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Volume 23, Issue 3
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The behavior of multiple independent managers and ecological traits interact to determine prevalence of weeds

Shaun R. Coutts

Corresponding Author

E-mail address: s.coutts@uq.edu.au

1University of Queensland, School of Biological Sciences, Brisbane, Queensland 4072 Australia

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E-mail address: s.coutts@uq.edu.au

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Hiroyuki Yokomizo

2Center for Environmental Risk Research, National Institute for Environmental Studies, 16-2 Onogawa, Tsukuba-City, Ibaraki 305-8506 Japan

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Yvonne M. Buckley

1University of Queensland, School of Biological Sciences, Brisbane, Queensland 4072 Australia

3CSIRO Ecosystem Sciences, GPO Box 2583, Brisbane, Queensland 4001 Australia

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First published: 01 April 2013
Citations: 11

Corresponding Editor: E. A. Newell.

Abstract

Management of damaging invasive plants is often undertaken by multiple decision makers, each managing only a small part of the invader's population. As weeds can move between properties and re‐infest eradicated sites from unmanaged sources, the dynamics of multiple decision makers plays a significant role in weed prevalence and invasion risk at the landscape scale. We used a spatially explicit agent‐based simulation to determine how individual agent behavior, in concert with weed population ecology, determined weed prevalence. We compared two invasive grass species that differ in ecology, control methods, and costs: Nassella trichotoma (serrated tussock) and Eragrostis curvula (African love grass). The way decision makers reacted to the benefit of management had a large effect on the extent of a weed. If benefits of weed control outweighed the costs, and either net benefit was very large or all agents were very sensitive to net benefits, then agents tended to act synchronously, reducing the pool of infested agents available to spread the weed. As N. trichotoma was more damaging than E. curvula and had more effective control methods, agents chose to manage it more often, which resulted in lower prevalence of N. trichotoma. A relatively low number of agents who were intrinsically less motivated to control weeds led to increased prevalence of both species. This was particularly apparent when long‐distance dispersal meant each infested agent increased the invasion risk for a large portion of the landscape. In this case, a small proportion of land mangers reluctant to control, regardless of costs and benefits, could lead to the whole landscape being infested, even when local control stopped new infestations. Social pressure was important, but only if it was independent of weed prevalence, suggesting that early access to information, and incentives to act on that information, may be crucial in stopping a weed from infesting large areas. The response of our model to both behavioral and ecological parameters was highly nonlinear. This implies that the outcomes of weed management programs that deal with multiple land mangers could be highly variable in both space and through time.

Introduction

Damaging invasive plants (henceforth called weeds) cause yield losses in crops and livestock (e.g., Cousens 1985, Firn 2009) and alter ecosystem functions (Le Maitre et al. 1996) and disturbance regimes (D'Antonio and Vitousek 1992), and thus are worth direct expenditure on control. When the strategy and timing of weed management is controlled by one decision maker, e.g., within a single farm, there is a well‐developed body of literature relevant to local weed management (e.g., Doyle 1997, Hamill et al. 2004). However, every widespread weed in the world exists across property boundaries, and as a result, no single land manager has access to the entire population. This makes it imperative to understand how human behavior affects coordination of local management efforts, and subsequently, persistence and spread of weeds at landscape scales (Shogren 2000). We use the term “decentralized weed control” to refer to situations where the decisions of many independent actors affect the weed population.

Previous work has conceptualized decentralized weed control as a problem of external costs (e.g., Jones and Campbell 2000), where the actions of one individual impose uncompensated costs on others. Because these studies are mainly from the economic literature, they focus on how ecological processes, such a spread, affect what actions people take and how much benefit they derive from those actions. We turn this question on its head and ask how people's actions, and the benefits they get from those actions, influence the spread of a weed and its prevalence in the landscape.

Due to the complexity and relatively large scale of decentralized weed control, models have often been used to gain general insights. Decentralized weed control can be modeled as a weakest link public goods game, where each land manager can pay to adopt good weed control practices that lower the infestation risk for everybody, and those that do not (the “weak links”) have the potential to infest many others (Perrings et al. 2002). Both laboratory experiments and models consistently find that groups playing weakest link public goods games are strongly attracted to one of two stable equilibria, either everybody cooperates or no one does (Knez and Camerer 1994, Devetag and Ortmann 2007, Hennessy 2008). More complicated bio‐economic models that incorporate human decision making and forest dynamics show similar behavior. In these models, the actions of one decision maker can cascade through the whole system, causing synchronized deforestation, even when keeping the forest would have benefited everyone (Satake et al. 2007). Both these results suggest that social behavior in relation to weed control might be highly nonlinear.

Nonlinear behaviors like positive feedbacks and multiple stable states imply that under the right conditions a few land managers who do not control weeds can have dramatic impacts at scales far larger than the area they manage. Because different land managers have different economic goals and/or production systems, some land managers will be less motivated to control a given weed than others. Also, because weed populations generally exist at scales far greater than individual properties, even if all agents are willing and able to control the weed, failure to synchronize efforts can allow the weed to persist (Gonzalez‐Andujar et al. 2001). Social pressure between decision makers can encourage them to act more uniformly by reinforcing the socially approved behavior (Milinski et al. 2006, Iwasa et al. 2007, 2010). Anecdotally, land managers report that social pressure influences weed control decisions (Anonymous 2009a, Holman et al. 2010), suggesting that this may be an important (but often unquantified) force shaping weed dynamics at the landscape level.

