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Modeling Responses of Leafy Spurge Dispersal to Control Strategies
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Zewei Miao
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Grant F. Walton Center for Remote Sensing & Spatial Analysis, Cook College, Rutgers
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University, 14 College Farm Road, New Brunswick, NJ 08901-8551
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To whom correspondence should be addressed. Telephone: 732-932-1583; Email:
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zmiao@crssa.rutgers.edu
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Received: 14 Jan. 2006 Accepted 2 Oct. 2006
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Handling editor: Da-Yong Zhang
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Abstract
Leafy spurge (Euphorbia esula L.) has substantial negative effects on grassland
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biodiversity, productivity, and economic benefit in North America. To predict these negative
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impacts, we need an appropriate plant-spread model which can simulate the response of an
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invading population to different control strategies. In the study, using a stochastic map lattice
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approach we generated a spatially explicitly stochastic process-based model to simulate dispersal
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trajectories of leafy spurge under various control scenarios. The model integrated dispersal
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curve, propagule pressure, and population growth of leafy spurge at local and short-temporal
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scales to capture spread features of leafy spurge at large spatial and long temporal scales. Our
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results suggested that narrow-, medium-, and fat-tailed kernels did not differ in their ability to
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predict spread in contrast to previous works. For all kernels, Allee effects were significantly
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present and could explain the lag phase (three decades) before leafy spurge spread accelerated.
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When simulating from the initial stage of introduction, Allee effects were critical in predicting
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spread rate of leafy spurge, because the prediction could be seriously affected by the low density
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period of leafy spurge community. No Allee effects were not able to simulate spread rate well in
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this circumstance. When applying control strategies to the current distribution, Allee effects
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could stop the spread of leafy spurge; No Allee effects, however, were able to slow but could not
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stop the spread. The presence of Allee effects had significant ramifications on the efficiencies of
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control strategies. For both Allee and no Allee effects, the later control strategies were
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implemented, the more efforts had to be input to achieve similar control results.
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Key words: invasive species, non-indigenous, propagule pressure, dispersal curve, stochastic
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process-based model, Allee effects.
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Z. Miao
Introduction
Invasions of exotic species have been proposed as one of largest components of habitat
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destruction, biodiversity losses, and economic damage (Cain et al., 2000; Leistritz et al. 1992;
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Rouget et al. 2003). As a perennial dicotyledonous herbaceous plant, leafy spurge (Euphorbia
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esula L.) adapts to a wide variety of habitats. (Selleck et al. 1962; Dunn 1979; Watson 1985),
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and has become a great threat to rangeland productivity, to species diversity, to the quality of
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wildlife habitats, and to land values in the Northern United States and the prairie provinces of
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Canada since its discovery in North America in 1827 (Selleck et al. 1962). 1.6 million ha of land
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in the Upper Great Plains (North Dakota, South Dakota, Montana, and Wyoming) of the United
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States were infested by 1994 (Everitt et al. 1995; Bangsund et al. 1999). In the past decade, leafy
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spurge infestations have been estimated to result in an annual economic loss of US$130 million
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in the four-state region (Leitch et al. 1994).
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There is a need to simulate dispersal features of leafy spurge under the influences of
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weed management practices. Field experiments demonstrate that control strategies of leafy
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spurge, including chemical, multi-species grazing, and biological efforts, are able to reduce
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population density (i.e., propagule pressure) to a large extent (Selleck et al. 1962; Watson 1985;
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Leitch et al. 1994). This likely constrains spatial spread through reduced propagule pressure. So
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far, few studies have attempted to simulate the response of dispersal process and patterns of leafy
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spurge invasion to various weed management practices. It is not yet known with certainty how
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ecological dispersal features of leafy spurge will be affected by weed management practices at
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large spatial and long temporal scales.
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Process-based models provide us a mechanistic understanding of invasion and dispersal
features of exotic species at large scales responding to management practices and environmental
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heterogeneities. Analyses of the spread of invading organisms frequently start with a reaction-
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diffusion (R-D) model with exponential growth and Fickian diffusion (Higgins et al., 1996; Kot
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et al., 1996). When the dispersal pattern is Brownian and the trapping rate by the environment is
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uniform, the dispersal pattern can be described as follows (in simple form):

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N t 1 ( x )   k( x, y)f [ N t ( y)]dy
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or
(1)

N t 1 ( x )
N (x) 
 2 N t ( y)

