On reducibility and ergodicity of population projection matrix models

Authors

  • Iain Stott,

    1. Centre for Ecology and Conservation, School of Biosciences, College of Life and Environmental Sciences, University of Exeter Cornwall Campus, Tremough, Treliever Road, Penryn, Cornwall TR10 9EZ, UK
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  • Stuart Townley,

    1. College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, North Park Road, Exeter, Devon EX4 4QF, UK
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  • David Carslake,

    1. Department of Biological Sciences, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
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  • David J. Hodgson

    Corresponding author
    1. Centre for Ecology and Conservation, School of Biosciences, College of Life and Environmental Sciences, University of Exeter Cornwall Campus, Tremough, Treliever Road, Penryn, Cornwall TR10 9EZ, UK
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Correspondence author. E-mail: d.j.hodgson@exeter.ac.uk

Summary

1. Population projection matrices (PPMs) are probably the most commonly used empirical population models. To be useful for predictive or prospective analyses, PPM models should generally be irreducible (the associated life cycle graph contains the necessary transition rates to facilitate pathways from all stages to all other stages) and therefore ergodic (whatever initial stage structure is used in the population projection, it will always exhibit the same stable asymptotic growth rate).

2. Evaluation of 652 PPM models for 171 species from the literature suggests that 24·7% of PPM models are reducible (parameterized transition rates do not facilitate pathways from all stages to all other stages). Reducible models are sometimes ergodic but may be non-ergodic (the model exhibits two or more stable asymptotic states with different asymptotic stable growth rates, which depend on the initial stage structure used in the population projection). In our sample of published PPMs, 15·6% are non-ergodic.

3. This presents a problem: reducible–ergodic models often defy biological rationale in their description of the life cycle but may or may not prove problematic for analysis as they often behave similarly to irreducible models. Reducible–non-ergodic models will usually defy biological rationale in their description of the both the life cycle and population dynamics, hence contravening most analytical methods.

4. We provide simple methods to evaluate reducibility and ergodicity of PPM models, present illustrative examples to elucidate the relationship between reducibility and ergodicity and provide empirical examples to evaluate the implications of these properties in PPM models.

5. As a prevailing tool for population ecologists, PPM models need to be as predictive as possible. However, there is a large incidence of reducibility in published PPMs, with significant implications for the predictive power of such models in many cases. We suggest that as a general rule, reducibility of PPM models should be avoided. However, we provide a guide to the pertinent analysis of reducible matrix models, largely based upon whether they are ergodic or not.

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 λmax– 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).

Figure 1.

 Example stage-structured life cycles. (a) An irreducible life cycle. Every stage is connected to every other stage via at least one pathway. (b) A reducible, post-reproductive life cycle. Sufficient transitions to facilitate pathways from every stage to every other stage are lacking: once in stage 4, there are no connections to the rest of the life cycle. Such a life cycle, although reducible, is biologically plausible. (c) A reducible life cycle that is missing a necessary transition rate. Sufficient transitions to facilitate pathways from every stage to every other stage are lacking: once in stages 1 and 2, an individual cannot then contribute to stages 3 and 4. Such a model clearly defies biological rationale, with portions of the life cycle isolated from the rest of the cycle.

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:

inline image

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, λmax. 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.

Illustrative models

To show how reducible non-negative matrix structure relates to ergodicity of the model, we present simulations of illustrative PPM models. These demonstrate the conditions under which a reducible model will be differentially ergodic or non-ergodic and, at a greater resolution, reveal how matrix structure relates to the number of different asymptotic states that the model may achieve. We use, e.g. λ(A1) to refer to the dominant eigenvalue of a constituent block of a reducible matrix, whereas λmax or λ(A) refers to the dominant eigenvalue of the overall matrix.

Illustrative models: methods

A theoretical, irreducible, six-stage matrix model was constructed. This model was designed to be similar to one that may be parameterized for a tree species (Fig. 2a). Transition rates were ‘knocked out’ of this model and replaced with zero values to create reducible models. The first of these may be subdivided to form two irreducible blocks on the diagonal (the two-block reducible model; Fig. 2b). The second may be subdivided to form three irreducible blocks on the diagonal (the three-block reducible model; Fig. 2c).