Our goal was to test how the prevalence of weeds was affected by three aspects of human behavior (social pressure, profit seeking, and ability to perceive and act on expected returns) and their interaction with ecological traits affecting a weed's ability to persist and spread to new properties. We also tested if a small minority of agents with less motivation to control at the property scale could affect weed spread at the landscape scale. We used a spatially explicit agent‐based simulation where each agent was a property manager. Agent‐based simulations provided a flexible and intuitive framework in which to combine spatially explicit decision making and ecological processes. We also developed a mean field approximation as a check that the results of our simulation were not due to stochasticity or arbitrary modeling decisions such as process scheduling and landscape size. We compared the behavior of the model for two different types of weeds with varying impact and control costs: Nassella trichotoma (serrated tussock), which is very damaging, and Eragrostis curvula (African lovegrass), which is less damaging.

Methods

Study species

We parameterized economic impact and the management costs and effectiveness for two widespread and damaging weeds of Australian pastoral areas, where weed control is often decentralized: Nassella trichotoma (serrated tussock) and Eragrostis curvula (African lovegrass). Both species can dominate pastures and are relatively unpalatable, significantly reducing stocking rates. There are, however, important differences between them. N. trichotoma has been recognized as a serious problem since at least 1937 (Campbell and Vere 1995), and there are now established strategies for its control (Michelmore 2003). It often does not flower in its first year (Parson and Cuthbertson 2001), giving managers a relatively long period to find and spot spray new plants before they produce seed. Eradication is possible (Miller 1998), but likely to take decades as seed banks may last up to 23 years (Lamoureaux and Bourdot 2004). Quarantine and preventative measures for N. trichotoma appear to be relatively effective (Campbell and Vere 1995).

In contrast, E. curvula was repeatedly introduced as a pasture species up until the 1980s (Johnston et al. 1984), and with intensive management it does have some value as feed (Roberts and Carbon 1969). However, in many areas, E. curvula is a damaging weed (Firn 2009). As it has only recently been recognized as a serious problem, management strategies for E. curvula are much less developed. Less effective management combined with the fact that it can produce seed only a few months after germination means that eradication of established populations is currently not feasible (Firn 2009). A short time to seed production also means that spot spraying new plants to stop the weed from establishing is very difficult. However, the use of quarantine areas has been shown to be effective as the seeds disperse very poorly if unassisted by animals or machinery (Firn 2009). There are, therefore, some effective preventative measures land managers can take to reduce the chances of establishment, but they appear to be limited compared to those employed against N. trichotoma. We included these differences between N. trichotoma and E. curvula through differences in probability of eradication, θ, effectiveness of preventative measures, l, and the utility of being in state j, uj, which defines impact and management costs (see Table 1 for values and Table 2 for further definitions).

Table 1. Costs of being in each state relative to being susceptible (represented by uXk), probability of eradication (θ), and effectiveness of vigilance (l) for each case study (Anonymous 2009a, b.
table image
Table 2. Table of parameters, their values (if constant across time steps), and their interpretation.
table image

We used a series of case studies from Meat and Livestock Australia and Australian Wool Innovation Limited (Anonymous 2009a, b) to parameterize the impact and management costs for these two species. Individual property managers who had one of these weeds on their property were asked how much they believed it reduced yield, how much it cost to control or mitigate, and how optimistic they were about eradicating the weed. These case studies were not a random sample; they were anecdotal accounts from a subset of property managers with a proactive approach to weed control. The case studies take into account the costs and benefits of simultaneous management strategies, such as using herbicides along with grazing management, growing supplementary feed through cultivation, and quarantine measures. Costs from the case studies are shown in Table 1.

For all case studies, controlling was more profitable than being invaded because control efforts allowed stocking rates to partially recover, which more than offset the costs of control. Being vigilant and eradicating populations early was always more profitable than controlling established populations. This was because most vigilance strategies entailed similar elements to control strategies (e.g., checking paddocks and spot spraying any new seedlings), and as there were far fewer plants to control when the population was very new and not yet established, herbicide and labor costs were lower.

Model development

We modeled interactions between the ecological, economic, and social forces that shape weed invasions by simulating a lattice of simple decision‐making agents. Each agent could change state through the ecological processes of invasion and eradication, or by deciding to control or taking preventative measures. All parameter combinations tested were simulated 1000 times. We developed a mean field approximation of the simulation that holds when dispersal is random with respect to distance (see Appendix). Unlike the simulation, the mean field approximation was not stochastic, used an infinite landscape, and was solved in continuous time.