 rN t ( x )1  t

D

t
K 
x 2

(2)
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where Nt+1(x) and Nt (y) are the population densities at generation t+1 and t at locations x and y,
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respectively; k(x,y) is the one-dimensional redistribution kernel, i.e., the probability density
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function for propagule dispersing to destination x from a source position y; r is the species’
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intrinsic rate of increase; and s is the diffusion constant (Higgins et al., 1996; Kot et al., 1996).
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This kind of R-D models assume that the invading organisms spread in a given redistribution
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kernel regardless of environmental heterogeneities and weed management practices. The precise
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shape of redistribution kernel k(x, y) is extremely important to describe the speed characteristics
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of invasion (Kot et al., 1996). The R-D models, however, is usually limited by the absence of
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suitable redistribution kernels and/or by inadequate parameter estimates for such kernels. For
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instances, ecological data is incomplete to generate the redistribution kernel for leafy spurge R-D
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model (Selleck et al., 1962; Cain et al., 2000).The R-D equations also assume that the invading
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populations are large enough that stochastic effects are not important, which, in nature, is not
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always the case (Hengeveld, 1994; Kot et al., 1996). Field experiments show that leafy spurge
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invasion are mainly caused by stochastical biological and environmental factors such as flooding
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water, wind, anthropogenic disturbance (e.g., vehicle tracks, overgrazing, road construction, and
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fire guards), birds, insects (e.g., ants), and wild and domestic animals (Selleck et al., 1962; Dunn,
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1979; Watson, 1985; Belcher and Wilson, 1989). All together, there is a need to establish a
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process-based model to simulate stochastic processes and patterns of leafy spurge invasion which
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responds to various control strategies. Over a certain spatial scale (stands to landscapes),
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stochastic process-based models can integrate biological attributes, spatial considerations,
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invasion stochasticity, and environmental heterogeneity into modelling species’ distributions
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(Rouget et al., 2003).
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Interactions among dispersal curve, propagule pressure, and intrinsic population growth
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are central to establishing a stochastic process-based spread model of exotic species’ invasion. In
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the invasion process (i.e., introduction, colonization, and naturalization), many factors such as
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propagule pressure, distance from the source, and external environment, determine spread range
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and abundance of non-indigenous species at a given locality. Dispersal curves are frequently
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fitted with a negative exponential curve or a negative power function (Wallace 1966; Taylor
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1978; Kot et al., 1996), and the probability density function of a dispersal curve may be normally
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distributed for some species (Kot et al., 1996). The role of propagule pressure or mass effects is
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clearly observable in many invasions. As availability of propagule pressure increase, chances of
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establishment, persistence, naturalization, and invasion, will increase as well, especially at the
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introduction stage (Rouget et al., 2003). Population growth dynamics affect propagule yield and
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dynamics of leafy spurge ecological communities; hence population growth will affect the
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probability of establishment in uninfested areas. Recent works indicate that the combined
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influence of Allee dynamics and stochastic processes strongly determines the successful
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establishment of alien species (Liebhold and Bascompte, 2003). However, the influences of
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Allee effects are still unknown in the process of leafy spurge invasion.
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In this paper, we link the data available across two different scales - dynamic density data
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at a local (plot) and short temporal scales, and long time series of infested area at a regional scale
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– to construct our dispersal model. We simulate, with minimum parameter requirements, the
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stochastic processes and patterns of spread of leafy spurge in response to control strategies. We
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test whether the form of the dispersal kernel (i.e., narrow- vs. fat-tailed kernels) and Allee effects
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are important for explaining the observed pattern of invasion, including the lag period before
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leafy spurge becomes invasive. We simulate efficiencies of four typical control scenarios of leafy
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spurge management in North America that have been applied at local spatial and short temporal
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scales. This work will be useful to future research on risk analysis, management options, and
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optimal control theory.
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Results
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Prediction of leafy spurge invasion vs. different redistribution kernels
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There were no differences in predicting infested area among the narrow-, medium-, and
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fat-tailed redistribution kernels, but their best-fitting parameters, β, α, and c, were different
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(Table 4). With the different forms of redistribution kernels and their own best fitting parameters,
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similar dispersal trajectories were obtained for all corresponding control strategies, regardless of
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Allee or no Allee effects (Figures 1 and 2). For the no control scenario and with Allee effects, all
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the three kernels gave good predictions in leafy spurge dissemination against the infested survey
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area (Table 1). For no control and with Allee effects, the narrow-, medium- and fat-tailed kernels
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predicted the infested area of 507787.2, 505371.2, and 496539.9 ha at year 81, respectively,
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while the survey leafy spurge area was 526091.3 at year 81. The relative differences between the
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prediction and the corresponding survey data per iteration were 0.29, 0.24, and 0.26,
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respectively.
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For the three kernels, the best-fit parameters usually consisted of low β (i.e., greater
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maximum dispersal distance ranging from 4200 to 6300 m), low α (i.e., low probability of
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establishment of individual propagules), and high c values (Allee effect coefficients, ranging
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from 1.5 to 2.5) (Table 4), whose estimates matched well with the survey data. The best-fitting β,
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α, and c values were 0.1804232, 0.0002869, and 2.440226 for narrow-tailed kernel, 0.00309,
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0.00166, and 1.707093 for medium-tailed kernel, 1.203162, 0.001102, and 2.06686 for fat-tailed
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kernel, respectively. Because of lack of differences between the three kernels with respect to
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predictions, the remainder of the analyses was presented only for narrow-tailed kernel.
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Influences of Allee and no Allee effects on lag phase in leafy spurge invasion
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The dispersal trajectory predicted with Allee effects (i.e., c>1 in Eq. 3) remained very
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different from no Allee effects (i.e., c=1 in Eq. 3) (Figure 1). When simulations started from the
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beginning of introduction, prediction with Allee effects was significantly better than no Allee
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effects. Allee Effects were essential in capturing the lag phase during leafy spurge invasion,
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regardless of kernel forms (Fig. 1). The relative differences between prediction and survey were
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0.28 for Allee effects and 0.51 for no Allee effects of narrow-tailed kernel, 0.24 for Allee effects
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and 0.48 for no Allee effects of medium-tailed kernel, and 0.26 for Allee effects and 0.50 for no
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Allee effects of fat-tailed kernel, respectively (Table 4). Allee effects reduced the probability of
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establishment of an individual propagule at the beginning of the invasion, permitting the
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observed lag phase of 3 decades, and increased the probability at the latter period in comparison
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to no Allee effects (Fig. 1). Leafy spurge spread rate of Allee effects was almost two times that
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of no Allee effects in the last simulating year (year 81).
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When simulations leaped over the initial phase of introduction, there were no significant
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differences between Allee and no Allee effects. When the simulations began from the current
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distribution (e.g., year 53, the 1st surveyed year of leafy spurge area), the model gave good
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predictions in leafy spurge dispersal compared to survey data (Fig. 2), no matter what Allee or no
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Allee effects. The relative differences between the prediction and the corresponding survey data
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per iteration were 0.13 and 0.15 for Allee and no Allee effects of the narrow-tailed kernel,
