#### Study species and area

*Artemisia ordosica* Krasch (Asteraceae) is a shrub with plumose, linearly lobate leaves (Kobayashi, Liao & Li 1995). Its root system is mainly distributed in the upper 30 cm of the sand, while its main roots may reach 1–3 m deep (Li *et al.* 2010b). This species is overwhelmingly dominant in the Mu Us Sandland on semi-fixed dunes, fixed dunes and fixed dunes covered with microbiotic crust. Recruitment is generally realized by reproduction from seed (Huang & Gutterman 2000), although plants may occasionally split into clonal fragments (Schenk 1999). Plants start reproducing at the age of 2–3 years. Seeds set in August and mature in October. Reproductive shoots die in winter, while woody vegetative shoots survive the winter and generate new vegetative or reproductive shoots in the next spring (Li *et al.* 2010b).

This study was conducted at Ordos Sandland Ecological Station (OSES, 39°29’37.6’’ N, 110°11’29.4’’ E) of Institute of Botany of Chinese Academy of Sciences, located in the north-eastern Mu Us Sandland in Inner Mongolia, China. Mu Us Sandland is a semi-arid area of 39 800 km^{2}, with a mean annual precipitation of 260–450 mm, which is mainly concentrated in summer (Zhang 1994). Annual rainfall during our two census periods was quite similar, but the rainfall during the main growing season in the first period was higher by 30% than that in the second period (data from KNMI Climate Explorer, http://climexp.knmi.nl/get_index.cgi). Mean annual temperature is 7.5–9.0 °C, with a maximum of 20–24 °C in July and a minimum of −8 to −12 °C in January (Zhang 1994). Most plant species in this area are forbs and grasses, although woody plants may dominate vegetations locally. The main shrub species are *A. ordosica*, *Hedysarum laeve*, *Salix psammophila* and *Sabina vulgaris* (Li *et al.* 2010a). These species comprise a large proportion of the total vegetation coverage and fix the sand dunes to different extents (Bai *et al.* 2008). Microbiotic crust is commonly found on the surface of well-fixed dunes in this study area.

#### Study design and data collection

Three dune fixation stages, i.e. semi-fixed dunes (SF), fixed dunes (F) and fixed dunes covered with microbiotic crust (FC), were selected. These habitats were located at 2–4 km distance from each other. The total vegetation coverage was 40.1 ± 1.4% (mean ± SE), 62.04 ± 1.0% and 59.26 ± 1.0% in SF, F and FC habitat, respectively. In July 2007, three permanent plots were established in each habitat. The distance between plots within habitats ranged from 500 to 1000 m. Plots measured 20 × 20 m in SF and F habitats and 40 × 40 m in FC habitat, where seedling density was quite low. All plots were then divided into 4 × 4 m subplots. Plants shorter than 10 cm in SF and F habitats and plants taller than 20 cm in F and FC habitats were measured in 4, 4, 10 and 9 randomly selected subplots, respectively. Plants of other sizes were measured in the entire plots. Sub-sampling was done to gain an even distribution of all plant sizes in each plot. In total, data from 6939 individuals were collected during three censuses, one each in July of 2007, 2008 and 2009.

At the first census, total height was measured for each individual. For individuals taller than 20 cm, the two largest perpendicular diameters of the plant’s crown were also measured. Reproductive status was recorded for each individual. The number of inflorescences was assessed into five categories, with the average amounting to 5, 15, 35, 75 and 150 inflorescences, respectively. Upon first measurement, each plant was labelled and its coordinates within the plot were recorded. In 2008 and 2009 the survival of all the labelled plants was checked, and the surviving plants were re-measured, reproductive status was recorded and new recruits were searched and measured.

#### Statistical analyses

We performed preliminary statistical analyses to compare the suitability of three measures of size for our models: height, crown area and crown volume. As plant height explained more variation in vital rates (survival, growth and reproduction) than the other two measures, we used height to characterize plant size in all analyses and models.

We tested for differences in vital rates among the three plots per habitat for each period using ancova for height growth and logistic regressions for survival and flowering probabilities (height as covariable in both cases). As we found no differences in plant growth among plots (*P* > 0.05), and just two and five cases of significant differences (*P* < 0.05) in flowering and survival among plots, respectively (during second census period), we decided to pool the data from all plots within each habitat for subsequent analyses.

We also tested for temporal differences in vital rates in each habitat, using paired *t*-tests for growth, and logistic regressions for survival and flowering probabilities. Differences between periods were found for all vital rates (paired *t*-tests and logistic regressions, *P* < 0.01), except for growth in fixed dunes. We therefore fitted statistical models for both periods separately.

We used multiple regression models to relate future height (*y*, at *t* + 1), survival and reproduction to the current height (*x*, at *t*) of an individual and to habitat (coded as dummy values with SF habitat as control). The probabilities of survival *s*(*x*) and flowering *p*_{f}(*x*) were modelled as logistic regressions, with height (*x*) and habitats as independent variables. Size change (*g*(*y*, *x*)) was tested using multiple linear regression models with future height (*y*) as independent variable against current height (*x*) and habitat. Subsequently, variances of that regression were again related to *x* and habitats in another multiple linear regression. The number of inflorescences *f*_{n}(*x*) was related to *x* and habitats in a multinomial logistic regression, as it was recorded as categories (Poorter *et al.* 2005).

