Spatial Bioinvasion Externalities with Heterogeneous Landowner Preferences: A Two-Agent Bioeconomic Model

Bioeconomic Model of Externality Dispersal and Control

The bioinvasion disperses on a network composed of two grids managed independently (decentralized management [DM]) or jointly (central planner management [CP]). The grids represent forestlands of landowners A and B and are linked through the SDD and LDD of the invasive plant. Landowner A’s recreation value is the product of their consumer surplus (CS) per recreation day and their average user days (UDs) (Walsh and Olienyk 1981; Rosenberger et al. 2013). Both CS and UD are quadratic functions of the longest connected subnetwork of uninvaded trees, representing an accessible forest recreation trail. In Section 4, I consider the case of other, nonrecreation NMES. The bioinvasion affects landowner B’s net revenue through the inhibited or delayed growth of young trees in invaded stands: adult trees are not affected, but their ability to regenerate is inhibited or delayed based on whether control is from the mechanical removal of the shrub. The control by each landowner determines their net benefits and those of the other landowner because they are connected through a biophysical network of invaded and uninvaded trees.

In Figure 1, grid G_A represents forestland A and is the set of I × J cells, denoted by their row and column position (i, j). Each cell (i, j) represents a pine tree. Similarly, grid G_B represents forestland B and consists of M × N cells, denoted by their row and column position (m, n). Each pine tree updates its age-diameter (i.e., timber yield and quality) and infestation states in discrete time steps (t) based on its own current age-diameter and infestation state and the infestation state of its neighbors (i.e., SDD process), that of non-neighboring pine trees (i.e., LDD2 process), and random arrivals of dispersed seeds from outside the landscape (i.e., LDD1 process). Each tree’s infestation state transitions are governed by a discrete-time Markov chain model (i.e., the probability distribution of the next state depends only on the current state and not on the sequence of events that precede it). A tree in a cell (i, j) or (m, n) and state Healthy (H) transitions to state Invaded−undetectable (I_u) once seeds are dispersed to it, and they successfully germinate. This transition probability depends on the number and location of invaded pine trees on the landscape. This state is unobservable to the landowner because it consists of seed arrival and germination with no recognizable above-ground plant parts. Subsequently, a tree transitions to the third state, Invaded−detectable (I_d), which is only observable to a landowner who conducts surveillance and is trained to identify the characteristic stem and leaves of the shrub. The transition to state Invaded−moderate (I_m) and later to state Invaded−high (I_h) occurs as the invasive plant grows, forms a clump, and produces berries that can then be dispersed to the landscape via the SDD and LDD2 processes. In the baseline model, I consider that landowners can only observe states I_m and I_h. In Section 4, I consider the case of surveillance and early detection where landowners are trained to identify and treat trees in state I_d).

An externality emerges when the privately optimal management strategy in one forestland is lower than optimal from a landscape perspective, which causes the bioinvasion to spread to the neighboring parcel, thus affecting the neighboring landowner’s net benefits (see Figure 1).

Economic Model

Each landowner’s objective is to maximize the present value of net benefits by choosing a time-varying invasive plant control strategy, 𝒲_t. The optimal control strategy is a set of cell-level control variables {z_{i, j, t}} equal to one if control (i.e., invasive plant removal through repeated stem cutting) takes place in cell (i, j) and year t and zero otherwise. For the baseline model description, I assume that landowner A has NMES preferences and landowner B has MES preferences. I consider later situations where they both have the same preferences. For simplicity of notation, I limit the model description to one game stage here and describe the sequential, repeated game below (“Decentralized Management and Central Planner Problems and Solution Procedure”).

The objective of landowner A is to choose the set of binary control variables 𝒲_t that maximize the following expected discounted sum, over 25 years (T), of net recreation values, where ρ is a yearly discount factor:

