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Keywords:

  • capture–recapture models;
  • closed population models;
  • plant census;
  • plant demography

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  • 1
    Most plant demographic studies follow marked individuals in permanent plots. Plots tend to be small, so detectability is assumed to be one for every individual. However, detectability could be affected by factors such as plant traits, time, space, observer, previous detection, biotic interactions, and especially by life-state.
  • 2
    We used a double-observer survey and closed population capture–recapture modelling to estimate state-specific detectability of the orchid Cleistes bifaria in a long-term study plot of 41.2 m2. Based on AICc model selection, detectability was different for each life-state and for tagged vs. previously untagged plants. There were no differences in detectability between the two observers.
  • 3
    Detectability estimates (SE) for one-leaf vegetative, two-leaf vegetative, and flowering/fruiting states correlated with mean size of these states and were 0.76 (0.05), 0.92 (0.06), and 1 (0.00), respectively, for previously tagged plants, and 0.84 (0.08), 0.75 (0.22), and 0 (0.00), respectively, for previously untagged plants. (We had insufficient data to obtain a satisfactory estimate of previously untagged flowering plants).
  • 4
    Our estimates are for a medium-sized plant in a small and intensively surveyed plot. It is possible that detectability is even lower for larger plots and smaller plants or smaller life-states (e.g. seedlings) and that detectabilities < 1 are widespread in plant demographic studies.
  • 5
    State-dependent detectabilities are especially worrying since they will lead to a size- or state-biased sample from the study plot. Failure to incorporate detectability into demographic estimation methods introduces a bias into most estimates of population parameters such as fecundity, recruitment, mortality, and transition rates between life-states. We illustrate this by a simple example using a matrix model, where a hypothetical population was stable but, due to imperfect detection, wrongly projected to be declining at a rate of 8% per year.
  • 6
    Almost all plant demographic studies are based on models for discrete states. State and size are important predictors both for demographic rates and detectability. We suggest that even in studies based on small plots, state- or size-specific detectability should be estimated at least at some point to avoid biased inference about the dynamics of the population sampled.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Demography is the key to understanding spatial and temporal variation in abundance of plants. Longitudinal studies yield the most detailed knowledge. Most often, the plant life cycle is divided into discrete states, and marked or mapped plants in permanent plots are followed through time (e.g. Tamm 1972; Hutchings 1987; Oostermeijer et al. 1996; Pascarella & Horvitz 1998). The sample of marked plants in a plot is assumed to be representative of the entire population studied. Once a year or at some other appropriate time interval all individuals in the plots are ‘censused’. The presence or absence of each individual, and attributes such as life-state, size, and number of fruits are then recorded. This permits estimation of survival and recruitment rates, and transition rates between life-states, which may be integrated into matrix population models to project the asymptotic population dynamics (Caswell 2001).

Cross-sectional studies may compare several populations at one point in time. Population structure is often sampled in plots thought to be representative of the entire population. A snap-shot of population structure (e.g. the numbers of seedling, vegetative, and flowering plants), may contain useful information on population dynamics. For instance, Oostermeijer et al. (1994) surveyed population structure of the rare Gentiana pneumonanthe in the Netherlands and distinguished three types of populations. ‘Dynamic’ populations had a large proportion of young life-states (seedlings and juveniles), ‘senile’ populations contained only adult states, while ‘stable’ populations contained members of all life-states. There was a strong positive relationship between the percent of bare soil surface and litter cover and the density of seedlings and juveniles. This finding indicated that the status of a population could be assessed by one quick visit to a site.

Plant demographic plots are often small (0.1–100 m2); so most studies implicitly assume that every individual is indeed found, i.e. that detectability equals 1. However, detectability even in small plots could be affected by various factors. Possible sources of variation in detectability are the size of a plant, its life-state, and phenologic state in the reproductive cycle (e.g. individuals with flower buds, flowers in anthesis, and fruits). Furthermore, its previous ‘detection history’ may affect its detectability. An individual recorded in earlier surveys may be less likely to be overlooked than a new recruit, of whose location the demographer has no prior knowledge. There may be temporal variation in detectability associated with varying light conditions or with changes in the vegetation matrix. Often there is small-scale heterogeneity in cover, structure and size of the vegetation matrix in a plot, and this could influence detectability. Detectability is likely to be different for different observers and is likely to be a function of experience for a particular observer. Finally, biotic interactions such as herbivory may decrease or increase detectability, e.g. by reducing the size of a plant or causing it to wilt and turn brown so that it becomes more conspicuous in surrounding green vegetation.

