Manuscript for Journal of Integrative Plant Biology Z. Miao
Modeling Responses of Leafy Spurge Dispersal to Control Strategies
Zewei Miao
GrantF. Walton Center for Remote Sensing & Spatial Analysis, Cook College, Rutgers University, 14 College Farm Road, New Brunswick, NJ 08901-8551
To whom correspondence should be addressed. Telephone: 732-932-1583; Email:
Received: 14 Jan. 2006 Accepted 2 Oct. 2006
Handling editor: Da-Yong Zhang
Abstract
Leafy spurge (Euphorbia esula L.) has substantial negative effects on grassland biodiversity, productivity, and economic benefit in North America. To predict these negative impacts, we need an appropriate plant-spread model which can simulate the response of an invading population to different control strategies. In the study, usinga stochastic map lattice approach we generated a spatially explicitlystochastic process-based model to simulatedispersaltrajectoriesof leafy spurge under various control scenarios. The model integrateddispersal curve, propagule pressure, and population growth of leafy spurge at local and short-temporal scales to capture spread features of leafy spurge at large spatialand long temporal scales. Our results suggested that narrow-, medium-, and fat-tailed kernels did not differ in their ability to predict spreadin contrast to 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 effectswere critical in predicting spread rate of leafy spurge, because the prediction could be seriously affected by the low density period of leafy spurge community. NoAllee effects were not able to simulate spread rate well in this circumstance. When applying control strategies to the current distribution, Allee effects couldstop the spread of leafy spurge; NoAllee effects, however, were able to slow but couldnot stop the spread. The presence of Allee effects had significant ramifications on the efficiencies of control strategies. For both Allee and noAllee effects, the later control strategies were implemented, the more effortshadto be input to achieve similar control results.
Key words: invasive species, non-indigenous, propagule pressure, dispersal curve, stochastic process-based model, Allee effects.
Introduction
Invasions of exotic species have been proposed as one of largest components of habitat destruction, biodiversity losses, and economic damage (Cain et al., 2000; Leistritz et al.1992; Rouget et al. 2003). As a perennial dicotyledonous herbaceous plant, leafy spurge (Euphorbia esula L.)cancadapts to a wide variety of habitats. (Selleck et al. 1962; Dunn 1979; Watson 1985), and has become a great threat to rangeland productivity, to species diversity, to the quality of wildlife habitats, and to land values in the Northern United States and the prairie provinces of Canada since its discovery in North America in 1827 (Selleck et al. 1962). 1.6 million ha of land in the Upper Great Plains (North Dakota, South Dakota, Montana, and Wyoming) of the United States were infested by 1994 (Everitt et al. 1995; Bangsund et al. 1999). In the past decade, leafy spurge infestations have been estimated to result in an annual economic loss of US$130 million in the four-state region (Leitch et al. 1994).
There is a need to simulate dispersal features of leafy spurge under the influences of weed management practices. Field experiments demonstrate that control strategies of leafy spurge, including chemical, multi-species grazing, and biological efforts, are able toreduce population density (i.e., propagule pressure) to a large extent (Selleck et al. 1962; Watson 1985;Leitch et al. 1994). This likely constrains spatial spread through reducedpropagule pressure. So far,few studies have attempted to simulate the response of dispersal process and patterns of leafy spurge invasion to various weed management practices.It is not yet known with certainty how ecological dispersal features of leafy spurge will be affected by weed management practices at large spatial and long temporal scales.