1

2

3

In equation [1], as in previous theoretical (e.g., Koskela and Ollikainen 1999) and empirical (e.g., Walsh and Olienyk 1981; Rosenberger et al. 2013) models of forest recreation, CS is a concave function of the recreation services or the trees providing these services. I follow Walsh and Olienyk (1981) and Rosenberger et al. (2013) to represent the total recreation value as the product of CS per day and the average UD, which are contingent on the number of healthy trees. However, to recognize that bioinvasion damages can be spatially nonuniform, I model the ES damages and, as a result, the recreation ES production function in a spatially explicit way. Accordingly, in equation [3], for any healthy tree (v_{i, j, t}=1) to count as part of a recreation trail in period t, both of its neighboring trees on that trail (v_{i+1, j, t} and v_{i, j+1, t}) need to be healthy. This spatially explicit damage is different from that of an invasive species on an NMES, such as air purification, where the location of the damage does not matter for the benefits, only for the total number of live trees (e.g., Walsh and Olienyk 1981; Rosenberger et al. 2013). In Section 4, I relax this constraint to consider the case of nonrecreation NMES. The CS and UD terms of the net benefit function are quadratic in the trail length. Mathematically, , where a₀<0, a₁<0, a₂<0 is the CS per recreation day, and , b₀>0, b₁>0, b₂>0 is the average UD. In equation [1], z_{i, j, t} is one if control takes place in cell (i, j) in year t, with unit treatment cost c_A and zero otherwise.³ In equation [2], v_{i, j, t} is one if cell (i, j) is in state Healthy (H) in year t and zero otherwise. Equation [3] is the bioinvasion equation of motion, specified as a cell-level infestation state transition equation, where s_{i, j, t} is the infestation state of cell (i, j) in period t, and P is the infestation state transition matrix defined in the following section. Because it incorporates the LDD2 mechanism, this infestation state transition captures the effect of the neighbor’s actions.

Landowner B’s benefits are from timber revenues, which are a linear function of the number of pine trees, their age-diameter, and timber values, realized in year T.⁴ The objective of a type B landowner is as follows:

4

5

In equation [4], revenue r_d,m,n,t in cell (m,n) in year t depends on timber yield y_m,n,t, which depends on tree age (juvenile, young, mature): r_d,m,n,t=0 for juveniles, r_d,m,n,t=py_m,n,t for young and mature trees. Juveniles have no commercial value, while young and mature trees command net price p and produce yields y_m,n,t,young and y_m,n,t,mature, respectively, where y_m,n,t,young<y_m,n,t,mature. The transition from juvenile to young occurs after τ_young years if a tree in cell (m,n) is healthy and the eight immediately neighboring trees are also healthy. The juvenile tree does not transition to young unless the invasive shrub is removed, which represents the process by which the invasive shrub can inhibit natural regeneration. The transition from young to mature occurs after τ_mature years and is not impacted by the bioinvasion (because the shrub cannot shade young trees). The damage of the bioinvasion for G_B occurs through the inhibition of the transition from age-diameter state τ_young to age-diameter state τ_mature in the absence of control and through the delay of this transition depending on whether, how much, and when control takes place in cell (m,n) and in its immediate neighborhood. As opposed to the NMES case, I assume that the MES damage is not spatially explicit (i.e., the MES value of a tree and bioinvasion damages do not depend on its location on the forestland). As in equation [1], z_m,n,t is one if control takes place in cell (m,n) in period t, with unit treatment cost c_B and zero otherwise. I define model parameters and report their baseline values in Table 1.

Model of Spatial-Dynamic Externality Dispersal

The invasive plant is introduced to the forestlands by bird-dispersed seeds according to a recurrent random exogenous long-distance dispersal event (LDD1). Subsequently, a transition from state Healthy (H) to state Invaded−undetectable (I_u) occurs according to probability α₁ that drives both the SDD and LDD2 dispersal processes (equations [6] and [7]). According to the SDD process, in each time step, a healthy pine tree can receive invasive plant seeds at time t+1 from any of its eight neighboring trees if they are in state Invaded−moderate (I_m) or state Invaded−high (I_h) (i.e., rate Nα in equation [7], where N is the number of neighboring invaded trees). Seeds successfully germinate if and do not germinate if , where u_t is a random draw from U ~ (0, 1). An LDD mechanism that is endogenous to the model (LDD2), causes trees in state H to transition to state I_u with a distance and density-dependent probability with rate γ_B,A,t (equation [7]).

The transition to states Invaded−detectable (I_d), I_m, and I_h occur with probabilities α₂, α₃, and α₄, respectively. I define these probabilities and their associated parameters in Table 1. First, I focus on probability α₁, which drives the distance and density-dependent specification of the externality. This probability depends on the number and location of trees in state I_m or I_h within the same forestland and those in the neighboring parcel. The distance and density-dependence of this probability capture the impact of a landowner’s private bioinvasion control actions, within their parcel, on the spatial damages borne by their neighbor at the border of and within the adjacent parcel. The transition matrix P in equations [3] and [5] govern the SDD and LDD mechanisms. It can be expressed as follows:

6

In equation [6], limiting the notation to G_A for exposition purposes, α₁ can be expressed as