Detection probabilities < 1 introduce a bias into most estimates of population parameters such as abundance, recruitment, mortality rates, and transition rates between life-states. If they are not accounted for they can produce misleading conclusions from a plot-based demographic study. We believe that the assumption that every plant within a plot is found is an important one that should be tested. If it is found to be false, then estimation of demographic parameters must account for detection probabilities, as in multistate capture–recapture models (Nichols & Kendall 1995).

The aim of this study is twofold: (1) to provide one counterexample to the widespread implicit assumption in virtually all plant demographic studies that every plant is always found, and (2) to show that detectability can readily be estimated within the rigorous statistical framework of capture–recapture models. We illustrate this by an example using a double-observer approach. We identify the factors that affect detectability in the medium-sized terrestrial orchid Cleistes bifaria, and draw attention to the potential biases incurred when one wrongly assumes that detectability equals 1. We believe that we point out a widespread issue for plant demography and show how it can be addressed. We further believe that our approach, results, and conclusions are relevant to plant demographic studies in which parameters such as survival and recruitment rates or population growth rates are estimated from a sample of marked individuals or from simple counts of detected plants. We emphasize that capture–recapture models are useful not only for detecting variable detection probabilities but also for developing models to properly estimate demographic parameters in the face of such variation.

Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

study species

Cleistes bifaria Catling and Gregg (Catling & Gregg 1992) is a perennial, self-compatible but naturally outcrossing, bee-pollinated orchid (Gregg 1989) that is widely distributed in the SE United States (Luer 1975). It occurs in a variety of acidic habitats including pine savannahs, meadows, and open oak-pine forests, at elevations from sea-level to 1000 m. In West Virginia, C. bifaria flowers in late June to early July.

C. bifaria reproduces sexually by seeds and asexually by root shoots; hence the ramet was the demographic unit in our study (we use this term interchangeably with ‘plant’). Ramets could assume five states: vegetative with one, two or occasionally three leaves, flowering/fruiting, or without above-ground parts. Mean ramet heights (SD) measured in June 2002 for a sample of 140 one-leaf, 74 two-leaf, and 51 flowering plants were 16.2 cm (5.8), 23.3 cm (5.0), and 37.5 cm (6.0), respectively. Vegetative plants are similar in appearance to flowering ones but smaller.

study site

We conducted the study at Beavers’ Meadow, a flat, acidic, seasonally wet, c. 9-ha meadow in NE Barbour County, West Virginia (39°16′45″N, 79°55′58″W). Mowed annually in late summer or early fall, the meadow is dominated by grasses, with scattered, almost pure patches of shrubs and occasional saplings from the adjacent forest (Gregg 1989). Throughout the plot there is a ground cover of Rubus hispidus and numerous stems of Smilax glauca. Although the vegetation in the plot is not particularly dense, stems of Cleistes often grow underneath the cover of Pteridium aquilinum (bracken fern), Vaccinium vacillans (late low blueberry), and Baptisia tinctoria (wild indigo).

background of the demographic study

From 1990 to 1999, KBG annually conducted an intensive survey of an 83.7-m2 plot at fruiting time (July–September) to locate all ramets of C. bifaria, and mapped their location within the plot. She marked each ramet with numbered tags and recorded life-state (one-leaf vegetative, two-leaf vegetative, flowering, or below-ground), fruiting, and mortality. Plastic tags (approx. 4 × 0.75 cm of various colours) were pushed into the soil beside each ramet with about 1 cm left above ground. An attempt was made to locate all previously marked ramets as well as to discover new recruits.

survey to estimate detectability

On 8–9 September 2001, we conducted a double-observer survey to estimate detectability for C. bifaria in our long-term survey plot. Two observers each made an independent survey of the same area. We were especially interested in estimating detectability for observer 1 (KBG) for the purpose of demographic estimation and modelling of the 1990–99 survey data. Observer 2 (MK) had several years of experience in plot-based plant demographic studies, but his experience with C. bifaria was limited to a 1-hour training lesson before the survey. We selected the southern half (41.2 m2) of the original long-term survey plot. All previously marked ramets were still tagged from the earlier studies.

