Monte Carlo analysis of parameter uncertainty in matrix models for the weed Cirsium vulgare

Authors


*Correspondence author. E-mail: btenhumberg2@unl.edu

Summary

  • 1Parameter uncertainty challenges the use of matrix models because it violates key assumptions underlying elasticity analyses. We have developed a matrix model to compare Monte Carlo methods with elasticity analyses for estimation of the relative importance of factors in the asymptotic population growth rate, λ, of Cirsium vulgare (spear thistle) in Nebraska, USA.
  • 2We calculated λ for a base model using 11 parameter estimates available for Nebraska populations plus eight extracted from the literature, causing parameter uncertainty. We then calculated λ for 10 000 alternative models using Monte Carlo parameter estimation; parameters were drawn from the full range of each parameter in the literature and partial rank correlation analysis (PRCC) was used to order the parameters by the magnitude of their effect on λ.
  • 3Monte Carlo analysis found that insect floral herbivory, affecting the regeneration transition, was the most important parameter affecting λ, whereas elasticity analyses suggested that the transition from small to medium size was the most significant. Statistical comparison, using PRCC vs. lower level elasticity (LLE), showed that the Monte Carlo analysis provided a more accurate assessment.
  • 4As λ > 1 in 99% of the model runs even with significant floral herbivory, we added two parameters influenced by weed management (probability of large thistles dying without producing seed and proportion of seeds that failed to germinate). Simulations that included reductions in these parameters, along with floral herbivory, led to λ < 1 in 17% of the runs, suggesting these three factors interact to produce the low densities observed for this invasive thistle in our study area.
  • 5Synthesis and applications. This study demonstrates the utility of the Monte Carlo approach for modelling weed dynamics with parameter uncertainty and multiple, potentially interacting, parameters. Invasive population growth by C. vulgare could be limited by a combination of weed management practices and the biotic resistance imposed by native floral herbivores.

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.

Methods

matrix model structure

We constructed stage-classified, pre-breeding census (late summer), birth-pulse matrix models (Caswell 2001) using annual time steps (Fig. 1). We modelled C. vulgare reproduction as a birth-pulse process because seed release occurs during a relatively short period near the end of the growing season (Barkley 1986). We assumed seed was produced and released after the population census. Seeds either die or overwinter; surviving overwintered seeds germinate and grow into small plants by the next census or they enter the seed bank. As survival of seeds in the soil for more than two winters is very low (van Breeman & van Leeuwen 1983), the seed bank consisted only of seeds that stayed in the soil for a second winter, after which we assumed that they either germinated or died. The model had four size-stage classes: seed bank (SB), small-sized plants (S; diameter < 10 cm, comprising mainly seedlings), medium-sized plants (M; diameter 11–20 cm) and large-sized plants (L; diameter > 21 cm), following Bullock, Hill & Silvertown (1994).

Figure 1.

Life cycle graph of the Cirsium vulgare population model. The arrows indicate transitions between the size-stage classes.

parameter estimation, base model

To calculate the elements, aij, of matrix A, we used 19 parameters suggested by the natural history of C. vulgare (Table 1 and Table 2). For example, the transition rate for plants from the small (S) to the medium (M) size class was calculated by multiplying four parameters: winter survival of small plants (i), proportion of small plants growing to a medium size (a), summer survival of medium-sized plants (d) and the proportion of medium-sized plants surviving insect foliage herbivory (g).