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0.132939 and 0.175604 for Allee and no Allee effects of the medium-tailed kernel, and 0.186954
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and 0.135012 for Allee and no Allee effects of the fat-tailed kernel, respectively.
Allee effects magnified the importance of early control strategies of leafy spurge. With
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Allee effects, responses of leafy spurge spread to control strategies were significantly greater
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than for no Allee effects, no matter what control scenarios (Fig. 2). For example, for the CC
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scenario, when control efforts were greater than 30%, leafy spurge was contained during the 81
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simulation years for Allee effects. In contrast, for no Allee effects, leafy spurge continued to
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spread even though the control level was 60% (Fig. 2). For a given control effort, the response of
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no Allee effects to control strategies was considerably less than with Allee effects, especially at
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the initial period.
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Responses of leafy spurge invasion to control strategies
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The later control strategies were initiated, the more effort was needed to get similar
control achievements. Similarly, when a given control practice was conducted, the smaller the
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population size, the better achievement were reached. Because no Allee effects were not able to
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simulate well leafy spurge invasion, the model with Allee effects was mainly used to analyze the
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effectiveness of control strategies. For the narrow-tailed kernel, by comparing the CC to CCC
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scenarios, when control strategies were conducted from year 0 for the CC scenario, a control
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level of 30% was able to contain leafy spurge spread during the 81 simulation years. When
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control was implemented at year 53 and 81 for CCC scenario, however, over 50% and 70%
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control levels were needed to stop leafy spurge dispersal, respectively (Fig. 3). A control level of
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30% could slow but not stop the spread of leafy spurge in the CCC scenario. Table 5 also
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indicated that the bigger the population size was, the lower control effectiveness would be for a
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given control effort.
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Our results suggested that the IC scenario was not able to contain leafy spurge spread in
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81 simulation years, except for 100% eradication. Even though less than 10% of propagules had
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an opportunity to spread, leafy spurge would ultimately disseminate during 81 simulating years,
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but the spread rate slowed down along with an increase of control efforts (Fig. 3). In other words,
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for IC scenario, one had to eradicate leafy spurge patches at the beginning of invasion to fully
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terminate leafy spurge dispersal.
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In contrast to other scenarios, the results of the IIC scenario suggested that more control
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efforts were required to contain leafy spurge spread than with the CC scenario. For example, for
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the IIC scenario, more than 70% of control levels had to be devoted to stop leafy spurge spread,
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while 30% of control efforts were enough to contain the spread for the CC scenario (Fig. 3). For
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the CC scenario, frequency of control application was higher than IIC and this had to be
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incorporated in the computation of control efforts. The IIC scenario worked more effectively
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than IC, since IC was almost not able to stop dissemination even at a control level of 90%. IIC
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control effectiveness was similar to the CCC scenario starting from the current distribution at
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year 81, but worse than that of the CCC scenario starting from the current distribution at year 53.
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As a common practice in the integrated pest management (IPM) of leafy spurge invasion, more
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input per control for CCC scenario was needed than the CC scenario to achieve similar results,
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despite the fact that control efforts were intermittent and the numbers of control practices were
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fewer than in the CC scenario.
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Discussion
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Dispersal of a spreading population and redistribution kernels
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Redistribution kernels are probability-density functions that describe the possibility that
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individuals will be found at specific spatial coordinates relative to the original, colonizing
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location (MacIsaac et al., 2004). Insight into the process and spatial patterns of dispersal of a
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nonindigenous species can be gleaned from its redistribution kernel (Taylor, 1978; Kot el al.,
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1996; Clark et al, 2003; MacIsaac et al., 2004). Kot el al. (1996) reported that the speed of
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invasion of a spreading population is extremely sensitive to the precise shape of redistribution, in
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particular, to the tail of the distribution. They reported that different shape kernels with similar
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coefficients of determination will yield dramatically different speeds in a spreading population,
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and fat-tailed kernels can generate accelerating invasions rather than constant-speed waves (Kot
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el al., 1996). Our results found that the short-, medium-, and fat-tailed kernels with different
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best-fit parameters are able to get similar trajectories for leafy spurge, and this results are not
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affected by control strategies.
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Dispersal distance coefficients are critical parameters for the stochastic process-based
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model prediction. In our study, values of β (dispersal coefficients) all are very low for the three
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kernels, with a maximum spread distance of around 4200 and 6300 m (for cumulative
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probabilities of 0.9995). These dispersal distances conform to previous dispersal observations of
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leafy spurge, which suggest that leafy spurge dissemination occurs in both local diffusion (short
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distance) and jump dispersal patterns (at medium and large distance) (Herbert and Rudd, 1933;
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Selleck et al., 1962). Our sensitivity analysis shows that a slight change in the β value can induce
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a large change in the dispersal distances of introduced new colonies from parental colonies.
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Therefore, paired field data linking population density to dispersal distance should be collected
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for empirical validation of leafy spurge invasion in the future.
The spread rate was strongly affected by propagule pressure and Allee effects.
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Biologically, these two factors have quantitative (i.e., number of individuals moved) and/or
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qualitative (i.e., the condition, sex ratio, or size structure of individuals moved) components that
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may influence whether an invasion succeeds (Kot el al., 1996; Seaman and Powell, 1996; Clark
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et al., 2003). These two factors are tightly combined in our model.
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Allee effects in leafy spurge invasion
Allee effects can cause the invasion of an alien species to fail, especially in the
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introduction and establishment of invading populations (Grevstad, 1999; Engen et al., 2003). Our
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study shows that regardless of kernel shapes, the c values in Eq. 3 are consistently much greater
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than unity, which suggests Allee effects are significantly present in a leafy spurge invasion. In
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contrast, no Allee effects cannot predict the trajectory of a leafy spurge invasion well, since no
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Allee effects predictions are too high to capture the lag phase at the initial stage, and too low at
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the later.
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Allee effects are indispensable in simulating the lag phase of a leafy spurge invasion
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(about 2 or 3 decades in North Dakota) (Hobbs and Humphriest 1995; Sakai et al., 2001). These
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lag times are expected if environmental or evolutionary change is an important part of the
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colonization process. This process may include the adaptations of invasive species to a new
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habitat, the evolution of invasive life history characteristics, or the purging of a genetic load
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responsible for an inbreeding depression (Sakai et al., 2001). Our predictions suggest that
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population dynamical processes, such as Allee effects, are sufficient to account for the lag phase
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observed in leafy spurge. There are also evidences from field experiments for Allee effects in
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leafy spurge, where seed germination is about 20–30% and the percentage of seedlings surviving
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is much lower at the initial stage (Herbert and Rudd, 1933; Selleck et al., 1962). Once a patch is
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established, leafy spurge can thrive in a variety of conditions, producing a large number of
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propagules (seeds) (Stroh et al., 1990; Leitch et al., 1994; Bangsund et al., 1999).
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Allee effects take an important part in leafy spurge management. With Allee effects, the
