THIS ARTICLE HAS BEEN RETRACTED Modeling Responses of Leafy Spurge Dispersal to Control Strategies

Journal of Integrative Plant Biology 2007, 49 (12): 1681–1692

THIS ARTICLE HAS BEEN RETRACTED

Modeling Responses of Leafy Spurge Dispersalto Control Strategies

Zewei Miao∗

(Grant F. Walton Center for Remote Sensing & Spatial Analysis, Cook College, Rutgers University,

New Brunswick, NJ 08901-8551, USA)

Abstract

Leafy spurge (Euphorbia esula L.) has substantial negative effects on grassland biodiversity, productivity, and economicbenefit in North America. To predict these negative impacts, we need an appropriate plant-spread model which can simulatethe response of an invading population to different control strategies. In this study, using a stochastic map lattice approachwe generated a spatially explicitly stochastic process-based model to simulate dispersal trajectories of leafy spurge undervarious control scenarios. The model integrated dispersal curve, propagule pressure, and population growth of leafy spurgeat local and short-temporal scales to capture spread features of leafy spurge at large spatial and long-temporal scales. Ourresults suggested that narrow-, medium-, and fat-tailed kernels did not differ in their ability to predict spread, in contrastto previous works. For all kernels, Allee effects were significantly present and could explain the lag phase (three decades)before leafy spurge spread accelerated. When simulating from the initial stage of introduction, Allee effects were criticalin predicting spread rate of leafy spurge, because the prediction could be seriously affected by the low density period ofleafy spurge community. No Allee effects models were not able to simulate spread rate well in this circumstance. Whenapplying control strategies to the current distribution, Allee effects could stop the spread of leafy spurge; no Allee effectsmodels, however, were able to slow but not stop the spread. The presence of Allee effects had significant ramificationson the efficiencies of control strategies. For both Allee and no Allee effects models, the later that control strategies wereimplemented, the more effort had to be input to achieve similar control results.

Key words: Allee effects; dispersal curve; invasive species; non-indigenous; propagule pressure; stochastic process-based model.

Miao Z (2007). Modeling responses of leafy spurge dispersal to control strategies. J. Integr. Plant Biol. 49(12), 1681–1692.

Available online at www.jipb.net

Invasions of exotic species have been proposed as one oflargest components of habitat destruction, biodiversity losses,and economic damage (Leistritz et al. 1992; Cain et al. 2000;Leung et al. 2002; Rouget et al. 2003). As a perennial di-cotyledonous herbaceous plant, leafy spurge (Euphorbia esulaL.) adapts to a wide variety of habitats (Selleck et al. 1962;Dunn 1979; Watson 1985), and has become a great threatto rangeland productivity, to species diversity, to the quality of

Received 14 Aug. 2006 Accepted 2 Oct. 2006

This research was funded by the Integrating Economics and Biology for Bioe-

conomic Risk Assessment/Management of Invasive Species in Agriculture

(Economic Research Service/USDA).∗Author for correspondence.

Tel: +1 732 932 1583;

E-mail: <[email protected]>.

C© 2007 Institute of Botany, the Chinese Academy of Sciences

doi: 10.1111/j.1744-7909.2007.00601.x

wildlife habitats, and to land values in the northern USA and theprairie provinces of Canada since its discovery in North Americain 1827 (Selleck et al. 1962; Wallace et al. 1992). An area of1.6 millionha of land in the Upper Great Plains (North Dakota,South Dakota, Montana, and Wyoming) of USA were infestedby 1994 (Everitt et al. 1995; Bangsund et al. 1999). In the pastdecade, leafy spurge infestations have been estimated to resultin an annual economic loss of US$130 million in the four-stateregion (Leitch et al. 1994).

There is a need to simulate dispersal features of leafyspurge under the influences of weed management practices.Field experiments demonstrate that control strategies for leafyspurge, including chemical, multi-species grazing, and bio-logical efforts, are able to reduce population density (i.e.propagule pressure) to a large extent (Selleck et al. 1962;Watson 1985; Leitch et al. 1994). This likely constrains spa-tial spread through reduced propagule pressure. So far, fewstudies have attempted to simulate the response of dispersalprocess and patterns of leafy spurge invasion to various weed

1682 Journal of Integrative Plant Biology Vol. 49 No. 12 2007

management practices. It is not yet known with certainty howecological dispersal features of leafy spurge will be affected byweed management practices at large spatial and long temporalscales.

Process-based models provide us with a mechanistic under-standing of invasion and dispersal features of exotic speciesat large scales responding to management practices and envi-ronmental heterogeneities. Analyses of the spread of invadingorganisms frequently start with a reaction-diffusion (R-D) modelwith exponential growth and Fickian diffusion (Higgins et al.1996; Kot et al. 1996). When the dispersal pattern is Brownianand the trapping rate by the environment is uniform, the disper-sal pattern can be described as follows (in simple form):

Nt+1(x) =∫ ∞

−∞k(x, y)f [Nt(y)]dy (1)

or

∂Nt+1(x)∂t

= rNt(x)(

1 − Nt(x)K

)+ D

∂2Nt(y)∂x2

(2)

where Nt+1(x) and Nt(y) are the population densities at gen-eration t + 1 and t, at locations x and y, respectively; k(x,y) isthe one-dimensional redistribution kernel, that is, the probabilitydensity function for propagule dispersing to destination x from asource position y; r is the species’ intrinsic rate of increase; ands is the diffusion constant (Higgins et al. 1996; Kot et al. 1996).This kind of R–D model assumes that the invading organismsspread in a given redistribution kernel regardless of environ-mental heterogeneities and weed management practices. Theprecise shape of the redistribution kernel k(x, y) is extremelyimportant to describe the speed characteristics of invasion (Kotet al. 1996). The R–D model, however, is usually limited by theabsence of suitable redistribution kernels and/or by inadequateparameter estimates for such kernels. For instance, ecologicaldata is incomplete to generate the redistribution kernel for leafyspurge R–D model (Selleck et al. 1962; Cain et al. 2000).TheR–D equations also assume that the invading populations arelarge enough that stochastic effects are not important, which, innature, is not always the case (Hengeveld 1994; Kot et al. 1996).Field experiments show that leafy spurge invasion are mainlycaused by stochastical biological and environmental factorssuch as flooding water, wind, anthropogenic disturbance (e.g.vehicle tracks, overgrazing, road construction, and fire guards),birds, insects (e.g. ants), and wild and domestic animals(Selleck et al. 1962; Dunn 1979; Watson 1985; Belcher and Wil-son 1989). All together, there is a need to establish a process-based model to simulate stochastic processes and patterns ofleafy spurge invasion that responds to various control strategies.Over a certain spatial scale (stands to landscapes), stochasticprocess-based models can integrate biological attributes, spa-tial considerations, invasion stochasticity, and environmentalheterogeneity into modelling species’ distributions (Rouget et al.2003).

