Matrix projection models are among the most widely used tools in plant ecology. However, the way in which plant ecologists use and interpret these models differs from the way in which they are presented in the broader academic literature. In contrast to calls from earlier reviews, most studies of plant populations are based on < 5 matrices and present simple metrics such as deterministic population growth rates. However, plant ecologists also cautioned against literal interpretation of model predictions. Although academic studies have emphasized testing quantitative model predictions, such forecasts are not the way in which plant ecologists find matrix models to be most useful. Improving forecasting ability would necessitate increased model complexity and longer studies. Therefore, in addition to longer term studies with better links to environmental drivers, priorities for research include critically evaluating relative/comparative uses of matrix models and asking how we can use many short‐term studies to understand long‐term population dynamics.

Review of matrix models We attempted to census all matrix models for plant populations, published in peer‐reviewed English‐language journals through April 2009. To create this list of papers, we started with citation lists from previous reviews and meta‐analyses conducted by members of our group (e.g., Menges 2000a; Burns et al. 2010; J.L. Williams, M.M. Ellis, M.C. Bricker, J.F. Brodie and E.W. Parsons, unpublished data) and with a prior database (Menges 2000b), updating these resources through additional literature searches and personal knowledge. We also searched Web of Science (http://www.isiknowledge.com) for all papers that cited reviews of matrix models, scanned all issues of key journals that publish large numbers of matrix model papers and updated this merged list with the reference libraries of all 16 working group members. We screened more than 400 candidate papers and found 396 published modelling studies for plant populations (a full citation list for these papers is accessible at: http://knb.ecoinformatics.org/knb/metacat/nceas.961/nceas). Of these, 355 papers included stage‐, size‐ or age‐based matrix models, highlighting the broad importance of matrix models as the dominant approach in plant population ecology. These studies include 20 papers on 25 species of annual plants, 194 papers on 189 species of herbaceous perennial plants and 139 papers on 154 species of woody plants. Of the 355 matrix model papers, 341 included enough information to be categorized systematically (see Table 1; our classifications of these papers are also available at: http://knb.ecoinformatics.org/knb/metacat/nceas.961/nceas). These papers, as well as our interpretation of models (Box 1), provide the basis for our systematic review. Table 1. Information assessed in the literature survey. Depending on the survey item, each responder entered specific information or could select from among specified options*. All the data are available at: http://knb.ecoinformatics.org/knb/metacat/nceas.961/nceas General information Specific options or definitions Author and year Maximum # annual transitions Study wide, summed over populations (e.g., 5 populations × 2 matrices per population = 10) Maximum # sites Number of physical sites or populations, as defined by matrices. May be fewer than number of matrices if multiple matrices are constructed for a site (e.g., different treatments or years at one location) Species name(s) Life form Woody, herbaceous perennial or annual Overall objective if management (one only) Management of rare/endangered/at‐risk spp.†, management of invasive pest species, other management (e.g., harvest)† Overall objective if no management (one only) Basic research in invasion/conservation biology, life history/population ecology, community ecology or other basic research Additional management objectives (many possible) Population status/viability for species of conservation concern, possible invasiveness of non‐native species, prioritizing populations for protection or control, assessing impacts of species interactions (e.g., herbivory, competition, pollination) on population viability, assessing impacts of biological control, assessing impacts of interactions with invasive species on native (plant) species, assessing impacts of grazing on native plants, other species interactions, evaluate sustainable harvest; link population viability to specific abiotic environmental variables [climate change, fires, other extreme disturbances (hurricanes, floods, drought)]; evaluate life stages to target for management (e.g., LTRE, sensitivity/elasticity analysis); identify which of multiple threats is more important to viability; assess effects management options (numerical experiments; simulation study); assess effects management options (actual experiments/natural experiments/monitoring); monitor establishment of translocated plants/viability of restored populations Metrics presented in the publication Estimates of DETERMINISTIC population growth rate, estimates of STOCHASTIC population growth rate, estimates of extinction risk, sensitivity/elasticity, LTRE, transient behaviour of populations, optimal return intervals/inflection points for periodic disturbances Author’s interpretation of results (choose one only)‡ 1. Authors explicitly state that quantitative results SHOULD NOT be taken literally 2. Authors lean away from/tend to de‐emphasize a literal interpretation of results 3. Unclear whether interpretation is qualitative (not literal) or quantitative (literal) 4. Authors lean towards/tend to emphasize a literal interpretation of results 5. Authors explicitly state that quantitative results SHOULD be taken literally Uses of matrix models First, we categorized the overall goals and objectives of each published paper. Half (170) of the studies used matrix models for basic research and half (171) used matrix models to address management questions. Through time, however, the use of matrix models has shifted from basic to management questions (logistic