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efficiencies of control strategies are significantly greater than without Allee effects; because with
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Allee effects, control strategies can effectively keep densities below the critical point where
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population size explodes. With Alee effects, predictions are in accordance with field experiments
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that demonstrate that early detection and treatment of invasions before explosive spread occurs
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will prevent many future dispersal problems (Hobbs and Humphriest, 1995; USDA-ARS TEAM
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leafy spurge area-wide IPM program, 2002a, 200b, 2002c). No Allee effects are not able to delay
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the lag time of a leafy spurge invasion, thus, leafy spurge spreads earlier and more quickly.
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Without Allee effects, more control efforts will be required to achieve the same results as Allee
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effects.
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Simulation of response of invasive species to control strategies
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In the field of ecological modeling, tremendous efforts are devoted to the study of
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mechanisms of introduction, establishment, dissemination, and propagule pressure, but few
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studies focus on the response of exotic species to management strategies. Many field
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experiments of control strategies have been fairly effective at the field plot and short temporal
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scales for leafy spurge (USDA-ARS TEAM leafy spurge area-wide IPM program. 2002a, 2002b,
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2002c). Little is known about the consequences of control strategies at large spatial and long
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temporal scales (for example, across several states and several decades). This is important given
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the labor, time, and cost needed to implement such strategies. Weed control generally falls into
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four categories: prevention, eradication, reduction, and containment. In the study, containment
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control scenarios are only taken into consideration in terms of the continuous spread of leafy
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spurge. We assume that control strategies reduce density and propagules, and ultimately decrease
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the probability of establishment of individual propagules. Our simulations confirm that the later
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control strategies begin, the more effort that one has to devote to get similar control results,
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which are in accordance with previous field experimental studies that containment and
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eradication of small infestations, at early stages, should be a major component of a leafy spurge
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management program in North America (Lym, 1997; Bangsund et al., 1999; Rouget and
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Richardson, 2003).
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For a given control level, efficiencies of various control strategies are different. Four
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scenarios in this study rank as: CC > CCC > IIC > IC. For different control levels, different
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control scenarios may have similar effectiveness. For example, the effectiveness of the CCC
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scenario starting from the current leafy spurge area at year 53 is similar to the IIC scenario, at
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some control levels. When the effectiveness of different strategies is similar, the choice between
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them should depend on further social, economic, and ecological welfare analysis of these
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management strategies.
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Future directions: model structure and environment heterogeneity
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The map lattice model combines dispersal curves, propagule pressure, population
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intrinsic growth dynamics, and Allee effects. The model takes into account the randomness of
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establishment of non-indigenous species. One of its most important advantages is the integration
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of population growth dynamics with stochastic dissemination across different scales (i.e.,
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combine density growth data set at field level for a few years with time series data set of leafy
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spurge at regional scales across decades), which is a challenging topic in ecological modeling
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(Higgins et al., 1996; Hobbs et al., 1995).
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Future development of the model should include competition between exotic and native
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species and environmental heterogeneity of various landscape fragments (e.g., landscape slope,
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soil, water, sunlight, wind, etc). Competition among ecologically alien and native species can be
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a major factor that determines the structure of animal and plant communities. When the
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competition is considered, for example, equation 12 will become:
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dN alien
N
 w1 N native
 ralien N alien (1  alien
)
dt
K alien
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where Nalien and Nnative are population densities (stems/ha) of alien and native species,
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respectively; ralien is intrinsic growth rate of the alien species population; Kalien is carrying
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capacity of population density of the alien species in a given area; and w1 is a competition
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weight, w1≤1. As w1 approaches to 1, the competition may influence population growth
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dynamics in the model predictions. Elements such as landscape composition and configuration
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are important in the introduction, establishment, and movement rates of dispersing individuals,
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because environmental variables usually are different across various landscapes. For instance,
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leafy spurge is more common in river areas and along roadsides (except for railways
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embankments), prefers sandy-loam soil, and can easily flourish in dry meadows associated with
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grass species (Selleck et al., 1962). If spatial heterogeneities are large, they could have
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considerable implications on the estimation of emigration and immigration parameters in
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population dispersal models (Bélisle and Clair, 2002).
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Materials and methods
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Model developments
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Using a map lattice approach, we integrated dispersal curves, the probability of propagule
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establishment, and population growth dynamics to develop the stochastic process-based dispersal
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model.
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Progagule pressures
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Propagule pressures play a significant role in biological invasion. If propagules interact
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with each other, in such a manner that Allee Effects (AE) are present, the total probability of
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establishment (E) can be described using the Weibull distribution (Dennis 2002, Leung et al.
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2004):
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E( N l,t )  (1  e
 ( Nl , t )c
Z. Miao
(3)
)
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where N is the number of propagules arriving at location l at time t, and α is a scalar for
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propagule pressure and c is a shape coefficient. When c equals to 1, Allee effects are absent (i.e.,
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no Allee effects) and each propagule has an independent chance of establishment.
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Leafy spurge propagules are composed of seeds and vegetation parts (e.g., roots and
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shoots) (Herbert and Rudd, 1933; Selleck et al., 1962). In this study, we focused on seed
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dissemination, because the infestation of a new area (e.g., abandoned fields and roadside) is
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typically initiated by seeds with the exception of crop fields (Herbert and Rudd, 1933). In the
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meantime, seed invasions are more stochastic and difficult to control than invasions via
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vegetative pathways (e.g., roots and shoots) (Herbert and Rudd, 1933; Selleck et al., 1962;
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Watson, 1985).
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Dispersal curve
Since the redistribution kernel was lacking for leafy spurge invasion, we employed three
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redistribution kernels to test short, medium, and long possible spread distances, respectively. For
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one dimension, probability density function (PDF) of the three kernels can be written as:
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PDF  ae x
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PDF  ae x
2
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PDF  ae 
x
(4)
(5)
(6)
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where PDF is the probability density function, x is the dispersal distance from infested original
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source, β is the dispersal coefficient, and a is the intercept of probability density. To conveniently
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compare with previous studies, equations 4, 5, and 6 match Taylor’s (1978) and Kot et al.’s
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(1996) redistribution kernels 1 (narrow-tailed), 2 (medium-tailed), and 5 (fat-tailed),
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respectively. Hereinafter, we called the three kernels as narrow-, medium-, and fat-tailed kernels
2
respectively. These redistribution dispersal kernels were extrapolated from distance-density
3
insect models:
4
N( x )  e x
5
N(x)  ex
2
6
N(x)  e
x
(7)
(8)
(9)
7
where N is the density number found at a distance x from the centre of dispersal, β is the
8
dispersal coefficient, and a is interception (Taylor, 1978; Kot et al., 1996).
9
10
For Eq. 7 the probability density function within a given distance 0~L is:
PDF 
e x