Interactions among dispersal curve, propagule pressure, andintrinsic population growth are central to establishing a stochas-tic process-based spread model of exotic species invasion. Inthe invasion process (i.e., introduction, colonization, and natu-ralization), many factors, such as propagule pressure, distancefrom the source, and external environment, determine spreadrange and abundance of non-indigenous species at a givenlocality. Dispersal curves are frequently fitted with a negativeexponential curve or a negative power function (Wallace 1966;Taylor 1978; Kot et al. 1996), and the probability density functionof a dispersal curve may be normally distributed for somespecies (Kot et al. 1996). The role of propagule pressure or masseffects is clearly observable in many invasions. As propagulepressure increases, chances of establishment, persistence,naturalization, and invasion will increase as well, especially atthe introduction stage (Rouget et al. 2003). Population growthdynamics affects propagule yield and dynamics of leafy spurgeecological communities; hence population growth will affectthe probability of establishment in uninfested areas. Recentworks indicate that the combined influence of Allee dynamicsand stochastic processes strongly determines the successfulestablishment of alien species (Liebhold and Bascompte 2003).However, the influences of Allee effects are still unknown in theprocess of leafy spurge invasion.

In this paper, we link the data available across two differentscales – dynamic density data at local (plot) and short temporalscales, and long time series of infested areas at a regional scale– to construct our dispersal model. We simulate, with minimumparameter requirements, the stochastic processes and patternsof spread of leafy spurge in response to control strategies. Wetest whether the form of the dispersal kernel (i.e. narrow-versusfat-tailed kernels) and Allee effects are important for explainingthe observed pattern of invasion, including the lag period beforeleafy spurge becomes invasive. We simulate efficiencies offour typical control scenarios of leafy spurge management inNorth America that have been applied at local spatial and shorttemporal scales. This work will be useful to future research onrisk analysis, management options, and optimal control theory.

Results

Prediction of leafy spurge invasion versus differentredistribution kernels

There were no differences in predicting infested area among thenarrow-, medium-, and fat-tailed redistribution kernels, but theirbest-fitting parameters, β, α, and c, were different (Table 4). Withthe different forms of redistribution kernels and their own best-fitting parameters, similar dispersal trajectories were obtainedfor all corresponding control strategies, regardless of Allee or noAllee effects (Figures 1 and 2). For the no control scenario andwith Allee effects, all the three kernels gave good predictions

Modeling Spatial Dispersal 1683

0

300 000

600 000

900 000

1 200 000

0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100

ObsAENAE

Years

Infe

sted

are

a (h

a) Narrow-tailed kernel

0

300 000

600 000

900 000

1 200 000ObsAENAE

Years

Infe

sted

are

a (h

a) Medium-tailed kernel

0

300 000

600 000

900 000

1 200 000ObsAENAE

Years

Infe

sted

are

a (h

a) Fat-tailed kernel

Figure 1. Simulation of leafy spurge dispersal from the beginning of

introduction, with different redistribution kernels.

Obs, observed data points, AE, simulating with Allee effects, and NAE,

simulating without Allee effects.

in leafy spurge dissemination against the infested survey area(Table 1). For no control and with Allee effects, the narrow-, medium- and fat-tailed kernels predicted the infested area of507 787.2, 505 371.2, and 496 539.9 ha at year 81, respectively,while the survey leafy spurge area was 526 091.3 ha at year 81.The relative differences between the prediction and the corre-sponding survey data per iteration were 0.29, 0.24, and 0.26,respectively.

For the three kernels, the best-fit parameters usually con-sisted of low β (i.e. greater maximum dispersal distance rangingfrom 4 200 to 6 300 m), low α (i.e. low probability of estab-

0

300 000

600 000

900 000

1 200 000

0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100

ObsAENAE

Infe

sted

are

a (h

a)

Years

Narrow-tailed kernel

0

300 000

600 000

900 000

1 200 000ObsAENAE

Infe

sted

are

a (h

a)

Years

Medium-tailed kernel

0

300 000

600 000

900 000

1 200 000ObsAENAE

Infe

sted

are

a (h

a)

Years

Fat-tailed kernel

Figure 2. Simulation of leafy spurge dispersal from current distribution

(year 53 since 1909) with different redistribution kernels.

Obs, observed data points, AE, simulation with Allee effects, and NAE,

simulation without Allee effects.

lishment of individual propagules), and high c values (Alleeeffect coefficients, ranging from 1.5 to 2.5) (Table 4), whoseestimates matched well with the survey data. The best-fitting β,α, and c values were 0.180 423 2, 0.000 286 9, and 2.440 226for narrow-tailed kernel, 0.003 09, 0.001 66, and 1.707 093 formedium-tailed kernel, 1.203 162, 0.001 102, and 2.066 86 forfat-tailed kernel, respectively. Because of lack of differencesbetween the three kernels with respect to predictions, theremainder of the analyses are presented only for narrow-tailedkernel.


Table 1. Leafy spurge infestation in North Dakota State

Years Time interval since Infested % of infestedthe initial infestation (a) area (ha) land to farmland

1909† 0 0 (the first 0reported year)

1962 53 80 900.00 0.495 597

1973 64 171 400.00 1.050 004

1982 73 348 800.00 2.136 764

1987 78 404 685.64 2.479 121

1990 81 526 091.33 3.222 858

† The first reported year of leafy spurge invasion in North Dakota.

Data from Leistritz et al. 1992; Lym et al. 1993; Leitch et al. 1994;

Bangsund et al. 1997.

Influences of Allee and no Allee effects on lag phasein leafy spurge invasion

The dispersal trajectory predicted with Allee effects (i.e. c > 1in Equation 3) remained very different from no Allee effects(i.e. c = 1 in Equation 3) (Figure 1). When simulations startedfrom the beginning of introduction, prediction with Allee effectswas significantly better than no Allee effects. Allee effectswere essential in capturing the lag phase during leafy spurgeinvasion, regardless of kernel forms (Figure 1). The relativedifferences between prediction and survey were 0.28 for Alleeeffects and 0.51 for no Allee effects of narrow-tailed kernel, 0.24for Allee effects and 0.48 for no Allee effects of medium-tailedkernel, and 0.26 for Allee effects and 0.50 for no Allee effectsof fat-tailed kernel, respectively (Table 4). Allee effects reducedthe probability of establishment of an individual propagule at thebeginning of the invasion, permitting the observed lag phase of3 decades, and increased the probability at the latter period incomparison to no Allee effects (Figure 1). Leafy spurge spreadrate with Allee effects was almost two times that of no Alleeeffects in the last simulating year (year 81).