regression of the proportion of studies directed at management questions: β = 0.064, N = 341 studies, d.f. = 1, Z = −5.57, P = 0.0003). Most applications to basic research (78%) used matrix models to study life history and population ecology. A substantial minority (11%) of basic research studies used matrix models for conservation biology or invasion biology, such as studies of effects of habitat fragmentation, studies of the rate of spread of non‐native species or studies of how to best construct population viability models. This proportion has also increased through time (logistic regression: β = 0.108, n = 170 basic research studies, d.f. = 1, Ζ = 2.02, P = 0.0165). Management studies most frequently modelled at‐risk (38%), harvested (26%) and invasive (18%) plant populations; matrix models were also occasionally used to assess management of populations affected by fire (9%), grazing (6%) or restoration (3%) (Fig. 2). Goals of application to management have tended to shift through time, primarily due to increases in the proportions of studies investigating at‐risk and invasive species (multinomial logistic regression, χ2 = 23.9, d.f. = 6, P < 0.001). For management‐oriented papers, we also recorded the ways in which models were used to guide management (‘additional management objectives’ in Table 1). Forty‐six per cent of papers measured some aspect of population status or population viability (defined broadly to include trends in population size over time as well as extinction risk per se; we discuss this issue further, below), 40% used matrix models to identify which life stages were most important to target for management and 27% used models to conduct simulation ‘experiments’ asking what‐if questions about possible consequences of changes in vital rates, species interactions or the environment. Studies that made explicit relationships between vital rates and environmental factors were less common: 28% of studies used models to relate population dynamics to environmental drivers, such as fires, floods or climate factors, 19% of studies used demographic models to interpret consequences of short‐term experiments and 8% of studies explicitly compared status of different sites or populations. Only 16% of studies used matrix models to explore consequences of species interactions; half of these looked at some aspect of biological control of invasive species. Figure 2 Open in figure viewer PowerPoint Number of matrix model papers, in relation to study objectives and model interpretation. Basic research studies are plotted on the left horizontal axis and management studies are plotted on the right axis. Spatial and temporal extent of data in matrix models Menges (2000a) and Morris & Doak (2002) have previously noted that matrix models for plants tend to be based on relatively few years of data, across relatively few populations and over relatively small proportions of species’ ranges. For our census of matrix models, we recorded the number of distinct matrices and the number of sites or populations in each study. For studies that included only one site and treatment, the number of matrices is a measure of study length. Thirty‐four percent of matrix model papers for plant populations included only one projection matrix and 48% used data from only one site and/or treatment. The mean number of matrices per site or treatment was two; this is partly because one‐third of the studies combined data from multiple sites to obtain one matrix, hence had < 1 matrix per site. Studies including data from only one site or treatment had a mean of 3.1 annual projection matrices (SD = 3.0, range 1–15). There has been some increase over time in the number of matrices per study, although this trend was only marginally statistically significant (Table 2A), and was largely driven by management‐oriented studies (Fig. 3). There was no trend over time in the number of sites or treatments per study (Table 2B). Table 2. Analysis of data used in matrix models, in relation to study goals and year of publication Metric Factor Estimate* SE Z† P A. Number of matrices Management 0.145 0.096 1.68 0.093 Fire 0.357 0.206 1.73 0.084 Grazing 0.256 0.242 1.06 0.288 Harvest −0.279 0.139 −2.00 0.045 Invasion −0.045 0.166 −0.27 0.787 Restoration 0.313 0.314 1.00 0.318 Viability 0.353 0.105 3.35 0.001 Year 0.009 0.006 1.44 0.151 B. Number of sites or treatments Management 0.085 0.099 0.83 0.259 Fire 0.477 0.229 2.18 0.029 Grazing 0.190 0.272 0.70 0.484 Harvest −0.336 0.156 −2.14 0.032 Invasion −0.198 0.192 −1.03 0.302 Restoration 0.703 0.349 2.02 0.044 Viability 0.226 0.125 1.80 0.072 Year 0.007 0.001 1.00 0.318 Figure 3 Open in figure viewer PowerPoint Temporal trends in the number of matrices used to construct matrix models in the peer‐reviewed literature for (a) basic research studies and (b) management studies. For this figure, studies are separated into stochastic models, defined as studies that reported either λ S or extinction risk, vs. deterministic models (all other studies). Lines are fitted regression curves for each subgroup (negative binomial generalized linear model). For visual clarity, points are slightly jittered, and one study is not shown on the graph, a stochastic, management‐oriented study with 28 matrices, published in 2005. Management‐oriented studies tended to draw on more matrices than basic research studies, even after accounting for trends through time (Table 2A; Fig. 3). However, the number of matrices used also differed among management studies with different goals. Relative to basic research, studies focused on population viability and, to a lesser extent, fire management, used more matrices, and studies focused on harvest used fewer matrices (Table 2A). The number of sites did not differ significantly between management‐oriented and basic research studies, but did differ among management goals. Studies requiring an understanding of changing dynamics due to disturbance and/or recovery (fire, restoration and at‐risk species) included more sites, whereas studies focused on harvest had fewer sites (Table 2B).