L
0
e
 x
= ae x
(10)
dx
11
where the coefficient a equals:
12
 2 e 
PDF=
2

 2
. Similarly, we derived PDF= 2 e x from Eq. 8 and
 L
1 e

x
from Eq. 9.
13
14
1.3 Population intrinsic growth dynamic of leafy spurge
15
As for population growth dynamics, the logistic intrinsic growth curve, the most
16
traditional density-dependent growth model (Getz, 1996), was adapted to our model:
17
N t 1  N t  rN t (1 
18
or
Nt
)
K
(11)
dN
N
 rN (1  )
dt
K
(12)
17
Manuscript for Journal of Integrative Plant Biology
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1
where Nt+1 and Nt are population density (stems/ha) at generation t+1 and t; r is intrinsic growth
2
rate of the population; and K is carrying capacity of population density in a given area.
3
4
5
Map lattice approach
We used a map lattice approach to integrate dispersal curves, the probability of propagule
6
establishment, and population growth dynamics in our model. At first, the map lattice approach
7
divided the total farm lands of the study area into large numbers of uniform cells (i.e., 4000 *
8
4000 cells). When cells are infested by propagules at time ti, the population begins to grow in a
9
logistic intrinsic growth curve and to produce propagules (i.e., seeds) at the next generation, i.e.,
10
ti+1. Propagules in the infested cells will spread in a negative exponential dispersal curve, i.e.,
11
equations 4, 5, or 6. In this way, we used existing data on population growth at the local and shot
12
temporal scale (within a cell) to extrapolate patterns across large and long-temporal scales
13
(multiple States).
14
15
16
Critical distances of redistribution kernels
Theoretically, distance of propagule dissemination may be from 0 to infinity. In practice,
17
to simulate propagule dispersal effectively, we had to consider spread within a certain critical
18
distance (L). For the narrow-tailed kernel, the critical distance was derived as follows:
19

p

L
0

0
20
21
e x dx
e x dx
i.e., p 
a
( 1 / )e x
(13)
L
0
( 1 / )
a
(14)
So, the critical distance L is:
18
Manuscript for Journal of Integrative Plant Biology
Z. Miao
L  ln( 1  a) /(   )
1
(15)
2
where p is the cumulative probability of propagule dispersal within the distance L, x is the
3
distance of a cell from the infested origin, a is the chosen cumulative probability, and β is the
4
dispersal coefficient. We set the cumulative probability of propagule dispersal to a = 0.9995 in
5
this study.
6
For the medium- and fat-tailed kernels, a simulation method was employed to calculate L
7
by two steps: (1) the maximum radius of the study area (i.e., 2000 cell unit lengths) was used to
8
substitute for infinity (∞) to calculate the cumulative probability: 
9
the medium-tailed kernel and
10
0
Once

p

L
e  x dx
0
2000
0
11
2000

2
e  x dx
2
 a and
2000
0
ae  x dx ≈

p

L

0
e   x dx
0
2000
0

e   x dx
ae  x dx ≈
2


0
ae  x dx for
2
ae  x dx for the fat-tailed kernel; and (2)
 a (hereafter, a=0.9995), L will be regarded as
the critical distance.
12
13
14
15
Individual cell-based propagule spread and establishment of invasion
Within a given area (radius 0~L), propagules disseminate to cell s (a new cell) from
surrounding infected cells, i, according to:
n
16
N s ,t   ( N i ,t * s1 * pi s )
(16)
i 1
17
where Ns,t and Ni,t are the density of plants in cells s and i at generation t, s1 is seeds per plant, n
18
is number of infested cells surrounding a new cell s within a given radius (0~L), and pi→s is
19
probability of propagule reaching new cell s from infested cell i. For instance, for the narrow-
20
tailed kernel, p is calculated as:
19
Manuscript for Journal of Integrative Plant Biology
1
p
Z. Miao
e  x