When simulations leaped over the initial phase of introduction,there were no significant differences between Allee and noAllee effects. When the simulations began from the currentdistribution (e.g., year 53, the first surveyed year of leafy spurgearea), the model gave good predictions in leafy spurge dispersalcompared to survey data (Figure 2), regardless of Allee or noAllee effects. The relative differences between the predictionand the corresponding survey data per iteration were 0.13 and0.15 for Allee and no Allee effects of the narrow-tailed kernel,0.132 939 and 0.175 604 for Allee and no Allee effects of themedium-tailed kernel, and 0.186 954 and 0.135 012 for Alleeand no Allee effects of the fat-tailed kernel, respectively.

Allee effects magnified the importance of early control strate-gies of leafy spurge. With Allee effects, responses of leafyspurge spread to control strategies were significantly greaterthan for no Allee effects, no matter what the control scenarios(Figure 2). For example, for the consecutive control (CC)scenario, when control efforts were greater than 30%, leafy

spurge was contained during the 81 simulation years for Alleeeffects. In contrast, for no Allee effects, leafy spurge continuedto spread even though the control level was 60% (Figure 2).For a given control effort, the response of no Allee effects tocontrol strategies was considerably less than with Allee effects,especially at the initial period.

Responses of leafy spurge invasion to control strategies

The later that control strategies were initiated, the more effortwas needed to get similar control achievements. Similarly,when a given control practice was conducted, the smaller thepopulation size, the better the control achieved. Because theno Allee effects model was not able to simulate leafy spurgeinvasion well, the model with Allee effects was mainly used toanalyze the effectiveness of control strategies. For the narrow-tailed kernel, by comparing the CC to consecutive controlbegun from current distribution (CCC) scenarios, when controlstrategies were conducted from year 0 for the CC scenario, acontrol level of 30% was able to contain leafy spurge spreadduring the 81 simulation years. When control was implementedat year 53 or year 81 for CCC scenario, however, over 50% and70% control levels were needed to stop leafy spurge dispersal,respectively (Figure 3). A control level of 30% could slow, but notstop, the spread of leafy spurge in the CCC scenario. Table 5also indicates that the bigger the population size, the lower thecontrol effectiveness would be for a given control effort.

Our results suggest that the initial single control (IC) scenariowas not able to contain leafy spurge spread in 81 simulationyears, except for 100% eradication. Even though less than10% of propagules had an opportunity to spread, leafy spurgewould ultimately disseminate during 81 simulating years, butthe spread rate slowed down with an increase of control efforts(Figure 3). In other words, for the IC scenario, one had toeradicate leafy spurge patches at the beginning of invasion tofully terminate leafy spurge dispersal.

The results of the initial intermittent control (IIC) scenariosuggest that more control efforts are required to contain leafyspurge spread than with the CC scenario. For example, for theIIC scenario, more than 70% of control levels were needed tostop leafy spurge spread, while 30% of control efforts wereenough to contain the spread for the CC scenario (Figure 4).For the CC scenario, frequency of control application was higherthan for IIC and this had to be incorporated in the computationof control efforts. The IIC scenario worked more effectively thanIC, as IC was almost unable to stop dissemination even ata control level of 90%. IIC control effectiveness was similarto the CCC scenario starting from the current distribution atyear 81, but worse than that of the CCC scenario starting fromthe current distribution at year 53. More input was needed forthe CCC scenario (a common practice in the integrated pestmanagement of leafy spurge invasion) than for the CC scenarioto achieve similar results, despite the fact that control efforts


0

150 000

300 000

450 000

600 000

0 20 40 60 80

0 20 40 60 80

ObsCL=0%CL=10%CL=20%CL=30%

Years

Infe

sted

are

a (h

a)

Allee effects

0

150 000

300 000

450 000

600 000ObsCL=0%CL=10%CL=20%CL=30%CL=50%CL=70%

Years

Infe

sted

are

a (h

a)

No Allee effects

Figure 3. Allee effects versus no Allee effects in simulation of response

of leafy spurge invasion to CC scenario from the beginning of introduction

in North Dakota. The simulations were done with the narrow-tailed

kernel.

Obs, observed data points, CC, consecutive control scenario, and CL,

control level (%).

were intermittent and the numbers of control practices werefewer than in the CC scenario.

Discussion

Dispersal of a spreading population and redistributionkernels

Redistribution kernels are probability-density functions that de-scribe the possibility that individuals will be found at specificspatial coordinates relative to the original, colonizing location(MacIsaac et al. 2004). Insight into the process and spatialpatterns of dispersal of a non-indigenous species can begleaned from its redistribution kernel (Taylor 1978; Kot et al.1996; Clark et al. 2003; MacIsaac et al. 2004). Kot et al. (1996)reported that the speed of invasion of a spreading populationis extremely sensitive to the precise shape of redistribution, inparticular, to the tail of the distribution. They reported that differ-ent shaped kernels with similar coefficients of determination willyield dramatically different speeds in a spreading population,

0

150 000

300 000

450 000

600 000

0 20 40 60 80

0 20 40 60 80

0 20 40 60 80 100

ObsCL=0%CL=10%CL=20%CL=30%

Years

Infe

sted

are

a (h

a)

CC scenario

0

150 000

300 000

450 000

600 000 ObsCL=0%CL=10%CL=20%CL=30%CL=50%

Years

Infe

sted

are

a (h

a)

CCC scenariosince year 53

0

300 000

600 000

900 000

1 200 000 ObsCL=0%CL=10%CL=20%CL=30%CL=50%CL=70%

CCC scenariosince year 81

Infe

sted

are

a (h

a)

Years

0 20 40 60 80

0 20 40 60 80

0

150 000

300 000

450 000

600 000 ObsCL=0CL=10%CL=20%CL=30%CL=50%CL=70%

Infe

sted

are

a (h

a)

Years

IIC scenario

0

150 000

300 000

450 000

600 000 ObsCL=0%CL=10%CL=30%CL=50%CL=70%CL=90%

Infe

sted

are

a (h

a)

Years

IC scenario

Figure 4. Predictions of efficiencies of various control strategies with

Allee effect. The simulations were done with the narrow-tailed kernel.

