## Introduction

Matrix population models, where individuals are divided into distinct classes based on their size, age or life history stage, are widely used in plant demographic studies to assess population performance. These models allow the long-term population growth rate (λ) and other population parameters to be estimated from individual-level data on survival, growth and fecundity (Caswell 2001). In addition to the traditional use of matrix models for population viability analyses (Menges 2000), matrix models have been used to analyse complex population dynamics and species interactions (e.g. Smith, Caswell & Mettler-Cherry 2005; Ramula, Toivonen & Mutikainen 2007), to produce harvest and restoration recommendations (e.g. Freckleton *et al.* 2003; Linares, Coma & Zabala 2008), and to assess alternative management strategies for invasive plant species (reviewed in Ramula *et al.* 2008).

Despite the popularity of matrix models, they have some limitations, which may decrease the accuracy and precision of estimated population parameters. First, all individuals within the same class are assumed to have identical demographic rates (Caswell 2001). This simplification makes matrix models sensitive to the selected matrix structure and therefore, different matrix structures may produce divergent estimates of λ (Ramula & Lehtilä 2005). The second limitation of matrix models is the large number of individuals required in each class to estimate demographic rates accurately. Small sample sizes often lead to the pooling of individuals with very different demographic rates within classes.

An alternative to matrix models is the integral projection model (IPM), which has many properties similar to matrix models, and can be parameterized from the same data using regression equations (Easterling, Ellner & Dixon 2000). For IPMs, demographic rates are modelled as a continuous function of an individual’s state rather than dividing individuals into distinct classes. IPMs were introduced to plant ecology a decade ago (Easterling *et al.* 2000) and their use is increasing rapidly (e.g. Rees & Rose 2002; Childs *et al.* 2004; Rose, Louda & Rees 2005; Williams & Crone 2006; Hesse, Rees & Müller-Schärer 2008; Kuss *et al.* 2008).

Using a large (*n* > 600 individuals) data set Easterling *et al.* (2000) found that matrix models and IPMs produced identical estimates of λ for a perennial herb, and recent applications of IPMs are usually based on data sets with hundreds of individuals (Childs *et al.* 2004; Ellner & Rees 2006, 2007; Hesse *et al.* 2008; Kuss *et al.* 2008). For endangered or invasive plant species, such large data sets are often lacking and population dynamics and management strategies must be assessed based on the available small data sets (Menges 2000; Simberloff 2003; Buckley *et al.* 2005). Hence, there is a need for models which reliably predict population dynamics from small demographic data sets, making the application of IPMs of great interest. We might generally expect IPMs to be more reliable than matrix models because they require fewer parameters to be estimated and these are estimated from the complete data set, rather than by dividing the data into classes. However, the magnitude of this effect has not been quantified, and we currently have no evidence that IPMs are more suitable for small data sets than matrix population models.

Many factors affect the accuracy and precision of λ estimates, including distance from the stable stage distribution, the variability of demographic rates and sample size (Caswell 2001). Small sample size may produce inaccurate estimates of λ because of large sampling error (Fiske, Bruna & Bolker 2008). One possibility to minimize sampling error for matrix model estimates is to focus the greatest sampling effort on the life stage(s) to which λ is most sensitive (Gross 2002). If no *a priori* knowledge of the importance of different demographic transitions to λ is available, the best accuracy for matrix models is achieved by sampling an equal number of individuals for all matrix classes (Münzbergová & Ehrlén 2005). In addition to model accuracy, the precision of the model is important. A model that produces precise but biased estimates is still useful if the magnitude of the bias is known and can therefore be corrected.

We explore the accuracy and precision of matrix population models and IPMs in relation to the amount of demographic data using two perennial herbs with different life histories (*Cirsium palustre* and *Primula veris*). We also compare two different techniques to parameterize an IPM, first using a constant regression model structure derived from the full data set and second, allowing the regression model structure to vary according to the data set at hand, which is sub-sampled from the full data set. We start by constructing a matrix population model and an IPM from the full data sets. We then reduce the number of individuals by sub-sampling the full data sets with and without replacement, and compare the accuracy and precision of λ in relation to full data sets. For the matrix models we use two alternative matrix structures and two different sampling techniques, a random sampling from the observed stage distribution and an equal sampling for all matrix classes. We concentrate on λ because it is commonly used to quantify population performance.