State variables and scales

Our landscape was a 64 × 64 cell lattice with wrap‐around boundaries; each cell represented a single property controlled by a decision‐making agent. Xk was a state variable that held the state of agent k. Each agent could be in one of four states; infested (Xk = I), controlling (Xk = C), susceptible (Xk = S), and vigilant (Xk = V). Infested agents had the weed and did not control it. We assumed that once established, the weed never went extinct without management, thus agents could only move from I to C. Controlling agents had the weed and undertook management to reduce and/or eventually eliminate the weed. Susceptible agents were weed free and carried out no management. Vigilant agents were weed free and took steps, such as paddock inspections and spot spraying with herbicides, to reduce the probability of the weed establishing (See Fig. 1 for possible transitions). Agents changed their state by either becoming invaded, eradicating the weed, or deciding to control or be vigilant based on their own state and the state of others in the lattice. We assumed that control activities stopped the weed from spreading, for example, by greatly reducing seed set.

figure image

Transition probabilities between weed states: susceptible (S), infested (I), controlling (C), and vigilant (V). Within one time step, agents can: (a) change state through ecological processes, making one move up (with probability of eradication, θ) or down (with probability of infestation, inline image); or (b) decide whether to change their behavior and move from state j with probability aj→i,k. Agents can also remain in any of the states (not shown). Further details of transition probabilities and term definitions are given in Table 2.

The simulation was run for 200 one‐year time steps; considered an appropriate timescale for these pastoral systems because eradication on individual properties can take decades. After 200 time steps, the proportion of agents in each state was averaged over the last five time steps and recorded.

Design concepts

Agents used the profit of being in each state, the probability of invasion or eradication, and social pressure to calculate the expected benefit of moving to a new state. Social pressure had three components: (1) internal social pressure to “do the right thing” (q), which was independent of the weed's frequency; (2) because reinforcement from multiple peers makes changing behavior more likely (Centola and Macy 2007), external social pressure, applied from weed‐free neighbors, was added to internal social pressure, after being scaled by the social pressure coefficient, y; and (3) we assumed an agent's neighbors were more likely to apply social pressure when they were themselves threatened with infestation. We made the amount of social pressure each neighbor applied proportional to the frequency of weeds among their neighbors, similar to Eq. 1 in Iwasa et al. (2007; See Fig. 2 for an explanation of neighbor's neighbors).

figure image

Schematic representation of neighbors and neighbors' neighbors. The neighborhood is a very important concept in our model. When agents chose to control or be vigilant about weeds, they took into account the social pressure applied to them by their neighbors. We called the set of agent k's neighbors Nk in this figure; Nk is indexed by the variable g. How much social pressure each neighbor applied to agent k depended on the frequency of weeds among their own neighbors. Following the same notation, the set of each neighbor's neighbors was called Ng. You can see that agent k (black box) is also in Ng=1's neighborhood, and although not shown, agent k is also in the neighborhoods of all its neighbors, because any agent is its neighbors' neighbor. We can see that agents also share neighbors; for example, agent g = 2 is a neighbor to agent k (i.e., it is in the set Nk), and is also a neighbor to agent g = 1 (i.e., it is also in the set Ng=1). For clarity, only two neighborhoods are shown, Nk and Ng=1, but all g have neighborhoods like that of Ng=1. Here we show a neighborhood size |Nk| = 8; the simulation was also tested with |Nk| = 4095.

In weed management there is an incentive to “free ride,” i.e., put less effort into weed prevention because the weed management of others lowers one's invasion risk. The free‐rider incentive was included in the model by incorporating invasion risk into the cost benefit calculation for being vigilant or susceptible (Eqs. 5 and 6). At each time step, weed‐free agents decided to be vigilant or susceptible, and if the actions of their neighbors lowered the invasion risk enough they decided to be susceptible, which was more beneficial when invasion risk was lower.

Imperfect ability to perceive and act on economic and social information can be approximated by making the decision rule more stochastic (Satake et al. 2007). Differences in production systems and economic goals mean that some land managers will see more value in being weed free than others. We modeled differences in the perceived value of weed management by shifting “unmotivated” agents' decision curves relative to the decision curves of “motivated” agents. Unmotivated agents' decision curves for control and vigilance were shifted to the right (Fig. 3), and their decision curves for being susceptible and infested to the left. Thus, unmotivated agents needed more benefit to have the same probability of deciding to control or be vigilant and less benefit to have the same probability of deciding to be susceptible or infested. Generally, we ran our model using two social settings: “good neighbors,” where there were no unmotivated agents and “small minority,” where 10% of agents were unmotivated. We also tested the model with 1% of agents unmotivated, but the results did not differ substantially from the good‐neighbor setting. Model runs that did not use these two social settings are made clear. Agents' transitions between states were decided stochastically using a biased coin toss.

figure image

Examples of the logit decision model in Eq. 4. The probability of an agent moving from state j to state i is a function of the benefit of making such a move (Eq. 5), and the difference in perceived value of weed control between motivated and unmotivated agents, hk.

We examined three dispersal scenarios: random, nearest‐neighbor, and combined dispersal. Under random dispersal, infested agents could spread weeds to any other weed‐free agent in the lattice. This describes situations such as long‐distance seed movement by road, which is a factor in the spread of N. trichotoma and E. curvula (Parson and Cuthbertson 2001, Firn 2009). In the nearest‐neighbor scenario, infested agents could only spread weeds to their eight immediate neighbors. This is applicable when seed dispersal is more closely linked with distance (i.e., natural dispersal vectors predominate). In reality, some seeds go to neighbors and others travel long distances. In the combined scenario, infested agents spawned 90% of new infestations in their eight immediate neighbors, and spawned 10% in randomly chosen agents. The 90/10 split is arbitrary due to a lack of data about seed transport for our two focal species. We tested other nearest‐neighbor/random dispersal splits and runs that did not use a 90/10 split are made clear. Our goal was to examine how social and economic pressure, in combination with different types of weed dispersal, might influence invasion dynamics.