Process-based models provide us a mechanistic understanding of invasion and dispersal featuresof exotic species at large scales responding to management practices and environmental heterogeneities.Analyses of the spread of invading organisms frequently start with a reaction-diffusion (R-D) model with exponential growth and Fickian diffusion (Higgins et al., 1996; Kot et al., 1996). When the dispersal pattern is Brownian and the trapping rate by the environment is uniform, the dispersal pattern can be described as follows (in simple form):
(1)
or (2)
where Nt+1(x) and Nt (y) are the population densities at generation t+1 and t at locations x and y, respectively; k(x,y) is the one-dimensional redistribution kernel, i.e., the probability density function for propagule dispersing to destination x from a source position y; r is the species’ intrinsic rate of increase; and s is the diffusion constant (Higgins et al., 1996; Kot et al., 1996). This kind of R-Dmodelsassume that the invading organismsspread in a given redistributionkernelregardless ofenvironmental heterogeneities and weed management practices. The precise shape of redistribution kernel k(x, y) is extremely important to describe the speed characteristics of invasion (Kot et al., 1996). The R-D models, however, is usually limited by the absence of suitable redistribution kernels and/or by inadequate parameter estimates for such kernels.For instances, ecological data is incomplete to generate the redistribution kernel for leafy spurge R-D model (Selleck et al., 1962; Cain et al., 2000).The R-D equations alsoassume that the invading populations are large enough that stochastic effects are not important, which, in nature, is not always the case (Hengeveld, 1994; Kot et al., 1996). Field experiments show that leafy spurge invasion are mainly caused by stochastical biological and environmental factors such 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 Wilson, 1989). All together, there is a need to establish aprocess-based model to simulate stochastic processes and patterns of leafy spurge invasion which responds to various control strategies. Over a certain spatial scale (stands to landscapes), stochastic process-based models can integrate biological attributes, spatial considerations, invasion stochasticity, and environmental heterogeneity into modellingspecies’ distributions (Rouget et al., 2003).
Interactions among dispersal curve, propagule pressure, and intrinsic population growth are central to establishing a stochastic process-based spread model of exotic species’ invasion. In the invasion process (i.e., introduction, colonization, and naturalization), many factors such as propagule pressure, distance from the source, and external environment, determine spread range and abundance of non-indigenous species at a given locality. Dispersal curves are frequently fitted with a negative exponential curve or a negative power function (Wallace 1966; Taylor 1978; Kot et al., 1996), and the probability density function of a dispersal curve may be normally distributed for some species (Kot et al., 1996). The role of propagule pressure or mass effects is clearly observable in many invasions. As availability of propagule pressure increase, chances of establishment, persistence, naturalization, and invasion, will increase as well,especially at the introduction stage (Rouget et al., 2003). Population growth dynamics affect propagule yield and dynamics of leafy spurge ecological communities; hence population growth will affect the probability of establishment in uninfested areas. Recent works indicate that the combined influence of Allee dynamics and stochastic processes strongly determines the successful establishment of alien species (Liebhold and Bascompte, 2003). However, the influences of Allee effects are still unknown in the process of leafy spurge invasion.
In this paper, we link the data available across two different scales - dynamic density data at a local (plot) and short temporal scales, and long time series of infested area at a regional scale – to construct our dispersal model. We simulate, with minimum parameter requirements, the stochastic processes and patterns of spread of leafy spurge in response to control strategies. We test whether the form of the dispersal kernel (i.e., narrow- vs.fat-tailed kernels) and Allee effects are importantfor explaining the observed pattern of invasion, including the lag period before leafy spurge becomes invasive. We simulate efficiencies of four typical control scenarios of leafy spurge management in North America that have been applied at local spatial and short temporal scales. This work will be useful to future research on risk analysis, management options, and optimal control theory.
Results
Prediction of leafy spurge invasion vs. different redistribution kernels
There were no differences in predicting infested area among the narrow-, medium-, and fat-tailed redistribution kernels, but their best-fitting parameters, β, α, and c, were different (Table 4). With the different forms of redistribution kernels and their own best fitting parameters, similar dispersal trajectories were obtained for all corresponding control strategies, regardless of Allee or no Allee effects (Figures1 and 2). For the no control scenario and with Allee effects, all the three kernels gave good predictions in leafy spurge dissemination against theinfested survey area(Table 1). For no control and with Allee effects, the narrow-, medium- and fat-tailed kernels predicted the infested area of 507787.2, 505371.2, and 496539.9 ha at year 81, respectively, while the survey leafy spurge area was 526091.3 at year 81. The relative differences between the prediction and the corresponding survey data per iteration were 0.29, 0.24, and 0.26, respectively.