#### Integral projection models

We analysed the population dynamics of *A. ordosica* using IPMs, which describe how a continuously size-structured population changes in discrete time (Easterling, Ellner & Dixon 2000). In IPMs, the state of the population at time *t* is described by a distribution function *n*(*x*, *t*) and *n*(*x*, *t*)d*x* represents the number of individuals with size in the range [*x*, *x* + d*x*]. The population dynamics is then written as:

- ((eqn 1))

where [*L*, *U*] is the range of all possible sizes, *p*(*y*, *x*) represents survival and growth from size *x* to size *y* and reproduction *f*(*y*, *x*) represents the number of new births of size *y* at *t* + 1 produced by an adult of size *x* at *t*. *p*(*y*, *x*) was calculated as *p*(*y*, *x*) = *s*(*x*)*g*(*y*, *x*) and *f*(*y*, *x*) as *f*(*y*, *x*) = *p*_{f}(*x*)*f*_{n}(*x*)*p*_{e}*f*_{d}(*y*), where *p*_{e} is the mean number of seedlings produced per inflorescence, and *f*_{d}(*y*) is the size distribution of seedlings. *p*(*y*, *x*) + *f*(*y*, *x*) is called the kernel, *k*(*y*, *x*), a non-negative surface representing all possible transitions from size *x* to size *y*.

The kernel *k*(*y*, *x*) can be transformed into a large transition matrix **K**(*y*, *x*) with *w* categories, using the midpoint rule (Easterling, Ellner & Dixon 2000). The dynamics of the population can then be described as in a classical matrix model: **n**(*t* + 1) = **K** **n**(*t*), that can yield the same output as matrix models: population growth rate (*λ*), sensitivity and elasticity (Ellner & Rees 2006). We used 100 mesh points, because *λ* values in all populations hardly changed any more when further increasing the number of mesh points. Because we found significant differences between periods and habitats, we constructed IPMs for each census period and habitat, resulting in six IPMs.

Confidence intervals for *λ* were calculated by bootstrapping (Jongejans *et al.* 2010). For each bootstrap estimate, we resampled with replacement from the data sets (*n* = 2487, 2438 and 2014 in SF, F and FC habitats, respectively), re-calculated regression coefficients, established the kernel and calculated *λ*. This was repeated 5000 times and the 95% confidence intervals for *λ* were obtained from the frequency distribution of these values.

To examine how far the observed size distribution was from the expected, stable stage structures resulting from IPMs were compared to observed population structures (mean of three plots for each habitat), using the percentage similarity index (PS; Horvitz & Schemske1995): PS = Σ(min[obs_{i}, ssd_{i}]) × 100, where obs_{i} and ssd_{i} are vectors of observed population structures and stable size distributions, respectively (both vectors scaled to sum to 1). High values of this index indicate a high level of similarity (Zuidema, de Kroon & Werger 2007).

To examine the relative importance of transition elements and vital rates to population growth *λ*, we conducted elasticity analyses (de Kroon, van Groenendael & Ehrlen 2000). We first performed elasticity analysis of matrix elements, in which survival is contained in growth and shrinkage transitions (de Kroon, van Groenendael & Ehrlen 2000; Caswell 2001). Then, to specifically examine the importance of each vital rate (survival, growth, shrinkage and reproduction), we performed vital rate elasticity analyses (Zuidema & Franco 2001). Sensitivities of vital rates in each size category can be calculated as: Survival:

- ((eqn 2))

Positive growth (for *i* > *j*):

- ((eqn 3))

Negative growth (for *i* < *j*):

- ((eqn 4))

Fecundity:

- ((eqn 5))

where *σ*_{j}, *γ*_{ij}, *ρ*_{ij} and *f*_{ij} represent the vital rates of survival, positive growth, negative growth and fecundity, respectively, while *P*_{j}, *G*_{ij}, *R*_{ij} and *F*_{ij} represent transition elements for stasis, progression, retrogression and fecundity, respectively. Note that in eqns 3 and 4, the summation over *i* is done to take into account growth from category *j* to all categories *i* that represent positive growth (*i* > *j*) or negative growth (*i* < *j*). In eqn 5, the summation over *i* is done to include new seedlings of all sizes. Note also that eqns 3 and 4 allow negative values for sensitivity in the case that the element sensitivity of stasis (*P*_{j}) is larger than that of growth (*G*_{ij} for eqn 3) or shrinkage (*R*_{ij} for eqn 4, Zuidema & Franco 2001). Elasticities of vital rates can then be calculated as:

- ((eqn 6))

where *x*_{ij} is the value of vital rate under consideration (*σ*_{j}, γ_{ij}, *ρ*_{ij} and *f*_{ij})_{.}

Differences in population growth rates among habitats or periods may be caused by variation in vital rates across habitats and periods. Analysis of life table response experiment (LTRE) allows quantification of the contribution of each element or vital rate to the observed difference in population growth rate (Caswell 2001; Jongejans & de Kroon 2005). We conducted a fixed-design LTRE on vital rates to evaluate the contribution of variation in vital rates to differences in population growth rate (Yamada *et al.* 2007). The two-factor LTRE analysis is as follows:

- ((eqn 7))

where *λ* of habitat *m* and period *n* is calculated as the sum of *λ* for the overall mean matrix, *λ*^{(··)}, the effect of habitat *m*, *α*^{(m)}, the effect of period *n*, *β*^{(n)}, and the residual ‘interaction’ effect (*αβ*)^{(mn)}. First, the main effects were estimated separately, while ignoring the interaction term (Caswell 2001; Jongejans & de Kroon 2005):

- ((eqn 8))

- ((eqn 9))

where differences between the value of a vital rate *a*_{ij}^{(m.)} of the mean-habitat matrix *K*^{(m.)} or *a*_{ij}^{(.n)} of the mean-period matrix *K*^{(.n)} and the overall mean vital rate *a*_{ij}^{(..)} of matrix *K*^{(··)} are multiplied by the sensitivity values of the matrix halfway between the matrix of interest and the overall mean matrix (Yamada *et al.* 2007). The interaction effect (*αβ*)^{(mn)} is then calculated as (Jongejans & de Kroon 2005):

- ((eqn 10))