Table 1.  Life-history parameters of Cirsium vulgare (S, small plants; M, medium plants; L, large plants). Base values were used to calculate the transition rates in a matrix model of C. vulgare populations in eastern Nebraska, USA. Parameters derived directly from Nebraska populations of C. vulgare are in bold and from the related native C. altissimum are underlined (parameter values are rounded to 3 decimal points); other base parameters are the midpoint of the parameter range in the literature. The low and high points of the reported parameter range in the literature were used as lower and upper limits in the Monte Carlo simulations. Where we could not extract a parameter range from the literature (n = 5; indicated by *), we created a range by adding and subtracting 0·1 from the mean values; larger ranges would have resulted in some rates > 1·0. Estimates of summer survival (c – d) in the literature did not exclude leaf herbivory, making them conservative estimates that cause some underestimation of λ calculated in the Monte Carlo simulations. However, experiments used to estimate the base value for summer survival in Nebraska controlled for insect herbivory (insecticide application)
  Range*
VariablesSymbolLowHighBase values
Probability of live plants graduating from S to M1,13a0·40·90·9
Probability of live plants remaining M (from M to M)1*b0·2240·4240·324
Summer survival of S 1,2,5,6,14c0·20·940·516
Summer survival of M1d0·50·780·64
Summer survival of L1,2e0·70·990·85
Proportion of S surviving insect foliage herbivory5f0·250·630·515
Proportion of M surviving insect foliage herbivory7*g0·750·950·85
Proportion of L surviving insect foliage herbivory7*h0·650·850·75
Winter survival of S3,13i0·2481·00·248
Winter survival of M3*j0·81·00·9
Winter survival of L3*k0·81·00·9
Flowering probability of L6,8l0·50·90·7
Flowering probability of M, as proportion of l1,8,13m0·20·650·44
Seed production of L1,2,6,8,10n800030 00022 119
Seed production of M, as a proportion of that by L14o0·30·70·5
Seed survival of floral herbivory2,8,9,10,11,13,14p0·0030·60·058
Survival of post-dispersal predation1,8q0·40·80·6
Germination rate1,5,12,13,14r0·060·80·214
Seed bank germination rate1,4,8,12s0·00·10·03
Table 2.  Parameters multiplied to calculate each size-class transition rate in the matrix model for Cirsium vulgare. Symbols are defined in Table 1
 Seed bankSmallMediumLarge
Seed bank00l m n o p q (1 – r)l n p q (1 – r)
Smallc f s0l m n o p q r c fl n p q r c f
Medium0a d g ib d g j (1 – l m)0
Large0(1 – a) e h i(1 – b) e h j (1 – l m)e h k (1 – l)

Where available, we used parameter estimates derived directly from C. vulgare populations in Nebraska to derive base values (Table 1; bold values). Values from the similar, co-occurring, native Cirsium altissimum, were used as second choice (Table 1; underlined values). Failing that, C. vulgare populations in other regions, both native and non-native, were used, taking the midpoint of the large range of values observed (Table 1). Our perturbation analysis to determine the effect of this parameter uncertainty assumed that there was no covariance between parameters; thus our results were biased towards an overestimation of the effect on λ.

Size-class transitions (a, b)

Local data on transition rates were supplemented with data for C. vulgare in Britain (Bullock, Hill & Silvertown 1994). It was found that, for plants that survived, 90% of the small-sized ones graduated into medium-sized plants, while 10% graduated into large-sized plants; 37·5% of the medium-sized plants remained in the medium size class, while 62·5% graduated to large-sized plants. Plant size reductions were not observed, so size regression was not included in our model; the transition from medium to small plants (Fig. 1) represented small-sized plants recruited from seed produced by medium-sized plants, seed that overwintered, germinated, survived and grew into the small size class by the next census date. All small-sized plants that did not die or grow into medium-sized plants were assumed to grow into large-sized plants.

Survival rates (c–k)

Survival of C. vulgare rosettes generally increases with size (de Jong & Klinkhamer 1986). For small plants (S), we recorded survival for 480 small seedling plants that germinated in 2004 in a seed-addition experiment in a prairie restoration (M. Takahashi & S. Louda, unpublished data) and 8066 seedlings that germinated in 2005 in an experiment to quantify spatial dynamics of the C. vulgare–insect herbivore interaction (J. Eckberg, S. Louda & B. Tenhumberg, unpublished data). For the 2004 cohort, 30·0% of the small (seedling) plants survived the summer; in addition, 24·8% of those remaining seedlings survived over the following winter. These survival rates included the effects of foliage herbivory. For the 2005 cohort, we found that, of the initial 3979 small (seedling) plants exposed to insect foliage herbivory (controls = no insecticide), 26·6% survived the summer; 51·6% in the insecticide-treated seedlings (n = 4087) survived the summer. So the proportion of small plants that died from foliage herbivory was 1 – the survival ratio of both experiments (1 – 0·266/0·516 = 0·485). For convenience, in this model we used the survival ratio (f = 0·266/0·516 = 0·515) such that the survival rate of plants exposed to herbivores was calculated as 0·516 × 0·515 = 0·266. We set the range of f-values equal to the variation among subplots in 2005 (mean ± SD = 0·25–0·63).