L
(17)
e  x dx
L
2
where x is the distance from the new cell to an infested cell, L is the critical distances described
3
in Eq. 15. Inserting the results from equation 16 into equation 3, we obtained the probability of
4
establishment for new cell s and simulated stochastic invasions. Once a cell becomes invaded,
5
the population in the infested cell s will grow at intrinsic growth rate following Eq. 11 at the next
6
time interval (t+1). The model assumed that once a cell was infested, it would not be infested
7
again (i.e., population dynamics of the infested area are dominated by local population growth
8
rather than propagule dispersal from other sites), but it would become a propagule source
9
capable of invading other uninfested cells.
10
11
Study area
12
13
Our study was based upon leafy spurge invasion in the Upper Great Plains (North
14
Dakota, South Dakota, Wyoming, and Montana) of the United States of America. The Upper
15
Great Plains constitutes a vast belt predominantly rolling plains sloping gradually eastward from
16
about 1200m above sea level (masl) to 300 masl. Wide valleys, isolated hills and badlands
17
occasionally interrupt the monotonous characteristic of the plains (Padbury et al., 2002). The
18
climate of the region is continental, characterized by long, cold winters; short but warm
19
summers; large diurnal ranges in temperatures; and frequent strong winds. Annual precipitation
20
varies mostly from 300 mm to 500 mm, but extreme year-to-year variations are common and
21
long spells of hot, dry weather characterize much of the summer (Selleck et al., 1962; Padbury et
22
al., 2002). Native vegetation of the region is largely open grassland characterized by drought-
20
Manuscript for Journal of Integrative Plant Biology
Z. Miao
1
tolerant short and medium grasses. In the southern and eastern parts, woody vegetation is
2
confined mostly to valley bottoms and along major streams. Agricultural crops are composed of
3
wheat, oat, barley, canola, corn, sorghum, soybean, sugarbeet, and sunflower (Padbury et al.,
4
2002). In the region, the first report of leafy spurge was 1909 in North Dakota. By 1940s, leafy
5
spurge was prevalent in the four states (Watson, 1985). Leafy spurge area increased in sigmoid
6
curve from 1950 to 1995 (Bangsund et al., 1999). From 1950 to 1990, the acreage of leafy
7
spurge infestations roughly doubled each decade. From 1990 to 1996, the rate of growth in leafy
8
spurge acreage slowed somewhat, but the total acreage in the Upper Great Plains still increased
9
by about 24% over the 6-year period (Lym et al., 1993; Bangsund et al., 1997; Bangsund et al.,
10
1999). In North Dakota, for example, from 1953 to 1993, the invaded area followed both
11
exponential and logistic curves at significant confidence levels (95%) (Table 1).
12
Field experiments on the population density growth of leafy spurge were conducted in the
13
province of Saskatchewan, Canada, at plot levels (Selleck et al., 1962) (Table 2). The carrying
14
capacity and intrinsic growth rate of density (i.e., K and r in Eqs. 11 and 12) were set to 353 (the
15
maximum density in the concerning leafy spurge literatures) and 0.4532708, respectively,
16
following the field experiments (Table 2).
17
18
Coefficient fitting and simulation experiments
19
20
21
Parameter fitting
Given the complexity of the response surface, the model parameters (dispersal
22
coefficients β in Eqs. 5, 6, and 7 and coefficients α and c in Eq. 3) were fitted with leafy spurge
23
survey data by using multiple optimization approaches to more thoroughly search the response
21
Manuscript for Journal of Integrative Plant Biology
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1
surface (Table 1). Our approaches included even-distribution sampling plus SIMPLEX (Bixby,
2
1992.) (i.e., applying SIMPLEX at each of 100 evenly distributed starting values, but
3
randomizing the combination of parameter values, , , and c as in Latin Hypercube sampling
4
(Mckay et al., 1979)), Simulated Annealing (Goffe et al. 1994), and grid search. Where
5
appropriate, after visual inspection, we used finer gradations in the grid-wise search. The best-
6
fitting parameters obtained by the above multiple methods were used to analyze the influences of
7
control strategies on leafy spurge spread rate and processes.
8
9
When results obviously would fit the empirical data poorly, three assumptions were used
to terminate simulations so that we could save simulation time: (i) at the initial stage (0 – 30
10
years at invasion, we set 1909 as year 0 of the invasion, which was the 1st reported year of the
11
leafy spurge invasion in North Dakota), if model prediction of an infested area was greater than 3
12
times the prediction at year 30, we assumed that the trajectory was wrong and terminated the
13
simulation. We used year 30, which had a sufficiently large infested area, to avoid the
14
stochasticity associated with small infested cells; (ii) similarly, during the period of years 31 to
15
53, if any estimate was greater than 3 times the predicted infested area (estimated using linear
16
regression), we terminated the simulation; and (iii) if model prediction of the infested area was
17
greater than 1,100,000 ha at year 81 (about 2 times of the maximum infested area from the
18
empirical survey), we terminated the simulation.
19
In the study, the relative differences (RD) between the actual and the corresponding
20
predicted values of leafy spurge area were used as our metric of fit. The lower the RD values, the
21
better the fit.
22
RD 
 sim   obs
 max obs
(18)
22
Manuscript for Journal of Integrative Plant Biology
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1
