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Take the steps.Transportation Research Research.Knowledge.Innovative Solutions!2007-42Management Practices for Weed Control in Roadway Rights-of-WayTechnical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. MN/RC 2007-42 4. Title and Subtitle 5. Report Date October 2007 6. Management Practices for Weed Control in Roadway Rights-of-Way 7. Authors 8. Performing Organization Report No. C. Arika, D. Wyse, J. Nieber, and R. Moon 9. Performing Organization Name and Address 10. Project/Task/Work Unit No. 11. Contract (McWhorter) or Grant (McWhorter) No. Department of Bioproducts and Biosystems Engineering University of Minnesota 1390 Eckles Avenue St. Paul, Minnesota 55108 (c) 81655 (wo) 124 12. Sponsoring Organization Name and Address 13. Type of Report and Period Covered Final Report 14. Sponsoring Agency Code Minnesota Department of Transportation 395 John Ireland Boulevard Mail Stop 330 St. Paul, Minnesota 55155 15. Supplementary Notes http://www.lrrb.org/PDF/200742.pdf 16. Abstract (Limit: 200 words) By law, Departments of Transportation are required to control noxious weeds along highway rights-of-way (ROWs). Since 2000, District 4 (D4) of Minnesota Department of Transportation (Mn/DOT) adopted a survey design consisting of n= 7, 3-mi segments to quantify infestations of Canada thistle (Cirsium arvense (L.)(Scop.), leafy spurge (Euphorbia esula L.), and poison ivy (Toxicodendron radicans) in chosen regions of the district. In 2004 and 2005, a second survey design was added to see if stratification by ecozone in D4, and greater numbers of 1/4-mi segments could improve precision. Comparison of matching sample statistics from the 3-mi and 1/4-mi plans in each year indicated the two plans yielded equivalent estimates of mean acres per roadway mile of each weed (α = 0.05). However, precision at the district level was much greater in all cases with the 1/4-mi plan. In addition, weed abundances varied substantially among ecozones (α ∂∂+−−=∂∂ττxutxutxutuu(x,t) is the birth rate at position x, and ),(1)(,(τ−−txutxu is the death rate at position x The authors have also noted that another major problem with adoption of either the Fisher equation or the delayed Fisher equation in population dynamics modeling: that it is very difficult to justify the birth rate, u(t) and death rate u(t)u(t-τ), in the delayed fisher equation, and that models of these kind are inappropriate for maturation or developmental delays (Al-Omari and Gourley, 2002). The Fisher equation, with modification as given below, was adopted as the basic equation governing the bacterial population u(x,t) at position x and time t, in respect of growth rate parameter (a), competition parameter (b), and diffusion coefficient (D) (Kenkre and Kuperman 2003). ),(),(),(),(2 22 txbutxauxtxuDttxu−+∂∂=∂∂For simplicity, the authors have considered only the one-dimensional situation, which is appropriate to some experiments that have been carried out with moving masks. In one such experiment (A.L. Lin, B. Mann, G. Torres, B. Lincoln, J. Kas, and H.L. Swinney, unpublished), a mask shades bacteria from harmful ultraviolet light that kills them in regions outside the mask but allows them to grow in regions under the mask. It was assumed that the growth rate has a positive constant value a inside the mask, and a negative value outside the mask. The peak value of the profile, um , will decrease as the mask size is decreased (alternatively as the diffusion coefficient is increased). The authors offered suggestion that parameters extracted in this manner may be used subsequently for the analysis of moving mask experiments with greater confidence in the reliability of the parameter values. 52Partial Differential Equations (Homes et al. 1994) Partial differential equations (PDE) have been used to model a variety of ecological phenomena. Holmes et al (1994) have stated that while PDEs that are sufficiently realistic to be used in ecological models are usually more difficult to solve than ordinary differential equations (ODE), there are advantages in their use as they allow modelers to incorporate both temporal and spatial processes simultaneously into equations governing population dynamics. In the above cited study, the authors have explored the application of the PDEs to dispersal, ecological invasions, the effect of habitat geometry and size (critical patch size), dispersal-mediated coexistence, and the emergence of diffusion-driven spatial patterning. Organisms are assumed to have Brownian random motion, the rate of which is invariant in time and space. This assumption leads to the diffusion model (Okubo 1980, Edelstein-Keshet 1986, Murray, 1989) () ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ∂∂+∂∂=∂∂2222,, yu xuDttyxuwhere u(x,y,t) is the density of organisms at spatial coordinates x, y and time t, and D is the diffusion coefficient that measures dispersal rate, with