#### study system

The nodding or musk thistle *C. nutans* is of Eurasian origin but in the last centuries has invaded the Americas, southern Africa, Australia and New Zealand, where it is a noxious weed in pastures and roadsides (Kelly & Popay 1985; Popay & Medd 1995). It is a monocarpic perennial plant and does not reproduce clonally. Bolting plants grow stems up to 2 m tall, on which capitula are formed through indeterminate growth. The wind-dispersed seeds can persist for at least a decade when buried in the soil (Burnside *et al*. 1981) but only a few years when soils are disturbed (Roberts & Chancellor 1979). Seedlings require open microsites for establishment and are sensitive to summer droughts (Medd & Lovett 1978; Wardle, Nicholson & Rahman 1992).

In order to model the population dynamics in the native range of this species, we used demographic data from Sheppard *et al*. (1989), Sheppard, Cullen & Aeschlimann (1994) and A. W. Sheppard *et al.* (unpublished data) from the Massif Central, southern France. This region was selected because its climatological conditions resemble those of the region of Australia where *C. nutans* is a major economic pest. Two of the three populations studied (populations 1 and 3) were in relatively infertile limestone depressions, while population 2 was in a more fertile and more actively managed pasture (Sheppard, Cullen & Aeschlimann 1994). The populations were well established at the beginning of the study (several thousands plants at each site; Table 1). All sites were used for grazing by sheep, and specialist herbivores were present at each site.

Table 1. Description of the three *Carduus nutans* populations near La Cavalerie (Département Aveyron, France; Sheppard *et al*. 1995). The predispersal seed loss data are from Sheppard, Cullen & Aeschlimann (1994) | *Carduus nutans* population |
---|

1 | 2 | 3 |
---|

Soil fertility | Low | High | Low |

Population area | 1200 m^{2} | 960 m^{2} | 900 m^{2} |

Quadrat size | 1 × 0·5 m^{2} | 1 × 1 m^{2} | 1 × 1 m^{2} |

Number of quadrats | 20 | 35 | 20 |

Density of plants (plants m^{−2}) | 3·1 | 3·3 | 21·0 |

First census | 2 May 1986 | 19 June 1986 | 22 October 1987 |

Last census | 4 May 1989 | 6 October 1987 | 4 May 1989 |

Number of censuses | 39 | 26 | 14 |

Predispersal seed loss to specialist herbivores (%) | 92 | 84 | 91 |

#### matrix construction

To study the dynamics of the French populations and the relative effects of specialist herbivores, generalist herbivory and microsite limitation, we re-analysed the demographic data and constructed population projection matrices. Transition matrices, which contain the probabilities of survival, growth and contributions via reproduction from every discrete class of individual plants to each class in the next time step, are regularly used to analyse population dynamics (Caswell 2001). We used the year-to-year transition matrix structure of Shea & Kelly (1998) for *C. nutans*, recognizing four stage classes: SB, seeds in the seed bank; S, small plants; M, medium-sized plants; L, large plants. Size was based on the maximal rosette area in a year, which was estimated by measurements of the widest rosette diameter, *d*_{1}, and its perpendicular rosette diameter, *d*_{2}. Rosette area, *a*, was calculated with the following formula:

- (eqn 1)

The size class boundaries between the S and the M classes, and between M and L, were based on the bolting probabilities of the plants in these classes. Plants with 20% or less probability of bolting the next year were placed in the S category; plants with 80% or more probability of bolting were assigned to the L category; and the remainder were assigned to the M class (Shea & Kelly 1998; Shea *et al*. 2005). To determine the rosette area values of the class boundaries, we aggregated all data from the three populations and fitted the bolting probability, *p*(*b*), with a binary logistic regression model with all plants that survive until the next year (*n* = 630, *P* < 0·001):

- (eqn 2)

The boundary rosette areas that match *p*(*b*)-values of 0·2 and 0·8 were 26·3 cm^{2} and 60·7 cm^{2}, respectively.

