## Introduction

Matrix projection models are a prevailing tool for analysing the dynamics of stage-structured populations (Seno & Nakajima 1999; Ehrlén 2000; Caswell 2001; Mandujano *et al*. 2001). To be realistic, however, such models require multiple parameters, and one constraint on wider use is the availability of sufficient data to estimate model parameters. Integral projection models (Ellner & Rees 2006) often require fewer parameters, and maximum likelihood and Bayesian methods can estimate missing or incompletely known parameters, using time series data (Hilborn & Mangel 1997; Gross, Craig & Hutchinson 2002). These methods, however, are difficult to implement; so additional ways to resolve high parameter uncertainty are needed for models to contribute to the management of weedy plants.

Perturbation analyses, used to rank the relative importance of factors influencing population growth rate, currently examine elasticity and sensitivity of matrix transition rates or parameter values (Caswell 2001). Such local perturbation analyses should be confined to examining the consequences of very small perturbations of single, well-known, independent parameters (Horvitz & Schemske 1995; Caswell 2001). Thus alternative methods are needed if there is parameter uncertainty, values vary widely or the effect of perturbation of one parameter is not independent of other values. Nevertheless, many authors suggest that elasticities give robust predictions of the effect of large changes in demographic parameters on the asymptotic population growth rate, λ (Caswell 2000; de Kroon, van Groenendael & Ehrlén 2000). For example, Caswell (2001) argued that ‘although elasticities are local slopes, they do a good job of predicting the results of even relatively large (± 50% at least) perturbations’. As a consequence, the results of sensitivity or elasticity analyses are used to infer the effect of large perturbations and to derive management recommendations (Crooks, Sanjayan & Doak 1998; Fisher, Hoyle & Blomberg 2000; Hunt 2001).

We have used Monte Carlo methods, to assess the effect of large parameter uncertainty on matrix model predictions of λ, and partial rank correlation analysis (PRCC), to determine the relative importance of each contributing variable (Blower & Dowlatabadi 1994). PRCC results are comparable to elasticity but the Monte Carlo/PRCC approach is a global perturbation analysis, successfully applied to complex ecological models (Blower & Dowlatabadi 1994; Hilborn & Mangel 1997; Rushton *et al*. 2000a,b; Tenhumberg *et al*. 2004) but not previously to matrix models.

We focused on the relative importance of factors influencing the population growth rate of the Eurasian thistle *Cirsium vulgare* (Savi) Ten., a highly invasive monocarpic thistle (Julien & Griffiths 1998) and a noxious weed in nine USA states (http://plants.usda.gov/, accessed November 2005). Despite its presence for more than 50 years, *C. vulgare* occurs only at low densities in western tallgrass prairie in eastern Nebraska, USA, along rural roadsides and in perennial pastures (Stubbendieck, Friisoe & Bolick 1994; Andersen & Louda 2007). A high level of floral herbivory significantly reduces seed production in Nebraska (Louda 1999; Louda & Rand 2002) and weed management practices probably affect its demography in rural areas. Roadside vegetation is generally mowed early and late in the growing season, and intensive row-crop agriculture involves cultivation and herbicide application.

Our overall aim was to understand the factors that lead to the observed population stasis in this invasive thistle. Our first goal was to evaluate the relative contribution of floral herbivory to the *C. vulgare* population growth rate and to identify parameters still requiring additional local data. The parameters for the base matrix model were extracted from studies of local populations performed over the last 15+ years, supplemented by parameter estimates from the literature. As eight of the estimates had to be derived from foreign populations, parameter uncertainty was high; thus a second goal was to compare a global perturbation analysis using Monte Carlo simulations to the usual local sensitivity and elasticity analysis for evaluating relative parameter importance in this situation. Our third goal was to explore the consequences of weed management practices on λ, by including the proportions of bolting thistles that die before producing seed (increased by mowing) and seed germinating successfully (reduced by dispersal into intensively managed cropland) in the model.