units of distance squared per time. The study concludes that implicit to the formulation of a PDE model is the assumption that the rates of birth, death, and movement can take a continuous range of values in both space and time, failing which, alternative mathematical models are more appropriate. The authors cite an example of situations where organisms reproduce continually and move between discrete patches, recommending that in such a situation may be better modeled by a system of coupled ODEs, citing Levin (1974) and Tilman (1994). They conclude that PDE are useful for examining the interaction between habitat geometry and competitive coexistence. Although diffusion models have been used successfully to model animal populations with reasonable accuracy, estimates on spread by plant population using reaction diffusion models have largely underestimated the area being invaded, often by orders of magnitude (Radosevich, 2003). Geometric Mean Population Growth Model (Freckleton and Watkinson, 1998) To be able to adequately predict weed population dynamics and how they respond to management, it is important to relate changes in population size to a range of demographic variables such as fecundity, germination and mortality. It is essential to describe both the effects of changing mean population parameters on the number of individuals in populations, and how spatial and temporal variations in such parameters may affect population numbers (Gonzalez-Andujar and Perry 1995, Cousens and Mortimer 1995). Freckleton and Watkinson (1998) have argued that population models that are used to predict weed population dynamics typically ignore temporal variability in life-history parameters and control measures, utilizing mean arithmetic population growth rates to predict population abundance. Further, weed populations are subject to intrinsic and extrinsic sources of variability. The intrinsic sources of variability include intraspecific competition and plant-to-plant variability in performance, and the latter being due to competition from crop, variability in weather, control measures, together with a variety of biotic and abiotic agents (Freckleton, 1998). 53In the current study, the authors have developed a simple analytical principle assessing how temporal variability may affect weed population persistence and mean abundance. It is their opinion that temporal variability around mean population parameters is likely to decrease population sizes relative to predictions based on arithmetic mean, and that such reductions are potentially large enough, which if ignored will compromise the predictive power of a model. In this developed model, the density of weeds at a time T is related to the initial density at time zero (No) through the product of population growth rates from time t = 0 to time t = T-1: ∏−==10Tttt oTlNNγWhere lt/γt is the net population growth rate from time t to time t + 1. lt is composed of parameters in the life-cycle that when increased tend to increase population growth rate (e.g fecundity, germination, survivorship), while γt is composed of parameters in the life-cycle that, when increased, tend to depress population growth rates (e.g. competitive effects, or the effects of control measures). For the population to persist, the overall geometric (not arithmetic) mean population growth rate must be greater than zero, as presented below: 0lnln1010≥−∑∑−=−=TttTttlγ Noteworthy here is that the geometric mean population growth is determined by the product of a series of rates, rather than their sum (Cohen 1996, 1993; Lewontun and Cohen 1969l Tuljapurkar and Orzack 1980). The authors conclude that predictive models of weed control will over-estimate weed abundance unless they are based on geometric mean population growth rates or include estimates of growth rate variability. It is important that estimation of spatial and temporal variability in weed population growth rates are obtained, or at least an indication given on how inherent variability could affect predictions by models based on arithmetic mean population growth rates. Matrix models - Population modeling approach for evaluating leafy spurge Watson (Watson, 1985)) used a transition matrix model to describe the population dynamics of leafy spurge. This approach has been used successfully to develop other weed population models (Cousens 1986, Mortimer 1983, Mortimer et al 1980, Sagar and Mortimer 1976.). An important step in developing transition matrix models is in forming a column vector. In the study on leafy spurge model by Watson (1985) illustrated below, the column vector, N, represents the number of plants in each of the indicated stages of the population: 54The values in the column vector changes as population size and structure changes over time. Figure 4.1: Diagrammatic model of a leafy spurge population: the boxes represent stages in the life cycle, arrows indicate processes, and valve symbols represent