Survival, growth, bolting, flowering and seed production were studied in each population in at least 20 randomly placed permanent 1-m^{2} quadrats (Table 1; Sheppard *et al*. 1989; Sheppard, Cullen & Aeschlimann 1994). In population 1, all plants in these quadrats were monitored every 6 weeks, and fortnightly during the flowering season, from 1986 to 1989 (i.e. 3 year-to-year transitions) (Sheppard *et al*. 1989). Population 2 was monitored from 1986 to 1987 (one transition) until the site owner mowed the population (Woodburn & Sheppard 1996). Population 3 was studied from 1987 to 1989 (two transitions; A.W. Sheppard *et al*. unpublished data). For each population, demographic data from different years were lumped to increase sample sizes. For each size class in each population, we sequentially calculated the following vital rates: σ_{j}, the probability that a rosette or seed survived till the next year; β_{j}, the probability that a surviving rosette bolted; γ_{ij}, the probabilities that a surviving rosette or seedling (γ_{i}) grew into the different size classes. Bolting plants flowered and almost invariably died afterwards, whether they succeeded in flowering and seed production or not. Seed production was estimated by measuring the diameter of all flower heads that opened on every bolted plant. We distinguished four size classes of flower heads (diameter < 13, 13–20, 21–30 and > 30 mm). Average seed production without insect herbivores, π_{j} (respectively 56, 172, 439 and 701 viable seeds for the four head size classes), was determined by spraying 57 heads on additional plants at site 2 with insecticides (Sheppard, Cullen & Aeschlimann 1994). These potential seed numbers were then compared with the seed production of unsprayed heads of similar flower head sizes in all three sites to determine the site-specific proportion of potential seeds surviving insect floral herbivory, ϕ.

A seed-addition experiment was conducted at sites 1 and 3 in December 1987 to estimate seedling establishment. In a factorial block design, 8 of 16 pasture plots of 0·125 m^{2} were each sown with 625 viable seeds mixed in a fine sand mixture. The establishment probability per seed, ɛ, was calculated as the number of seedlings that recruited during 1988 in the seeded plots (corrected for background germinations by subtracting the seedling numbers of unseeded plots) divided by the number of added seeds, and was on average 0·09 (SD = 0·06) and 0·11 (SD = 0·04) for sites 1 and 3, respectively. We assumed that seedling establishment was comparable (0·10) in population 2.

Seed survival in the soil was estimated from the same experiment using two 3·2-cm diameter × 10-cm deep soil cores per plot. The survival of new seeds was determined by counting the number of viable seeds in cores from seeded plots in July 1988 and October 1988. The proportion of the seeds that survived over this period was on average 0·32. This value was used in the matrix models as the probability that new seeds are incorporated into the seed bank (i.e. not germinating or dying), ν. As no data were available on the year-to-year survival in the seed bank, σ_{1}, by seeds older than 1 year, we used the proportion of seeds that were still viable 1 year later (July 1989) in the seed-addition experiment as an estimate. This value was on average 0·26. In summary, each vital rate was represented by a symbol (Fig. 1 lists all vital rates and symbols) and groups of vital rates comprised the elements of the 4 × 4 (SB, S, M and L) annual transition matrix:

- (eqn 3)

Each of the 16 matrix elements *a*_{ij} represented the average contribution (through survival and growth or through reproduction) of an individual in that stage class in year *t* (column *j*) to the number of individuals in a stage class in the next (*t* + 1) year (row *i*).