For survival of medium- and large-sized plants, we used the midpoint of the published range to estimate these unknown rates. Guretzky & Louda (1997) found that foliage herbivory on C. altissimum reduced the survival of large rosettes by 25% but affected the survival of the medium-sized established rosettes less. To include such herbivory in the base model, we used a 15% reduction in survival for medium-sized plants and a 25% reduction for large-sized plants as our parameter estimates; we handled these as we did f for small plants (above).

Seed production rates (l–p)

The probability of flowering generally increases with C. vulgare rosette size and, in the indigenous environment, plants can take several years to reach flowering size (Klinkhamer, de Jong & van der Meijden 1988; Bullock, Hill & Silvertown 1994). However, in non-indigenous pastures in Queensland, Australia, C. vulgare was biennial (Forcella & Wood 1986). In Nebraska, 30·8% of the medium-sized rosettes in our experimental seed addition flowered in their second year (M. Takahashi & S. Louda, unpublished data). As no Nebraska data were available on the probability of flowering by large rosettes, we used the midpoint of the published range (Table 1 ; 70%, l= 0·7). To ensure that medium-sized plants always had a smaller probability of flowering than large ones in the Monte Carlo simulations, we created a parameter, m, and set m= 0·44, such that the flowering probability of medium-sized plants was constrained to 44% of the proportion of large-sized plants flowering (i.e. flowering of medium-sized plants: lm = 0·7 × 0·44 = 0·308), consistent with the data available. Similarly, we also used the parameter o (Table 1) to constrain the calculation of seed production of medium-sized plants that flowered (below).

To estimate production of viable seed, we used data on seed initiation, floral herbivory and viable seeds recorded from 1997 to 1999 in the region (Louda 1999; Louda & Rand 2002; S. Louda, unpublished data). Individual C. vulgare plants initiated an average of 22 119 florets (SE 3521) and produced an average of 1285 viable seeds (SE 247); thus we set n= 22 119. Reduction of seed by insect floral herbivory averaged 18 601 seeds plant−1 (84·1%) when strict criteria of definitive evidence of insect feeding were used (Louda & Rand 2002), and 94·2% if all of the evidence of probable damage by herbivores was included (S. Louda, unpublished data); we used the latter value (P = 0·058) as the maximum estimate of the effect of floral herbivory on viable seed production of populations in eastern Nebraska.

Germination rates (q–s)

Viable C. vulgare seeds falling on the ground are readily consumed or removed by seed predators such as rodents and ants; this post-dispersal seed predation and loss can be as high as 68% (Klinkhamer & de Jong 1988; Klinkhamer, de Jong & van der Meijden 1988). We used the midpoint of the range of values in the literature in our base model, leaving 60% of viable seeds escaping post-dispersal predation.