Initialization

At the beginning of each run of 200 time steps, 1% of cells were randomly selected to be infested, and the rest were set to susceptible. The specified number of unmotivated agents (generally 0 or 10% of agents) were randomly assigned. Agents that were set to unmotivated stayed unmotivated for all 200 time steps.

Process overview and scheduling

(1) First, the ecological processes of infestation and eradication took place; susceptible and vigilant agents got infested with probability inline image , and controlling agents eradicated the weed with probability θ (Fig. 1a). (2) Based on the resulting landscape, agents with the weed decided if they controlled or not, and those without the weed decided to be susceptible or vigilant (Fig. 1b).

Sub‐models

Both decision and infestation probabilities were dynamic, but for convenience we drop the time subscript as these probabilities only ever referred to the current time step. The simulation was written with Java SE 6 (Oracle Technology Network 2006).

Infestation and eradication

Susceptible or vigilant agents became infested with the following probabilities:

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where inline image was the probability agent k got invaded if they were susceptible (Xk = S; Eq. 1a) or vigilant (Xk = V; Eq. 1b), where there were M other agents in the landscape (M = 64 × 64 − 1 = 4095). Parameter l (Eq. 1b) determined how effectively vigilance stopped infestation, and 0 ≤ l ≤ 1 was always true. When l = 0, it meant that vigilance always stopped infestation, and l = 1 meant that vigilance had no effect on infestation risk. Kronecker's delta:
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ensured that only infested agents could give the weed to agent k. Recall that for infested agents, Xk = I, thus inline image = 1 only when agent m is infested and is 0 if agent m is susceptible, vigilant, or controlling (i.e., Xk = S, V, or C, respectively). Parameter βk,m (Eq. 3) is the probability that if agent m had the weed they would infest agent k:
urn:x-wiley:10510761:media:eap2013233523:eap2013233523-math-0004
where β (note no subscript) is the probability that an infested agent will infest one susceptible agent in one turn. For example, β = 0.5 meant each infested agent would create, on average, one new infestation every two years. Holding β constant across the three dispersal scenarios stopped the distribution of dispersal (i.e., where new infestations occurred) being confounded with the number of new infestations per infested agent. The proportion of infestations that occur in nearest neighbors (evenly divided among the eight nearest neighbors) is 1 − w, and w is the proportion of infestations that occur in any other agent in the landscape (evenly divided among the M − 8 other agents in the landscape). We made βk,m a simple‐step function, which is appropriate at the scale of properties (in this context often hundreds to thousands of hectares in size). Weeds can “jump the fence” to a neighbor without human help, using dispersal mechanisms like wind and water (top line in Eq. 3). For our focal species, seed dispersal of more than one kilometer by these mechanisms is rare. Further, because dispersal kernels are flat in the tails, the destination of far‐dispersing seeds is almost independent of distance. Thus, it is unlikely a seed will be dispersed all the way across a neighboring property without human help, and if it does, it could disperse almost anywhere in the landscape. Also, very long‐distance dispersal (tens to hundreds of kilometers) is often the result of accidental seed movement by road; farm machinery and transported stock are commonly cited vectors (Parson and Cuthbertson 2001, Firn 2009). To capture these processes, we modeled very long‐distance dispersal seed dispersal as random with respect to distance (i.e., the seed could go anywhere on the landscape; bottom line of Eq. 3). For the nearest‐neighbor dispersal scenario w = 0, for combined dispersal, we assumed w = 0.1, and for random dispersal, we assumed each agent had an equal chance of being infested, thus w = (M − 8)/M (note that w ≠1, because nearest neighbors have a small chance of infestation when dispersal is random).

If Xk = C, then agent k could eradicate the weed with probability θ. A biased coin toss was used to apply θ.

Decision model

Following Iwasa et al. (2007), we modeled the probability that agent k decided to move from their current state j to state i, aj→i,k (Eq. 4), using a logit decision model (example shown in Fig. 3). There is a long history of using logit decision models to describe how groups of people change their decisions in economics (McFadden 1981) and game theory (Hofbauer and Sigmund 2003). Logit decision models have the flexibility to allow all agents to make the best response to their current invasion risk (large value of α in Eq. 4), or for some agents to stochastically make a suboptimal choice (lower value of α in Eq. 4) (Iwasa et al. 2010). To make the problem tractable, each agent only knows his current invasion risk and so decides as though invasion risk (and by extension, the state of the landscape) will stay constant:

urn:x-wiley:10510761:media:eap2013233523:eap2013233523-math-0005
The shape‐constant α controlled how deterministic the decision model was: If α was large, agents were more likely to choose the option with the highest benefit (dashed line in Fig. 3), as α got smaller, agents' decisions became more random with respect to benefit (solid lines in Fig. 3), α was always non‐negative. The benefit to agent k for deciding to move from state j to state i, bj→i,k (Eq. 5), was calculated as the difference between the expected return of moving to new state i, E[Rk|Xk = i], and the expected return of staying in the current state, E[Rk|Xk = j]:
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If agent k expects a bigger return for deciding to change state than staying in its current state, then bj→i,k will be positive and the probability that they will change state, aj→i,k (Eq. 4), will be ≥0.5. How much greater than 0.5 will depend on the size of bj→i,k and α. Vice versa, if agent k expects a bigger return for staying in its current state than changing state, then bj→i,k will be negative and aj→i,k, will be ≤0.5. For simplicity, when calculating the expected economic return, we assumed that agents were myopic, only taking into account the expected economic benefit from the next time step given the current state of the landscape.