For the three kernels, the best-fit parameters usually consisted of low β (i.e., greater maximum dispersal distance ranging from 4200 to 6300 m), low α (i.e., low probability of establishment of individual propagules), and high c values (Allee effect coefficients, ranging from 1.5 to 2.5)(Table 4), whose estimates matched well with the survey data. The best-fitting β, α, and c values were 0.1804232, 0.0002869, and 2.440226 for narrow-tailed kernel, 0.00309, 0.00166, and 1.707093 for medium-tailed kernel, 1.203162, 0.001102, and 2.06686 for fat-tailed kernel, respectively.Because of lack of differences between the three kernels with respect to predictions, the remainder ofthe analyses was presented only fornarrow-tailed kernel.
Influences of Allee and noAllee effects on lag phase in leafy spurge invasion
The dispersal trajectory predicted with Allee effects (i.e., c>1 in Eq. 3) remained very different from no Allee effects (i.e., c=1 in Eq. 3) (Figure 1). When simulations started from the beginning of introduction, prediction with Allee effectswas significantly better than noAllee effects. Allee Effects were essential in capturing the lag phase during leafy spurge invasion, regardless of kernel forms (Fig. 1). The relative differences between prediction and survey were 0.28 forAllee effects and 0.51 for noAllee effectsofnarrow-tailed kernel,0.24 for Allee effects and 0.48 for no Allee effectsofmedium-tailed kernel, and 0.26 for Allee effects and 0.50 for no Allee effectsoffat-tailed kernel, respectively (Table 4).Allee effects reducedthe probability of establishment of an individual propagule at the beginning of the invasion, permitting the observed lag phase of 3 decades, and increased the probability at the latter period in comparison to no Allee effects (Fig. 1).Leafy spurge spread rate of Allee effectswas almost two times that of no Allee effectsin the last simulating year (year 81).
When simulations leaped over the initial phase of introduction, there were no significant differences between Allee and no Allee effects. When the simulations began from the current distribution (e.g., year 53, the 1st surveyed year of leafy spurge area), the model gave good predictions in leafy spurge dispersal compared to survey data (Fig. 2), no matter what Allee or no Allee effects.The relative differences between the prediction and the corresponding survey data per iteration were 0.13 and 0.15 for Allee and no Allee effects of the narrow-tailed kernel, 0.132939 and 0.175604 for Allee and no Allee effects of the medium-tailed kernel, and 0.186954 and 0.135012 for Allee and no Allee effects of the fat-tailed kernel, respectively.
Allee effects magnifiedthe importance of early control strategies of leafy spurge. With Allee effects, responses of leafy spurge spread to control strategies were significantly greater than for noAllee effects, no matter what control scenarios (Fig. 2). For example, for the CC scenario, when control efforts were greater than 30%, leafy spurge was contained during the 81 simulation years for Allee effects. In contrast, for no Allee effects, leafy spurge continued to spread even though the control level was 60% (Fig.2). For a given control effort, the response of no Allee effects to control strategies was considerably less than with Allee effects, especially at the initial period.
Responses of leafy spurge invasion to control strategies
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 population size, the better achievement were reached. Because no Allee effectswere not able to simulate well leafy spurge invasion, the model with Allee effects was mainly used to analyze the effectiveness of control strategies. For the narrow-tailed kernel, by comparing the CC to CCC scenarios, when control strategies were conducted from year 0 for the CC scenario, a control level of 30% was able to contain leafy spurge spread during the 81 simulation years. When control was implemented at year 53 and 81 for CCC scenario, however, over 50% and 70% control levelswere needed to stop leafy spurge dispersal, respectively (Fig. 3). A control level of 30% could slow but not stop the spread of leafy spurge in the CCC scenario. Table 5 also indicated that the bigger the population size was, the lower control effectiveness would be for a given control effort.
Our results suggested that theIC scenario was not able to contain leafy spurge spread in 81 simulation years, except for 100% eradication. Even though less than 10% of propagules had an opportunity to spread, leafy spurge would ultimately disseminate during 81 simulating years, but the spread rate slowed down along with an increase of control efforts (Fig. 3). In other words, for IC scenario, one had to eradicate leafy spurge patches at the beginning of invasion to fullyterminate leafy spurge dispersal.