The recorded germination rate of C. vulgare in this region was highly variable. In 2004, for seeds planted in 50 × 50-cm plots along a habitat gradient, 3–42% of the seeds germinated (M. Takahashi & S. Louda, unpublished data). In 2005, for seeds planted in larger plots (2 × 2 m) across a larger spatial scale and multiple sites (70 plots, ~538 seeds added per plot), the estimated germination by C. vulgare across all plots was 21·4% (J. Eckberg, S. Louda & B. Tenhumberg, unpublished data). Because of the larger sample size and broad spatial representation, we used the latter estimate for our base model. The expected number of seeds produced per plant that survive to germinate in spring is quite low (npq = 22 119 × 0·058 × 0·6 = 769), so it is unlikely that competition affects germination and early seedling survival (J. Eckberg, S. Louda & B. Tenhumberg, unpublished data). Therefore we assumed that recruitment was density independent.

perturbation analyses

We compared three methods for calculating the sensitivity of C. vulgare population growth rate to perturbation of model parameters. First, we calculated elasticity matrices (E) to assess the relative importance of small linear perturbations of individual matrix transitions (aij) for asymptotic population growth rate, λ, when other parameters were held constant (Caswell 2001). So, Eij = (aij/λ)(viwj/<w, v>), where v is the vector of scaled reproductive values, w is the scaled age distribution, and the bracket < > indicates the scalar product.

Second, because the calculation of each matrix element used multiple parameters, we also calculated the elasticity for specific components of the matrix entries, the lower level elasticities (LLE; Caswell 2001). The lower level elasticity of parameter X (LLEX) is calculated as the weighted sum of the sensitivities (Sij=viwj/<w, v>) of those matrix elements that are influenced by X; thus, inline image Lower level elasticity values also assess the effect of small linear perturbations of single parameters; however, in contrast to elasticity values of the matrix entries, lower level elasticity values are not required to add up to 1.

Thirdly, we performed a Monte Carlo sensitivity analysis, given the uncertainty imposed by multiple estimated parameters and the possibilities of parameter interactions and non-linearities in response to larger perturbations associated with weed population dynamics. We used a Latin Hypercube (Blower & Dowlatabadi 1994) to produce 10 000 random parameter combinations, with each parameter drawn (sampling without replacement) from a uniform distribution bounded by the lowest and highest values in the C. vulgare literature (Table 1). According to Blower & Dowlatabadi (1994), the minimum number of simulations required for Latin Hypercube sampling is 3/4K, given K equals the number of uncertain variables. In our case K= 19 or 21, so the number of simulations we used is orders of magnitude higher. No information exists on the true distribution of parameter values, and with the lack of such knowledge our uniform distribution is justifiable (Hilborn & Mangel 1997). Alternate distributions led to similar results (see Tables S1 and S2 in the Supplementary material). The only exception in this parameterization was that we excluded an unusually low estimate for seed reduction by floral herbivory (19%) from a non-native locale, Australia (Forcella & Wood 1986).

Thus the Monte Carlo analysis consisted of 10 000 unique parameter combinations in which all parameters varied simultaneously; we calculated λ-values for each parameter combination. Because input variables were not normally distributed and the outcome variables were generally non-linear functions of the input variables, non-parametric tests of ranked data were necessary (Conover 1980). We calculated partial rank correlation coefficient (PRCCs) as in Blower & Dowlatabadi (1994) to determine the statistical relationship of each parameter to the estimate of λ (Conover 1980) and the relative importance of each variable (Blower & Dowlatabadi 1994). This procedure enabled us to determine the independent effect of each parameter, even with unknown levels of correlation among parameters (Conover 1980).

examination of variation in two parameters affected by weed management

The initial analyses above fell short of fully explaining the low population densities of C. vulgare observed in eastern Nebraska; therefore, we asked whether reduction in seed production and regeneration, the aim of local land management practices, could interact with biological factors to limit λ. We considered the probability of large plants surviving to flower and set seed (e), which is reduced by roadside mowing, and the probability of successful seed germination (r), which is reduced in croplands by intensive weed management.

Thus, as a first approximation, we explored how much e and r would have to be reduced in order to produce λ≤ 1 at various levels of floral herbivory. To do so, we defined two new parameters, the proportion of large flowering plants dying early (u), for example in response to mowing, and the proportion of seed lost to unsuitable habitat (v), for example into cropland, such that the new proportion of large plants surviving the summer was e(IM) and the new germination rate was T(IV). We varied u and v simultaneously between 0 and 1·0, assuming all other parameters were those in the base model (where u and v= 0, by definition). By varying these two parameters between 0 (no effect) and 1·0 (where all bolting thistles died or no seeds germinated), we could quantify the effects, examine the parameter interactions and calculate the relative importance of such variation to the estimated λ, given the parameter uncertainties, for varied levels of floral herbivory.