Obs, observed data points, CL, control level (%), IC, initial single

control scenario, CC, consecutive control scenario, IIC, initial intermittent

control, and CCC, consecutive control begun from current distribution.


and fat-tailed kernels can generate accelerating invasions ratherthan constant-speed waves (Kot et al. 1996). Our results foundthat the short-, medium-, and fat-tailed kernels with differentbest-fit parameters are able to achieve similar trajectories forleafy spurge, and these results are not affected by controlstrategies.

Dispersal distance coefficients are critical parameters forthe stochastic process-based model prediction. In our study,values of β (dispersal coefficients) are all very low for the threekernels, with a maximum spread distance of around 4 200 and6 300 m (for cumulative probabilities of 0.999 5). These dispersaldistances conform to previous dispersal observations for leafyspurge, which suggest that leafy spurge dissemination occursin both local diffusion (short distance) and jump dispersal (atmedium and large distance) patterns (Herbert and Rudd 1933;Selleck et al. 1962). Our sensitivity analysis shows that aslight change in the β value can induce a large change in thedispersal distances from parental colonies of introduced newcolonies. Therefore, paired field data linking population densityto dispersal distance should be collected for empirical validationof leafy spurge invasion in the future.

The spread rate was strongly affected by propagule pressureand Allee effects. Biologically, these two factors have quanti-tative (i.e. number of individuals moved) and/or qualitative (i.e.the condition, sex ratio, or size structure of individuals moved)components that may influence whether an invasion succeeds(Kot et al. 1996; Seaman and Powell 1996; Clark et al. 2003).These two factors are tightly combined in our model.

Allee effects in leafy spurge invasion

Allee effects can cause the invasion of an alien species to fail,especially in the introduction and establishment of invading pop-ulations (Grevstad 1999; Engen et al. 2003). Our study showsthat regardless of kernel shapes, the c values in Equation 3are consistently much greater than unity, which suggests Alleeeffects are significantly present in a leafy spurge invasion. Incontrast, a no Allee effects model cannot predict the trajectoryof a leafy spurge invasion well, as no Allee effects predictionsare too high to capture the lag phase at the initial stage, and toolow at the later.

Allee effects are indispensable in simulating the lag phase ofa leafy spurge invasion (about 2 or 3 decades in North Dakota)(Hobbs and Humphriest 1995; Sakai et al. 2001). These lagtimes are expected if environmental or evolutionary change isan important part of the colonization process. This process mayinclude the adaptation of invasive species to a new habitat, theevolution of invasive life history characteristics, or the purgingof a genetic load responsible for inbreeding depression (Sakaiet al. 2001). Our predictions suggest that population dynamicsprocesses, such as Allee effects, are sufficient to account for thelag phase observed in leafy spurge. There is also evidence fromfield experiments for Allee effects in leafy spurge, where seed

germination is about 20–30% and the percentage of seedlingssurviving is much lower at the initial stage (Herbert and Rudd1933; Selleck et al. 1962). Once a patch is established, leafyspurge can thrive in a variety of conditions, producing a largenumber of propagules (seeds) (Stroh et al. 1990; Leitch et al.1994; Bangsund et al. 1999).

Allee effects have an important part in leafy spurge manage-ment. With Allee effects, the efficiencies of control strategiesare significantly greater than without Allee effects; because withAllee effects, control strategies can effectively keep densitiesbelow the critical point where population size explodes. Predic-tions with Allee effects are in accordance with field experimentsthat demonstrate that early detection and treatment of invasionsbefore explosive spread occurs will prevent many future disper-sal problems (Hobbs and Humphriest 1995; USDA-ARS TEAMleafy spurge area-wide IPM program 2002a, 2002b, 2002c). NoAllee effects models are not able to increase the lag time of aleafy spurge invasion, thus, leafy spurge spreads earlier andmore quickly. Without Allee effects, more control efforts will berequired to achieve the same results as with Allee effects.

Simulation of response of invasive speciesto control strategies

In the field of ecological modeling, tremendous efforts aredevoted to the study of mechanisms of introduction, establish-ment, dissemination, and propagule pressure, but few stud-ies focus on the response of exotic species to managementstrategies. Many field experiments of control strategies havebeen fairly effective at the field plot and short temporal scalesfor leafy spurge (USDA-ARS TEAM leafy spurge area-wideIPM program. 2002a, 2002b, 2002c). Little is known about theconsequences of control strategies at large spatial and longtemporal scales (for example, across several states and severaldecades). This is important given the labor, time, and costneeded to implement such strategies. Weed control generallyfalls into four categories: prevention, eradication, reduction,and containment. In this study, containment control scenariosare only taken into consideration in terms of the continuousspread of leafy spurge. We assume that control strategiesreduce density and propagules, and ultimately decrease theprobability of establishment of individual propagules. Our sim-ulations confirm that the later that control strategies begin, themore effort has to be devoted to get similar control results.This is in accordance with previous field experimental studiesshowing that containment and eradication of small infestationsat early stages should be a major component of a leafy spurgemanagement program in North America (Lym 1997; Bangsundet al. 1999; Rouget et al. 2003).

For a given control level, efficiencies of various control strate-gies are different. The efficiencies of the four scenarios in thisstudy rank as: CC > CCC > IIC > IC. For different control levels,different control scenarios may have similar effectiveness. For


example, the effectiveness of the CCC scenario starting from thecurrent leafy spurge area at year 53 is similar to the IIC scenario,at some control levels. When the effectiveness of differentstrategies is similar, the choice between them should dependon further social, economic, and ecological welfare analysis ofthese management strategies.

Future directions: model structure and environmentheterogeneity

The map lattice model combines dispersal curves, propagulepressure, population intrinsic growth dynamics, and Allee ef-fects. The model takes into account the randomness of estab-lishment of non-indigenous species. One of its most importantadvantages is the integration of population growth dynamicswith stochastic dissemination across different scales (i.e. com-bines density growth dataset at field level for a few years withtime series dataset of leafy spurge at regional scales acrossdecades), which is a challenging topic in ecological modeling(Hobbs et al. 1995; Higgins et al. 1996).

Future development of the model should include competitionbetween exotic and native species and environmental hetero-geneity of various landscape fragments (e.g. landscape slope,soil, water, sunlight, wind, etc). Competition among ecologicallyalien and native species can be a major factor that determinesthe structure of animal and plant communities. When com-petition is considered, for example, Equation 12 will become:dNalien

dt = ralienNalien(1 − Nalien+w1NnativeKalien

)where Nalien and Nnative are population densities (stems/ha)of alien and native species, respectively; ralien is the intrinsicgrowth rate of the alien species population; Kalien is the car-rying capacity of population density of the alien species in agiven area; and w1 is a competition weight, w1 ≤ 1. As w1

approaches 1, competition may influence population growthdynamics in the model predictions. Elements such as landscapecomposition and configuration are important in the introduction,establishment, and movement rates of dispersing individuals,because environmental variables usually are different acrossvarious landscapes. For instance, leafy spurge is more commonin river areas and along roadsides (except for railway embank-ments), prefers sandy-loam soil, and can easily flourish in drymeadows associated with grass species (Selleck et al. 1962). Ifspatial heterogeneities are large, they could have considerableimplications on the estimation of emigration and immigrationparameters in population dispersal models (Belisle and Clair2002).