If agent k did not have the weed it could decide to be susceptible or vigilant. Susceptible agents (Eq. 6a) received the highest utility (recall that uS = 1 > uV > uC > uI), but also had a higher invasion risk and felt social pressure. Vigilant agents had a lower utility, but also a lower risk of invasion and felt no social pressure (Eq. 6b). Agents balanced these competing factors using Eq. 5 and E[Rk|Xk = j]. E[Rk|Xk = j], was composed of two parts: E[Rk|Xk = j] = [economic return for agent k in state j] − [social pressure on agent k in state j]. The expected economic return for being susceptible (first set of fences in Eq. 6a) was the utility for being susceptible, uS (highest possible utility), weighted by the probability that if agent k was susceptible, they would not get invaded ( inline image ), plus the utility for invaded agents, uI (lowest possible utility) weighted by the probability that agent k would get invaded if it was susceptible ( inline image ). The economic return for agent k deciding to be vigilant was calculated in a similar manner (Eq. 6b); the utility of being vigilant, uV, weighted by the probability of not getting invaded if they were vigilant ( inline image ), plus the utility of being invaded, uI, weighted by the probability of agent k getting invaded if they were vigilant ( inline image ). To reflect that some agents were more motivated than others to be vigilant, agent k perceived hk less utility for being in state V; for motivated agents, hk = 0, and for unmotivated agents, hk > 0 (difference between solid black and gray lines in Fig. 3). Agents are designated as either motivated or unmotivated at the start of a run, and keep that designation through the whole run (i.e., hk is set for each agent at the start of a run and is constant). The final set of brackets in Eqs. 6a and 7a is the social pressure felt by each agent for not taking action against the weed:

urn:x-wiley:10510761:media:eap2013233523:eap2013233523-math-0007
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We assume social pressure was felt by susceptible agents, but not vigilant ones. Thus, the expected return for being susceptible (Eq. 6a) contains a social pressure term in the second set of brackets, while the expected return for being vigilant (Eq. 6b) has no social pressure term. The social pressure term in Eq. 6a reduced the economic benefit of being susceptible. The internal social pressure to control the weed irrespective of the weeds prevalence is termed q. This may be due to a social norm to control exotic species, even if people do not know how damaging they are. Managers' motivation to control a weed may also change with that weed's prevalence. The rationale is that when a weed is very rare, fewer people will be aware of the weed and problems it causes. As the weed increases in prevalence, more people learn to identify it and what damage it can do, and so become more concerned about it and place higher social pressure on their neighbors to control it. To model this effect, we assumed that social pressure increased as a linear function of total concern about the weed among an agent's neighbors (the two summation terms in Eqs. 6a and 7a), with intercept q and slope y. The slope, y, controls how strongly agents respond to social pressure imposed by neighbors. Total concern among neighbors had two components: the number of agents who could apply social pressure (first summation term) and how concerned each of those neighbors were (second summation term). Nk was the set of all agent k's neighbors, with each agent in set Nk indexed by the subscript g. Similarly, in the second summation term, Ng is the set of each of agent g's neighbors; that is, agent k's neighbors' neighbors (see Fig. 2). Recall that δi,j (Eq. 2) is 1 if i = j and 0 otherwise. Thus, the double summation term makes the amount of social pressure each vigilant neighbor (counted with δXg,V) applied to agent k proportional to the number of their neighbors with weeds (counted with δXn,I and δXn,C in the second summation term; note the change in subscript from Xg to Xn). |Nk| is the number of elements in set Nk, i.e., the number of neighbors agent k has. We assume the neighborhood size is the same for every agent, which means that |Nk| = |Ng|. We used two neighborhood sizes throughout the paper: |Nk| = |Ng| = 8 (nearest neighbors) and |Nk| = |Ng| = 4095 (the entire landscape). The division term in Eq. 6a converts the count from the double summation to a frequency.

If agent k had the weed, then it could decide between controlling or doing nothing and being infested. For our parameterization with N. trichotoma and E. curvula, controlling always had a larger benefit than doing nothing because the utility of controlling was greater than the utility of being infested (uC > uI), and we assumed agents that were controlling did not feel social pressure, but infested agents did. The calculation of the expected return for being infested (Eq. 7a) or controlling (Eq. 7b) is similar in form to the calculation of the expected return for being susceptible (Eq. 6a) and vigilant (Eq. 6b), with a few minor differences, as follows:

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We assumed that the weed never went extinct without control, so if agent k decided not to control and be infested, then the expected economic return was simply the utility of being infested (first term in Eq. 7a is just uI). E[Rk|Xk = C] was the expected return for a controlling agent (Eq. 7b) given that agents get uC if control is unsuccessful and 1 if they succeed, which happened with probability θ. We assumed that controlling agents do not feel social pressure, so E[Rk|Xk = C] (Eq. 7b) does not have the social pressure term. For infested agents, we assumed that both vigilant and susceptible agents could apply social pressure, so the second summation term contains both δXg,V and δXg,S; recall the social pressure term in Eq. 6a only included δXg,V. Like vigilance, some agents were more motived than others to control, and agent k perceived hk less utility for being in state C (for motivated agents hk = 0, and for unmotivated agents hk > 0). All agents were either motivated or unmotivated, and hk was constant for each agent, whether they were deciding to be vigilant or controlling.

