away from the center of the circle, with a weight of one at the center and a weight of zero at 5 km. Any cell $(i,j)$ with $CR{T}^{(i,j)}<5$ that was aerially surveyed is considered a candidate for LDD. LDD candidate cells are updated each year as the outbreak spreads.

We use a logistic regression model and 3 separate y (2000, 2005, and 2010) of USDA FS ADS data to estimate the probability that an LDD candidate cell will become an outlier. The sample size or total potential number of cells per year of observation for LDD is $N=\mathrm{430,280}$. Each LDD candidate cell either contained redtops [$y^{\left(i,j\right)}=1$] or had no redtops [$y^{\left(i,j\right)}=0$]. According to ADS data, only 26,833 cells contained redtops and are actual outliers. The probability that cell $\left(i,j\right)$ will harbor an outlier population is

$$\text{Pr}\left(y^{(i,j)}=1|{\mathbf{u}}^{(\mathit{i},\mathit{j})\mathit{\text{'}}}\mathit{\beta }\right)={\pi }^{(i,j)}=\frac{{\mathit{e}}^{{\mathbf{u}}^{(\mathit{i},\mathit{j})\mathit{\text{'}}}\mathit{\beta }}}{1+{\mathit{e}}^{{\mathbf{u}}^{(\mathit{i},\mathit{j})\mathit{\text{'}}}\mathit{\beta }}},$$ [10]

where ${\mathbf{u}}^{(\mathit{i},\mathit{j})}$ is a vector of cell-specific factors that influences the probability of MPB outlier outbreaks (SI Appendix, Table S3), and $\mathit{\beta }$ is a vector of logistic coefficients.

The signs of all coefficients match our expectations (SI Appendix, Predicting Outlier Outbreaks). Using the estimated coefficients (SI Appendix, Table S3) and the values of the explanatory variables on each cell, Eq. 10 gives predicted probability ${\widehat{\pi }}^{(i,j)}$ for LDD on each candidate cell. The predicted probabilities are compared with a threshold probability, $\overline{\pi }$, to determine whether cell $(i,j)$ experiences LDD. Predicted values ${\widehat{\pi }}^{(i,j)}>\overline{\pi }$ result in LDD and successful MPB attack [${\Omega }^{(i,j)}=1$]; predicted values ${\widehat{\pi }}^{(i,j)}<\overline{\pi }$ result in no LDD [${\Omega }^{(i,j)}=0$]. We also incorporate specific local and global processes that cause LDD to evolve as redtops occupy more cells (SI Appendix, Predicting Outlier Outbreaks). The logit LDD model performs significantly better than a completely random LDD process. The logit model correctly predicts about 28% of actual outlier cells, whereas the random process correctly predicts only 6% of actual outlier cells.

After a cell is identified as supporting successful LDD, it is seeded with a portion of the emerging beetles. Specifically, the proportion of all emerging beetles from across the grid that engage in LDD is evenly distributed to all cells identified as LDD recipients:

$$P_t^{\left(i,j\right)\leftarrow \text{long}}={\Omega }_{ij}F\left(1-\theta \right){\displaystyle \sum }_{\left(l,k\right)}{R}_t^{\left(l,k\right)}/{\displaystyle \sum }_{\left(l,k\right)}{\Omega }^{\left(l,k\right)},$$ [11]

where ${\displaystyle {\sum }_{\left(l,k\right)}{R}_t^{\left(l,k\right)}}$ is the aggregate number of redtops in year $t$, and ${\displaystyle {\sum }_{\left(l,k\right)}{\Omega }^{\left(l,k\right)}}$ is the total number of new outlier spots.

Parameterizing Dispersal and Model Validation.

Parameters governing LFD, attack success, and reproductive success (SI Appendix, Table S2) were determined by fitting model predictions to ADS observations in the Sawtooth National Recreation Area of central Idaho for 1995–2005 using maximum likelihood to optimize correspondence between predicted and observed presence on 25-ha blocks. LFD was predicted under the assumption that individual beetles were only capable of 1 d of travel. Simulations using selected parameters for ${\mu }_0$, ${\mu }_1$, $F$, and $z$ reproduced observed spatial patterns in the Sawtooth National Recreation Area with 85% accuracy, and population growth rate dynamics were resolved with 92% accuracy (23). These parameters were also used to test model validity in the Lake Chelan region of the eastern Cascades. Without any additional parameterization, model correspondence with observations in the Chelan study area was over 85% in space, and growth rate dynamics were 93% accurate. Thus, the model captures fundamental mechanisms of MPB attack and dispersal and not simply details specific to the area in which it was parameterized.

To balance LDD and short-distance dispersal, we set $\theta =0.975$ based on evidence that up to 2.5% of a population may engage in LDD (38). Rate of natural tree mortality ($d$) was set at 2% based on the work by Runkle (49). The juvenile age class scale parameter ($b$) was selected to ensure that no more than 90 juvenile trees were in any point on the grid.