Materials and Methods

Model developments

Using a map lattice approach, we integrated dispersal curves,the probability of propagule establishment, and population

growth dynamics to develop the stochastic process-based dis-persal model.

Progagule pressures

Propagule pressures play a significant role in biological inva-sion. If propagules interact with each other, in such a mannerthat Allee effects are present, the total probability of establish-ment (E) can be described using the Weibull distribution (Dennis2002; Leung et al. 2004):

E(Nl,t) = (1 − e−(αNl,t)c) (3)

where N is the number of propagules arriving at location l attime t, and α is a scalar for propagule pressure and c is a shapecoefficient. When c equals 1, Allee effects are absent (i.e. noAllee effects) and each propagule has an independent chanceof establishment.

Leafy spurge propagules are composed of seeds and veg-etative parts (e.g. roots and shoots) (Herbert and Rudd 1933;Selleck et al. 1962). In this study, we focused on seed dissemi-nation, because the infestation of a new area (e.g. abandonedfields and roadside) is typically initiated by seeds, with the ex-ception of crop fields (Herbert and Rudd 1933). Seed invasionsare more stochastic and difficult to control than invasions viavegetative pathways (e.g. roots and shoots) (Herbert and Rudd1933; Selleck et al. 1962; Watson 1985).

Dispersal curve

Since the redistribution kernel was lacking for leafy spurgeinvasion, we employed three redistribution kernels to test short,medium, and long possible spread distances. For one dimen-sion, probability density function (PDF) of the three kernels canbe written as:

PDF = ae−βx (4)

PDF = ae−βx2(5)

PDF = ae−β√

x (6)

where PDF is the probability density function, x is the disper-sal distance from infested original source, β is the dispersalcoefficient, and a is the intercept of probability density. Toconveniently compare with previous studies, Equations 4, 5, and6 match Taylor’s (1978) and Kot et al.’s (1996) redistributionkernels1 (narrow-tailed), 2 (medium-tailed), and 5 (fat-tailed),respectively. Hereinafter, we will call the three kernels narrow-,medium-, and fat-tailed kernels, respectively. These redistribu-tion dispersal kernels were extrapolated from distance-densityinsect models:

N(x) = eα−βx (7)

N(x) = eα−βx2(8)


N(x) = eα−β√

x (9)

where N is the density number found at a distance x fromthe centre of dispersal, β is the dispersal coefficient, and a isinterception (Taylor 1978; Kot et al. 1996).

For Equation 7, the probability density function within a givendistance 0–L is:

PDF = eα−βx∫ L0 eα−βxdx

= ae−βx (10)

where the coefficient a equals: β

1−e−βL . Similarly, we derived

PDF = 2√

β

πe−βx2

from Equation 8 and PDF = β2e−β√

x

2 fromEquation 9.

Population intrinsic growth dynamic of leafy spurge

With respect to population growth dynamics, the logistic intrinsicgrowth curve, the most traditional density-dependent growthmodel (Getz 1996), was adapted to our model:

Nt+1 = Nt + rNt

(1 − Nt

K

)(11)

ordNdt

= rN(

1 − NK

)(12)

where Nt + 1 and Nt are population density (stems/ha) at gener-ation t + 1 and t; r is intrinsic growth rate of the population; andK is carrying capacity of population density in a given area.

Map lattice approach

We used a map lattice approach to integrate dispersal curves,the probability of propagule establishment, and populationgrowth dynamics in our model. At first, the map lattice approachdivided the total farm lands of the study area into large numbersof uniform cells (i.e. 4 000 × 4 000 cells). When cells are infestedby propagules at time ti , the population begins to grow in alogistic intrinsic growth curve and to produce propagules (i.e.,seeds) at the next generation, that is, t i+1. Propagules in theinfested cells will spread in a negative exponential dispersalcurve, that is, Equations 4, 5, or 6. In this way, we used existingdata on population growth at the local and short-temporal scale(within a cell) to extrapolate patterns across large and long-temporal scales (multiple States).

Critical distances of redistribution kernels

Theoretically, distance of propagule dissemination may befrom 0 to infinity. In practice, to simulate propagule dispersaleffectively, we had to consider spread within a certain criticaldistance (L). For the narrow-tailed kernel, the critical distancewas derived as follows:

p =∫ L

0 e−βxdx∫ ∞0 e−βxdx

= a (13)

i.e. p = (−1/β) e−βx∣∣L0

(−1/β)= a (14)

So, the critical distance L is:

L = ln(1 − a)/(−β) (15)

where p is the cumulative probability of propagule dispersalwithin the distance L, x is the distance of a cell from theinfested origin, a is the chosen cumulative probability, and β

is the dispersal coefficient. We set the cumulative probability ofpropagule dispersal to a = 0.999 5 in this study.

For the medium- and fat-tailed kernels, a simulation methodwas employed to calculate L by two steps: (1) the maximumradius of the study area (i.e. 2 000 cell unit lengths) wasused to substitute for infinity (∞) to calculate the cumula-tive probability:

∫ 20000 ae−βx2

dx ≈ ∫ ∞0 ae−βx2

dx for the medium-tailed kernel and

∫ 20000 ae−β

√xdx ≈ ∫ ∞

0 ae−β√

xdx for the fat-tailed

kernel; and (2) once p =∫ L

0 e−βx2dx∫ 2000

0 e−βx2 dx≥ a and p =

∫ L0 e−β

√xdx∫ 2000

0 e−β√

xdx≥

a (hereafter, a = 0.999 5), L will be regarded as the criticaldistance.