Finally, we used a Monte Carlo perturbation analysis to explore the sensitivity of λ to variation in u and v, and determine the relative contribution of these parameters to the population growth rate in the presence of uncertainty in the 19 other model parameters. The outcomes were viewed as predictions for further testing.

Results

comparative analysis of parameter influence on λ

We analysed the matrix model (Fig. 1) using base parameter values (Table 1) that represented the current best estimates of vital rates for C. vulgare populations in eastern Nebraska. The estimate of the asymptotic population growth rate (λ) for this model (Table 3) was 1·538, suggesting an annual population increase of 53·8%, much higher than observed. With floral herbivory set to zero, λ was even higher (5·20; analysis not shown).

Table 3.  Matrix projection model of Cirsium vulgare in eastern Nebraska, USA, using base values (Table 1). Estimated asymptotic population growth rate λ= 1·538
 Matrix transition ratesElasticity matrix
Seed bankSmallMediumLargeSeed bankSmallMediumLarge
Seed bank0·0000·00093·149423·4040·0000·0000·0100·016
Small0·0080·000 6·752 30·6890·0260·0000·1460·219
Medium0·0000·121 0·110  0·0000·0000·2950·0230·000
Large0·0000·016 0·267  0·1710·0000·0960·1390·029

Elasticity analysis suggested that small changes in the transition from small- to medium-sized plants had the greatest impact on the base model estimate of λ. This transition was influenced by the proportion of plants growing from small to medium size (a), winter survival of small-sized plants (i), summer survival of medium-sized plants (d) and the proportion of medium-sized plants surviving insect foliage herbivory (g) (Table 2). Two of these parameter estimates (a, i) were from C. vulgare populations in eastern Nebraska and one (g) was an estimate from the co-occurring native C. altissimum. The second most important transition was recruitment, represented by the large-plant to small-plant transition (Table 3). This transition involved seven parameters (Table 2), of which five (n, p, r, c, f) were based on direct data from C. vulgare populations in Nebraska (Table 1) whereas two (l, q) had been estimated from the literature.

We then analysed the effect of perturbing single-component parameters using LLE. The sign of the LLE values indicates whether λ increases or decreases as a parameter increases; the larger the absolute value, the higher its influence on λ. The highest absolute LLE value was a=−0·571 (Table 4), suggesting growth from small to medium size influenced λ the most. Further, λ decreased as the proportion of plants growing from small to medium size (a) increased (1 − a is the proportion of plants growing to large size, with the highest seed production). Other parameters with high (and identical) absolute LLE values were: seed surviving floral herbivory (p); summer and winter survival of small-sized plants (c, i); seed production of large plants (n); proportion of small plants surviving insect foliage herbivory (f); and survival of post-dispersal predation (q). These parameters are all in life-history loops that determine recruitment (M → SB, M → S, L → SB, L → S), but other parameters in these loops had much smaller LLE values (0·251 and 0·085 for l and m).