7

Probability α₁ is determined by the SDD (i.e., αN_i,j,t term), LDD1 (i.e., ε_t term), and LDD2 (i.e., γ_B,A,t term) mechanisms. Parameter α represents the rate at which seed arrival and germination occurs in a cell (i, j). In each time step, seed arrival in a cell leads to germination and establishment if 1−e_−α is greater than a random draw from U​(0, 1). The SDD term also depends on the state of the SDD neighborhood of cell (i, j), which is denoted by and describes the number of cells in the immediate neighborhood that can act as a bioinvasion source (i.e., in state I_m or I_h). Accordingly, and the first term of the exponential rate will have {0,...8α}, depending on the number of immediate neighbors that are in state I_m or I_h. The LDD2 mechanism in equation [7] is represented by the dynamic rate γ_B,A,t, which is a power-law dispersal parameter specified by the spatial-dynamic, distance- and density-dependent dispersal function defined in equation [8a] (Atallah, Gómez, and Conrad 2017). Indicator variables x and y, which equal one if a pine tree in row m and column N is in state I_m or I_h and zero otherwise are used to calculate the total number of invaded pine trees in each period that contribute to the LDD2 process from G_B to each cell in G_A. If x=0, the corresponding forestland rows that have y=1 contain pine trees in state I_m or I_h that contribute to the LDD2 from G_B to G_A. If x=0 for all columns N (i.e., there are no cells on G_B that have glossy buckthorn producing berries that can be dispersed to G_A), γ_B,A,t is not defined, and no dispersal occurs from these columns:

8a

The transition rate γ_B,A,j,t is inversely proportional to the distance from the bordering column (i.e., distance from column j on G_A to column N on G_B, regardless of its row position in column j). I chose a power-law specification because it allows the bioinvasion LDD to have new foci emerging beyond the bioinvasion front, which is appropriate for modeling bird-mediated seed dispersal. Dynamic parameter γ_B,A,j,t is also proportional to the total number of pine trees that are in state I_m or I_h on G_B, weighted by their column position j (the numerator in equation [8a]) so that invaded cells closer to the bordering column contribute more to the externality than cells situated farther from the boundary.

Similarly, dispersal from G_A to G_B is given by

8b

This specification of LDD2 is adapted from the externality specification in Atallah, Gómez, and Conrad (2017), the only differences being the cell neighborhood type and the addition of the LDD1 term and is in contrast with fixed externality dispersal rates or dispersal from cell to cell only that are common in the extant resource and environmental economics literature.

Model Initialization

All pine trees are initialized as state Healthy (H). In each forestland G_A and G_B, half of the trees are initialized in the juvenile state and half in the mature state, reflecting a mixed-age forest structure. At the first-time step and the beginning of each year, 2% of each parcel is invaded through an exogenous inflow of the invasive plant’s seeds (LDD1, or ε_t in equation [7]). That is, cells in each grid are chosen at random to transition from state Healthy (H) to state Invaded−undetectable (I_u). The invasive plant grows, and the affected trees transition to Invaded−detectable (I_d). In this state, the invasive plant produces seeds that can be dispersed to neighboring and non-neighboring cells according to equations [3] and [5], and the dispersal occurs both within (SDD) and between (LDD2) forestland parcels.

Decentralized Management and Central Planner Problems and Solution Procedures

I first solve the DM problem as a forward-looking, noncooperative, five-stage, sequential, repeated game. In each stage, a landowner decides on the timing and intensity of bioinvasion control that maximizes their expected discounted net benefits as defined in equations [1] and [4]. Landowners make their decisions public and consider each other’s decisions when making their own. The bioinvasion starts on one forestland, and the landowner affected first makes bioinvasion control decisions on the timing and intensity of control over the entire time horizon, considering that in stage 1, there is no bioinvasion and therefore no control on the neighboring forestland. In stage 2, the bioinvasion arrives at the neighboring forestland (at a level that depends on the control actions of the first landowner), and the second landowner makes control decisions on the timing and intensity of control on their property for the remaining time horizon, taking into consideration the other landowner’s stated plan for future bioinvasion control. The game is repeated in stages 3, 4, and 5, representing five years each and a time horizon of 25 years. To illustrate, in a repeated game where the bioinvasion starts on G_A, and landowner A moves first, A moves at t = 1 and makes a discounted expected net benefit maximizing management plan for the entire time horizon. At t = 6, landowner B moves and makes a management plan for the remaining time horizon (t = 6–25), considering landowner A’s previous actions and their stated plan for the remaining time horizon. At the beginning of stages 3 (t = 11–25), 4 (t = 16–25), and 5 (t = 21–25), each landowner updates their optimal time path, taking into consideration past payoffs and updated future control plan of their neighbor. This game setting assumes that players share information by making their bioinvasion management plan known to the other player and abide by it in each stage but can change the plan for the subsequent stages.