On the first day, the plot was surveyed by observer 1 during 4 h 35 min and on the second, by observer 2 during 3 h 5 min. The weather was sunny and cloudless on both days. For previously marked ramets, we recorded tag number and life-state (one-leaf, two-leaf, and flowering/fruiting). Ramets without a tag (untagged plants of previously unknown location) were tagged in the same manner as in previous years (Gregg 1991).

For each individual of C. bifaria, the survey gave rise to one of three possible detection histories, 11, 10, or 01 (Table 1), where 1 denotes detection and 0 non-detection, and the first numeral refers to KBG, and the second, to MK. These data allowed us to estimate detectability of C. bifaria plants in the study plot and test for differences related to observer, life-state and previous tagging history.

Table 1.  Numbers of Cleistes bifaria individuals per detection history and combination of the factors life-state and tagging history (stratum). The first numeral in detection history refers to detection (1) or non-detection (0) by observer 1; the second numeral refers to observer 2
Detection historyStratum j, kTotal
Untagged 1Untagged 2untagged FTagged 1Tagged 2Tagged F
  1. ‘Untagged’ denotes plants that were found during this survey; ‘tagged’ denotes plants that had already been tagged during the demographic surveys conducted in the same plot 1990–99. 1, 2, and F denote the life-states one-leaf, two-leaf vegetative, and flowering.

01 411 7 0013
10 40014 2020
11203025 8965
Total28414610998

model selection and estimation

We analysed the survey with closed population capture–recapture models (Otis et al. 1978) that contain parameters for detectability (p) and abundance (N). For each combination of state and tagging history (stratum), the expected numbers of detection histories can be written as functions of the model parameters. Maximum-likelihood estimates of parameters are obtained using the observed numbers of each detection history and the associated cell probabilities.

Let j = life-state and k = tag status prior to the study, with 0 denoting previously untagged plants, and 1 previously tagged plants. The data consist of inline image = number of plants of state j and previous tagging status k that were detected by both observer 1 and observer 2, inline image number of plants in j and k that were detected by observer 1 but not by observer 2, andinline image= number of plants in j and k that were not detected by observer 1 but detected by observer 2.

We writeinline image= detectability for observer 1 of j and k, inline image= detectability for observer 2 of j and all k combined, and Njk = numbers of plants detected in j and k. For our most general model, the xjk can then be viewed as random variables with expectations

  • image
  • image
  • image

We term this model the independence model, because parameter p1 is independent of parameter p2. Modelling consisted of further reducing the complexity of this model by constraining some of the parameters p to be equal. For instance, a model may be specified where parameter p is the same for one-leaf and two-leaf plants, and another, where it is allowed to be different between the groups. A comparison of the fit of these two models is equivalent to testing the hypothesis that one-leaf and two-leaf plants have the same detectability.

We based model selection on Akaike's information criterion (AIC) to identify the best model in a set of a priori candidates (Anderson & Burnham 1999). AIC is calculated as minus twice the maximized log-likelihood of a model plus twice its number of parameters (Anderson & Burnham 1999). AIC is an information theoretic optimality criterion that trades off bias (decreasing with the number of parameters in a model) with variance (increasing with the number of parameters in a model) to identify a most parsimonious model with a minimal value of AIC. AIC allows the simultaneous comparison and ranking of multiple models, including non-nested ones. In contrast to significance tests, AIC avoids the problems encountered when testing multiple and non-nested models (Burnham & Anderson 1998). We used AICc (Anderson & Burnham 1999; Burnham & Anderson 1998), a small-sample version of AIC. For each model, AICc is a dimensionless number and only defined up to a constant. Thus, to allow a better comparison between the models, we present ΔAICc. For each model, this is the distance in AICc units from the AICc value of the best model, which has by definition the lowest AICc and therefore ΔAICc = 0.

The likelihood of a model can be expressed as exp(–ΔAICc/2) (Burnham & Anderson 2001). Normalizing these likelihoods such that they sum to 1 over the set of models considered yields the Akaike weight wi for each model i. Akaike weights measure the relative support by the data for each model and can be used to obtain model-averaged estimates of detectability for each life-state. They incorporate model-selection uncertainty (Buckland et al. 1997) and therefore are not conditional on just one particular selected model. Estimates and their standard errors are averages over models weighted by the Akaike weights of each model (Burnham & Anderson 1998).