Figure 2.

 The matrices used for modelling. In each case, the life cycle described by the matrix is also shown for clarity. (a) The irreducible projection matrix upon which the reducible matrices are based. This is similar to one that may be parameterized for a tree species: fecundity and survival with stasis increase with size, whilst survival with growth decreases with size. Each stage also has a small probability of regression. (b) The two-block reducible matrix in block-permuted form. An × indicates the ‘knocking-out’ of transition a5,4 from the irreducible matrix and its replacement with a zero value in order to create the reducible matrix. The matrix may be subdivided with irreducible blocks A1 and A2 on the diagonal as indicated. ‘0’ indicates a zero block (i.e. all elements of the block are zero). (c) The three-block reducible matrix in block-permuted form. An × indicates the ‘knocking-out’ of transitions a4,3 and a6,5 from the irreducible matrix and their replacement with zero values in order to create the reducible matrix. The matrix may be subdivided with irreducible blocks on the diagonal as indicated. Again, ‘0’ indicates a zero block. Note: in all cases, only irreducible blocks on the diagonal are important in the modelling process. Henceforth in this study, blocks A1,2 and A2,3 can be disregarded.

Ergodic properties of reducible non-negative matrices are determined by the relative sizes of the dominant eigenvalues of the constituent blocks of the matrix – specifically, it is the relative sizes of the blocks on the diagonal that are important (Bapat 1998). We manipulated the relative sizes of the dominant eigenvalues of these diagonal blocks by varying transition rates on the diagonal of the block (i.e. varying survival with stasis, the transition rate that measures the rate of survival without growth or regression to a different stage class). As a result, sets of matrices were generated with eigenvalues that varied on a continuous scale. Each single matrix in these sets was projected using six stage-biased initial population vectors, with all individuals in a single stage and a density of one, in a manner similar to that employed by Townley & Hodgson (2008). In practice, these are vectors of zeroes except for a one in one row only. This enabled an assessment of growth rate according to life cycle stage: as reducibility is related to connectivity between stages, it was important that the ‘fate’ of each individual stage was assessed independently. The actual, realized asymptotic growth rate of each stage-biased projection was calculated by dividing population size at time t + 1 [N(t+1)] by population size at time t [N(t)] after the population had reached a stable growth rate. An ergodic model will exhibit the same realized asymptotic growth rate for every stage-biased projection, whereas a non-ergodic model will exhibit two or more growth rates among its set of stage-biased projections.

For the two-block reducible matrix, the modelling process was simple. λ(A1) was fixed at its ‘natural’ value (0·738), whilst λ(A2) was varied over the range 0·66–0·81. For the three-block reducible matrix, each of the block-specific eigenvalues had to be varied in order to assess every permutation of relative eigenvalue size. As such, in part 1 of the process λ(A1) and λ(A2) were fixed at their ‘natural’ values (0·711 and 0·611 respectively), whilst λ(A3) varied over the range 0·53–0·78. In part 2 of the process, λ(A3) was fixed at its highest value from part 1 (0·78), λ(A1) remained fixed at its ‘natural’ value (0·611) and λ(A2) was varied over the range 0·61–0·86. In part 3, λ(A2) was fixed at its highest value from part 2 (0·86), λ(A3) remained fixed at its highest value from part 1 (0·78) and λ(A1) was varied over the range 0·70–0·95. These values may seem somewhat arbitrary but were chosen so that all transition rates remained within biologically realistic limits, whilst still allowing all permutations of relative eigenvalue size to be assessed.

All modelling was carried out using R version 2.9.2 (R Development Core Team 2009). Appendix S2 contains the code used in our models to calculate realized λ values using simulation.

Illustrative models: results

Models showed that for both the two- and three-block reducible matrices, the models only exhibited ergodicity where λ(A1) was the largest eigenvalue, with all other cases exhibiting non-ergodicity (Figs 3 and 4).

Figure 3.