Individual cell-based propagule spread and establishmentof invasion

Within a given area (radius 0–L), propagules disseminate to cells (a new cell) from surrounding infected cells, i, according to:

Ns,t =n∑

i=1

(Ni,t · s1 · pi→s) (16)

where Ns,t and Ni,t are the density of plants in cells s and i atgeneration t, s1 is seeds per plant, n is number of infested cellssurrounding a new cell s within a given radius (0–L), and pi→s

is probability of propagule reaching new cell s from infested celli. For instance, for the narrow-tailed kernel, p is calculated as:

p = e−βx∫ L−L e−βxdx

(17)

where x is the distance from the new cell to an infested cell,L is the critical distance described in Equation 15. Inserting theresults from Equation 16 into Equation 3, we obtained the prob-ability of establishment for new cell s and simulated stochasticinvasions. Once a cell becomes invaded, the population inthe infested cell s will grow at intrinsic growth rate followingEquation 11 at the next time interval (t+1). The model assumedthat once a cell was infested, it would not be infested again(i.e. population dynamics of the infested area are dominatedby local population growth rather than propagule dispersal fromother sites), but it would become a propagule source capable ofinvading other uninfested cells.

Study area

Our study was based on leafy spurge invasion in the UpperGreat Plains (North Dakota, South Dakota, Wyoming, and


Table 2. Shoots per square meter of leafy spurge in various habitats during 1951–1957

Habitat Year Mean

1951 1952 1953 1954 1955 1956 1957

Leafy spurge 1–35 per m2 at the beginning 19.1 38.0 56.9 69.4 109.3 121.2 144.2 79.7

Leafy spurge 36–99 per m2 at the beginning 61.5 84.5 102.0 112.0 162.9 179.9 199.4 128.9

Leafy spurge 100 + per m2 at the beginning 122.5 125.5 141.3 156.7 202.8 241.6 207.7 171.1

(Data from Selleck et al. 1962).

Montana) of USA. The Upper Great Plains constitutes a vastbelt of predominantly rolling plains sloping gradually eastwardfrom about 1 200 m to 300 m above sea level. Wide valleys, iso-lated hills and badlands occasionally interrupt the monotonouscharacteristic of the plains (Padbury et al. 2002). The climateof the region is continental, characterized by long, cold winters;short but warm summers; large diurnal ranges in temperatures;and frequent strong winds. Annual precipitation varies mostlyfrom 300 mm to 500 mm, but extreme year-to-year variations arecommon and long spells of hot, dry weather characterize muchof the summer (Selleck et al. 1962; Padbury et al. 2002). Nativevegetation of the region is largely open grassland characterizedby drought-tolerant short and medium grasses. In the southernand eastern parts, woody vegetation is confined mostly to valleybottoms and along major streams. Agricultural crops are wheat,oat, barley, canola, corn, sorghum, soybean, sugarbeet, andsunflower (Padbury et al. 2002).The first report of leafy spurgein the region was in 1909 in North Dakota. By the 1940s, leafyspurge was prevalent in the four states (Watson 1985). Leafyspurge area increased in a sigmoid curve from 1950 to 1995(Bangsund et al. 1999). From 1950 to 1990, the acreage ofleafy spurge infestations roughly doubled each decade. From1990 to 1996, the rate of growth in leafy spurge acreage slowedsomewhat, but the total acreage in the Upper Great Plains stillincreased by about 24% over the 6-year period (Lym et al. 1993;Bangsund et al. 1997; Bangsund et al. 1999). In North Dakota,for example, from 1953 to 1993, the invaded area followed bothexponential and logistic curves at significant confidence levels(95%) (Table 1).

Field experiments on the population density growth of leafyspurge were conducted in the province of Saskatchewan,Canada, at plot level (Selleck et al. 1962) (Table 2). Thecarrying capacity and intrinsic growth rate of density (i.e. Kand r in Equations 11 and 12) were set to 353 (the maximumdensity in the relevant leafy spurge literature) and 0.453 270 8,respectively, following the field experiments (Table 2).

Coefficient fitting and simulation experiments

Parameter fittingGiven the complexity of the response surface, the model

parameters (dispersal coefficients β in Equations 5, 6, and 7and coefficients α and c in Equation 3) were fitted with leafy

spurge survey data by using multiple optimization approachesto more thoroughly search the response surface (Table 1). Ourapproaches included even-distribution sampling plus SIMPLEX(Bixby 1992.) (i.e. applying SIMPLEX at each of 100 evenlydistributed starting values, but randomizing the combination ofparameter values, α, β, and c as in Latin Hypercube sampling(Mckay et al. 1979)), Simulated Annealing (Goffe et al. 1994),and grid search. Where appropriate, after visual inspection, weused finer gradations in the grid-wise search. The best-fittingparameters obtained by the above multiple methods were usedto analyze the influences of control strategies on leafy spurgespread rate and processes.

When results obviously would fit the empirical data poorly,three assumptions were used to terminate simulations so thatwe could save simulation time: (i) at the initial stage (0–30 yearsat invasion, we set 1909 as year 0 of the invasion, which was thefirst reported year of the leafy spurge invasion in North Dakota),if model prediction of an infested area was greater than 3 timesthe prediction at year 30, we assumed that the trajectory waswrong and terminated the simulation. We used year 30, whichhad a sufficiently large infested area, to avoid the stochasticityassociated with small infested cells; (ii) similarly, during theperiod of years 31 to 53, if any estimate was greater than 3 timesthe predicted infested area (estimated using linear regression),we terminated the simulation; and (iii) if model prediction of theinfested area was greater than 1 100 000 ha at year 81 (about2 times of the maximum infested area from the empirical survey),we terminated the simulation.

In the study, the relative differences (RD) between the actualand the corresponding predicted values of leafy spurge areawere used as our metric of fit. The lower the RD values, thebetter the fit.

RD = |θsim − θobs|θmax −obs

(18)

where θsim is the simulated leafy spurge area (ha) of the ithyear, θobs the survey leafy spurge area (ha) of the ith year,and θmax−obs is the maximum of the leafy spurge survey area(ha).

Simulated cell scaleWe had to consider larger cell sizes in our map lattice to keepthe simulations logistically feasible. To analyze consequences


Table 3. Best-fitting β, α, and c values of the narrow-tailed redistribution kernel at three cell scales and the relative differences between observed and

predicted infestations (Equation 18)

Best-fitting parameters Cell sizes (ha)

4 9 16

Dispersal coefficients (β) 0.180 423 2 0.180 423 2 0.180 423 2

Propagules pressure coefficients (α) 0.000 765 9 0.000 286 9 0.000 127 9

Allee-effect coefficients (c) 2.440 226 0 2.440 226 0 2.440 226 0

Relative differences per iteration 0.372 089 0 0.398 259 0 0.379 972 0

Table 4. Best-fitting β, α, and c values for the narrow-, medium, and fat-tailed redistribution kernels and the relative differences between observed

and predicted infestations (Equation 18)

Best-fitting parameters Narrow-tailed kernel Medium-tailed kernel Fat-tailed kernel

NAE† AE‡ NAE AE NAE AE

Dispersal coefficients (β) 1.161 449 5 0.180 423 2 0.090 100 0 0.003 090 0 2.733 676 0 1.203 162 0

Propagules pressure coefficients (α) 0.122 505 0 0.000 286 9 0.039 700 0 0.001 660 0 0.464 685 0 0.000 966 4

AE coefficients (c) 1.000 000 0 2.440 226 0 1.000 000 0 1.707 093 0 1.000 000 0 2.066 860 0

Relative difference per iteration 0.509 560 0 0.285 842 0 0.475 546 0 0.239 015 0 0.504 994 0 0.258 396 0

†NAE = no Allee effects; ‡AE = Allee effects.