Box 1 Self‐survey The peer‐reviewed literature does not necessarily include all of the ways in which models are useful in practice. For example, not all models are published in peer‐reviewed journals. Similarly, the review process is known to bias the results that get published, and could also bias the ways in which authors present their results. We evaluated our own opinions of how matrix models are most useful and reliable, and how much we agree on the strengths, limitations and most important improvements for matrix models. Three of the authors (Crone, Menges and Ellis) assembled a working group of scientists (the authors of this paper) of varying backgrounds, but with these commonalities. All of us had: collected demographic data on plant populations, assembled projection matrices, analysed and/or modelled population dynamics based on these matrices and published their results and discussed the implications of this modelling with land managers and other conservation professionals. All literature reviews include some element of the authors’ opinions; by presenting ours quantitatively, we identify (at least in part) where we agree and where there is little consensus, in spite of shared experiences and background. We quantified our opinions using a combination of survey and discussion. This survey assayed each person’s view of how well matrix models would predict population dynamic and management metrics such as growth rates and extinction probabilities (Fig. 5). Following standard risk assessment procedures for eliciting expert opinion (Burgman 2005), we began with an initial ‘naive’ survey, meant to help us articulate our own opinions. We then discussed areas where our answers differed, and the rationale behind our opinions. After group discussion, individuals retook the same survey. The results of this second survey are expected to identify true areas of agreement and disagreement, as opposed to individual differences in the interpretation of survey language. The survey questions (Fig. 5) focused on subjective model interpretation in terms of how we think models are most useful (a) and how they might be best improved (b). Our responses show some areas of clear consensus [see Fig. 5a.8, b.1 (extinction risk) and b.5 (extinction risk)]. However, we were divided on many issues. Overall, we tended to agree more about how models are best interpreted as generally applied (a) than about why they fail (a measure of how they should be improved) (b).