Second, I solve the CP problem. I consider the case of a fully informed CP who values the sum of the net benefits by landowners A and B and determines the timing and intensity of invasive shrub removal in forestlands G_A and G_B. In this sense, the CP solution also represents the point of view of the sole owner of both forestlands or that of the CP of both forestlands. From a landscape perspective, the CP solution achieves the optimal level, location, and timing of control by accounting for bioinvasion externalities across the two forestlands. The CP chooses two sets of bioinvasion management strategies, and (one for each forestland), that maximize the present value of total expected payoffs, π_CP, defined as the sum of the present value of expected payoffs of G_A and G_B, π_A and π_B, respectively.⁵ The CP solves the following maximization problem, subject to equations [2], [3], and [5]:

I define the social cost of the externality between the two forestlands as the difference between the CP expected payoff, E(π_CP) and the aggregate expected noncooperative payoff . I consider the effect of subsidies and rank them based on the net aggregate payoff they generate, which I define as aggregate payoff net of total subsidy costs (i.e., unit subsidy × realized total amount of control).

I obtain the optimal timing and intensity of control using a metaheuristic optimization engine OptQuest, which uses an algorithm combining Tabu search, scatter search, integer programming, and neural networks (Kleijnen and Wan 2007). (Tabu search helps avoid the trap of local optimality.) For each landowner and the CP, I report the optimal expected value and standard deviation for the net benefits and the number of treatments, obtained from 100 simulation runs. Outcome realizations for a run in a Monte Carlo simulation differ by the random location of initial arrivals of the bioinvasion, the random location of the yearly exogenous arrivals (i.e., LDD1 process), and the random success of seed germination and establishment conditional on arrival within (i.e., SDD process) and between (i.e., LDD2 process) forestlands. The simulation and optimization models are written in Java using the AnyLogic software, which integrates the OptQuest optimization engine.

Externalities under the Preference Homogeneity Case

When both landowners are the NMES type, neither controls the bioinvasion in a repeated sequential game (Table 2). Expectedly, in the absence of control, the payoff is lower on the forestland where the bioinvasion starts, even though they value the resource similarly and face the same damages and control costs. In contrast to the NMES landowner’s no control strategy, a CP’s optimal strategy is to control continuously on both forestlands, starting control simultaneously on the parcels but at a higher intensity where the bioinvasion starts (783 vs. 682 treatments).⁶ The DM aggregate payoff is 48% lower than that of a CP and is caused by the landowners’ privately optimal decisions not to control (see Table 2). This result of no control by the NMES landowner is partly driven by the magnitudes of the bioinvasion damages (constraint in equation [4] and equation of motion in equation [5]) and control costs.

When both landowners are the MES type, they control the bioinvasion in stage 3 (years 13 and 14) and stage 4 (years 15–20) (Figure 2). The control timing of the MES landowners is determined by their net benefit and damage functions: their objective is to maximize profits by “freeing” young trees from the invasive shrub so they can reach the higher-valued mature age classes. The DM aggregate payoff is 11% lower than that of a CP (Table 3). While the 48% difference between CP and DM in the case where both landowners have NMES preferences stems from undercontrol (see Table 2), the 11% difference here stems from overcontrol (2,165 vs. 1,688 treatments; Table 3), which is caused by the sequential nature of the DM case and the resulting misalignment of the timing of control on both forestlands, despite their identical preferences. Landowners update their plans in response to the realized plan of their neighbor, which results in increases in their payoffs compared with their expected payoffs at the beginning of the game. For example, updating strategies as a result of the repeated nature of the interaction decreases the CP-DM difference from 29% (in stage 1, not shown in Table 3) to 11% (by stage 5) in the repeated game with two MES landowners.

• Download figure
• Open in new tab
• Download powerpoint

Figure 2

Optimal Bioinvasion Control in Forestlands G_A and G_B with Market Ecosystem Service Preferences: left, Bioinvasion Starts on G_A, Landowner A Moves First; right, Bioinvasion Starts on G_B, Landowner B Moves First

Externalities under the Preference Heterogeneity Case

In the preference heterogeneity case, I consider that landowner A has NMES preferences and landowner B has MES preferences. When the bioinvasion starts on G_A, and A moves first, the DM solution consists of A not controlling at all and B controlling the invasive plant in years 14, 15, and 23 (Figure 3a.1) at a level that is higher than that of the CP (1,543 vs. 689 treatments; Figure 3a.1 vs. 3a.3; Table 4, panel B). Landowner A overcontrols relative to the CP because of landowner B’s absence of control in a DM setting. Both A’s and B’s actions are consistent with their actions under the preference homogeneity cases where B controls and A does not (see Tables 2 and 3). The social cost of the externality generated by A’s absence of control is $7,591 for the two acres (Table 4, panel B), a value that is statistically different from zero at the 1% level. The DM aggregate payoff is 40% lower than that under a CP case. Note that landowner A’s absence of control is due to the bioinvasion dispersal across properties. That is, when forestlands G_A and G_B are ecologically isolated (i.e., no LDD2), landowner A controls as much as landowner B (Table 4, panel A).