Table 4.  LLE and PRCC of each parameter with λ, where λ compared by PRCC were calculated in two Monte Carlo sensitivity analyses (10 000 different parameter combinations each). The absolute values of LLE and PRCC indicate the estimated relative importance of each variable to λ (absolute values in the upper 80th percentile in bold). In PRCC1, the first two parameters (u, v) were set to 0, whereas in PRCC2 these parameters were also varied (0–1) (see text)
VariablesLLEPRCC1PRCC2
Reproductive failure of L flowering plants (mowed) (u)  –0·743
Recruitment reduction by unsuitable habitat (crops) (v)  –0·736
Seed survival of floral herbivory (p) 0·391 0·871 0·736
Germination rate (r) 0·358 0·796 0·608
Summer survival of S (c) 0·391 0·680 0·481
Winter survival of S (i) 0·391 0·646 0·455
Seed production of L (n) 0·391 0·633 0·451
Proportion of S surviving insect foliage herbivory (f) 0·391 0·491 0·329
Survival of post-dispersal predation (q) 0·391 0·401 0·258
Flowering probability of L (l) 0·251 0·324 0·166
Summer survival of L (e) 0·264 0·186 0·140
Probability to graduate from S to M (a)–0·571–0·458–0·133
Proportion of L surviving insect foliage herbivory (h) 0·264 0·145 0·106
Seed production of M as a proportion of L (o) 0·156 0·142 0·068
Seed bank germination rate (s) 0·026 0·022 0·048
Summer survival of M (d) 0·318 0·077 0·047
Flowering probability of M as proportion of L (m) 0·085 0·141 0·038
Proportion of M surviving insect foliage herbivory (g) 0·318 0·065 0·034
Winter survival of M (j) 0·161–0·003 0·021
Probability of remaining in M (from M to M) (b)–0·044 0·007–0·011
Winter survival of L (k) 0·029–0·003–0·004

Results from the PRCC1 of the Monte Carlo Latin Hypercube perturbation are shown in Table 4 and Fig. 2. Only 1% of the simulations, which encompassed 10 000 different random parameter combinations across the reported range of each parameter in the literature, predicted a population growth rate of λ= 1. In this analysis, λ was most sensitive to insect floral herbivory (PRCC1 = 0·871) but it was not sufficient to completely halt C. vulgare population growth in the model. Other parameters with high PRCC1 values (> 0·5) were: germination rate (r, PRCC1 = 0·796); summer and winter survival of small plants (c, i; PRCC1 = 0·680 and 0·646, respectively); and seed production of large plants (n, PRCC1 = 0·633). We had field-based estimates for all of the important parameters emerging from this analysis (Table 1). In contrast to LLE, PRCC analysis differentiated among parameters in the important life-history loops; for example, the PRCC values of the parameters in the transition L → S are as follows: 0·871 (p), 0·796 (r), 0·68 (c), 0·633 (n), 0·491 (f), 0·401 (q), 0·324 (l).

Figure 2.

Population growth rate (λ) as a function of seed mortality (proportion of seeds destroyed) as a result of insect floral herbivory (1 – p), germination rate (r), summer survival of small plants (c) and seed production of large plants (n). The corresponding PRCC are in the upper corner of each plot. Smooth spline fits through the cloud of points illustrate the pattern in effect of each parameter on λ (dashed lines, λ= 1).

Finally, we compared the LLE from the local analysis with PRCC1 values from the global analysis. The eight parameters with the highest PRCC1 values also had very high LLE values. However, the relationship between PRCC1 and LLE was non-linear, with LLE appearing to loose resolution at the upper end of variation. PRCC1 indicated substantial differences in impact between the most important parameters ranked by LLE. For example, germination rate had the second highest PRCC1 value but was ranked eighth by the LLE analysis. Also, the parameter with the highest LLE value was ranked seventh by PRCC1. Thus both statistical analysis and direct comparison showed that, when parameter uncertainty occurred and the effect of disturbance on multiple parameters was not independent, elasticity analysis erroneously identified parameters as highly important (false positive).

Conclusions of the PRCC analysis did not rely on the choice of parameter distribution. The PRCC values from Monte Carlo analyses using beta and normal distributions were similar to the ones assuming uniformly distributed parameter values (see Table S1, S2, and Fig. S1 in the Supplementary material). The parameter ranges for the most important parameter values (high PRCC values) were large; consequently, we explored the effect of variation in the range of parameter values on model predictions (see Figs S2–S4 in the Supplementary material). In general, the larger the parameter range, the smaller the average predicted population growth rates (see Fig. S2 in the Supplementary material). Furthermore, as expected, if the range was very small (± 0·0001% of the nominal value), the agreement with the LLE was very good (see Fig. S3a in the Supplementary material). However, there were two exceptions. Winter survival of small plants had the second highest elasticity value (LLE = 0·391) but a relatively small PRCC value (0·548); and the parameter with the highest absolute LLE value (LLE of the ‘probability of live plants graduating from S to M’ = −0·571) had only a medium-high absolute PRCC value (–0·759). These discrepancies indicated that local slopes can be dependent on the values of other model parameters (in the Monte Carlo analysis parameter values were varied simultaneously). Thus we found that the agreement between PRCC and LLE weakens with increasing range.