Concluding remarks To an outsider, it might seem puzzling that plant ecologists continue to rely on matrix models, more often than not with the explicit caveat that output from these models should not be taken literally. Why do we use this technique if we do not trust model predictions? Perhaps the most important message from this apparent contradiction is that the primary function of models is not to forecast the future. Because they are based strongly in basic biology, matrix models force us to outline how different aspects of the life cycle interact, and what aspects of natural history are known vs. unknown. It is tempting to speculate that the overwhelming popularity of matrix models in plant ecology stems from the fact that these models take biology that we can see and measure in the short term (stage‐specific birth, growth and death rates), and calculate the logical long‐term consequences of that biology. Setting up matrix models also emphasizes that management of one life stage, or of one vital rate, only affects population dynamics through its effect on the full life cycle. For example, a key and longstanding insight from matrix models is that not all vital rates are equally important for population dynamics or fitness (e.g., Crouse et al. 1987; Crone 2001). Therefore, users of matrix models can calculate the logical outcomes of observed environmental conditions and experimental manipulations, and use models to examine what‐if scenarios that explore possibilities outside the range of observed conditions. These benefits do not require that matrix models make accurate predictions, simply that they integrate field data in a biologically meaningful way. Model predictions are also likely to be more robust and less biased than management assessments based on subjective judgment (McCarthy et al. 2004). A second insight from the state‐of‐the‐art of plant demography is that plant ecologists define ‘population viability’ broadly as the future fate of plant populations (Menges 2000a). In contrast, most previous assessments of population viability analysis have focused narrowly on testing our ability to forecast extinction risk (Brook et al. 2000; Coulson et al. 2001; Ellner et al. 2002; Holmes et al. 2007; Ellner & Holmes 2008). These reviews include very few data for plant populations, so it may be that animal ecologists – for whom it is easier to define ‘individual’ organisms and who may also tend to draw on longer data sets for fewer species – take the concept of extinction risk much more literally than do plant ecologists. This does not mean that plant ecologists are not concerned about extinction. Rather, we tend to use population growth rates, rather than extinction risk per se, as a metric of relative viability: rapidly declining populations are likely to become extinct (even if they are large now), rapidly growing populations are likely to expand until resources and space limit growth, and populations near λ = 1 require more biological understanding to accurately predict their fates. Expected minimum population sizes (modified from quasi‐extinction calculations) may be an equally robust or superior metric of relative population viability, at least over short time frames. However, we do not expect that plant ecologists will be able to calculate or precisely predict extinction risk for typical plant populations. Our review revealed considerable variation in how much we trust the predictions of matrix models, both among published studies and within our working group. Most of us felt that models should be approached in a relative/comparative way, but a substantial minority of papers present results as literal predictions, and there has been no trend away from literal model interpretation through time. If we intend to use models to make predictions that are as accurate as possible, then it is also disturbing that recent studies also do not make use of the most sophisticated tools available. For example, methods for transient analysis have been around for decades, and they are likely to be relevant at the time scales over which we evaluate management, but these are rarely used in practice. However, it may be plant ecologists use matrix models specifically because these models lead to simple metrics that are consistent across studies, such as the asymptotic population growth rate, λ. These simple metrics are often omitted from or buried in more sophisticated analyses. Given limited data, it is also not clear that making quantitative predictions is realistic in most cases, even with the best models. Therefore, rather than recommending use of more sophisticated models, many (but not all) of us tend towards recommending more caution regarding literal interpretation of model outputs. Finally, in spite of differences of opinion in some areas (Box 1), we agreed strongly that better understanding of environmental drivers of population dynamics would be likely to substantially improve models and model predictions. Correct understanding of drivers of stochasticity is also central to expectations for long‐term population dynamics and evolution. Use of stochastic population models is growing in plant ecology, but these rely on very short data sets, and stochasticity is not often linked to environmental factors. For example, a number of recent studies have begun to use published matrix models for plant populations to explore patterns of stochasticity in natural populations (Morris et al. 2008; Jongejans et al. 2010; Buckley et al. 2010). Because most published matrix models are for very short time series, these comparative analyses use studies with as few as two or three estimates of each vital rate for stochastic modelling. In this context, our review of plant demography implies two key needs for future research. First, we need better study designs and more long‐term demographic data, particularly in relation to drivers of variation in vital rates. This call cannot be repeated too often, but it is also important to remember that more detailed and long‐term studies may come at a cost of breadth across populations and species. Our review shows, at best, modest increases study length through time (Table 2; Fig. 3). Although we have been conducting demographic studies for decades, plant ecologists have tended to accumulate short‐term studies for more species, rather than longer term studies. Some constraints to study length are obvious, but others are probably much less so. In practice, we will always have more short‐term studies than long‐term studies. Therefore, the second need is to investigate critically how we can use these short‐term studies. We need to focus on developing methods and tests for dealing with and quantifying the limitations of shorter term studies, as well as emphasizing the need for long‐term data. It would also be interesting to know whether we can use short‐term studies from hundreds of different species to understand long‐term stochastic population dynamics.

Acknowledgements This work was conducted as a part of the ‘Testing Matrix Models’ Working Group supported by the National Center for Ecological Analysis and Synthesis, a centre funded by NSF (Grant #EF‐0553768), the University of California, Santa Barbara, and the State of California. We would also like to thank three anonymous referees, and, especially, numerous land managers and other conservation professionals for crucial interactions over many years that have profoundly affected how we view the utility of matrix projection models for management.