 Axes at the top of the graph show the modelling process for the simple two-block reducible matrix. λ(A1) is fixed (=0·738) and λ(A2) varies. Relative positions of λ(A1) and graphs (a) and (b) are indicated on the scale at the top of the figure. The y-axes of graphs (a) and (b) follow a log scale. Population size [N(t)] is standardized by λ(A) to clearly indicate differences in growth rates, therefore stages that follow λ(A) converge to horizontal lines on the graph. The number of different asymptotic growth rates that may be achieved by the model depends on the relative magnitudes of the dominant eigenvalues of its constituent blocks: (a) λ(A2) < λ(A1), therefore all stages follow λ(A) = λ(A1) = 0·738. (b) λ(A1) < λ(A2), therefore stages 5 and 6 follow λ(A) = λ(A2) = 0·775. Stages 1–4 continue to follow λ(A1) = 0·738.

Figure 4.

 Axes at the top of the graph show the modelling process for the three-block reducible matrix. In part 1, λ(A2) and λ(A1) are fixed (=0·611 and 0·711 respectively) and λ(A3) varies. In part 2, λ(A1) and λ(A3) are fixed (=0·711 and 0·780 respectively) and λ(A2) varies. In part 3, λ(A3) and λ(A2) are fixed (=0·780 and 0·860 respectively) and λ(A1) varies. In all cases, relative positions of block-specific eigenvalues and of graphs (a)–(g) are indicated where appropriate on the axes at the top of the figure. The y-axes of graphs (a)–(g) follow a log scale. Population size [N(t)] is standardized by λ(A) to clearly indicate differences in growth rates, therefore stages that follow λ(A) converge to horizontal lines on the graph. The number of different asymptotic growth rates that may be achieved by the model depends on the relative magnitudes of the dominant eigenvalues of its constituent blocks: (a) λ(A3) < λ(A2) < λ(A1), therefore all stages follow λ(A) = λ(A1) = 0·711. (b) λ(A2) < λ(A3) < λ(A1), therefore all stages follow λ(A) = λ(A1) = 0·711. (c) λ(A2) < λ(A1) < λ(A3), therefore stage 6 follows λ(A) = λ(A3) = 0·780, while stages 1–5 follow λ(A1) = 0·711. (d) λ(A1) < λ(A2) < λ(A3), therefore stage 6 follows λ(A) = λ(A3) = 0·780, while stages 4 and 5 follow λ(A2) = 0·740 and stages 1–3 follow λ(A1) = 0·711. (e) λ(A1) < λ(A3) < λ(A2), therefore stages 4–6 follow λ(A) = λ(A2) = 0·860, while stages 1–3 follow λ(A1) = 0·711. (f) λ(A3) < λ(A1) < λ(A2), therefore stages 4–6 follow λ(A) = λ(A2) = 0·860, while stages 1–3 follow λ(A1) = 0·820. (g) The model is returned to its original state of λ(A3) < λ(A2) < λ(A1), and all stages follow λ(A) = λ(A1) = 0·920.

More specifically, the models showed that the number of different asymptotic growth rates that may be achieved by a reducible PPM model depends on the relative magnitudes of the dominant eigenvalues of its constituent blocks. The two-block reducible matrix can only exhibit a maximum of two different growth rates. This will occur when λ(A1) < λ(A2) (Fig. 3). The three-block reducible matrix can be thought of as a two-block reducible matrix within another two-block reducible matrix. Therefore, λ(A3) must be greater than both λ(A2) and λ(A1) to exhibit asymptotic independence (i.e. to exist as a potential asymptotic growth rate of the model). However, λ(A2) must only be greater than λ(A1) to exhibit asymptotic independence. As such, it is only when λ(A1) < λ(A2) < λ(A3) that the model exhibits three different long-term growth rates (Fig. 4). However, the model may exhibit two different long-term growth rates where λ(A2) < λ(A1) < λ(A3), where λ(A1) < λ(A3) < λ(A2) or where λ(A3) < λ(A1) < λ(A2) (Fig. 4). The same logic would therefore apply to any reducible model – for example, in a four-block reducible model, λ(A4) would have to be greater than λ(A3), λ(A2) and λ(A1) to exhibit asymptotic independence, and so on.