monte carlo analysis of variation in two regeneration parameters

In the second Monte Carlo analysis (10 000 parameter combinations), in which the additional parameters u and v varied 0–1, we found that increases in pre-reproductive mortality (u) and decreases in seed germination rates (v) could help limit population growth rate (λ = 1) over a range of parameter combinations at realistic levels of floral herbivory (p). In Fig. 3, the combinations of parameter values that restrict λ to < 1 are represented by the area above each contour line, representing a specific level of seed escape from floral herbivory (from half that observed, p/2, to three times observed, 3p). In total, 17% of the 1000 simulation runs predicted population stasis or decline (λ = 1). In this analysis, the parameters u, v and p had the highest PRCC2 values (Table 4) and the similar values suggested that they contributed equally to the limitation of C. vulgare population growth rate in the tallgrass prairie region of Nebraska.

Figure 3.

Contour plots of the predicted combined effect of three parameters, floral herbivory, reproductive failure (increased by mowing) and reduced germination (increased by weed management in croplands), hypothesized to be important in determining the asymptotic population growth rate (λ) of C. vulgare thistle populations in Nebraska, USA. The contour lines represent parameter combinations where λ= 1 for different proportions of seeds escaping floral herbivory (p), with the observed proportion (P = 0·058) as well as half (p/2 = 0·029), twice (2p = 0·116) or triple (3p = 0·174) that base value. The area above each line indicates λ < 1, whereas the area below each line indicates λ > 1.

Discussion

demographic analysis with parameter uncertainty

In theory, all of the transition rates needed to build a demographic matrix can be measured directly (16 transitions for a 4 × 4 matrix). However, in this study, as many others, we had to supplement local field data on transitions, and on processes affecting those transitions, with information from the literature. As data in the literature were from studies carried out for completely different purposes and in different locations (different countries), parameter uncertainties were large; also, we could not evaluate potential correlation among parameter values.

The reliability of local sensitivity and elasticity analyses given such parameter uncertainty is questionable. The range of possible parameter values is too large to meet the underlying assumptions, such as infinitesimally small changes in one parameter while all other parameters are held constant and independence of transitions. In fact, using methods adapted from robust control theory (transfer functions), Hodgson & Townley (2004) and Hodgson, Townley & McCarthy (2006) have demonstrated that the interpretation of sensitivity analysis can be quite misleading. One reason is that the response of λ to larger changes in parameter values is frequently non-linear. Further, even if the response of λ is linear for all parameters, there can be interplay between the uncertainties in two parameters, such that management recommendations will depend upon whether there is uncertainty in one or both parameters (Deines et al. 2007).

As an alternative, we used a numerical global analysis (Monte Carlo) to deal practically with potentially large parameter uncertainty. Such analyses have been used for estimation of confidence intervals for λ but rarely for perturbation analyses. However, adding PRCC, as a statistical evaluation of the importance of each parameter to λ in a Monte Carlo analysis, allowed strong inference on the relative contribution of each parameter, even within important life-history loops. The results in this study show that Monte Carlo sensitivity analysis with PRCC, which does not require assumptions about parametric correlation coefficients, provides a quantitative determination of the relative importance of matrix parameters to λ and of their contribution to the limitation of a biological invasion.

