Simple rules to contain an invasive species with a complex life cycle and high dispersal capacity

Authors


Summary

1. Designing practical rules for controlling invasive species is a challenging task for managers, particularly when species are long-lived, have complex life cycles and high dispersal capacities. Previous findings derived from plant matrix population analyses suggest that effective control of long-lived invaders may be achieved by focusing on killing adult plants. However, the cost-effectiveness of managing different life stages has not been evaluated.

2. We illustrate the benefits of integrating matrix population models with decision theory to undertake this evaluation, using empirical data from the largest infestation of mesquite (Leguminosae: Prosopis spp) within Australia. We include in our model the mesquite life cycle, different dispersal rates and control actions that target individuals at different life stages with varying costs, depending on the intensity of control effort. We then use stochastic dynamic programming to derive cost-effective control strategies that minimize the cost of controlling the core infestation locally below a density threshold and the future cost of control arising from infestation of adjacent areas via seed dispersal.

3. Through sensitivity analysis, we show that four robust management rules guide the allocation of resources between mesquite life stages for this infestation: (i) When there is no seed dispersal, no action is required until density of adults exceeds the control threshold and then only control of adults is needed; (ii) when there is seed dispersal, control strategy is dependent on knowledge of the density of adults and large juveniles (LJ) and broad categories of dispersal rates only; (iii) if density of adults is higher than density of LJ, controlling adults is most cost-effective; (iv) alternatively, if density of LJ is equal or higher than density of adults, management efforts should be spread between adults, large and to a lesser extent small juveniles, but never saplings.

4.Synthesis and applications. In this study, we show that simple rules can be found for managing invasive plants with complex life cycles and high dispersal rates when population models are combined with decision theory. In the case of our mesquite population, focussing effort on controlling adults is not always the most cost-effective way to meet our management objective.

Introduction

Invasive species have negatively impacted on biodiversity and ecosystem functions worldwide (Mooney and International Council for Science 2005). Because of this detrimental impact, a large amount of time and effort is spent trying to control existing invader populations and prevent further spread. Controlling invasive species with complex life cycles is difficult and expensive to achieve in practice (Simberloff 2009), particularly if species produce a lot of propagules that easily disperse (Lockwood, Cassey & Blackburn 2005; Epanchin-Niell & Hastings 2010). Theory suggests the most basic and effective control strategy is to kill individuals at a stage during the life cycle that maximizes reductions in population growth rates and dispersal (Caswell 2001). Using matrix population models (MPM) that capture species complex life cycles, recent studies have focused on identifying these sensitive life stages (Buckley et al. 2005; Shea, Sheppard & Woodburn 2006; Ramula et al. 2008; Shea et al. 2010).

However, focusing on the most sensitive life stages is not necessarily the most cost-effective strategy to eradicate or to contain a species in the long term (Buhle, Margolis & Ruesink 2005; Hastings, Hall & Taylor 2006). For example, seedlings may cost more to find than adults depending on density and adults may be more expensive to control. Therefore, designing control strategies through time and space requires finding practical and cost-effective rules of thumb to aid with allocation decisions such as whether to focus on one or two key life stages or whether to spread efforts across all life stages. Responding to this need to consider both management costs and effectiveness, other studies have combined decision theory with population models to find the most cost-effective control strategies (Shea & Possingham 2000; Regan et al. 2006; Hauser et al. 2007; Bogich, Liebhold & Shea 2008; Firn et al. 2008; Rout, Salomon & McCarthy 2009; Yokomizo et al. 2009; Moore et al. 2011; Regan, Chadès & Possingham 2011). However, these studies tend to include less detail concerning the population dynamics of the invader and may in some cases be regarded as being overly simplistic.

Guidelines for managing long-lived plant invaders are emerging from these various approaches. Studies based on MPM suggest control effort should focus on killing adult life stages to reduce regeneration locally and spread to new sites (Ramula et al. 2008). Theoretical work suggests, however, that if the cost of control differs between life stages, the most effective control strategies may change from a focus on adults to including other life stages (Hastings, Hall & Taylor 2006). In addition, a variety of models developed for invasive animals using stochastic dynamic programming (SDP) predict that the most cost-effective control strategy will change depending on the level of propagule pressure (Travis & Park 2004; Kotani et al. 2009).