• Download figure
• Open in new tab
• Download powerpoint

Figure 3

Optimal Bioinvasion Control with Nonmarket Ecosystem Service Preferences for Landowner A and Market Ecosystem Service Preferences for Landowner B: (a) Bioinvasion Starts on G_A, A Moves First; (b) Bioinvasion Starts on G_B, B Moves First

When the bioinvasion starts on G_B, and B moves first, landowner B controls earlier (year 11 vs. 14; see Figure 3b.1) and less intensely (785 treatments in Table 4, panel C vs. 1,543 treatments in panel B) than they do when landowner A moves first. The change in the first-move order affects the social cost of the externality, which is reduced by 31% (from $7,591 to $5,187), and the aggregate payoff increases by 7% (from $11,263 to $12,060) (see Table 4, panels B and C). Landowner B’s first move confers benefits at the landscape level through bioinvasion control on their property that consist of an increase in landowner A’s payoff from $831/acre to $1,569/acre (see Table 4, panels B and C), even if no control occurs on G_A in either case. The increase in landowner A’s payoffs amounts to $797/acre and is statistically different from zero at the 1% level. The DM aggregate payoff is 30% lower than that of a CP, compared with 40% in the case where the bioinvasion starts on G_A, and A moves first. In sum, the social cost of the externality is lowest when the MES landowner is affected first, in which case they control earlier and less intensely than when they move second.

Effect of Preference Heterogeneity on Welfare

To measure the effect of preference heterogeneity on the sum of landowner payoffs and the relative social cost of the externality, I compare the results in Tables 2 and 3 (preference homogeneity) with those in Table 4 (preference heterogeneity case). I report the comparisons for the cases where the bioinvasion starts on the forestland of the MES landowner and where it starts on the property of the NMES landowner.

The MES landowner’s payoff decreases sizably in the presence of an NMES neighbor due to the underprovision of control on the NMES forestland. Furthermore, this decrease is larger when the NMES landowner is affected first by the bioinvasion. Compared with the preference homogeneity case, when the MES landowner has a neighbor with NMES preferences, and they are affected first by the bioinvasion, their payoff decreases from $12,491/acre (Table 3, preference homogeneity case) to $10,490/acre (Table 4, panel B), which amounts to approximately $2,000/acre, equivalent to a 16% decrease in payoff in the presence of preference heterogeneity. When the MES landowner is affected second by the bioinvasion, their payoff decreases by a larger amount, $4,886/acre (from $15,318/acre in Table 3 to $10,432/acre in Table 4, panel B), which is equivalent to a 32% decrease in payoff in the presence of preference heterogeneity.

In contrast, the NMES landowner’s payoff is not meaningfully affected by the preferences of their neighbor. Compared with the preference homogeneity case, when the NMES landowner has a neighbor with MES preferences, and they are affected first by the bioinvasion, their payoff is negligibly lower: it decreases from $841/acre (Table 2, preference homogeneity case) to $831/acre (Table 4, panel C), which amounts to approximately $10/acre, equivalent to a 1% decrease in payoff. Similarly, when the NMES landowner is affected second by the bioinvasion, their payoff decreases by 1% ($22/acre) in the presence of preference heterogeneity. (It decreases from 1,591/acre in Table 2, preference homogeneity case, to $1,569/acre in Table 3, panel C.)

Results in this section suggest that a transition from a landscape with homogeneous preferences to one with preference heterogeneity under the baseline parameter values is detrimental for the MES landowner but has no meaningful effect on the NMES landowner. If forest parcelization increases as a result of a demographic transition of landowners, results from the preference heterogeneity case, when compared with results from the preference homogeneity case, give insights as to how possible transitions might contribute to the production of bioinvasion externalities. I observe that when I move from NMES only to NMES and MES preferences coexisting on a connected landscape, the difference between DM and CP payoffs decreases from 48% (Table 2) to 30%–40% (Table 4). In contrast, the CP-DM payoff difference increases from 11% (Table 3) to 30%–40% (Table 4) when I move from MES only to coexisting NMES and MES preferences. That is, an increasing number of NMES-type parcels on a landscape dominated by MES-type ownerships increases the total economic damages from invasive species when NMES landowners act as weaker links in the landscape-level provision of bioinvasion control, regardless of where the bioinvasion starts.

Nonuniform Subsidies for Nonuniform Preferences?

The social cost of the externality generated by the privately optimal decision of landowners can be reduced through a subsidy. Forest landowners in the United States have had access to technical and financial assistance to manage invasive species through many programs, notably the Environmental Quality Incentives Program (EQIP). A common feature among these programs is that the cost-share payments they provide are uniform in the sense that they are available to landowners regardless of landowner preferences or ownership motivations.