where  sim is the simulated leafy spurge area (ha) of the ith year,  obs the survey leafy spurge
2
area (ha) of the ith year, and  max obs is the maximum of the leafy spurge survey area (ha).
3
4
Simulated cell scale
5
We had to consider larger cell sizes in our map lattice to keep the simulations logistically
6
feasible. To analyze consequences of upscaling on model performance, we assigned cell sizes of
7
4, 9, and 16 ha per cell (i.e., cell length was 2, 3, and 4), respectively, for a subset of simulations.
8
As the cell size increased, total cell numbers simulated within the target study area were
9
decreased, thus the model efficiency was improved. During the parameter fitting, once model
10
parameters (β, α, and c) were optimized for one cell size scale, the parameter fitting for the other
11
two scales could be done through fixing β (dispersal distance coefficient) and c (Allee effect
12
coefficient), and adjusting α (propagules pressure coefficients) of the narrow-tailed redistribution
13
kernel (Table 3). Results suggested that there were no significant differences in the prediction of
14
spread processes and patterns among the three scales by using their own optimized parameters,
15
though the variability within different simulation iterations increased somewhat as cell size rose.
16
Hereinafter, we used the cell size of 9 ha to simulate responses of leafy spurge dispersal to
17
control strategies. This corresponded with a map lattice model with approximately 1.8 Million
18
cells.
19
20
21
Control scenario design
Four common alternative control scenarios were designed to represent major control
22
strategies in American leafy spurge management: initial single control (IC), consecutive control
23
(CC), initial intermittent control (IIC), and consecutive control begun from current distribution
23
Manuscript for Journal of Integrative Plant Biology
Z. Miao
1
(CCC) (USDA-ARS Team Leafy Spurge Area-Wide IPM Program, 2002a, 2002b, 200c). The IC
2
scenario meant that control strategies were implemented only once at the beginning of leafy
3
spurge invasion. The CC scenario represented control practices which were consecutively
4
executed as soon as the local area was invaded by leafy spurge. The IIC scenario described
5
control strategies that were intermittently executed per interval. Finally, the CCC scenario meant
6
that no control or prevention practices were executed at the beginning of introduction; control
7
efforts began to input consecutively with a significant increase of leafy spurge area. Each
8
scenario constituted 8 control levels: destruction of 0, 10, 20, 30, 50, 70, 90, and 100% leafy
9
spurge stems (i.e., the resulting proportional reduction in propagule supply, i.e., N in equations
10
11 and 16). 0% indicates no control; 100% indicates local eradication. These scenarios and
11
control levels came from control practices including chemical, biological, prescribed fire, multi-
12
species grazing controls, or integrated practice management, which have being implemented or
13
experimented at local scales for a couple of years (Selleck et al., 1962; USDA-ARS TEAM
14
Leafy Spurge Area-Wide IPM Program, 2002a, 2002b, 200c).
15
Twenty iterations were run for each simulation. In total, more than 250, 000 simulations
16
of invasion spread over 80 years were performed, which included best-fitting of model
17
parameters.
18
19
20
Acknowledgements
This research was funded by the Integrating Economics and Biology for Bioeconomic
21
Risk Assessment/Management of Invasive Species in Agriculture (Economic Research
22
Service/USDA). The author greatly thanks Prof. Brian Leung for guiding the model development
23
and Catherine Pirkle for editing the earlier manuscript.
24
Manuscript for Journal of Integrative Plant Biology
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1
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18
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19
(verified on July 15, 2004).
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(verified on July 15, 2004).
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2
leafy spurge- a comprehensive, easy-to-read manual on how to use biological control as an
3
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5
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6
Dakota Wildland. Agricultural Economics Report No. 281, North Dakota State University,
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Chapter 9(3):93−104.
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Figure Legends:
2
3
Figure 1. Simulation of leafy spurge dispersal from the beginning of introduction with different
4
redistribution kernels. Obs=observed data points, AE= simulating with Allee effects, and
5
NAE=simulating without Allee effects.
6
7
Figure 2. Simulation of leafy spurge dispersal from current distribution (year 53 since 1909) with
8
different redistribution kernels. Obs=observed data points, AE= simulating with Allee effects,
9
and NAE=simulating without Allee effects.
10
11
Figure 3. Allee effects vs. no Allee effects in simulation of response of leafy spurge invasion to
12
CC scenario from the begging of introduction in North Dakota. The simulations were done with
13
the narrow-tailed kernel. Obs=observed data points, CC=consecutive control scenario, and
14
CL=control level (%).
15
16
Figure 4. Predictions of efficiencies of various control strategies with Allee effect. The
17
simulations were done with the narrow-tailed kernel. Obs=observed data points, CL=control
18
level (%), IC=initial single control scenario, CC=consecutive control scenario, IIC=initial
19
intermittent control, and CCC = consecutive control begun from current distribution.