Capturing life cycle complexity and dispersal patterns of long-lived invasive plants along with the costs of management within an optimization framework has not yet been performed. Consequently, three important questions remain regarding how to efficiently manage such invaders: (i) Does variation in seed dispersal rates influence which life stage(s) we should manage?, (ii) Do we need to know the density of every life stage to predict which life stage(s) to control? and (iii) Can we focus management on controlling adults only (as MPMs have suggested) or do we need to spread management effort between different life stages (as decision theory models have suggested)?

Here, we investigate these three questions to find the simplest and most cost-effective way of controlling and preventing the spread of an Australian population of the woody perennial invader Prosopis spp (hereafter mesquite). Mesquite is considered one of the 100 worst invasive alien species in the world by the IUCN (Lowe et al. 2000) and is one of twenty weeds of national significance in Australia (Thorpe & Lynch, 2000). Dense mesquite infestations in arid and semi-arid regions of Australia can result in serious impacts on biodiversity, ecosystem functioning and the pastoral industry (Grice 2006; Martin, Campbell & Grounds 2006; van Klinken & Campbell 2009). However, low densities of infestation are often the focus of control efforts to prevent them reaching damaging densities and to prevent spread by seed into new areas (Martin & van Klinken 2006). Mesquite is a particularly efficient disperser with most pods being consumed upon maturation by a wide range of vertebrate herbivores (van Klinken & Campbell 2009).

In our case study, we examine how to supplement a biocontrol agent with herbicide control to better contain a hybrid swarm Prosopis pallida × P. velutina × P. glandulosa var. glandulosa (van Klinken & Campbell 2001) in the Pilbara Region of Western Australia. Based on economic and ecological data from this infestation, we model the population dynamics using a MPM framework that includes multiple life stages and the costs of herbicide spraying at different life stages. We use SDP to find the most cost-effective strategies for containing the infestation of mesquite below 1% canopy cover (CC), depending on the rate of seed dispersal outside the core infestation. Finally, we evaluate the benefits of combining complex MPM with SDP for developing simple rules of thumb for managers faced with the task of controlling high dispersal invasive species with complex life cycles.

Materials and methods

Case Study – the Invasion of Mesquite

Mesquite is a long-lived leguminous shrub or tree (up to 170 years, 3–15 m tall) native to the Americas. Current knowledge on the biology and control of mesquite has been synthesized by van Klinken & Campbell (2009). The general life cycle of mesquite is presented in Fig. 1. Reproduction is by seed alone. Once germinated, individual plants have a high survival rate within 12 months and grow and mature to become adults after a minimum of 3 years. Large adults can produce up to 30 000 seeds per plant. Vertebrate herbivores such as cattle and kangaroos are the primary vector of dispersal because the seeds are highly palatable, which can result in very high net dispersal rates across wide areas. The control techniques used to manage mesquite depend on the density of infestation. For instance, herbicide control is currently the most suitable option for controlling outlying and low-density populations. This technique involves traversing infestations by foot or vehicle searching for individual plants of different size classes. Once detected, each individual mesquite plant is sprayed with herbicide. In other words, every plant detected is controlled. Detectability is influenced by plant height and canopy volume, the latter being positively correlated with the stem diameter (Fig. S1, Supporting Information). Where infestations are dense to moderately dense, the most cost-effective control measure is to bulldoze with the aim of removing all above-ground life stages.

Figure 1.

 Life cycle of mesquite with six annual life stages: Seed bank (SB), sapling (Sap: <0·3 m tall and stem diameter <3 mm), small juveniles (SJ: <0·3 m tall and stem diameter 3–10 mm), large juveniles (LJ: ≥0·3 m <1·5 m tall or stem diameter 10–20 mm), small adults (SA: ≥1·5 m <3·0 m tall, stem diameter >20 mm), and large adults (LA: ≥3 m tall, stem diameter >20 mm). SA and LA together represent the canopy cover (CC). Arrows between stages correspond to annual transitions from SB to adult stages and to annual reproduction events from adults to SB or saplings. Self-loops represent stasis from year to year in the same life stage.