Table 5. Population size based-spread rate of leafy spurge under various control levels of the consecutive control (CC) scenario

Population Control levels (%)size at generation t (ha)

0† 10 20 30 50 70 90

550.4 (500.0)‡ 202.1 189.5 169.2 144.9 121.5 84.6 19.8

1017.0 (1000.0) 320.0 303.3 281.3 250.2 208.4 154.8 58.5

5356.8 (5000.0) 1 133.1 1 066.1 996.3 915.3 779.0 619.7 322.2

10956.6 (10000.0) 1 835.1 1 745.1 1 638.0 1 506.2 1 285.7 1 049.0 564.8

28819.8 (30000.0) 3 600.9 3 406.5 3 166.2 2 962.4 2 517.3 2 077.2 1 184.9

60288.3 (60000.0) 5 782.1 5 425.7 5 106.2 4 799.3 4 167.0 3 452.4 2 005.2

100147.1 (100000.0) 7 965.0 7 483.5 7 052.0 6 590.3 5 740.7 4 830.3 2 791.8

153689.4 (150000.0) 10 328.0 9 762.3 9 197.6 8 618.4 7 521.3 6 354.5 3 693.6

249613.2 (250000.0) 14 188.1 13 432.5 12 616.2 11 822.4 10 399.5 8 793.0 5 202.5

309457.8 (300000.0) 16 262.6 15 425.6 14 527.8 13 574.3 11 962.8 10 191.6 6 001.2

455518.8 (450000.0) 21 003.3 19 760.0 18 691.7 17 553.6 15 457.1 13 181.9 7 849.8

†0% = no control;‡Values in bracket refers to the expected population size basis at time t; owing to stochasticity of the model, we use the average of 20 iterations.

of upscaling on model performance, we assigned cell sizes of4, 9, and 16 ha per cell (i.e., cell length was 2, 3, and 4), respec-tively, for a subset of simulations. As the cell size increased,total cell numbers simulated within the target study area weredecreased, and thus the model efficiency was improved. Duringthe parameter fitting, once model parameters (β, α, and c)were optimized for one cell size scale, the parameter fittingfor the other two scales could be done through fixing β (dis-persal distance coefficient) and c (Allee effect coefficient), andadjusting α (propagules pressure coefficients) of the narrow-tailed redistribution kernel (Table 3). Results suggested thatthere were no significant differences in the prediction of spread

processes and patterns among the three scales by using theirown optimized parameters, though the variability within differentsimulation iterations increased somewhat as cell size rose.Hereinafter, we used a cell size of 9 ha to simulate responses ofleafy spurge dispersal to control strategies. This correspondedwith a map lattice model with approximately 1.8 million cells.

Control scenario designFour common alternative control scenarios were designed torepresent major control strategies in American leafy spurgemanagement: initial single control (IC), consecutive control


(CC), initial intermittent control (IIC), and consecutive controlbegun from current distribution (CCC) (USDA-ARS Team LeafySpurge Area-Wide IPM Program 2002a, 2002b, 2002c). TheIC scenario meant that control strategies were implementedonly once at the beginning of leafy spurge invasion. The CCscenario represented control practices which were consecu-tively executed as soon as the local area was invaded byleafy spurge. The IIC scenario described control strategiesthat were intermittently executed per interval. Finally, the CCCscenario meant that no control or prevention practices wereexecuted at the beginning of introduction; control efforts beganto input consecutively with a significant increase of leafy spurgearea. Each scenario constituted 8 control levels: destructionof 0, 10, 20, 30, 50, 70, 90, and 100% of leafy spurge stems(i.e. the resulting proportional reduction in propagule supply,i.e. N in Equations 11 and 16). No control is indicated by0%; 100% indicates local eradication. These scenarios andcontrol levels came from control practices including chemical,biological, prescribed fire, multi-species grazing controls, or inte-grated practice management, which have being implemented orexperimented with at local scales for a couple of years (Sellecket al. 1962; USDA-ARS TEAM Leafy Spurge Area-Wide IPMProgram 2002a, 2002b, 2002c).

Twenty iterations were run for each simulation. In total, morethan 250 000 simulations of invasion spread over 80 years wereperformed, which included best-fitting of model parameters.

Acknowledgements

The author greatly thanks Prof. Brian Leung for guiding themodel development and Catherine Pirkle for editing the earliermanuscript.

References

Bangsund DA, Leistritz FL, Leitch JA (1997). Predicted Future

Economic Impacts of Biological Control of Leafy Spurge in the

Upper Midwest. NDSU Agricultural Economics Report No. 382.

North Dakota Agricultural Experiment Station, North Dakota State

University, Fargo. pp. 1–71.

Bangsund DA, Leistritz FL, Leitch JA (1999). Assessing economic

impacts of biological control of weeds: the case of leafy spurge in the

northern Great Plains of the United States. J. Environ. Manage. 56,

35–43.

Belcher JW, Wilson SD (1989). Leafy spurge and the species com-

position of a mixed-grass prairie. J. Range Manage. 42, 172–

175.

Belisle M, St. Clair CC (2002). Cumulative effects of barriers on the

movements of forest birds. Conserv. Ecol. 5, 9.

Bixby RE (1992). Implementing the simplex method: the initial basis.

ORSA J. Comput. 4, 267–284.

Cain ML, Milligan BG, Strand AE (2000). Long-distance seed dispersal

in plant populations. Am. J. Bot. 87, 1217–1227.

Clark JS, Lewis M, McLachlan JS, Hille Ris Lambers J (2003).

Estimating population spread: what can we forecast and how well.

Ecology 84, 1979–1988.

Dennis B (2002). Allee effects in stochastic populations. Oikos 96, 389–

401.

Dunn PH (1979). The distribution of leafy spurge (Euphorbia esula) and

other weedy Euphorbia spp. in the United States. Weed Sci. 27,

509–516.

Engen S, Lande R, Saether BE (2003). Demographic stochasticity and

allee effects in populations with two sexes. Ecology 84, 2378–2386.