20
21
22
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Table 1. Leafy spurge infestation in North Dakota State (Leistritz et al., 1992; Lym et al., 1993;
2
Leitch et al., 1994; Bangsund et al., 1997)
Years
Time interval since
Infested area (ha)
the initial infestation
3
% of infested land to
farms land
1909†
0
0 (the 1st reported year)
0
1962
53
80900.00
0.495597
1973
64
171400.00
1.050004
1982
73
348800.00
2.136764
1987
78
404685.64
2.479121
1990
81
526091.33
3.222858
†
The 1st reported year of leafy spurge invasion in North Dakota.
4
5
6
7
8
9
10
11
12
13
14
15
16
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Table 2. Shoots per square m of leafy spurge in various habitats during 1951–1957 (Selleck et
2
al., 1962)
Habitat
Leafy spurge 1-35 per m2
Year
Mean
1951
1952
1953
1954
1955
1956
1957
19.1
38.0
56.9
69.4
109.3 121.2 144.2 79.7
61.5
84.5
102.0 112.0 162.9 179.9 199.4 128.9
at the beginning
Leafy spurge 36-99 per
m2 at the beginning
Leafy spurge 100+ per m2
122.5 125.5 141.3 156.7 202.8 241.6 207.7 171.1
at the beginning
3
4
5
6
7
8
9
10
11
12
13
14
15
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Table 3. Best-fitting β, α, and C values of the narrow-tailed redistribution kernel at three cell
2
scales and the relative differences (RD) between observed and predicted infestations (equation
3
18)
Best-fitting parameters
Cell sizes (ha)
4
9
16
Dispersal coefficients (β)
0.1804232
0.1804232
0.1804232
Propagules pressure coefficients (α)
0.0007659
0.0002869
0.0001279
Allee-effect coefficients (C)
2.440226
2.440226
2.440226
Relative differences per iteration
0.372089
0.398259
0.379972
4
5
6
7
8
9
10
11
12
13
14
15
16
17
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1
Table 4. Best-fitting β, α, and C values for the narrow-, medium, and fat-tailed redistribution
2
kernels and the relative differences (RD) between observed and predicted infestations (equation
3
18)
Best-fitting parameters
Narrow-tailed kernel
Medium-tailed kernel
Fat-tailed kernel
NAE†
AE‡
NAE
AE
NAE
AE
Dispersal coefficients (β)
1.1614495
0.1804232
0.0901
0.00309
2.733676
1.203162
Propagules pressure
0.122505
0.0002869
0.0397
0.00166
0.464685
0.0009664
1
2.440226
1
1.707093
1
2.06686
0.50956
0.285842
0.475546
0.239015
0.504994
0.258396
coefficients (α)
AE coefficients (C)
Relative difference per
iteration
4
†
NAE = No Allee effects; ‡AE = Allee effects
5
6
7
8
9
10
11
12
13
14
15
16
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1
2
Table 5. Population size based-spread rate of leafy spurge under various control levels of the CC
3
scenario
Population
Control levels (%)
0†
10
20
30
50
70
90
550.35 (500)‡
202.05
189.45
169.2
144.9
121.5
84.6
19.8
1017 (1000)
319.95
303.3
281.25
250.2
208.35
154.8
58.5
5356.8 (5000)
1133.1
1066.05
996.3
915.3
778.95
619.65
322.2
10956.6 (10000)
1835.1
1745.1
1638
1506.15
1285.65
1048.95
564.75
28819.8 (30000)
3600.9
3406.5
3166.2
2962.35
2517.3
2077.2
1184.85
60288.3 (60000)
5782.05
5425.65
5106.15
4799.25
4167
3452.4
2005.2
100147.1 (100000)
7965.0
7483.5
7051.95
6590.25
5740.65
4830.3
2791.8
153689.4 (150000)
10327.95
9762.3
9197.55
8618.4
7521.3
6354.45
3693.6
249613.2 (250000)
14188.05
13432.5
12616.2
11822.4
10399.5
8793.0
5202.45
309457.8 (300000)
16262.55
15425.55
14527.8
13574.25
11962.8
10191.6
6001.2
455518.8 (450000)
21003.3
19759.95 18691.65
17553.6
15457.05 13181.85
7849.8
size at generation t
(ha)
4
†
0% = No control;
5
‡
Values in bracket refers to the expected population size basis at time t; owing to stochasticity of the model, we use
6
the average of 20 iterations.
7
8
9
10
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Z. Miao
Infested area (ha)
1200000
Obs
AE
NAE
900000
Narrow-tailed kernel
600000
300000
0
0
20
40
1
60
Years
80
100
2
3
Infested area (ha)
1200000
Obs
AE
NAE
900000
Medium-tailed kernel
600000
300000
0
0
4
20
40
60
80
Years
5
6
7
8
36
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Infested area (ha)
1200000
Obs
AE
NAE
900000
Z. Miao
Fat-tailed kernel
600000
300000
0
0
1
20
40
60
Years
80
2
3
4
Figure 1
5
6
7
8
9
10
11
12
13
14
15
16
37
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1
Infested area (ha)
1200000
Obs
AE
NAE
900000
Narrow-tailed kernel
600000
300000
0
0
20
40
2
60
Years
80
100
3
4
Infested area (ha)
5
Obs
AE
NAE
1200000
Medium-tailed kernel
900000
600000
300000
0
0
6
20
40
60
Years
80
7
8
9
38
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1
Infested area (ha)
1200000
Obs
AE
NAE
900000
Fat-tailed kernel
600000
300000
0
0
2
20
40
60
Years
80
3
4
Figure 2
5
39
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1
2
Infested area (ha)
600000
Obs
CL=0%
CL=10%
CL=20%
CL=30%
450000
Allee effects
300000
150000
0
0
20
3
40
Years
60
80
4
Infested area (ha)
600000
Obs
CL=0%
CL=10%
CL=20%
CL=30%
CL=50%
CL=70%
450000
300000
No Allee effects
150000
0
0
5
20
40
Years
60
6
7
Figure 3
8
9
40
80
Manuscript for Journal of Integrative Plant Biology
Infested area (ha)
600000
Obs
CL=0%
CL=10%
CL=20%
CL=30%
450000
Z. Miao
CC scenario
300000
150000
0
0
20
40
Years
1
60
80
2
Infested area (ha)
600000
Obs
CL=0%
CL=10%
CL=20%
CL=30%
CL=50%
450000
300000
CCC scenario
since year 53
150000
0
0
20
40
Years
3
60
80
4
Infested area (ha)
1200000
Obs
CL=0%
CL=10%
CL=20%
CL=30%
CL=50%
CL=70%
900000
600000
CCC scenario
since year 81
300000
0
0
5
20
40
60
Years
80
41
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1
2
Infested area (ha)
600000
Obs
CL=0
CL=10%
CL=20%
CL=30%
CL=50%
CL=70%
450000
300000
IIC scenario
150000
0
0
20
3
40
Years
60
80
4
Infested area (ha)
600000
Obs
CL=0%
CL=10%
CL=30%
CL=50%
CL=70%
CL=90%
450000
300000
IC scenario
150000
0
0
5
20
40
Years
60
80
6
7
8
Figure 4
9
42