The only exception to this rule will occur when two diagonal blocks are not fixed relative to one another. Take a reducible matrix with the structure:

inline image

The matrix can be re-permuted so that blocks A2 and A3 are swapped but without changing the overall structure of the matrix:

inline image

For λ(A2) to exhibit asymptotic independence as illustrated by our models, the condition λ(A1) < λ(A2) must be true, as usual. However, as there is no discernable difference between these two placements of A2 and A3, then for λ(A3) to exhibit asymptotic independence, the condition λ(A1) < λ(A3) is necessary, but λ(A2) < λ(A3) is no longer necessary. Note that whilst this affects the number of possible asymptotic growth rates where λ(A1) is not the largest eigenvalue, it does not affect the condition that for the model to exhibit ergodicity, λ(A1) must be the largest eigenvalue. Hence, if A1 is not fixed relative to the other diagonal blocks, the model will be non-ergodic. It is worth mentioning also that whilst we have only illustrated this case with matrices where λmax < 1, the rules we have described above apply to any non-negative matrix with λmax > 0.

Incidence of reducibility and non-erodicity in published PPMs

Incidence of reducibility and non-ergodicity in published PPMs: methods

Population projection matrices were collected from the literature, with the database at the time of analysis numbering 652 matrices (152 animals and 500 plants) for 171 species (57 animals and 114 plants) across a diverse range of taxa (see Appendix S1 in Supporting Information). In the first instance, we sought out large comparative analyses of PPM models of plants and animals and sourced many PPMs from the original articles cited by those analyses. Online searches of ecological literature provided additional PPMs, although these searches were not systematic. PPMs were not chosen with any bias regarding reducibility or ergodicity. We feel that the database of PPMs represents a fair sample of the entire population of published PPMs; hence, any result described here may be interpreted as applicable to the wider PPM literature. The database held by the authors is being continually added to, and contains extra information on the species and models present. It is producible at any time upon request.

Each matrix was tested for reducibility using the argument that a square matrix A is irreducible if, and only if, (I + A)s−1 is positive (i.e. every element of (I + A)s−1 is greater than 0), where I is the identity matrix and s represents the number of columns or rows in the matrix, equal to the number of stage classes in the life cycle model (Caswell 2001).

Each matrix was tested for ergodicity using the argument that a non-negative matrix A is, under certain natural conditions, ergodic if and only if the dominant left eigenvector v of A is positive (i.e. every element of v is greater than 0). Biologically, this vector is known as the ‘reproductive value’ vector, and satisfies the equation vTA = λmaxvT, where λmax is the dominant eigenvalue of A and vT denotes the transpose of v (i.e. from a column vector to a row vector). The specific conditions and mathematical proof for this can be found in Appendix S4, and the subject is discussed at a greater depth in Dietzenbacher (1991). As this is a new method in population biology for evaluating ergodicity, each result was double checked with simulation of population dynamics using stage-biased vectors in the same manner described here for the illustrative model (see Illustrative models: methods). In every case, results from simulation of the model agreed with results from evaluation of the dominant left eigenvector.

All tests were carried out using R version 2.9.2 (R Development Core Team 2009). Appendix S2 contains code to test reducibility and ergodicity of matrices using the above arguments.

Incidence of reducibility and non-ergodicity in published PPMs: results

Analysis of the matrix database showed that the incidence of reducibility in published PPMs is high, at 24·7%. Of the reducible models, 63·2% are non-ergodic, which makes 15·6% of our sample of published PPMs non-ergodic.

Some reducible models have been modelled with post-reproductive stage classes or have similar plausibly reducible life cycles, and thus do not defy biological rationale (as illustrated in Fig. 1b). However, most reducible models are lacking vital transition rates to complete the life cycle and as such defy biological rationale (as illustrated in Fig. 1c). Some reducible models showed a complete absence of certain stages in the model, and hence incorporated no transition rates whatsoever for those stages (so that the PPM contained columns of zeroes). In addition, many models included 100% rates of survival with stasis in adult stage classes, which although are not an origin of reducibility per se, effectively model immortal individuals and so clearly defy biological rationale.