Taking the baseline model case of no subsidy (i.e., c_A = c_B = $4/pine tree) as a baseline, I consider both uniform subsidies—for example, (c_A, c_B) = (2, 2)—and nonuniform subsidies—for example, (c_A, c_B) = (2, 4)—under the cases where the bioinvasion starts on the NMES (G_A) and the MES (G_B) forestlands. Considering nonuniform subsidies is particularly interesting given the results on the previous section: if there is a landowner type that would control without the subsidy (i.e., the MES landowner), then nonuniform subsidies based on preferences and bioinvasion order might generate higher net aggregate payoff compared with uniform subsidies. Also, targeting the weaker-link landowners might increase the subsidy’s cost-effectiveness by making control optimal for them and indirectly increasing returns to control for their neighbors through the complementarity of bioinvasion control on adjacent forestlands (Cornes 1993; Fenichel, Richards, and Shanafelt 2014; Atallah, Gómez, and Conrad 2017; Atallah et al. 2023). However, it might also be that targeting the forestland where the bioinvasion starts is more cost-effective, regardless of whether they are the weaker-link landowner.

Per-acre cost-share pay rates range from 40% of the average cost of a practice and cannot exceed 75% of the documented cost (USDA NRCS 2011). In this model, such rates would be equivalent to per-tree costs of (c_A, c_B) = (2, 2) and (c_A, c_B) = (1, 1). That is subsidies of (2, 2) and (3, 3), respectively. I consider these subsidy schemes in addition to all possible combinations of nonuniform schemes and compare them with the baseline case of no subsidy.

Regardless of the move order, I find that a nonuniform subsidy of (3, 0) (i.e., $3/treatment for A and no subsidy for B) maximizes the net aggregate payoff, defined as the aggregate payoff minus the total subsidy. In the case where the bioinvasion starts on G_A, and A moves first, this nonuniform subsidy increases the net aggregate payoff by 56% relative to the baseline case of no subsidy. Uniform subsidies (2, 2) and (3, 3) increase the net aggregate payoff by 51% and 45%, respectively, compared with the baseline of no subsidy (Table 5, panel A).

In the case where the bioinvasion starts on G_B, and B moves first, a subsidy of (3, 0) increases the net aggregate payoff by 21% relative to the baseline case of no subsidy, while uniform subsidies (3, 3) and (2, 2) generate increases of 18% and 16%, respectively (Table 5, panel B). The relative increase in the net payoff is lower when landowner B moves first partly because the baseline payoff is higher in that case ($12,060/acre; Table 5, panel B vs. $11,263/acre; Table 5, panel A) and because they control the bioinvasion early on and provide a benefit to landowner A as a result. Conversely, the relative increase in the net payoff is larger when A moves first because A does not control the bioinvasion in the baseline case.

The results in this section suggest that the subsidy scheme that maximizes the net aggregate payoff is a nonuniform subsidy consisting of no payment for the MES landowner and a 75% cost share for the NMES landowner, regardless of where the bioinvasion starts. Such nonuniform subsidy makes it privately optimal for the NMES landowner to control the bioinvasion. More interestingly, despite being targeted to the NMES landowner, it indirectly incentivizes the MES landowner to initiate control earlier (Figure 3a.2 vs. 3a.1) or to keep controlling later (Figure 3b.2 vs. 3b.1), compared with the no subsidy case. As a result, the optimal subsidy leads to less intense and more frequent control, which is characteristic of a CP control program (Figure 3.2 vs. 3.3) or that of a landowner who equally values NMES and MES.

Sensitivity to the SDD and LDD Parameters

I identify the threshold values for the parameters being varied that trigger a change in the control decisions of the landowners relative to the baseline results (Table 4, panel B). In other words, for all values between the baseline value (Table 1) and the threshold value for each parameter (Appendix Table A1), there is no change in whether landowners control the bioinvasion relative to the baseline results.

Scenarios S1–S4 consist of univariate (i.e., one-way) sensitivity analyses, relative to the baseline scenario (S0): S1 and S2 represent the case with the slower and faster SDD thresholds, respectively. S3 and S4 consider the case with the slower and faster LDD thresholds, respectively. Scenarios S5–S8 consist of bivariate (i.e., two-way) sensitivity analyses that represent cases where both dispersal mechanisms are slower (S5), where the SDD is slower but the LDD is faster (S6), where the SDD is faster and the LDD is slower (S7), and a case where both dispersal mechanisms are faster (S8) (Appendix Table A1).