Results

The mean field approximation always produced the same general pattern as the simulation, and there was good agreement between the simulation and mean field approximation over a wide range of parameter space for all the social scenarios (lines and points are closely matched in Figs. 4, 5, and 6). Suggesting that, for the random dispersal scenario at least, lattice boundary conditions, the way stochasticity was applied, and using discrete time did not change the general results or conclusions. While the general pattern remained similar, there was some quantitative departure of the mean field from simulation results for 7 out of the 32 parameter combinations where the mean field and simulation were compared. There was no general rule for when the departures occurred. In Fig. 4 the mean field and simulation only diverge when α = 5 (i.e., the agents' decisions are more deterministic). However, in Fig. 5, only one out of eight comparisons between the mean field and simulation diverged (solid black line and point‐up triangles in Fig. 5a) and that occurred when α = 1, for the good‐neighbor social scenario. For the 16 sets of parameter values tested in Fig. 6, only two showed much deviation between the mean field and simulation (dashed lines and point‐down triangles in Fig. 6d); both of these for the small minority social scenario.

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The proportion of agents either infested with or controlling weeds for different values of (a) eradication probability, θ, and (b) infestation probability, β, for the random dispersal scenario under more stochastic (α = 1) and more deterministic (α = 5) decisions (see Fig. 2). Vertical lines show the 95% confidence intervals for the simulation. For both panels, q = 0, y = 0.5, and l = 0.5. For panel (a), β is equivalent to each infested agent infesting, on average, one new susceptible agent every 10 time steps; for panel (b), θ is equivalent to a controlling agent eradicating the weed, on average, once every 20 time steps. On the x‐axis of panel (b), β is equivalent to each infested agent infesting, on average, one susceptible agent every 10 (minimum) and 2 (maximum) time steps. The cost structure used is from case study 1 for Eragrostis curvula, shown in Table 1.

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The effect of internal social pressure (q) on the proportion of agents with weeds (states I and C) for (a, b) random (w = [8 − M]/M), (c, d) combined (w = 0.1), and (e, f) nearest‐neighbor dispersal (w = 0) for both N. trichotoma and E. curvula, for the good‐neighbor (no unmotivated agents) and small minority (10% of agents unmotivated) social scenarios; y = 0.5 and β = 0.33 for all plots. Vertical lines show 95% confidence intervals. Values of θ and l, and the cost structures for case study 1 shown in Table 1 were used.

figure image

The proportion of agents with weeds (states I and C) across different values of α, for the random dispersal scenario for (a) N. trichotoma and (b) E. curvula. To show the effects of cost structure and ease of management separately, panel (c) shows results for the θ and l values of N. trichotoma and cost structure of E. curvula, and panel (d) shows the θ and l values of E. curvula and cost structure of N. trichotoma. The β values correspond to an infested agent infesting a susceptible neighbor, on average, once every three (black) and five (gray) time steps. The cost structures and values of θ and l used for each species are shown in Table 1 (case study 1); q = y = 0.5 in all plots.

Prevalence of the weed showed the expected relationship with both eradication and infestation probabilities (θ and β, respectively). The weed became less prevalent as it became easier to eradicate (higher θ) and more prevalent as infestations became more likely (Fig. 4).

Unsurprisingly, when there were more unmotivated agents, both E. curvula (Fig. 7a) and N. trichotoma (Fig. 7b) became more prevalent. There was an interaction between dispersal, species, and the proportion of unmotivated agents. As dispersal became more random, unmotivated agents had a greater effect on weed prevalence, and much more so for E. curvula than N. trichotoma (steeper lines for higher values of w in Fig. 7).

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The proportion of agents either infested or controlling weeds (y‐axis) under different proportions of unmotivated agents (x‐axis) for (a) Eragrostis curvula and (b) Nassella trichotoma. Points show the mean of 1000 simulations, vertical lines show 95% confidence limits, and joining lines connect points in the same series. Each series shows the results for a different split of random and nearest‐neighbor dispersal, and the proportion of random dispersal is shown in the right margin: w = 0 is the nearest‐neighbor scenario, w = 0.1 is the combined scenario, and w = (M − 8)/M (rounded up to 1, for display) is the random‐dispersal scenario. For all simulations β = 0.33; i.e., on average, each infested agent could infest one susceptible agent every three time steps. Values of θ and l, and the cost structures for case 1 shown in Table 1 were used, and q = y = 0.5.

Nassella trichotoma had a lower prevalence than E. curvula after 200 time steps for all dispersal and social scenarios (for example, compare the columns for each species in Fig. 5). For E. curvula internal social pressure, q, caused the proportion of agents with weeds to change from almost 1 to 0 when there was some random dispersal, and agents responded strongly to expected benefit (α = 5; Fig. 5b, d). Internal social pressure, q, had less effect on the prevalence of N. trichotoma (curves for N. trichotoma in Fig. 5 are relatively flat). When there was a significant difference between social scenarios, weeds had a higher prevalence in the landscape under the small minority scenario (point‐down triangles).