In contrast to other scenarios, the results of the IIC scenario suggested that more control efforts were required to contain leafy spurge spread than with the CC scenario. For example, for the IIC scenario, more than 70% of control levels had to be devoted to stop leafy spurge spread, while 30% of control efforts were enough to contain the spread for the CC scenario (Fig. 3). For the CC scenario, frequency of control application was higher than IIC and this hadto be incorporated in the computation of control efforts. The IIC scenario worked more effectively than IC, since IC was almost not able to stop dissemination even at a control level of 90%. IIC control effectiveness was similar to the CCC scenario starting from the current distribution at year 81, but worse than that of the CCC scenario starting from the current distribution at year 53. As a common practice in the integrated pest management (IPM) of leafy spurge invasion, more input per control for CCC scenario was needed than the CC scenario to achieve similar results, despite the fact that control efforts were intermittent and the numbers of control practices were fewer than in the CC scenario.
Discussion
Dispersal of a spreading population and redistribution kernels
Redistribution kernels are probability-density functionsthat describe the possibility that individuals will be found at specific spatial coordinates relative to the original, colonizing location (MacIsaac et al., 2004). Insight into the process and spatial patterns of dispersal of a nonindigenous species can be gleaned from its redistribution kernel (Taylor, 1978; Kot el al., 1996; Clark et al, 2003;MacIsaac et al., 2004). Kot el al. (1996) reported that the speed of invasion of a spreading population is extremely sensitive to the precise shape of redistribution, in particular, to the tail of the distribution. They reported that different shape kernels with similar coefficients of determination will yield dramatically different speeds in a spreading population, andfat-tailed kernels can generate accelerating invasions rather than constant-speed waves (Kot el al., 1996). Our results found that the short-, medium-, and fat-tailed kernels with different best-fit parameters are able to get similar trajectories for leafy spurge, and this resultsare not affected by control strategies.
Dispersal distance coefficients are critical parameters for the stochastic process-based modelprediction. In our study, values of β (dispersal coefficients) all are very low for the three kernels, with a maximum spread distance of around 4200 and 6300 m (for cumulative probabilities of 0.9995).These dispersal distances conform to previous dispersal observations of leafy spurge, which suggest that leafy spurge dissemination occurs in both local diffusion (short distance) and jump dispersal patterns (at medium and large distance) (Herbert and Rudd, 1933; Selleck et al., 1962).Our sensitivity analysis shows that a slight change in the β value can induce a large change in the dispersal distances of introduced new colonies from parental colonies. Therefore, paired field data linking population density to dispersal distance should be collected for empirical validation of leafy spurge invasion in the future.
The spread rate wasstrongly affected by propagule pressure and Allee effects. Biologically, these two factors have quantitative (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 el 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 populations(Grevstad, 1999; Engen et al., 2003).Our study shows that regardless of kernel shapes, the c values in Eq. 3are consistently much greater than unity, which suggestsAllee effects are significantly present in a leafy spurge invasion.In contrast, no Allee effects cannot predict the trajectory of a leafy spurge invasion well, since no Allee effects predictions are too high to capture the lag phaseat the initial stage, and too low at the later.
Allee effectsare indispensable in simulating the lag phase of a leafy spurge invasion (about 2 or 3 decades in North Dakota) (Hobbs and Humphriest 1995; Sakai et al., 2001). These lag times are expected if environmental or evolutionary change is an important part of the colonization process. This process may include the adaptations ofinvasive species to a new habitat, the evolution of invasive life history characteristics, or the purging of a genetic load responsible for an inbreeding depression (Sakai et al., 2001). Our predictions suggest that population dynamical processes, such as Allee effects, are sufficient to account for the lag phase observed in leafy spurge. There are also evidences from field experiments for Allee effects in leafy spurge, where seed germination is about 20–30% and the percentage of seedlings surviving is much lower at the initial stage (Herbert and Rudd, 1933; Selleck et al., 1962). Once a patch is established, leafy spurge can thrive in a variety of conditions, producing a large number of propagules (seeds) (Stroh et al., 1990; Leitch et al., 1994; Bangsund et al., 1999).