In general, I find that the model results are robust to large changes in the dispersal parameter values. I identify threshold parameter values beyond which the baseline results change to an outcome where either no bioinvasion control takes place on any forestland or both landowners control the bioinvasion. These threshold values that qualitatively change the results are extreme in the case of the SDD: the slower (α_slower = 0.01 year⁻¹) and faster (α_faster = 3.91 year⁻¹) SDD parameter threshold values correspond to environmental conditions or species where the probability of seed germination and establishment (conditional on one neighboring tree being infested) is 1% and 98%, respectively. (In the baseline case, this probability is 50% when α_base = 0.69 year⁻¹.) In the case of the slower LDD, the threshold parameter value that changes the optimal solution (γ_slower = 5) makes dispersal decay more rapidly over space so that long-distance arrivals reach only 19% of the neighboring forestland over 25 years (or 3 of the 16 grid columns). In the case of the faster LDD, the parameter threshold value (γ_faster = 1.5) makes dispersal decay over space so minimal that long-distance arrivals occur everywhere in the neighboring forestland. That is, the probability that a shrub on the westernmost border of one forestland causes an infestation on the easternmost border of the other forestland is greater than zero.⁷ The LDD threshold parameters seem plausible relative to the extreme SDD parameters. Slower and faster LDD, or conversely, more or less rapid decay over space, represent the cases of invasive species that are spread by vectors that travel shorter (e.g., mammals) or farther (e.g., birds) distances. Results from these simulations can illustrate the benefits of policies that seek to reduce SDD vs. LDD directly.

The results in Appendix Table A1 suggest that for a fast-enough LDD (S4, S6, S8) none of the forest landowners control the bioinvasion, even if the SDD is extremely slow (S6). In contrast, both landowners control the bioinvasion when both dispersal mechanisms are at their low threshold (S5) or where one is at the low threshold and the other is at the baseline value (S1, S3). The largest decrease in DM payoffs relative to CP payoffs (85%) occurs whenever the LDD is at its faster threshold (γ_faster = 1.5 in S4, S6, S8). In contrast, the difference between DM and CP payoffs is negligible (1%) when either the LDD or both types of dispersal are at their slower thresholds (S3 and S5) (Appendix Table A1). The DM-CP difference is intermediary (14% in S1, 21% in S7, and 49% in S2) whenever the LDD is at its baseline value or slower, regardless of the speed of the SDD. Generally, these results show that whether an invasive plant causes a trivial problem (i.e., one that is resolved by control on both forestlands) or a severe one (i.e., where no landowner controls the bioinvasion) depends disproportionately on the LDD. Given the large effects of moderate reductions in the LDD, relative to the moderate effects of extreme reductions in the SDD, it is surprising that there are few management actions, bioinvasion policies, or incentive programs that directly address the LDD by birds, similar to policies that reduce human-mediated LDD.⁸ Trakhtenbrot et al. (2005) suggest that this lack of appreciation of the significance of the LDD vs. SDD in conservation policy and management is partly due to difficulties in defining and quantifying LDD. There are conservation policy recommendations that seek to affect long-distance bird-mediated dispersal when it is desirable but insufficient: for example, Amezaga, Santamaría, and Green (2002) suggest considering the migration routes of waterfowl that vector aquatic invertebrates when constructing wetland reserve networks. I do not know of policies that are designed to reduce undesirable LDD of invasive plant seeds by birds. The sensitivity of the model results to the LDD parameter suggests that there might be economic benefits to investing in long-distance bird dispersal research that can inform bioinvasion management and policy. Management strategies and policies focused on reducing the source of the bioinvasion alone through shrub removal, holding dispersal constant, might have limited success in reducing externalities relative to slowing down dispersal, especially the long-distance kind.

The sensitivity results can also be interpreted as showing that the baseline results extend to other invasive shrubs with slower and faster SDD and LDD that fall between the baseline and threshold values. The results at and beyond the threshold values illustrate how the model can be adapted for other invasive species with extremely slow and fast SDD (S1 and S2), negligible slow LDD (S3), and relatively fast LDD (S4).

Sensitivity to the Recreation Benefit Parameters

In the baseline case, I considered that the NMES landowner derives trail-recreation benefits from healthy trees. Now I conduct sensitivity analyses to the parameters of the recreation benefit parameters (b₀, a₁, and b₁ in the UD and CS functions; see equation [1] and Table 1). I identify threshold parameter values beyond which the optimal solution changes such that both landowners control the bioinvasion, and the difference between DM and CP payoffs is decreased. The intercept of the UD function (b₀ in Table 1) represents the number of days an NMES landowner recreates on their forestland with no healthy trees. I find that for values of b₀ equal to or greater than 28 days a year, up from 9 days/year in the baseline case, the NMES landowner controls the bioinvasion. At this threshold parameter value, the CP-DM payoff difference is reduced to 11%, compared with 40% in the baseline case (Appendix Table A2, panel A). I also find that if parameter a₁ is equal to or greater than $0.36 per UD for each additional healthy tree per acre, and parameter b₁ is equal to or greater than 0.36 days per additional healthy tree per acre, the NMES landowner controls the bioinvasion. At these threshold values, the difference between DM and CP payoffs is 9%, down from 40% in the baseline case (Appendix Table A2, panel B). These results illustrate how the baseline results might change for NMES landowners who derive higher trail-recreation values from their trees.