Study Site

Our study site was located at the Mardie pastoral station (21°11′ N, 115°58′ E) in the semi-arid Pilbara region of Western Australia [average annual rainfall (1890–2005) = 298 mm, average yearly evaporation approximately 2500 mm, average min and max temperature: 25–40 °C]. The region is invaded by the hybrid swarm P. pallida × P. velutina × P. glandulosa var. glandulosa (van Klinken & Campbell 2001). It is the most severe infestation within Australia, where isolated and dense stands of mesquite cover more than 150 000 ha of pastoral land (van Klinken et al. 2007). More than 55% of the invaded area (approximately 80 000 ha) is composed of isolated plants that are preferentially controlled by spraying herbicide (van Klinken et al. 2007).

The regional long-term objective in the Pilbara is to contain the infestation (van Klinken, Graham & Flack 2006). To achieve this objective, a biological control agent, the leaf-tying moth Evippe sp. #1 (Gelechiidae), was released in 1998 and has significantly suppressed the local population. Notably, the control agent contributed to reducing seed production (approximately 500 seeds per tree), seed bank (SB) (100 seeds ha−1 in isolated infestations to 3000 seeds ha−1 in dense patches), and reducing tree growth (but not survival) (van Klinken & Campbell 2009). Despite the effectiveness of biological control, recruitment still occurs in high rainfall years suggesting the population will continue to expand within the core infestation and spread seeds to adjacent regions. Hence, there is a need to supplement the biocontrol agent with classic control techniques, such as herbicide control of isolated plants, bulldozing denser infestations and then killing with herbicide the new recruits the following years (van Klinken & Campbell 2009).

Control Objective

Our objective of containment is to minimize the expected cost over time of controlling, with herbicide, individuals within the infested region, as well as the future cost of control arising from infestation from seeds dispersed to adjacent areas. We chose containment as a goal rather than eradication, as it is more realistic in practice for naturalized species (Simberloff 2009). We chose 1% of adult CC as the local suppression threshold because population recovery has been found to be substantially reduced below this threshold (R. van Klinken & J.B. Pichancourt, unpublished data). Because our management objective is also to minimize spread to adjacent areas, we consider the effect of seed dispersal rates (d) out of the containment area, from no net dispersal (= 0%) to the extreme situation where all seeds are dispersed from the containment area by herbivores (= 100%).

Model of Population Dynamics

Usher transition matrix

A survey of the population in the Mardie station, from 1998 to 2007, allowed us to estimate the population vital rates at different life stages i (Table S1, Supporting information) and their relation to the climate conditions inside the core infestation (see details in Appendix S1, Supporting information). When there is dispersal (> 0%), the risk of invasion increases through the establishment of new foci, but also results in less seeds contributing to the population turnover within the core infestation (1 − d). We modelled the population dynamics inside the core infestation and the seed dispersal outside the core infestation, through a (6 × 6) Usher-like transition matrix U (Caswell 2001) such that:

image(eqn 1)

The transition matrix UG integrates the annual probability to survive (Si) within a life stage i (see nomenclature of the life stages in Table 1), to grow (Gi) from stage i to i + 1, to reverse (Ri) from i to i − 1 (e.g., in response to disturbances) or to stay within the same stage [(1 − Gi) (1 − Ri)]:

Table 1.  Searching actions and efforts for different scenarios of intensity of control, where the percentages represent the proportion of individuals of the population for each life stage that are searched, detected and then killed with herbicide (assuming a 100% kill rate). Searching actions always involve spraying herbicides but the searching effort and life stages targeted vary. Estimation of the costs and assumptions can be found in Appendix S2 (Supporting Information)
Life stagesSearching effort
Action a1
Do nothing
Action a2F
Poor (Focus)
Action a2S
Poor (Spread)
Action a3F
Good (Focus)
Action a3S
Good (Spread)
Action a4
Very good
Canopy cover (CC) i.e. large and small adults inline image inline image inline image inline image inline image inline image
Large juveniles (LJ) inline image inline image inline image inline image inline image inline image
Small juveniles (SJ) inline image inline image inline image inline image inline image inline image
Saplings (Sap) inline image inline image inline image inline image inline image inline image
Cost of searching ($ ha−1)$0$15$15$40$40$160
image(eqn 2)