Empirical examples

We chose three reducible matrices from the literature to illustrate how reducibility may (or may not) affect model outcomes and conclusions, depending on the ergodic properties of the model and the analyses performed. These matrices were block permuted (an R code to block permute a reducible matrix may be found in Appendix S2 and further information on block permuting a reducible matrix by hand may be found in Appendix S3), and the eigenvalues of their diagonal blocks were calculated. These were then compared with the realized asymptotic growth rates of their stage-biased projections.

Empirical examples: Nuttallia obscurata

Nuttallia obscurata (the varnish clam) is a marine invasive species in British Columbia, Canada. Dudas, Dower, & Anholt (2007) parameterize two matrices for the species, in order to evaluate the dynamics of invasive populations and identify which life-history stages are the best to target with control strategies. By performing asymptotic sensitivity and elasticity analyses, they concluded that reducing the survival of adult individuals would be the most beneficial strategy to curb population growth.

The PPM for the Robbers’ passage population of N. obscurata is shown in Fig. 5a. There was no observed growth of stage 1 individuals into stage 2 individuals – these instead all bypassed stage 2, becoming stage 3 individuals in 1 year. The authors noted this irregularity and corrected the estimates of transition rates accordingly but failed to note that the resulting matrix is reducible: individuals in stage 2 may grow and reproduce, but no individual in stages 1, 3, 4 or 5 can contribute in any way to stage 2 (whether through growth or reproduction).

Figure 5.

 Empirical examples of reducible matrices with varying numbers of asymptotic growth rates that may be achieved (see Fig. 6). Each matrix is shown in its original (left) and block-permuted (right) form. In each case, the life cycle described by the matrix is also shown for clarity. (a) Left: PPM parameterized for the Robber’s passage population of varnish clams Nuttallia obscurata (Dudas, Dower, & Anholt 2007). Right: the block-permuted matrix, with two irreducible blocks on the diagonal. (b) Left: PPM parameterized for marlberry Ardisia escallonoides (Pascarella & Horvitz 1998) under a 5% open canopy. Right: the block-permuted matrix, with three irreducible blocks on the diagonal. (c) Left: PPM parameterized for the Jeffrey Pine Pinus jeffreyi (van Mantgem & Stephenson 2005). Right: the block-permuted matrix, with five irreducible blocks on the diagonal.

Although reducible, the matrix is ergodic (Fig. 6a). Therefore, in this particular case, the conclusions of the study would not be greatly affected by the reducible structure of the model. Every population projection will eventually settle to a rate of geometric growth described by the dominant eigenvalue of the matrix, and so the asymptotic growth rate of the population can be safely defined as such. Therefore, the sensitivity and elasticity analyses conducted by Dudas, Dower, & Anholt (2007) correctly evaluate the effects of perturbations to vital rates on asymptotic population growth. This is not necessarily always the case, as other examples will show.

Figure 6.

 Projections of the example empirical reducible matrices from stage-biased initial populations. In all cases, y-axes follow a log scale and population size [N(t)] is standardized by λ(A) to clearly indicate differences in growth rates; therefore, stages that follow λ(A) converge to horizontal lines on the graph. In agreement with the illustrative models, the number of different asymptotic growth rates that may be achieved by the model (and therefore the ergodicity of the model) depends on the relative magnitudes of the dominant eigenvalues of its constituent blocks: (a) The Nuttallia obscurata model exhibits ergodicity, as λ(A2) < λ(A1), therefore all stages follow λ(A) = λ(A1)=0·727. (b) The Ardisia escallonioides model exhibits two different asymptotic growth rates, as λ(A2) < λ(A1) < λ(A3). Stage 8 follows λ(A) = λ(A3) = 1, whereas stages 1–7 follow λ(A1)=0·984. (c) The Pinus jeffreyi model exhibits three different growth rates, as λ(A1) < λ(A3) < λ(A4) < λ(A2) < λ(A5). Stage 5 follows λ(A) = λ(A5)=0·945 [although this projection does not completely converge within the 500 projection intervals plotted, it goes on to converge to λ(A)], stages 1–3 follow λ(A2) = 0·944 and stage 4 follows λ(A1) = 0·778.