Everitt JH, Anderson GL, Escobar DE, Davis MR, Spencer NR,

Andrascik RJ (1995). Use of remote sensing for detecting and

mapping leafy spurge (Euphorbia esula). Weed Technol. 9, 599–

609.

Getz WM (1996). A hypothesis regarding the abruptness of density

dependence and the growth rate of populations. Ecology 77, 2014–

2026.

Goffe WL, Ferrier GD, Rogers J (1994). Global optimization of statisti-

cal functions with simulated annealing. J. Econometrics 60, 65–99.

Grevstad FS (1999). Experimental invasions using biological control in-

troductions: the influence of release size on the chance of population

establishment. Biol. Invasion. 1, 313–323.

Hengeveld R (1994). Small-step invasion research. Trends Ecol. Evol.

9, 339–342.

Herbert CH, Rudd VE (1933). Leafy Spurge–Life History and Habits:

North Dakota Agricultural Experimental Station Bulletin No. 266.

North Dakota Agricultural Experimental Station. North Dakota Agri-

cultural College, Fargo, North Dakota.

Higgins SI, Richardson DM, Cowling RM (1996). Modeling invasive

plant spread: the role of plant environment interactions and model

structure. Ecology 77, 2043–2054.

Hobbs RJ, Humphriest SE (1995). An integrated approach to the

ecology and management of plant invasions. Conserv. Biol. 9, 761–

770.

Hobbs RJ, Richardson DM, Davis GW (1995). Mediterranean-type

ecosystems: opportunities and constraints for studying the function

of biodiversity. In: Davis GW, Richardson DM, eds. Mediterranean-

Type Ecosystems. The Function of Biodiversity. Springer, Berlin,

pp. 1–42.

Kot M, Lewis MA, Van den Drissche P (1996). Dispersal data and the

spread of invading organisms. Ecology 77, 2027–2042.

Leistritz FL, Thompson F, Leitch JA (1992). Economic Impact of Leafy

Spurge (Euphorbia esula) in North Dakota. Weed Sci. 40, 275–

280.

Leitch JA, Leistritz L, Bangsund D (1994). Economic Effect of Leafy

Spurge in the Upper Great Plains: Methods, Models, and Results.

Agricultural Economics Report 316. North Dakota State University,

Fargo, North Dakota, USA. pp. 1–7.

Leung B, Lodge DM, Finnoff D, Shogren JF, Lewis M, Lamberti G

(2002). An ounce of prevention or a pound of cure: bioeconomic risk

analysis of invasive species. Proc. Biol. Sci. 269, 2407–2413.


Leung B, Drake JM, Lodge DM (2004). Predicting invasions: propagule

pressure and the gravity of Allee effects. Ecology 85, 1651–1660.

Liebhold A, Bascompte J (2003). The Allee effect, stochastic dynamics

and the eradication of alien species. Ecol. Lett. 6, 133–140.

Lym RG (1997). The history of leafy spurge control in North Dakota.

North Dakota Agricultural Research, Winter 1997-Article. Available

at: http://www.ag.ndsj.nodak.edu/ndagres/ndagres.htm (verified on:

Aug. 23, 2004).

Lym RG, Messersmith CG, Zollinger R (1993). Leafy spurge identi-

fication and control. North Dakota State University, Fargo, W-765.

pp. 1–7.

MacIsaac HJ, Borbely JVM, Muirhead JR, Graniero PA (2004).

Backcasting and forecasting biological invasions of inland lakes.

Ecol. Appl. 14, 773–783.

Mckay MD, Beckman RJ, Conover WJ (1979). A comparison of

three methods for selecting values of input variables in the anal-

ysis of output from a computer code. Technometrics 21, 239–

245.

Padbury G, Waltman S, Caprio J, Coen G, McGinn S, Mortensen D

et al. (2002). Agroecosystems and land resources of the Northern

Great Plains. 94, 251–261.

Rouget M, Richardson DM, Cowling RM, Lloyd W, Lombard AT

(2003). Current patterns of habitat transformation and future threats

to biodiversity in terrestrial ecosystems of the Cape Floristic Region,

South Africa. Biol. Conserv. 112, 63–85.

Sakai AK, Allendorf FW, Holt JS, Lodge DM, Molofsky J, With KA

et al. (2001). The population biology of invasive species. Annu. Rev.

Ecol. Syst. 32, 305–332.

Seaman DE, Powell RA (1996). An evaluation of the accuracy of kernel

density estimators for home range analysis. Ecology 77, 2075–

2085.

Selleck GW, Coupland RT, Frankton C (1962). Leafy spurge in

Saskatchewan. Ecol. Monogr. 32, 1–29.

Stroh RK, Leitch JA, Bangsund DA (1990). Leafy Spurge Patch

Expansion. Staff paper no. AE9001, North Dakota State University,

1990. pp. 15–17.

Taylor RAJ (1978). The relationship between density and distance of

dispersing insects. Ecol. Entomol. 3, 63–70.

The USDA-ARS TEAM leafy spurge area-wide IPM program (2002a).

Biological control of leafy spurge– a comprehensive, easy-to-read

manual on how to use biological control as an effective leafy spurge

management tool. Available at: http://www.team.ars.usda.gov/ (ver-

ified on July 15, 2004).

The USDA-ARS TEAM leafy spurge area-wide IPM program (2002b).

Herbicide control of leafy spurge– a comprehensive, easy-to-read

manual on how to use biological control as an effective leafy spurge

management tool. Available at: http://www.team.ars.usda.gov/ (ver-

ified on July 15, 2004).

The USDA-ARS TEAM leafy spurge area-wide IPM program

(2002c). Multi-species grazing of leafy spurge- a comprehen-

sive, easy-to-read manual on how to use biological control

as an effective leafy spurge management tool. Available at:

http://www.team.ars.usda.gov/ (verified on July 15, 2004).

Wallace B (1966). On the dispersal of Drosophila. Am. Nat. 100, 551–

563.

Wallace NM, Leitch JA, Leistritz FL (1992). Economic impact of leafy

spurge on North Dakota Wildland. Agricultural Economics Report

No. 281, North Dakota State University, Fargo. North Dakota Farm

Res. 49, 9–13.

Watson AK (1985). Integrated management of leafy spurge. In: Weed

Science Society of America, ed. Leafy Spurge, Monograph Series of

the Weed Science Society of America. pp. 93–104.

(Handling editor: Da-Yong Zhang)

Documents

THIS ARTICLE HAS BEEN RETRACTED Modeling Responses of Leafy Spurge Dispersal to Control Strategies