The fertility matrix UF(d) integrates the potential number of seeds produced by an individual tree at a stage i (fi) during the summer season that will contribute to the turnover of the population within the core infestation. Therefore, we only accounted for the percentage of these seeds that are not dispersed (1 − d) and that will either germinate (Germ) or will enter the SB (1 − Germ). Following Caswell (2001), UF(d) can be defined as follows:

image(eqn 3)

We used U(d) to project the population state N at every life stage under six management actions a (see section ‘Control actions and costs’) and for eleven seed dispersal rates (d) such that:

image(eqn 4)

where inline image and inline image are, respectively, the population state vectors at time t (≥ 1) and zero, and a the type of control action ∈ A the finite set of actions (a and A should not be confused with conventional matrix notation used in Caswell 2001). We chose a management period t of 5 years to assess the impact of the management decisions in the presence of the biological control agent.

We modelled the effect of dispersal d by assuming a proportion of the seeds produced were dispersed outside the core infestation:

image(eqn 5)

In the absence of knowledge of the spatial pattern of the vectors that disperse seeds outside of the core infestation, we assume the seeds were spread randomly and therefore that the dispersal rates are controlled only by physical factors such as patch area and geometry (Englund & Hambäck 2004a). A proportion of these dispersed seeds can reach the adult stage outside the core infestation (nSA,OUT + nLA,OUT)/nseeds dispersed. Following the assumption of random dispersal (Englund & Hambäck 2004a), all the new adult trees are located far from each other, such that if they are not controlled, each adult tree will produce a new foci of infestation with density of infestation >1% CC. Furthermore, the number of seeds produced and exported from the nascent foci to the core infestation could also be assumed to be small enough not to be considered in the calculation. Note this is a worst-case scenario, because a non-random dispersal should lead to fewer new foci and therefore a lower rate of spread and lower associated cost of control. Analysis of vital rates across the distribution of this hybrid revealed no significant difference in demography (unpublished data). We therefore assumed that the local demography outside the core infestation is the same as that in the core infestation:

image(eqn 6)

From the population vector, inline image after a 5-year time step, we estimated the number of seeds dispersed that will become adults outside the core infestation (nadults out) in the long term and therefore produce nadults out new foci of infestation with more than 1% CC:

image(eqn 7)

where inline image.

Deriving a Markov transition matrix

Stochastic dynamic programming requires a finite set of states that identifies the state of the population at every life stage. We therefore discretized the population states ni at every life stage i into several density states (Table S2, Supporting Information). The nSA and nLA states within the core infestation were then aggregated into one nCC state (for adult CC). We defined an SDP state, inline image as a combination of the density states at each life stage. We determined by simulation the probability of transition between each state and action pair. We ran simulations 1000 times using eqn 1 with a 5-year time step for all the possible 4950 SDP states, 11 dispersal rates and six control actions (Table 1). We estimated the conditional probabilities p(Xt+5|Xt, a) of transitioning from one SDP state Xt to a new state inline image during the time step, given a control action is applied. Finally, we stored, for each action and dispersal rate, the transition probabilities between SDP states in 4950 × 4950 Markov transition matrices.

Economic model

Control actions

We considered six actions that capture two gradients of control effort (Table 1), where “what is searched and detected, is what is killed” (controlled). The first gradient represents an increasing spatial resolution of control effort per hectare, which allows searchers to detect more cryptic individuals from smaller size classes. We used four resolutions of searching effort: doing nothing (a1); poor: restrict search to 1–2 size classes [adults and large juveniles (LJ)] (a2); good: restrict search to 2–3 size classes [include small juveniles (SJ)] (a3) and a very good search for all size classes (include saplings) (a4). The second gradient represents the strategy within transects of either focussing search effort with the aim of finding all adults (>1·5 m) or, for the same resources, having a more generalist search strategy aimed at spreading the search effort between adults and juveniles (>0·3 m tall) (Table 1). This was applied to poor [a2s (spread) vs a2f (focus)] and good [a3s (spread) vs a3f (focus)] search strategies. The six control actions were determined in collaboration with the National Prickle Bush Coordinator (see Appendix S2, Supporting Information).