the rate at which a process occurs over a specific iteration time (Source: Watson, 1985) The variables of the model transition parameters are: S1 = proportion of seed that remain viable in the seed bank S2 = proportion of basal buds that remain viable S3 = proportion of seedlings that remain seedlings S4 = proportion of vegetative shoots (non-flowering mature) that remain vegetative S5 = proportion of flowering shoots that remain flowering G1 = proportion of seed that germinates to become seedling G2 = proportion of basal buds that grow to vegetative shoots F5 = Number of seeds produced per flowering shoot V4 = number of buds produced per vegetative shoot V5 = number of buds produced per flowering shoot The transition matrix summarizes the rates of transition between life history stages for an entire plant population. The transition matrix, M, for the leafy spurge model may be represented as: 55In this transition matrix, the S1 through S5 are the survival rates or the proportion of the individuals that remain in the same stage, F5 represents the production of seeds by flowering shoots, V4 and V5 are the production rates of basal buds on vegetative and flowering shoots, respectively, and G1 through G4 are the rates at which individuals graduate from one stage to another. Zero values mean no transition between stages is possible; for example basal buds do not produce seeds. The columns in the matrix show the fate of individuals which initiate at each of the five life history stages of leafy spurge, while the rows correspond to life history stages that result at a subsequent observation time (t) and show the sources of individuals in the population. The transition matrix and the population column vector are combined through matrix algebra to create an equation describing population changes over time: N(t+1) = MN(t). This equation indicates that at the next observation time (t+l), the population size and number of individuals in each life history stage [N(t+l)] is a result of the transitions (M) of individuals contained in life history stages at the current time [N(t)]. If the transition rates and population sizes are determined accurately, the procedure should predict the size of future plant population. (Watson, 1985). 4.2.2 Life history models These types of models utilize physiological and associated data recorded throughout the life history (growth stages) of a plant species. Spatially-Varying Growth Curve - Modeling weed growth (Banerjee et al. 2005) Recent advances in Geographical Information Systems (GIS) now allow geocoding of agricultural data enabling sophisticated spatial analysis for understanding spatial patterns. A common problem in use of this technique is capturing spatial variation in growth patterns over the entire experimental domain. Statistical modeling in these settings can be challenging because agricultural designs are often spatially replicated, with arrays of sub-plots, and agronomists are usually interested in capturing spatial variation at different resolutions. In this article the authors have developed a frame- work for modeling spatially-varying growth curves as Gaussian processes that capture spatial associations at single and multiple resolutions. Non-spatial growth curve models When spatial variation in the growth patterns is insignificant, simple non-spatial growth curve models may be adequate for data analysis. In the absence of spatial variation, the micro and macro level growth patterns may be indistinguishable; in this scenario subplots are treated as individual 56locations, observing the respective growth curves. Letting N = NsNr to represent the total number of locations, and Yit being the response (weed density in the log scale recorded for location i in time t), then applicable basic growth curve model is NitfxYitiT itit,.,1,)(=∈++=β where xit is a vector of covariates specific to the ith location, fi(t) is a function capturing the growth through time and єit is measurement error following N(0, τ2). Note that we will assume fi(t) to be linear in t, based upon the empirical evidence from the raw data. This Equation encompasses several non-spatial growth curve models by specifying fi(t). For example, if no variation in growth patterns is expected between locations, a uniform growth curve model, where fi(t) does not depend upon i will suffice. Taking α0 and α1 as independent normal coefficients or as correlated with covariance matrix Λ we have the following two models: Model 1a: )),(, 0(~;)(22 0102σσαααDiagNttfi+= Model 1b: ), 0(~;)(10Λ+=Nttfiααα Single resolution spatial growth curve models It is reasonable to expect similar growth patterns in locations of close proximity; such locations would share similar topographic and environmental conditions, hence would possess similar baseline growth (intercept) and growth rates (slope). In other words, the growth curves are likely to be spatially associated and should be modelled as a f