For AICc to be a valid means of selecting the best among a set of models, at least one of the models (usually the most general) needs to fit the data adequately. However, we had no direct way to test the goodness-of-fit of our most general model. Due to low expectations in some cells, Chi-square could not be applied to compare observed with expected frequencies. Therefore, we added the saturated model to our set of candidate model (J.D. Nichols, pers. comm.). It fits the data by definition, as it has as many parameters as data points and reproduces the data exactly. This means that any other model that is selected by AICc to fit the data better must also fit adequately. Model specification, model selection and model averaging were carried out using the freely available capture–recapture software MARK (White & Burnham 1999).

specific models/hypotheses tested

Based on the biology of C. bifaria and our sampling design, we considered 12 models with combinations of three sources of variation for the detectability of C. bifaria: life-state, tagging history, and observer. Each model represented a hypothesis about how these sources of variation interact to affect the probability of detecting an individual plant. Modelling consisted of finding a constrained version of the independence model that provided the most parsimonious representation of our data. It was achieved by setting detectabilities to be equal across levels of these sources of variation.

Table 2 describes the models considered in terms of their detectability parameters. All models in Table 2 except model 1 contain no tagging history effect for observer 2. The indices of the second p in each cell are equal across untagged and tagged plants for each state. The likelihood of observer 2 finding a plant was estimated regardless of whether a plant was tagged or not. Our interest focused on observer 1 who had conducted the long-term study in the same plot. The sampling protocol was simpler when we did not allow for the estimation of a separate tagging history effect in observer 2.

Table 2.  The set of models considered for the detectability (p) of Cleistes bifaria plants in our survey. All models are constrained versions of the independence model, the expectations of which areinline image,inline image, inline image, where j = life-state and k = tag status prior to the study, and p1 and p2 refer to detectability for observer 1 and 2, respectively. We describe the structure of these constrained models in terms of their detectability parameters. The first and second p in each cell denote detectability of the first (KBG) and the second observer (MK), respectively. Parameters with the same subscript are identical. Model 1 (not shown) is the saturated model. In model 3, c is a constant and means that detectability for observer 1 and tagged plants differs by a constant from that for untagged plants. Strata are named as in Table 1.
Stratum j, kModel
23456789101112
Untagged 1p1,p2p1,p2p1,p2p1,p2p1,p1p1,p2p1,p1p1,p2p1,p2p1,p1p1,p2
Untagged 2p3,p4p3,p4p3,p4p1,p2p2,p2p1,p2p1,p1p1,p2p1,p2p2,p2p3,p4
Untagged Fp5,p6p5,p6p5,p6p1,p2p3,p3p1,p2p1,p1p3,p4p3,p4p3,p3p5,p6
Tagged 1p7,p2(p1 + c),p2p1,p2p3,p2p1,p1p1,p2p1,p1p5,p2p1,p2p4,p1p2,p2
Tagged 2p8,p4(p3 + c),p4p3,p4p3,p2p2,p2p1,p2p1,p1p5,p2p1,p2p5,p2p4,p4
Tagged Fp9,p6(p5 + c),p6p5,p6p3,p2p3,p3p1,p2p1,p1p6,p4p3,p4p6,p3p6,p6

As an example, consider model 9 with an identical detectability for one-leaf and two-leaf vegetative plants. Detectability of vegetative plants untagged prior to our study is p1 for observer 1 and p2 for observer 2. Detectability of vegetative plants that already had a tag prior to our study is p5 for observer 1. Because we never estimated a separate detectability for untagged and tagged plants for observer 2, the indices for that observer are identical across the tagging history in all models except the saturated. Finally, for observer 1, p3 refers to detectability of untagged and p6 of tagged flowering plants, and p4 is the detectability of flowering plants for observer 2 regardless of tagging status.

Model 1 is the most general, fully saturated model. It was included in the model set to ensure that at least one model fitted the data adequately. Model 2 is the most general model that is consistent with our expectations and biological knowledge. Models 3–8 are anova type simplifications of model 2. Model 3 assumes an additive instead of an interactive combination of effects of state and history for observer 1. Model 4 and model 5 assume an observer and a single main effect for state and tagging history, respectively. Models 6 and 7 assume only a state or only an observer effect, respectively. Model 8 is the null model with the same detectability across all factors.