Beyond Recreation Benefits

I expand the bundle of NMES benefits and consider that in addition to trail recreation, the NMES landowner derives benefits from ES, such as carbon sequestration, pollutant removal from the air, and water filtration. I add a term in the NMES discounted benefit function that increases in the number of healthy and infested trees, excluding only those that are heavily infested. In this term, I make two modifications to accommodate these NMES types. First, I relax the spatial contiguity constraint in equation [3] that is not relevant for NMES whose provision and damage are not spatially defined (i.e., tree location on or away from a trail does not matter for carbon sequestration or air purification). Second, I include trees invaded at the Low and Moderate levels considering that while the shrub invasion at these levels affects aesthetics and hiking recreation benefits, it does not affect a tree’s ability to store carbon or remove fine particles from the air. I identify threshold parameter values beyond which the NMES landowner controls the bioinvasion. I find that this outcome occurs when the NMES landowner derives yearly benefits from NMES, such as carbon sequestration, air purification, and water filtration, that are equal to or greater than $350 per acre (or $0.45/tree). For example, if the NMES landowner considers carbon sequestration benefits in their private bioinvasion control decision making, then the threshold value identified here tells us how large this privately held value needs to be for the landowner to initiate bioinvasion control. At this threshold value, the difference between DM and CP payoffs is 3%, down from 40% in the baseline case (Appendix Table A2, panel C). In comparison, existing estimates of stated forest landowner carbon sequestration benefits range from $5 to $30/acre/year (Fletcher, Kittredge, and Stevens 2009; White et al. 2018).

Surveillance, Early Detection, and Management

In the baseline model, the landowners only observe the bioinvasion states Invaded−moderate (I_m) and Invaded−high (I_h), at which point 19% of the trees are Invaded−undetectable (I_u) or Invaded−detectable (I_d), 13% are moderately infested, and none are heavily infested. However, extension services or cooperative invasive species management areas provide technical assistance and training in identification and monitoring (Ingle 2013; Cronk 2017), which can give landowners the ability to detect the bioinvasion at the state I_u, before shrubs produce berries and form clumps. I consider that even with surveillance, landowners do not observe the invasive shrub when it is still at the seed germination stage I_u.

I find that, when surveillance is possible under baseline model parameters, the NMES landowner still does not find it optimal to control the bioinvasion even if the invasive shrub is discoverable earlier. The MES landowner, by contrast, starts controlling five years earlier because of surveillance and early detection and decreases the total number of treatments by 8% (the difference between 1,548 treatments in Table 4 and 1,416 treatments in Appendix Table A3, panel A). The possibility of surveillance and early detection benefits both landowners, even if only one of them engages in it. The benefits of earlier and less intense treatment by the MES landowner increase the payoffs of the NMES landowner by 98% (the difference between $831 in Table 4 and $1,648 in Appendix Table A3, panel A). The aggregate benefit of surveillance and early detection is $1,635/two acres, an increase of 15% relative to the baseline.

I also consider a case where the SDD is at its lower threshold (Appendix Table A1, panel S1). I find that if SDD is slow, surveillance and early detection are available, and the invasive shrub is instantly detectable (i.e., L₁ = 0), both landowners find it optimal to engage in earlier control and to conduct 9% fewer treatments in total, relative to the slow SDD baseline (1,520 vs. 1,866 treatments in total, 654 vs. 868 treatments for the NMES landowner, and 866 vs. 997 for the MES landowner; Appendix Table A1, panel S1 vs. Table A3, panel B). The benefits of earlier and less intense treatment consist of a 25% increase in the payoffs of the NMES landowner and a 6% increase in the payoffs of the MES landowner. (The percentage increases are the difference between $2,319 in Appendix Table A3, panel B and $1,859 in Appendix Table A1, panel S1 for the NMES landowner and the difference between $16,261 in Appendix Table A3, panel B and $15,287 in Appendix Table A1, panel S1 for the MES landowner, respectively). These results illustrate the benefits of early detection training on reducing the costs and damages generated by the invasive shrub. Expectedly, reductions are larger for faster-spreading invasive species (33% vs. 13%; Appendix Table A3, panel B).