## Introduction

Projection of demographic data has provided a tool by which the dynamics of populations can be explored in detail (Piñero *et al*. 1984; van Groenendael & Slim 1988; Svensson *et al*. 1993; Horvitz & Schemske 1995; Caswell 1997). The benefits include allowing the estimation of extinction probabilities, the prediction of the invasiveness of both native and non-native species, and the study of life-history evolution. Sensitivity (Caswell 1978, 1989) and elasticity analyses (de Kroon *et al*. 1986) allow the contribution of different demographic processes to be assessed, and this has improved our understanding of population dynamics as well as allowing comparisons to be made across taxa and populations (Silvertown *et al*. 1993, 1996; Franco & Silvertown 1996).

Projections of population growth must consider the temporal variation of demographic parameters (Tuljapurkar 1985; Nakaoka 1996). For long-lived perennials, the ways in which the effects of such variation over their lifetime influence their dynamics are not readily approximated by short-term demographic analyses (Cain & Damman 1997; Damman & Cain 1998). Environmental heterogeneity may interact with a sequence of intrinsic biological processes that only take place over long periods of time, and it is therefore necessary to assess population dynamics of perennial species over long time periods. This is especially true for species inhabiting extreme environmental conditions where specific demographic processes, such as growth and reproduction, may be limited to occasional ‘opportunity windows’(*sensu*Eriksson 1989). Several models have accounted for this variation by including a stochastic environmental component (Tuljapurkar 1985, 1989; Huenneke & Marks 1987; van Groenendael & Slim 1988; Nakaoka 1996; Damman & Cain 1998). We used periodic matrix models (a subset of stochastic models in which the assumption of time-invariance is relaxed; Skellam 1966) and their associated sensitivity and elasticity analyses (Caswell 1989; Caswell & Trevisan 1994). We assumed that the matrices reflect the range of environments encountered and that population growth can be described in a hypothetical habitat that cycles among the environments observed (Caswell & Trevisan 1994). We used a demographic study of *Prosopis glandulosa* var. *torreyana* over a period of 4 years to determine (i) how elasticity analysis of periodic matrix models provides information on the impact of different demographic processes at different stages of the environmental cycle; and (ii) how the use of this method in conjunction with annual and mean matrix models improves our understanding of the natural dynamics of populations.