As expected, random dispersal produced the highest prevalence of weeds for a given social scenario, followed by combined dispersal, with nearest‐neighbor dispersal having the lowest proportion of weeds. The social scenario used had a greater effect on the proportion of agents with weeds when the dispersal scenario became more random, compare Fig. 5b, d, and f (α = 5; gray).

There were strong interactions between decision parameters; for example, how deterministically agents responded to benefits, α, changed the way internal social pressure influenced the prevalence of weeds in the landscape for several scenarios, especially for E. curvula. In Fig. 5b, d, and f, when decisions were more stochastic (α = 1), internal social pressure had little effect on the proportion of agents with weeds. When decisions were more benefit driven (α = 5), increasing internal social pressure caused large declines in the proportion of agents with weeds.

Although management and impact costs (incorporated through uj) made some difference to weed prevalence (Fig. 6a is similar to Fig. 6c, and Fig. 6b is similar to Fig. 6d), it was differences in eradication probability, θ, and effectiveness of vigilance, l, that really drove the different outcomes for N. trichotoma and E. curvula (Fig. 6a and d are dissimilar, despite having the same impact and management costs, as are Fig. 6b and c).

For E. curvula (Fig. 6b), the prevalence of weeds was high as α increased from 0, then, when agents responded to benefit strongly enough the prevalence of the weed started to drop off. For the good‐neighbor scenario (point‐up triangles), E. curvula prevalence decreased quickly with increasing α, and reached a low level when agents responded strongly to benefit (α = 5). For the small minority scenario (point‐down triangles), the proportion of agents with E. curvula did not start to decrease until α was higher, and the decline was not as steep. The proportion of agents with N. trichotoma, on the other hand (Fig. 6a), dropped quickly as α increased from 0, and then leveled off.

There was an interaction between agents' response to benefit, α, infestation probability, β, and social scenario. In the good‐neighbor scenario, increasing infestation probability, β, pushed the curves in Fig. 6 to the right; thus, at higher infestation rates, the same proportion of infested agents could still be achieved, but agents had to respond to benefits more strongly. Under the small minority scenario, increasing β raised the curves in Fig. 6; thus, higher infestation rates meant a higher proportion of agents with the weed for every value of α tested.

Discussion

When a high proportion of dispersal was random, unmotivated agents could adversely affect every other agent in the landscape. As a result, unmotivated agents had a much larger effect on weed spread when dispersal was independent of distance (e.g., more seed transport by road). Long‐distance dispersal is common among invasive plants (Theoharides and Dukes 2007), suggesting that in many cases unmotivated decision makers will be important drivers of weed invasions. This supports existing theory, which predicts that outcomes of public‐goods problems are more strongly based on the “weakest links” when a large number of players interact (Perrings et al. 2002).

In contrast, we expect that if there is a strong incentive to manage a weed, then unmotivated land managers will have limited effects at the landscape scale. Unmotivated agents in our simulation had a much larger effect on the spread of E. curvula than N. trichotoma. Because N. trichotoma was much more damaging than E. curvula, and had more effective management options, agents acted against N. trichotoma more consistently. As a result unmotivated agents were less likely to be infested by N. trichotoma in the first place, and when they were infested their neighbors were more likely to control or be vigilant, reducing the impact of the unmotivated agents on the rest of the landscape.

Agents' behavior was an important influence on weed spread rates, and was influenced by the characteristics of the weed. In our example, N. trichotoma was economically damaging and management was relatively effective, thus, most decision makers took control or vigilance measures on economic grounds, and social pressure was less important. In contrast, decision makers had less economic incentive for either vigilance or control against E. curvula, and the extra motivation provided by social pressure was important in encouraging agents to manage the weed. This suggests that social pressure may only be a factor in the spread of a small subset of invasive species, those that are not so damaging, or easy to manage that people control them solely on economic grounds; or those so benign that people do not consider them weeds.

Early management is crucial for cost‐effective weed management (Davies and Sheley 2007). Social norms (represented in our model as internal social pressure) and economic cost were effective at encouraging early management. This is why increased internal social pressure was effective against E. curvula, even though this species could not be eradicated (recall we assumed controlling agents could not infest others). For decision makers to control a weed, even when it is at very low levels in the landscape, the weed must be identified as damaging before anyone in the local area has experienced it. This will be a reasonable assumption when a species is already a problem in other areas. Local governments routinely provide information on potential weeds and their control options to land managers; for example, California's Plant Health and Pest Prevention Services (information available online).5 However, just giving land managers information does not mean they will use it (Gardener et al. 2010), and when they do not, more proactive strategies will be required. One approach is to pay managers to be vigilant or control regardless of the weed's prevalence (Pokorny and Krueger‐Mangold 2007).

When the benefits of controlling a weed were obvious (as for N. trichotoma in our model), or when agents responded strongly to benefits, most agents undertook management synchronously, reducing the infested area. If, on the other hand, agents did not respond strongly to benefits, stochasticity in decision making ensured that there were enough susceptible and infested agents to sustain the invasion. This result shows that even temporarily uncontrolled infestations can sustain an invasion, provided there are enough of them at any one time. It should be noted that a landscape full of informed and responsive decision makers will not necessarily have few weeds. If the individual costs of weed control or prevention are greater than the benefit, then responsive decision makers will not act in the community interest (Satake et al. 2007).