State effects are a function of plant size and architecture. Size increased as one-leaf < two-leaf < flowering plant. A simplification was therefore to constrain the detection probabilities of vegetative plants to be equal and to contrast them with flowering individuals. Models 9 and 10 represent the hypotheses that the vegetative states (one- and two-leaf) do not differ in their detection probabilities. The former assumes a common detectability for the two vegetative states in addition to observer and tagging history effects. As a further simplification, the latter assumes that detectability does not depend on tagging status. To test whether the effect of life-state was mainly due to the different sizes of the life-states as opposed to their architecture, we added a linear size constraint on detectability (model not shown in Tables 2 and 3). Because we did not measure the individual size of plants during our study, we used mean heights calculated from measurements taken on another occasion.

Table 3.  Analysis of the effects of observer, life-state, and tagging history, on the detectability p of Cleistes bifaria plants. Model numbers refer to Table 2. For each model, the deviance, number of estimable parameters (no. params), AICc, ΔAICc (distance in AICc units from the lowest- AICc model) and Akaike weights wi are given
Model no.Terms included in the model and interpretationDevianceNo. paramsAICcΔAICcWeight wi
1Saturated model, reproduces the data, fit perfect 0.0015−281.933.370.06
2Independence model with effects of observer and state and tagging history for observer 1 9.1411−282.023.280.07
3Additive effects of state and tagging history for observer 1 and state effect for observer 217.62 8−280.215.100.03
4State effect for each observer18.38 7−281.623.690.05
5Tagging history effect for each observer24.91 5−279.375.930.02
6State effect identical across observers22.04 5−282.243.060.08
7Observer effect25.91 4−280.484.830.03
8Null model with only the grand mean27.59 3−280.884.430.04
9Same as model 2, but common detectability for one- and two-leaf vegetative plants for observer and tagging history13.89 8−283.931.370.18
10Same as model 9, but no effect of tagging history20.53 6−281.633.680.06
11Two sets of state effects: For each state, observer 1 finds untagged plant with the same probability as observer 2 (test of prior knowledge of location in the tagging history effect)17.22 8−280.614.700.03
12Two sets of state effects: For each state, observer 1 finds tagged plant with the same probability as observer 2 (test of presence of tags in the tagging history effect)12.52 8−285.3100.35

A tagging history effect may consist of two parts: the prior knowledge of the location of plants and the presence of tags that helped to locate a plant. Models 11 and 12 represent hypotheses that make different assumptions about these components. Model 11 represents the hypothesis that prior knowledge is more important. Observer 2 had no prior knowledge about location. Therefore his detectability is set equal to that for untagged plants in observer 1. Model 12 assumes the opposite, that the presence of tags is more important. The presence of tags would affect detectability for tagged plants for observer 1 in a similar way as for observer 2. The model thus assumes that detectabilities for observer 2 and for tagged plants for observer 1 are identical.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The most parsimonious model for detectability of C. bifaria in our study plot was model 12 (Table 3). It had a different set of detectabilities for each life-state for previously tagged vs. previously untagged plants. A comparison of model 12 with the less well fitting model 11 indicated that the tagging history effect was mainly due to the presence of tags and not to prior knowledge of the locations of the plants. Support for model 12 was almost twice as strong as that for the next best model (model 9), as shown by the ratio of the Akaike weights. However, model 9 was within two AICc units and hence, it provided a good approximation to the data as well (Burnham & Anderson 1998). This model suggests that detectability was indeed similar between the two vegetative states.

To test whether the effect of life-state in the most parsimonious model was mainly due to the different size of the life-states, we set detectability as a function of the average size of each life-state. This model fitted almost as well as our best model 12 (ΔAICc = 1.16). Therefore, the life-state effect appears to be largely a size-effect.