Perhaps in this case, individuals grew faster than anticipated. A re-definition of size classes to a lesser resolution might have produced an irreducible matrix that described population dynamics more effectively. This should be easy to do if data on sizes of individuals are available. When parameterizing models, it is worth bearing in mind that the life cycle model may have to be redefined in order to fit with the data collected. Information on parameterizing models according to data availability is available (Vandermeer 1978; Moloney 1986). Alternatively, for species such as the varnish clam that exhibit continuous state variables (e.g. where organism size is a strong determinant of survival and/or growth and/or fecundity), integral projection models (IPMs) offer a means of population projection that utilizes smooth, continuous relationships between such state variables and an organism’s vital rates, which may be more accurate than the discrete-class approximations to such relationships provided by a PPM (Easterling, Ellner, & Dixon 2000; Ellner & Rees 2006, 2007).

Empirical examples: Ardisia escallonioides

Ardisia escallonioides (Marlberry) is a perennial understorey shrub found in the subtropical forests of southern Florida, as well as on islands and in coastal regions of the Caribbean Sea and the Gulf of Mexico. Pascarella & Horvitz (1998) collected data from the Florida populations in order to parameterize models that would allow the assessment of the role of environmental variation in shaping population dynamics of the species. The environmental variation in question was the degree of forest canopy ‘openness’ as a result of hurricane damage. Patches were identified with varying degrees of canopy openness, and an individual matrix was parameterized for each. The individual matrices were then combined with a patch-transition matrix into a megamatrix that described dynamics of the population as a whole. Asymptotic growth rates, stable stage structures and sensitivity and elasticity analyses were conducted for individual matrices and the megamatrix and the results were compared and contrasted with one another.

The matrix representing the 5% open patch is presented in Fig. 5b. There is no growth of stage 7 or stage 8 individuals, and 100% stasis of stage 8 individuals. The authors noted that their original matrix parameterization was not ‘full rank’ (sensuCaswell 1989), so would not exhibit asymptotic behaviour comparable with other patch-specific matrices, and corrected it as such (by estimating fecundity values and juvenile survival rates). However, they did not note that the resulting model was reducible, and so may still not exhibit comparable asymptotic behaviour (although it is noteworthy that they realized and corrected this in later analyses that used the data – see Appendix A of Tuljapurkar, Horvitz, & Pascarella 2003).

The reducible model exhibits two possible asymptotic growth rates (Fig. 6b). Any projection that excludes stage 8 individuals in the initial population structure will follow λ = 0·984, and any projection that includes stage 8 individuals in the initial population structure will follow λ = 1. In this case, the asymptotic growth rate of the population has no single numerical definition, although it is assumed to equal the dominant eigenvalue of the matrix. Hence, it is not comparable with the other matrix models. Sensitivity and elasticity analyses measured the effect of perturbations on the dominant eigenvalue, but this is not necessarily equivalent to the effect that such perturbations have on asymptotic growth rate, as asymptotic growth rate is not always equal to the dominant eigenvalue. These analyses are therefore fundamentally flawed for the 5% open canopy patch. Having said that, the errors in this model are diluted by data from other patches in the megamatrix such that the megamatrix is irreducible, so that the conclusions of analyses based on the megamatrix will be little affected by the single reducible model.

The authors recognize that transition rates absent in such a model must occur in populations and that ‘a much larger sample size would be needed to detect them empirically’ (Pascarella & Horvitz 1998, p. 551). With constraints on the amount of data that can be collected, perhaps in such situations it may be necessary to estimate these transition rates (such as from historical data, or based upon observed rates in other (sub)populations where available) to ensure a reliable model. Again, IPMs may offer an alternative approach in such a situation, as they usually require less data than traditional PPM models (Easterling, Ellner, & Dixon 2000; Ellner & Rees 2006, 2007).

Empirical examples: Pinus jeffreyi

Pinus jeffreyi (the Jeffrey Pine) is a coniferous tree found in the south-eastern USA. van Mantgem & Stephenson (2005) constructed PPM models for a population of the species (along with numerous other species of coniferous tree) in the Sierra Nevada in order to assess the reliability of PPM models as predictors of the dynamics of populations of such species. They compared matrix model projections with empirical data on population size, and concluded that PPM models were good predictors of population size, short-term growth, survival and recruitment.