Cost

At time t, the cost of control CT corresponds to the sum of all the control expenditures at every life stage:

image(ean 8)

where the total cost of control CT is a function of the SDP state Xt and the action at. CS is the cost for searching (Table 1), where CS depends on the density of individuals within the core infestation and the type of control action used (see estimations and assumptions in Appendix S2, Supporting information). Ck is the cost of killing by herbicide control. Ck varies depending on the stem diameter of individual plants. Exact costs for mesquite were not available; however, it is reasonable to assume that costs are similar to those measured on the weed Parkinsonia aculeata (Fabaceae): $0·09 for adult trees, LJ $0·07, SJ $0·05 and sapling $0·02 (Vitelli & Madigan 2004). We applied a cost penalty Cp when the CC exceeds the 1% threshold (X>1%). We assume Cp is equal to the cost of the most expensive action (a4). In addition, we applied a cost for spreading to adjacent regions outside the core infestation Co, where Co depends on the number of adults produced outside the core infestation: for every new seed that produces an adult, we imposed a maximal cost, equivalent to the cost of doing a4 within the site. Here, we model the cost of on-ground control. We do not include the costs associated with accessing or remote sensing particular sites, which may be dependent on the particular site being controlled.

Optimal Decision Model

Optimal solution

We calculated a state-dependent cost-effective solution (hereafter called ‘optimal solution’) using SDP for each of the 11 dispersal values d. An optimal solution, π, is a function that matches a state Xt to an optimal management action a in A, defined by inline image. Formally, the optimal solution π minimizes the expected sum of the discounted cost over an infinite time horizon:

image(eqn 9)

with the discount factor, γ, set at 0·96. We determined the optimal solution by the SDP method of value iteration (Puterman 2005). We used the Matlab MDP toolbox v3.0 (Chadès et al. 2010) to perform the calculations.

Sensitivity analysis and simplification of the optimal solution

The 11 optimal solutions predict which control actions to use for a given dispersal rate in each SDP state (4950 SDP states). From these complex solutions, we developed a simple heuristic with fewer states that approximates the optimal solutions. We assessed the performance of the heuristic in answering our three management questions related to the influence of (i) dispersal, (ii) detection and (iii) control effort. To answer the first question, we examined for every state, the similarity of the optimal solutions between all pairs of dispersal rates (10% intervals from 0% to 100%). Dispersal rates with on average more than 95% similarity (i.e. sensitivity threshold) are grouped together and we used these groups to derive a management heuristic according to the level of seed dispersal. We then estimated for each group, the life stages that need to be known to predict the optimal actions and therefore detected (i.e. second question). Accounting for the most sensitive life stages, we reduced the state space to fewer dimensions (i.e. dispersal groups and life stages) and transformed a deterministic solution π: XA into an approximate stochastic solution inline image, where each state inline image in the reduced state space inline image determines a probability distribution over the set of actions A. Finally, we used this heuristic to derive a simple rule that predicts, for every dispersal rate, when to focus or spread the control effort on one or several life stages.

Performance of the approximate solution against the optimal solution

In practice, containment requires ongoing treatment through time. We therefore compared the performance of our stochastic heuristic to the deterministic optimal solution by evaluating the average sum of cost over time under the eleven scenarios of seed dispersal. We analysed simulations after 60 years of containment where the model reassesses the optimal treatment each 5-year time step and for a starting population structure >1% CC at t = 0, where the density structure per hectare at every life stage is inline image.

Results

Do We Need to Adapt the Control Strategy to the Level of Seed Dispersal?

Comparing the results for 11 different levels of seed dispersal (from 0% to 100% with a resolution interval of 10%: see Table 2), we found that three broad groups of seed dispersal changed which control strategies were optimal: (i) seed dispersal = 0%, (ii) = 10–50% and (iii) = 60–100%. Within these three dispersal groups, the control actions were on average similar by more than 95%.

Table 2.  Grouping similar deterministic opti- mal solutions for different seed dispersal rates (d) from 0% to 100%
d d
0%10%20%30%40%50%60%70%80%90%100%
  1. Each value inside the table represents the similarities between two deterministic optimal solutions for two seed dispersal rates. The similarities were calculated as the average similarity of probability for every type of control action for every combination of density class between life stages.