All of our results indicate that the response of spread to both human behavior and a weed's characteristics are nonlinear. Other ecological–economic models have shown similar threshold behavior caused by feedback between decision‐making agents (Iwasa et al. 2007, Satake et al. 2007). Nonlinear behaviors (particularly thresholds) appear to be a common feature in ecological–economic systems with multiple agents. This will make the outcome of decentralized weed management less predictable at landscape scales, as small management interventions (such as imposing fines) may or may not have a large effect depending on whether the factor being managed is near a threshold or not.

Weakest link public‐goods games (the general class of problem to which collective invasive management belongs) can give us some insight in to how such complex systems might react to management interventions. One general finding from weakest link public‐goods games is that there is an incentive to free ride. In a weed control context this occurs because if an agent's neighbors control the weed: It lowers the invasion risk for the agent and so they do not need to take preventative action (Hennessy 2008). Payments, or support to the agents with the least incentive to control a weed, can encourage everyone in the group to control. If everyone knows that the least motivated agents are likely to control, then the best choice for everyone else is also to control (Hennessy 2008). This sort of dynamic suggests that weed prevalence should show a threshold response to the number of unmotivated agents (paying the least motivated agents is equivalent to reducing the number of unmotivated agents). This is the pattern we see, especially when unmotivated agents can have a large effect due to more long‐distance dispersal.

A second finding from weakest link games is that if a member of the group puts in less effort early in the game, then the strongly attracting equilibrium is for everyone to put in the least amount of effort. There is experimental evidence that peoples' decisions not to put in effort is mostly deterministic once trust in the group has been broken (Knez and Camerer 1994). Thus, the weed prevalence predicted by our model could be optimistic when decision making is very random with respect to benefit, because weakest link theory suggests that once one neighbor refuses to control their weeds, an agent should always choose not to control (i.e., act deterministically). Balanced against this are numerous anecdotal accounts of people who do control weeds, despite their neighbors' failure to do the same. Our model may also over‐predict coordination because the agents are nonstrategic, being unable to predict and respond to the possible future actions of their neighbors. Strategic agents might free ride on their neighbors' future decisions to be vigilant. A strategic agent could decide not to be vigilant because they believe that future increases in invasion risk will cause their neighbors' to become vigilant, reducing the agent's invasion risk even if they stay susceptible. This type of strategic free riding will make agents less likely to take preventive action and make coordinated action harder to achieve than is predicted by our model. However, this type of strategic behavior assumes that land managers have a good idea of what the landscape will look like in the future and how their neighbors will respond to that future.

One limitation of this work is that our results are context dependent. Strong interactions between social parameters, dispersal scenarios, and/or decision parameters, suggest that when management is decentralized, weed invasion dynamics and opportunities for landscape‐scale control will depend on the type of weed (e.g., how easy it is to eradicate, how much it reduces yield), dispersal mode, the mix of production systems in a landscape, and how decision makers perceive and react to expected benefits. This context dependence was also evident when comparing the mean field approximation and simulation. In general, the mean field approximation did a good job of reproducing the results of the simulation, but sometimes the two approaches produced slightly different predictions, and there was no consistent pattern as to when this would occur. There were three differences between the mean field approximation and simulation, the simulation was run in discrete time, on a finite landscape, with stocasticity in agents' decisions and infestation, while the mean field approximation had continuous time, an infinite landscape, and was deterministic. It is possible that different combinations of these three factors contributed to each departure between the mean field approximation and simulation we found.

Finding general rules is difficult in a model with a large number of interacting parts. One approach is to develop a simplified analytical model, from which generalizations can be drawn so long as the assumptions of the model are met. Our failure to develop such an analytical model (even the simplified mean field approximation we developed could only be solved numerically) does not mean that this problem is analytically intractable. Tractable analytical models have been developed for a similar class of problems in eutrophication and forest harvest (Iwasa et al. 2007, 2010, Satake et al. 2007). We believe that developing an analytical model for decentralized weed management is an important goal for future research.

Despite the contextual dependence of our results, some general conclusions can be drawn. More damaging weeds with effective control strategies will be controlled by more decision makers and will spread less quickly across the landscape. One or two land mangers that are reluctant to control will not greatly increase weed prevalence, but a small minority (ca. >10%) who are unmotivated can affect the whole landscape if there is long‐distance dispersal. The way decision makers react to expected benefits can have a large effect on the extent of a weed because if all agents believe control is a good idea and act on it, they tended to act synchronously, reducing the pool of infested agents available to spread the weed.

Acknowledgments

This work was funded by the Rural Industries Research and Development Corporation (RIRDC). Y. M. Buckley is supported through an Australian Research Council Australian Research Fellowship (DP0771387). H. Yokomizo is supported by a Grant‐in‐Aid for Scientific Research of JSPS. Thanks to Ryan McAllister for helpful suggestions.

  1. 5 www.cdfa.ca.gov/phpps/
  2. Supplemental Material

    Appendix

    Mean field approximation of the simulation (Ecological Archives A023‐025‐A1).