The PPM model for P. jeffreyi is presented in Fig. 5c. No stage 4 individuals progressed to become stage 5 individuals within the time frame of the study. The authors noted this, and recognized that this missing transition would be a problem when projecting over long time periods. However, as the study only looked at short-term dynamics (with models projected over a maximum of two time intervals), they concluded that it should not have great adverse effects on their analyses. That said, reducibility per se was not explicitly discussed.

The reducible model exhibits three possible asymptotic growth rates (Fig. 6c). The matrix, in its block-permuted form, has five irreducible blocks on the diagonal. This is a good example of how a model missing just one transition rate then becomes highly mathematically constrained as a result: eigenvalues are constrained to be equal to the values of survival with stasis (on the diagonal of the matrix). Eigenvectors are equally constrained by these eigenvalues. Hence, an analysis that uses any eigenvalues of the matrix will be adversely affected. These include not only asymptotic analyses such as sensitivity and elasticity but also some measures of transient dynamics, such as the damping ratio of the population (Caswell 2001) and certain measures of transient sensitivity and elasticity (Fox & Gurevitch 2000; Yearsley 2004).

van Mantgem & Stephenson (2005) are correct in that the reducibility of the matrix has little impact on the conclusions of their analyses – they wished merely to compare empirical with predicted population dynamics, and the models they present are an accurate representation of the demographic rates of the species in the time period considered. This example illustrates a case in which a reducible matrix, despite being non-ergodic and having highly constrained eigenvalues, should not pose a problem for the specific analyses being conducted. That said, the supplementary material does provide dominant eigenvalues of the PPMs as values of asymptotic population growth, and elasticity values are used in the manuscript to support certain conclusions (although are not presented) – these analyses will be flawed as a result of the model structure.

Discussion

We have demonstrated that reducibility, and hence non-ergodicity, in published PPM models is common. We have illustrated how the structure of reducible models affects their ergodic properties, and provided some empirical examples showing how these properties may or may not affect model outcomes and conclusions, depending on the analyses implemented.

But why are so many published PPMs reducible? Whilst some models may plausibly be based upon a reducible life cycle (e.g. Brault & Caswell 1993), most are missing vital transition rates necessary for a complete life cycle. This leads to the question as to why such transition rates are missing from those models. The most likely explanation for this, as noted earlier by Pascarella & Horvitz (1998) in the Ardisia example, is that the relevant data required to fully parameterize the model are lacking. There are two possible explanations for this – either the data collected overlooked life cycle transitions that did occur, or the missing transitions did not occur during the time of study. Certain methods of data collection may result in poor estimates of transition rates, and perhaps a failure to capture a full set of transition rates –Münzbergová & Ehrlén (2005) note that commonly used methods may result in ‘poor representation of some stages’, and advise to collect data for an equal number of individuals per stage. However, even if the data collected do accurately represent the life cycle of the population during the time of study (with certain transition rates missing), this time is often limited to two consecutive years (Fieberg & Ellner 2001), comprising two population censuses: just enough to parameterize a single projection matrix. Such matrices will only represent demographic rates under the environmental conditions specific to that year. Under undesirable environmental conditions, it takes little stretch of the imagination to envisage that some life cycle transitions may not occur in a single year. Even under desirable environmental conditions, low transition rates (especially of growth) may be seen under certain life cycle models, especially for long-lived species (Enright, Franco, & Silvertown 1995). While the data collected in such an instance may adequately represent the demographics of that species in that particular year, it is clearly insufficient to describe the dynamics of the population over many years – this much seems intuitive. Additionally, as we have shown here with our illustrative and empirical examples, complications arise as reducible matrix models based upon such data have certain mathematical properties – such as eigenvalues that are constrained to equal those of the sub-blocks in all cases, and non-ergodicity in some cases – that may pose problems for analysis.

For these reasons, as a general rule it is best to avoid using reducible models. Having said that, if a reducible model is unavoidable, such a model will not always prove problematic for analysis (as was seen earlier in the Pinus example). It is important that demographers know how to: (1) identify reducible matrices, (2) evaluate the ergodic properties of those matrices and (3) discern whether the models will prove problematic for the analyses that are to be conducted. We propose that matrices should be defined as irreducible, reducible–ergodic or reducible–non-ergodic and that this information be used to guide analysis. Reducibility and ergodicity of a matrix can be easily evaluated using the methods we present (see the Incidence of reducibility and non-ergodicity in published PPMs: methods section; Appendices S2 and S4) and analyses appropriate to that model structure can then be chosen.