  2. *Indicates when the average similarities are significant, i.e. >95% similarities. Values in bold indicate the three largest and most distinctive groups of dispersal rates based on their similarities. These dispersal groups (0%, 10–50%, 60–100%) were therefore used to inform management.

0% 100*94939291908887868585
10%94 100* 98* 98* 97* 96*9493929190
20%93 98* 100* 99* 98* 96*9494929190
30%92 98* 99* 100* 99* 98*9595949392
40%91 97* 98* 99* 100* 99*97*96*959493
50%90 96* 96* 98* 99* 100*98*98*96*9594
60%8894949597*98* 100* 99* 98* 97* 96*
70%8793949596*98* 99* 100* 98* 97* 96*
80%869292949596* 98* 98* 100* 99* 98*
90%859191939495 97* 97* 99* 100* 99*
100%859090929394 96* 96* 98* 99* 100*

Do we Need to Know the Density of Every Life Stage?

Using the three dispersal groups, we found the infestation of just a few life stages contributed to predicting the optimal control strategy (Fig. 2a–c). When = 0%, the optimal control strategy was to do nothing (a1) or focus the control on adults only (a2F). These actions were completely dependent on the density of adults (i.e. CC) (Fig. 2a). In the presence of even low levels of dispersal (Fig. 2b), the risk of spreading to adjacent regions means that most control strategies were sometimes optimal. However, performing very good control (a4) that includes controlling saplings is never optimal. In this case, the specific control strategy to use is mainly determined by knowledge of the density of adults and LJ (Fig. 2b–c). Knowledge of the density of SJ, saplings and seeds in the SB was not as important in determining optimal control strategies.

Figure 2.

 Most important life stages to know for different groups of dispersal rates (d): (a) for 0%, (b) for 10–50%, c) for 60–100%. For every control action (see Table 2), the y-axis represents the relative sensitivity (percentage) of each optimal control action when changing the density of individuals at every life stage.

Using only the three groups of seed dispersal and the density of adults and LJ, we reduced the dimension of the full set of optimal solutions represented by 54 450 states (11 dispersal rates × 4950 SDP states) of infestation into a simpler heuristic solution that involves choosing control actions for only 144 possible invasion states (Fig. S2, Supporting information). Further comparison of the probability of using one of the six control actions given these 144 states shows that we only need to account for 21 states (Fig. 3) to approximate the set of optimal solutions (i.e. a reduction of more than 99·9% in the SDP states).

Figure 3.

 Approximate optimal solutions of control predicted from adult and large juvenile (LJ) densities. Most optimal actions and related probability p over the set of all possible actions (a) explained by adults (Ad) for 0% seed dispersal, explained by pairs of density levels between adults (Ad) and LJ for (b) 10–50% dispersal and (c) for 60–100% dispersal.

Do We Need to invest in Controlling Individuals from Every Life Stage?

Examination of the heuristic for the 21 states revealed when = 0% and the infestation is contained below 1% CC (Fig. 3a), the optimal control strategy is always to do nothing (a1). As soon as the infestation increased above 1% threshold, only adults should be controlled (a2F). When some seed dispersal is possible (Fig. 3b,d), doing nothing is again a good solution, especially for very low densities of adults and LJ. The intensity of control increases progressively as the densities of adults Ad and LJ increase and as seed dispersal increases, such that controlling adults needs to be progressively supplemented by controlling small and LJ (a3F and a3S), but never saplings (a4). In addition, the heuristic predicts that the control effort should focus on adults as the density of adults alone increases (a2F and a3F). In contrast, control effort should be spread between adults and LJ (a2S) and sometimes SJ (a3S) when the density of only LJ increases. When both the density of adults and LJ increases, control efforts should be more focused on adults (a2F and a3F) than other life stages (a2S and a3S). The most thorough control strategy (a4), where searching effort includes saplings, was never recommended.