In the global analysis, we found that λ was most closely correlated with floral herbivory (Table 4) rather than with the transition from small to medium size suggested by elasticity analyses (Table 3). Some parameters with low PRCC values had relatively high LLE values, demonstrating that LLE had a high false-positive error rate in identifying the predominant parameter(s) influencing λ. Thus this study supports suggestions that sensitivity and elasticity analyses are ambiguous in the presence of large parameter uncertainty (Hodgson & Townley 2004), which is often the case in weed management. Monte Carlo analysis provides an effective alternative approach to evaluating matrix model predictions of the relative importance of factors influencing the asymptotic population growth rate, λ, of weed populations.

evaluation of parameter influence on λ

The Monte Carlo analysis identified floral herbivore reduction of viable seed as the most significant naturally occurring constraint on C. vulgare population growth in the region (PRCC1 = 0·871; Table 4). The estimates of parameters with the highest PRCC1 values (> 0·41) were based on field data from Nebraska populations (compare Tables 1 and 2). Further, the actual parameter uncertainty was not as large as expected, as the small contribution of parameters with high uncertainty lowered the overall uncertainty in the predicted λ.

In addition, our analysis of the base model showed that observed levels of floral herbivory were not sufficient to explain fully the low population densities of C. vulgare observed in this region. If the eight parameter estimates from the literature (plus two from studies of the co-occurring native relative) were sufficient to model C. vulgare in Nebraska, seed loss to floral herbivores would have had to average 98·2% for λ= 1, more than that recorded (71–96·5%) (Louda & Rand 2002; S. Louda, unpublished data). Our model is deterministic and ignores temporal and spatial variation in parameter values, and deterministic models tend to overestimate population performance (Morris & Doak 2002). However, Doak, Gross & Morris (2005) found that, in the absence of ‘good’ data, deterministic models can provide better predictions than stochastic ones. One hypothesis to explain this discrepancy is that our initial matrix model missed at least one mechanism critical for explaining the observed population stasis. This led us to explore the potential interaction of variation in two regeneration parameters affected by weed management practices with the floral herbivory.

monte carlo analysis of variation in two regeneration parameters

In a second Monte Carlo analysis, we evaluated the conditions under which reductions in two regeneration parameters targeted by weed management practices (mowing, cropland weed management) could result in thistle population control. We found that increased proportions of large plants dying before setting seed and seeds lost prior to germination, in combination with floral herbivory, predicted λ≤ 1 for a wide range of values (Fig. 3). These results suggest that weed management practices that limit seed production and seedling establishment, added to the extensive floral herbivory, contribute to halting C. vulgare population growth in this region. Thus understanding the low C. vulgare abundance in eastern Nebraska, or the invasive potential of exotic plants in general, requires evaluation of the interaction of land management practices with other factors, such as biotic resistance. An important next step would be a direct test of the effect of land management practices on λ, and quantification of spatial patterns leading to the development of a spatially explicit model.

conclusion

Ideally, studies of plant demography should include all necessary life-history parameters. However, limited time and funding, as well as logistical constraints, often lead to data sets that fall short of ideal. As a consequence, parameter uncertainty and interdependencies are common, particularly for invasive plants. Monte Carlo sensitivity analysis, along with PRCC, provides a robust numerical alternative to elasticity analysis in these cases. We found that the interactions between floral herbivory and two parameters that can be influenced by local weed management practices (survival to flowering and successful seed germination) predict population stasis across a wide range of parameter values, and probably explain the limited population growth and invasiveness of C. vulgare in Nebraska. This finding is consistent with the suggestion that multiple, interacting factors are generally needed for effective weed control (Shea, Thrall & Burdon 2000; Grigulis et al. 2001; Huwer et al. 2005).

Acknowledgements

We are indebted to the colleagues and students who have discussed these issues with us or contributed to the data collection. We also thank Andrew J. Tyre and Joy Kogut for insightful suggestions to improve the manuscript. Partial funding for work on thistle–insect interactions in Nebraska was provided by grants from The NSF (DEB9616210, DEB0414777), The Nature Conservancy and the USDA National Research Initiative Program 2001-35320-09882; 2004–02252 to S. M. Louda, and the University of Nebraska Research Council Layman Funds to B. Tenhumberg.

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