#### matrix analysis

We calculated the projected population growth rates (λ, the dominant eigenvalue) for the three constructed 4 × 4 matrices. A λ-value below unity means that a population will decline in size if the vital rates stay constant through time. To estimate the uncertainty (95% confidence intervals) of each λ-value, we resampled all underlying data sets 3000 times using the bootstrap method (Efron 1982; Kalisz & McPeek 1992). In order to predict the impact of different biotic and abiotic factors, such as floral herbivory and drought, on λ, we quantified the importance of the life-history components that were affected by these factors. First, we evaluated how much each of the 16 matrix elements contributed to λ by calculating the elasticity (*e*_{ij}, the relative sensitivity) of λ to each element, i.e. the proportional change in λ in response to small, proportional perturbations of each element separately (de Kroon *et al*. 1986):

- (eqn 4)

where *a*_{ij} is the matrix element of the *i*th row and *j*th column. As the elasticity values of all elements in a matrix sum to 1, these elasticity values can be directly compared among different matrices (e.g. for different populations) to investigate whether populations differ in which transitions are most important for population growth.

All nine (out of 16) transitions from a rosette class (S, M or L) to the same or another rosette class theoretically consist of both a survival and a reproduction component. The transition S–S, for instance, contained both the survival of small plants that stayed small till the next year and the contribution of small plants that flowered and produced new small plants the next year. To be able to study the importance of these separate vital rates (i.e. life-cycle components), we also calculated the elasticity values of all vital rates (i.e. the proportional response of λ to small, proportional perturbations of each vital rate; Caswell 2001; Franco & Silvertown 2004):

- (eqn 5)

#### scenario analysis

We analysed the effect of three different factors (floral herbivory by insects, rosette damage by sheep and drought stress for seed germination) that may control the dynamics of native range *C. nutans* populations. We estimated these effects using the increase in population growth rate when one or more of the factors were omitted from the model. Sheppard *et al*. (1989) compared the survival of rosettes that were damaged by sheep with the survival of undamaged plants in these populations during the summer of 1986. Mortality of undamaged plants was about one-third lower than the mortality of damaged plants. Our first scenario was therefore to simulate the absence of sheep damage by reducing mortality (1 − σ_{j}) of S, M and L by one-third to investigate the impact of the damage by sheep on λ. Twenty-seven per cent of the observed plants had no signs of sheep damage. In the ‘no sheep damage’ scenario, the impact of sheep may therefore have been slightly overestimated as the scenario simulates the potential effect in the case when all plants are damaged.

The second factor we examined was seedling recruitment, which differed strongly between years (Sheppard *et al*. 1989). To investigate whether deviations from the best possible environmental conditions for recruitment limited the population growth, we fixed the recruitment rates (ɛ) at 0·2, which is approximately the observed maximum. This level was about twice the average of the observed recruitment rate in the sowed plots and similar to recruitment rates found in invasive populations (Woodburn & Sheppard 1996; Shea & Kelly 1998). In the model, this increased seed germination was compensated for by reducing both the proportion of seeds remaining (σ_{1}) and entering (ν) the seed bank (as they now germinated immediately).

For the third factor, we simulated population dynamics with no predispersal seed losses to insect floral herbivory (i.e. ϕ = 1). Nodding thistle capitula are attacked by a range of specialist insects including tephritid flies and weevils. The fly *Tephritis hyoscyami* L. is the first of these, attacking the early bud stage. Then the receptacle weevil *Rhinocyllus conicus* Frölich (a specialist frequently occurring on European *C. nutans* plants; Zwölfer & Preiss 1983; Zwölfer & Harris 1984) and gall fly *Urophora solstitialis* L. attack the flower heads later on in the bud stage, utilizing the flow of resources in the receptacle of the developing flower head. Other receptacle weevils, *Larinus jaceae* F. and *Larinus sturnus* Schaller, then oviposit on or in the bracts of the developing flower heads, where the larvae feed on the receptacle. The seed losses as a result of each particular insect species were variable in our populations, but the sum of the seed losses caused by all insects that fed on the flower heads was stable over the sites and years (90·2% on average, *n*= 6, SE = 2·7; Sheppard, Cullen & Aeschlimann 1994).

Finally, we modelled four additional scenarios in which we combined two or all three of the above scenarios (no damage by sheep, best establishment and no seed losses to floral herbivory). For each scenario we again bootstrapped the underlying databases to estimate 95% confidence intervals.