Performance of the Heuristic Against the Optimal Solution

Comparisons of the performance of the approximate stochastic solution (heuristic with 21 SDP states) against the deterministic optimal solution (54 450 SDP states) showed that a good level of containment is achievable within (Fig. 4a) and outside (Fig. 4b) the core infestation, using the simple heuristic. The solutions produced by the heuristic were also as cost-effective as the options produced using the exact solution (Fig. 4c–f). Both heuristic and optimal solutions predict a global increase in the strength of containment from <10 adult ha−1 (i.e. <1% CC) when there is no dispersal, to below 1 adult ha−1 (i.e. <0·1% CC) when all seeds are dispersed. This stronger containment results in a global increase of the searching costs CS (Fig. 4d), but constant killing costs Ck (Fig. 4e). Furthermore, both heuristic and optimal solutions predict containment of the level of infestation is possible outside the core infestation within a 60-year time frame (Fig. 4b) (approximately 0–1 adult produced per hectare of core infestation controlled vs. 20–300 adults ha−1 if not controlled), and therefore, the associated costs of infestation from seeds propagated to adjacent areas Co (Fig. 4f). Overall, the heuristic represents a good alternative to the optimal solution (Fig. 4).

Figure 4.

 Management statistics (per hectare) after 60 years of control for different levels of dispersal rates by using either the exact (black) or the approximate (red) solution. (a) Density of adults and associated canopy cover. (b) Number of adults outside the core infestation produced per hectare of land controlled within the core infestation. (c) Total cost of control (CT). (d) Cost of searching (CS). (e) Cost of killing (Ck). (f) Associated cost for the infestation outside the core infestation (CO). Solid and dashed lines represent respectively the average and SD 1000 replicates were used for the analysis.

Discussion

Population models and decision theory have been used to find rules of thumb for containing difficult to control invasive species. For long-lived invaders, these rules have generally been derived from models that either kept the complex details of the life stages of a species (Ramula et al. 2008) or used decision theory with simplified population models to find cost-effective strategies (Hastings, Hall & Taylor 2006; Firn et al. 2008). In this study, we couched a complex stochastic population model into a SDP algorithm, to find empirically the optimal approach for cost-effectively controlling an invasive mesquite population at different life stages and dispersal rates. Overall, we found a simple heuristic could reliably approximate the optimal solutions, but a complex population model was necessary to identify thresholds and narrow down the important life stages.

From the heuristic, we derived four new management rules to assist the allocation of management resources between different life stages when containing infestations: (i) When there is no seed dispersal, no action is required until adult density exceeds the control threshold and then only control of adults is recommended; (ii) when there is seed dispersal, the control strategy is dependent on knowledge of densities of adults and LJ and broad categories of dispersal only; (iii) if adult density is higher than density of LJ, killing adults is most cost-effective; (iv) alternatively, if density of LJ is equal or higher than adult density, management effort should be spread between adults, LJ and to a lesser extent SJ.

The Density of Every Life Stage is Not Needed to Predict the Most Cost-Effective Control Action

Recent studies suggest that designing cost-effective management strategies is highly sensitive to the detectability of cryptic species, cryptic life stages or cryptic ecological processes (Mehta et al. 2007; Bogich, Liebhold & Shea 2008; Hauser & McCarthy 2009; Moore et al. 2011; Regan, Chadès & Possingham 2011). For these studies, estimating the explicit impact of the detection probability on the control strategies has been answered using Partially Observable Markov Decision Process (POMDP) (Chadès et al. 2008; However, POMDP is impossible to implement when detectability involves many life stages because of computational complexity (Chadès et al. 2011). As a consequence, there is currently no clear understanding on whether the density of all the life stages needs to be perfectly detected to predict which life stage need to be controlled.

Here, we demonstrate that cost-effective control strategies may be achieved even if we are not sure of the density of the most cryptic life stages (i.e. SB, saplings and SJ). In our case deciding between management options requires knowledge of the density of adults and LJ only. We think this management option is realistic as the density of adults and LJ could be assessed by resource managers walking a small number of transects throughout the intended management zone. Walked-transects are currently unavoidable given that existing high-resolution remote sensing technologies remain expensive to use for detecting juveniles across large areas (Rango et al. 2006) or because cheaper options (e.g. historical archives of aerial photography or visual survey from aircraft) cannot readily identify juvenile mesquite (van Klinken et al. 2007; Robinson, van Klinken & Metternicht 2008). The resources not spent on detecting the most cryptic life stages or their control can be spent on surveying for new foci outside the core infestation, if necessary. Finding similar rules for other species may be extremely useful for adapting detection effort, because detection errors are recurrent problems when surveying individuals of different sizes or densities (Fleming & Tracey 2008), and ultimately when designing management decisions. Therefore, even if POMDP is necessary to estimate precisely the acceptable level of detection at every life stage, the methods used here provide an indirect way of solving problems of low-density invasion when species have cryptic life stages.