Based on the low-AICc model, detectability (followed by SE, where available) for previously tagged C. bifaria plants in the one-leaf, two-leaf vegetative, and flowering/fruiting state was estimated at 0.76 (0.05), 0.92 (0.06), and 1, respectively, for both observers (Fig. 1). For previously untagged plants in the same states, detectability was 0.84 (0.08), 0.75 (0.22), and 0, respectively. The unexpectedly low estimate for untagged flowering plants was due to a single plant that was overlooked. Judging from standard errors, sampling error is the most likely explanation for detectability change in opposite directions for plants that were, and were not, previously tagged.

image

Figure 1. Estimated detectability (95% CI) of Cleistes bifaria plants based on the most parsimonious model in Table 3 (model 12), which had life-state specific effects for previously tagged vs. untagged plants and no differences between observers. Tag1, tag2, and tagF denote previously tagged plants in the one-leaf, two-leaf, and flowering/fruiting state. Untag1, untag2, and untagF denote the same for previously untagged plants. The estimate for both kinds of flowering plants (asterisks) was at the boundary of parameter space, so no CI is available.

Download figure to PowerPoint

To incorporate the uncertainty in model selection into the estimate of detectability, we also obtained the unconditional estimates (Table 4). They support the result from the analysis of the most parsimonious model, which shows that we did not have sufficient information to reliably estimate detectability for untagged flowering plants (estimate for observer 1: 95% CI 0.010–0.947).

Table 4.  Model-averaged estimates of the detectability on a study plot of Cleistes bifaria. Estimates and their standard errors are obtained as weighted average of the estimates of each model with the weighting scheme based on Akaike weights in Table 3 (Burnham & Anderson 1998). Also shown are lower and upper 95% confidence limits
Life-stateTagging statusObserverDetectabilitySELower 95% CLUpper 95% CL
One-leafUntagged10.8260.0720.6400.927
One-leafUntagged20.7740.0640.6240.876
One-leafTagged10.8080.0650.6490.905
One-leafTagged20.7740.0640.6240.876
Two-leafUntagged10.8030.1530.3790.965
Two-leafUntagged20.8610.0920.5800.970
Two-leafTagged10.9090.0700.6580.981
Two-leafTagged20.8570.0880.5920.961
FloweringUntagged10.3000.4000.0100.947
FloweringUntagged20.9800.0430.5330.999
FloweringTagged10.9690.0530.4960.999
FloweringTagged20.9380.0980.3540.998

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Detectability for individuals of the orchid C. bifaria in a permanent plot used for a long-term demographic study was life-state dependent, and averaged only 0.82 for vegetative plants. As expected, detectability was almost 1 for the more conspicuous flowering plants, but even they were sometimes overlooked.

Only two studies have previously estimated detectability in plants. They did so at much larger spatial scales and for different demographic units. Detectability for patches of the milkweed Asclepias meadii in a 4.5-ha prairie patch averaged only 0.25 (Alexander et al. 1997). For clumps of the relatively conspicuous orchid Cypripedium calceolus var. parviflorum in an area of 0.044 ha, detectability was 0.91 (Shefferson et al. 2001).

Our study shows that even under excellent survey conditions (moderately large herbaceous plant, small, flat plot with relatively open vegetation), not all plants may be found. This may be typical for plant demographic studies. The least conspicuous life-state of C. bifaria, one-leaf plants, averaged 16.2 cm in height and was thus reasonably tall. Many demographic studies include seedlings and other much smaller states that probably have a much smaller detectability. We believe that an implicit assumption made by plant demographers, that all plants in a plot are always found, may often be wrong.

This gives cause for concern, because in demographic estimation detectability < 1 leads to biased estimates of survival probabilities, state-transition probabilities, and state-specific numbers of organisms (Nichols & Pollock 1983; Williams et al. 2002). Perhaps more important than simple bias is the possibility that covariate relationships may be determined by relationships involving detectability rather than the estimated demographic parameter. For instance, part of the positive relationship between percentage of bare soil surface and recruitment in populations of Gentiana pneumonanthe (Oostermeijer et al. 1994) and Salvia pratensis (Hegland et al. 2001) may perhaps have been due to the fact that fewer seedlings could have been overlooked where there was more bare ground.