For deterministic asymptotic analyses, a reducible–ergodic matrix can be analysed in very much the same way as an irreducible matrix. Analyses such as evaluation of the dominant eigenvalue and/or eigenvector and calculations of asymptotic sensitivity or elasticity will be largely unaffected by the model. On the other hand, deterministic asymptotic analyses of reducible–non-ergodic matrices will often prove spurious: there is no single definition of asymptotic stable population growth and structure. For an unavoidably reducible model, an alternative is to analyse the portion of the life cycle that one is interested in – for example, when analysing population growth rates of species with post-reproductive stage classes, one might want to only analyse the block for the reproductive part of the population (Caswell 2001, p. 89).

Reducible models have implications for stochastic analyses also. In stochastic analyses, sets of transition matrices are generated, with each single matrix representing transition rates under a certain environmental condition. This is usually done either by using matrices parameterized over different time intervals or by drawing transition rates from probability distribution functions with covarying parameters (Fieberg & Ellner 2001). A different matrix (emulating a unique set of environmental conditions) is chosen for each projection interval. In this case, it is clearly difficult to evaluate ergodicity of the model – each simulation is unique, as a different sequence of parameters or matrices is selected each time the simulation is run. However, reducibility of the combined matrix set can be evaluated. For the matrix set A(1), A(2),…, A(n), the arithmetic average matrix M=(1/n)(A(1) + A(2) +⋯+ A(n)). As ergodicity cannot be evaluated, reducibility of M should be avoided in stochastic analyses. Individual reducible matrices, while undesirable, should not usually be a problem provided that the missing transitions are present elsewhere in the matrix set. However, the impacts of individual reducible matrices on stochastic analyses warrant further exploration and we note that this rule of thumb may not always apply. An alternative to simulating stochastic models is to calculate Tuljapurkar’s approximation to the stochastic growth rate (Fieberg & Ellner 2001). This calculation should only be based on ergodic matrix models, as calculations utilize the dominant eigenvalue, which can only be properly defined for ergodic models.

Transient population dynamics are gaining increased attention from researchers. Methods for calculating transients are many and varied, and the impacts of reducibility and ergodicity of matrices on transient analyses will vary according to analyses used. For methods that implicate matrix eigenvalues in their calculation (e.g. Fox & Gurevitch 2000; Yearsley 2004), reducible matrices (whether ergodic or non-ergodic) should be avoided, as the eigenvalues of the matrix are constrained by the model structure. For methods that do not utilize matrix eigenvalues (e.g. Koons, Holmes, & Grand 2007; Townley & Hodgson 2008; Tenhumberg, Tyre, & Rebarber 2009), a general rule would be to avoid reducible–non-ergodic matrices: many analyses relate transient measures to asymptotic growth (Koons, Holmes, & Grand 2007; Townley & Hodgson 2008) and so the model used requires a robust definition of asymptotic growth. That said, in some cases (as exemplified in the Pinus example above), empirical calculation of population growth rates over the transient period may not be adversely affected by a non-ergodic model.

Conclusions

With reducible models so abundant in the literature, it is important for population biologists to know of the issues concerning their use, be able to identify whether a matrix is irreducible, reducible–ergodic or reducible–non-ergodic, and take the necessary precautions to alleviate potential problems associated with these model structures, either by redefining model structure if possible or by choosing the right analytical methods pertaining to the use of that matrix. The methods and models presented here should help to inform on these processes. Although biologists now have the necessary mathematical, statistical and computational tools at their disposal to conduct highly complex analyses, it is important not to forget about the foundations of PPM models and to remember to consider the basic life cycle model and matrix structure very carefully.

Acknowledgements

We thank Miguel Franco for supplying/collecting much of the database of PPM models for plants. This work was supported by NERC and European Social Fund grants to D.J.H. and a Leverhulme Visiting Fellowship to S.T.

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