## Introduction

Ecological dynamics are inherently complex. However, it is desirable in applications of population ecology to be able to accurately predict future dynamics of populations (Caswell 2001, 2007; Townley *et al.* 2007; Townley & Hodgson 2008). Population projection matrices (PPMs) are possibly the most often used empirical population models, and much research in population ecology focuses on the design of increasingly accurate matrix modelling techniques. Factors such as spatial heterogeneity (e.g. Day & Possingham 1995; Hunter & Caswell 2005), density dependence (e.g. Jensen 1993; Grant & Benton 2000; Armsworth 2002), exogenous stochastic influence (e.g. Nakaoka 1997; Fieberg & Ellner 2001; Tuljapurkar, Horvitz, & Pascarella 2003) and transient dynamics (e.g. Koons *et al.* 2005; Townley & Hodgson 2008; Tenhumberg, Tyre, & Rebarber 2009; Stott *et al.* 2010) are now commonly incorporated into modelling and analytical methods. However, for a building to be robust it must also have secure foundations. Equally, matrix model complexities are building blocks upheld by the basic grounding of the model itself – the life cycle.

Almost all life cycle models can be described as *irreducible*, meaning that they contain direct or indirect pathways from every stage class to every other stage class (Fig. 1a). A PPM associated with an irreducible life cycle can itself be described as irreducible. According to the Perron–Frobenius theorem (Perron 1907a,b; Frobenius 1912; for overviews, see Caswell 2001; Li & Schneider 2002; Elhashash & Szyld 2008), an irreducible, primitive matrix **A** will have a single, positive eigenvalue that is a simple root of the characteristic polynomial of **A** and whose modulus is greater than all other eigenvalues of the matrix (Caswell 2001; Li & Schneider 2002) – this is commonly known as the dominant eigenvalue of the matrix. It is this value – also known as *λ*_{max}– that describes the long-term (asymptotic) growth rate of the population and which has to date been studied in the majority of published PPM analyses (see, e.g. Esparza-Olguín, Valverde, & Vilchis-Anaya 2002; Grenier, McDonald, & Buskirk 2007; Zúñiga-Vega *et al.* 2007). It is simple to calculate, and conceptually easy to understand. Long-established methods of evaluating the effects of changes to vital rates on the long-term population growth rate exist in the forms of sensitivity and elasticity analyses (Caswell 2001), and these are often used to inform on population management and conservation decisions (Crowder *et al.* 1994; Crooks, Sanjauan, & Doak 1998; Runge, Langtimm, & Kendall 2004). Population projections from irreducible matrices are always *ergodic*– that is, they will always eventually exhibit the same outcome (i.e. the same stable growth as described by the dominant eigenvalue), irrespective of the initial conditions of the model (i.e. the initial population structure used in the projection).

By contrast, a *reducible* life cycle model is essentially not complete – transitions do not facilitate pathways from every stage class to every other stage class (Caswell 2001). Consequently, one or more portions of the life cycle are isolated from the rest of the cycle (Fig. 1b,c). Except in a few cases – for example, when modelling a species with post-reproductive stage classes (Caswell 2001; Fig. 1b) – a reducible life cycle model defies biological rationale. A reducible PPM associated with such a life cycle may undergo simultaneous row and column permutations (i.e. have its rows and columns simultaneously rearranged) so that it takes the form:

That is, it may be divided into a number of submatrices (blocks), with those on the diagonal being irreducible (Caswell 2001; Li & Schneider 2002), and where ‘0’ denotes a block of zeroes. The presence or absence of data in blocks at the top-right corner determines the position of the diagonal blocks relative to one another. We refer to this as the block-permuted matrix. Each diagonal block has a dominant eigenvalue, each of which will be an eigenvalue of the overall matrix, and the largest of these will be equal to the dominant eigenvalue of the overall matrix, *λ*_{max}. However, for some reducible matrices, the long-term stable growth rate may be described by eigenvalues other than the dominant (Caswell 2001), dependent on the initial stage structure of the population used in the projection. A reducible model that may exhibit more than one final state, dependent on initial conditions, may be described as *non-ergodic*. When using dominant eigenvalues or vectors to inform on management and conservation decisions, a reducible, non-ergodic model can hence prove problematic, as the asymptotic growth rate experienced by the population may be radically different from that predicted by the model. The ergodic properties of reducible non-negative matrices depend on the specific structure of the reducible model and the relative sizes of the dominant eigenvalues of its constituent blocks. This has been explored in the mathematical literature (e.g. Dietzenbacher 1991; Bapat 1998; Kolotilina 2004); however, its application has not yet extended to population biology. Indeed, the issue of reducibility *per se* is rarely explicitly considered in the biological literature (however, notable exceptions include Bierzychudek 1982; Caswell 2001; Ehrlén & Lehtilä 2002; Bode, Bode, & Armsworth 2006; Maron, Horvitz, & Williams 2010; Stott *et al.* 2010).

Here, we present methods to easily evaluate reducibility and ergodicity of matrices. Using these methods, we show that a considerable proportion (approximately one-quarter) of existing published PPMs are reducible, many of which are non-ergodic. We elucidate the relationship between reducibility and ergodicity using illustrative theoretical models of reducible PPMs. We explore the real-world implications of reducibility and non-ergodicity of PPMs using three empirical examples of reducible matrices employing different analytical techniques. Lastly, we encourage population demographers to carefully consider their life cycle model and advise that reducibility should be avoided where possible; however, we discuss analytical approaches to unavoidably reducible models, based upon their ergodic properties.