Theoretical Population Biology ( ) – Contents lists available at SciVerse ScienceDirect Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb Effects of demographic structure on key properties of stochastic density-independent population dynamics Yngvild Vindenes a,b,∗ , Bernt-Erik Sæther b , Steinar Engen c a Centre for Ecological and Evolutionary Synthesis (CEES), Department of Biology, University of Oslo, Norway b Centre for Conservation Biology (CCB), Department of Biology, Norwegian University of Science and Technology, Trondheim, Norway c Centre for Conservation Biology (CCB), Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway article info Article history: Available online xxxx Keywords: Reproductive value Dynamic heterogeneity Permanent heterogeneity Diffusion approximation Matrix model Integral projection model abstract The development of stochastic demography has largely been based on age structured populations, although other types of demographic structure, especially permanent and dynamic heterogeneity, are likely common in natural populations. The combination of stochasticity and demographic structure is a challenge for analyses of population dynamics and extinction risk, because the population structure will fluctuate around the stable structure and the population size shows transient fluctuations. However, by using a diffusion approximation for the total reproductive value, density-independent dynamics of structured populations can be described with only three population parameters: the expected population growth rate, the environmental variance and the demographic variance. These parameters depend on population structure via the state-specific vital rates and transition rates. Once they are found, the diffusion approximation represents a substantial reduction in model complexity. Here, we review and compare the key population parameters across a wide range of demographic structure, from the case of no structure to the most general case of dynamic heterogeneity, and for both discrete and continuous types. We focus on the demographic variance, but also show how environmental stochasticity can be included. This study brings together results from recent models, each considering a specific type of population structure, and places them in a general framework for structured populations. Comparison across different types of demographic structure reveals that the reproductive value is an essential concept for understanding how population structure affects stochastic dynamics and extinction risk. © 2011 Elsevier Inc. All rights reserved. 1. Introduction Natural populations can show complex dynamics, due to individual differences in survival and reproduction that arise from temporal and spatial variation in environmental conditions and/or genetic differences that affect expressed phenotypes (Coulson et al., 2001; Clutton-Brock and Coulson, 2002; Benton et al., 2006; Clutton-Brock and Sheldon, 2010). Yet the demography, growth and evolutionary changes of populations are often studied using deterministic models that assume a constant environment and no random variation between individuals. Such models have provided many results that are key to understanding properties of population growth and life history evolution (Lande, 1982; Charlesworth, 1994; Caswell, 2001). Nevertheless, deterministic models miss important aspects of the dynamics of natural ∗ Corresponding author. E-mail addresses: yngvild.vindenes@bio.uio.no (Y. Vindenes), Bernt-Erik.Sather@bio.ntnu.no (B.-E. Sæther), steinaen@math.ntnu.no (S. Engen). populations because they do not include stochastic variability of the vital rates. For instance, stochasticity tends to reduce the long-term population growth rate (Tuljapurkar, 1990; Lande et al., 2003), and results from deterministic sensitivity analyses of the relative effect of different vital rates on population growth may not hold when variability is included (Orzack and Tuljapurkar, 1989; Pfister, 1998; Morris et al., 2008; Jonzén et al., 2010). In order to understand how individual differences affect population properties we therefore need stochastic model frameworks that account for the main sources of random variation, in addition to demographic structure. Stochastic demography has been defined as the study of struc- tured populations living in randomly fluctuating environments, causing the vital rates to fluctuate in time (Nations and Boyce, 1997; Boyce et al., 2006). Assuming that individuals in a population share the same environment, such fluctuations will create covari- ation between individuals with respect to demographic processes like survival and reproduction (Engen et al., 1998). These processes are also inherently stochastic, so that even in a constant environ- ment there will be temporal fluctuations in the realized survival 0040-5809/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2011.10.002