Precise Estimates of Seed Dispersal Rates are Not Needed to Determine Control Effort

Whether or not control effort within a core infestation should be adjusted depending on propagule pressure remains unresolved (Travis & Park 2004; Kotani et al. 2009; Epanchin-Niell & Hastings 2010). For example, studies have found that a very good knowledge of the level of propagules dispersed outside the core infestation is important for adjusting the overall intensity of control (Travis & Park 2004; Kotani et al. 2009). Reliably calculating long-distance dispersal distances in the field is, however, difficult (Nathan, 2006).

The risk of contamination outside the core infestation increases as dispersal from the core area to outside increases. If the management objective is to minimize this risk and associated management costs, our findings suggest that we should intensify management within the core infestation. However, estimating the level of seed dispersal is very difficult when seeds are dispersed by fauna. We show an approximate knowledge of the rate of seeds dispersed does not necessarily decrease our ability to predict which control action to use. Furthermore, our results suggest reducing dispersal to or near 0%, or broadly from high (60–100%) to moderate levels (10–50%), can also reduce the cost and effort of controlling the core infestation. In this case, reducing dispersal is possible by altering farm design, fencing and livestock hygiene practices (Campbell & Grice 2000).

It is Not Necessary to Control Individuals from Every Life Stage

There are few management rules available for predicting which life stages should be killed to contain long-lived invaders such as mesquite. Ramula et al. (2008) found adults were the best life stage to focus control efforts on because seed production can be reduced at the same time. However, other studies have suggested that when the cost of control is included, the most cost-effective control strategy is expected to include killing adults and other life stages (Buhle, Margolis & Ruesink 2005; Hastings, Hall & Taylor 2006). Our results confirm that focusing control efforts on adult trees is a cost-effective strategy when both adult CC is above a 1% threshold and there is no seed dispersal, and if there is seed dispersal but the density of LJ is low. If the density of LJ increases relative to the density of adults, control of adults should be supplemented by the control of LJ, or supplemented by also controlling SJ (but never saplings), if both adult and LJ densities increase (Fig. 3). These management rules are robust to changes in dispersal rate. These results demonstrate that from complex models, simple rules of thumb can emerge for managers.

Implications for Controlling Mesquite and Other Invaders

At present, general management recommendations for long-lived weeds like mesquite are not based on the four rules we developed, but on a simple decision rule that consists of spending resources on controlling adults (Ramula et al. 2008). In practice, smaller individuals surrounding the adults are also opportunistically killed, because of the low additional search effort and expenditure in herbicide. Our results suggest for this mesquite infestation that (i) the most cost-effective control strategy may include targeting smaller size classes than adults, (ii) that knowledge of the density of larger trees is likely to be more important than smaller more cryptic size classes in determining the optimal management strategy and (iii) that this knowledge will help managers decide whether to focus their attention on controlling large reproductive plants (i.e., the management rule currently used) or spread their attention by controlling other smaller size classes (Fig. 3).

When a management rule is already used for such serious weeds, there will be an understandable reluctance to change (Panetta & Brooks 2008). We suggest the four new rules derived here can be used to evaluate within an adaptive management framework (Keith et al. 2011) whether changing the current practice will lead to better management of large mesquite infestations.

Optimal management of an invasive species will depend on many factors including the species life cycle and dispersal strategies, climate and landscape context, management type and resources and most importantly the management objective (conservation, containment, eradication). Encouragingly, we find that simple cost-effective rules can be derived by combining MPM with SDP. Initially, incorporating complexity is important to ensure key life stages and costs are not overlooked. Further rules may emerge by replicating our framework on other species.

Acknowledgements

J.B.P. was funded through an OCE post-doctoral fellowship and T.G.M. supported by a Julius Career Award. We thank S. Raghu and P. Caplat for useful comments on this manuscript. Thanks to Land and Water Australia (Defeating the weed menace program) and the CSIRO Climate Adaptation Flagship for joint funding. We thank Nathan March for costing scenarios.

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