## Introduction

Matrix models are a popular tool for demographic analysis of long-lived plant species. So far, these models have been constructed for > 100 tree and shrub species (reviews Franco & Silvertown 2004; R. Salguero-Gómez, unpublished data). The parameterization and construction of these models for long-lived species is very different from that for short-lived species. Transition values in matrices of short-lived species are typically based on observed transitions among categories. But for long-lived and slow-growing species – most trees, large palms and large shrubs – the annual change in size is (very) small relative to the maximum size and observing transitions among matrix categories is impossible. Thus, transitions in matrix models for trees and shrubs are typically calculated from growth rates of individuals or passage time through categories (e.g. Bernal 1998; Zuidema & Boot 2002; Chien, Zuidema & Nghia 2008).

There are two limitations of classical matrix models that have important implications for the output of models constructed for long-lived species. First, in their standard form matrix models offer limited possibilities to incorporate variation among individuals within a size category. Growth variation can be incorporated to distinguish between groups of individuals with fast growth, slow growth and those that shrink in size (e.g. Horvitz & Schemske 1995; Salguero-Gómez & Casper 2010), but this does not include variation between *all* individuals. Growth variation among individuals may strongly influence the dynamics and growth rate of populations (Pfister & Stevens 2003), especially for long-lived species (Wisdom, Mills & Doak 2000; Zuidema & Franco 2001; Zuidema *et al.* 2009). Second, the output (population growth rate λ, elasticities) of matrix models is highly sensitive to variation in the number of categories (Enright, Franco & Silvertown 1995, Ramula & Lehtilä 2005). In models for trees, it is customary to define rather wide categories, e.g. of 10 cm diameter from which typically 1–5% of the individuals progress to the next size category every year (e.g. Bernal 1998; Zuidema & Boot 2002; Chien, Zuidema & Nghia 2008). As the transition probabilities in a matrix model depend only on the current situation, there is no obstruction for unrealistically fast pathways through the life cycle. For instance, in matrix models with 10-cm-wide diameter categories and small progression probabilities, a small fraction may reach 50 cm diameter in five time steps, something that is clearly impossible biologically (and physically). This fraction contributes strongly to population growth and probably causes the high estimates of λ for small matrix models (Zuidema 2000; Ramula & Lehtil 2005). In a model with narrow categories such ‘leaps’ through the life cycle are impossible and such models produce lower values of λ, different elasticity patterns and higher age estimates (Zuidema 2000).

Integral Projection Models (IPMs) provide solutions for both above-mentioned problems. IPMs are extensions of matrix models that yield similar output (population growth, sensitivity, elasticity, age estimates), but use continuous relations of vital rates (growth, survival, reproduction) versus size (or age) as input, instead of category-specific values (Easterling, Ellner & Dixon 2000; Ellner & Rees 2006). The above-mentioned limitations do not apply to IPMs, as these models explicitly incorporate variation among individuals in growth rates (and any other vital rate), typically use large numbers of categories (or *mesh points*) and allow for transitions of individuals to any other size (be it larger or smaller). So far, IPMs have hardly been used for long-lived, slow-growing species (Metcalf *et al.* 2009), in spite of their potential to be used for these species groups.

Here, we propose and apply a new method to parameterize IPMs for trees. Specifically, we evaluated the sensitivity of λ, elasticity and ages to changes in matrix dimension (number of categories) of IPMs and validated the age estimates obtained from the models with tree ring data. In the following, we first briefly review approaches to parameterization of matrix models for trees and introduce a new way to parameterize IPMs for long-lived, slow-growing species. Next, we construct and analyse IPMs for six tree species from Vietnam, for which demographic analyses and matrix models have been published (Chien 2006; Chien, Zuidema & Nghia 2008). Finally, we discuss the suitability of IPMs for modelling tree population dynamics.