We illustrate the potential impact of ignoring detectability < 1 on the estimation of demographic parameters with the following hypothetical example. Consider a stable population of a theoretical plant species with three life-states, one- and two-leaf vegetative, and flowering plants. There is no seed bank. One-leaf plants become two-leaf plants the following year, flower, or die. Two-leaf plants either survive in that state, flower, or die. Flowering plants either survive in that state, become two-leaf plants, produce one-leaf plants via seed, or die. Recruitment can therefore be estimated by the ratio of the number of one-leaf plants in year 2 and the number of flowering plants in year 1. Survival and transition rates can be estimated by recording the fate of marked individuals in successive years. Matrix A describes the true dynamics of this population (Table 5a).

Table 5.  Illustration of the potential impact on a matrix model of imperfect detectability for a stable population of a hypothetical plant species with three states and no seed bank. (a) True transition matrix A (λ = 1.00). (b) Apparent transition matrix A′ (λ = 0.92). See text for derivation of A′
(a)One-leafTwo-leafFlowering
One-leaf000.3
Two-leaf0.40.30.2
Flowering0.240.50.7
(b)One-leafTwo-leafFlowering
One-leaf000.24
Two-leaf0.360.270.18
Flowering0.230.470.66

We then simulated the effect of imperfect detectability on the inference from a matrix model based on a 2-year survey. Levels of detectability were set at similar values to those recorded for tagged plants in our C. bifaria plot (0.77, 0.89, and 0.95, respectively, for one-leaf, two-leaf, and flowering plants).

The apparent transition rates are obtained as follows (J.D. Nichols, pers. comm.). The expectation of the apparent state transition rate from j in the first year to k in the second year equalsinline image, whereinline imageis the number of plants that are in state j in the first year, inline imageis the number among these that are in state k in the second year, and pk is the detectability of a plant in state k. The expectation of the apparent recruitment rate equals inline image, whereinline imageis the number of flowering plants in the first year,inline imageis the number of one-leaf plants in the second year, and pflowering and psmall are the respective detection probabilities of a plant in these states. Matrix A′ (Table 5b) contains the apparent transition rates for our hypothetical population.

The effect of imperfect detection can be seen by comparing the true transition matrix A with the apparent transition matrix A′. The hypothetical population was chosen to be stable (λ = 1.00). However, from matrix A′, it would be inappropriately concluded that the population is in strong decline (λ = 0.92). Elasticity estimates were biased on average by 6.4%. Imperfect detection in plant demographic studies may thus be a factor leading to underestimation of the true population growth rates based on matrix models. This may have been a reason why matrix models of three populations of the biennial forb Gentianella germanica projected precipitous declines, when in fact the number of flowering plants over 3 and 4 years remained stable or even increased (Kéry 2000).

There is a further bias that we have not addressed in this study. The sample of detected plants is unlikely to be representative of all plants in a plot. Detectability of C. bifaria was lower for smaller plants. However, plant size is usually correlated with survival, fecundity, and transition probability between states. The plants detected in a plot will therefore be biased towards larger and more conspicuous life-states. The unseen fraction of plants in a plot will differ from the detected plants in fecundity and recruitment. Then even when the re-detectability is high (i.e. when most marked plants are found), estimates will be biased, because the marked plants constitute a size-biased sample from all the plants present in a plot. Imperfect detection always distorts estimates of demographic parameters to some degree. Conclusions from demographic models (e.g. matrix population models) built with multiple estimates, each of which suffers a bias due to detection probabilities < 1, may therefore be even more strongly biased.

Some might claim that it is sufficient to standardize the survey protocol across time and space and increase survey effort. They would then be prepared to make the untested assumption that detectability equals 1. We argue that this is not a valid solution. Rather, the onus should be on those that make the assumption that their detectability is essentially equal to 1 to produce evidence based on a study such as ours to support their claim. Capture–recapture models are designed to achieve this.

Although we used two observers, a richer set of hypotheses about sources of variation in detectability can be formulated and tested with more observers. Care, however, has to be taken not to disturb the vegetation, as this might change both the demography and presumably also the detectability of a study species (Cahill et al. 2002). Possible factors whose effects can be estimated include observers, survey methods, and spatial and temporal differences between surveys. Where demographic surveys are carried out by multiple observers, it will be desirable to estimate detectability for each of them. A priori hypotheses about how observers differ in detectability could be readily tested. For instance, observer differences might only be relevant for small life-states and vegetative plants and not for flowering plants. Constraints could be used to achieve more parsimony. For instance, detectability could be modelled as a linear function of plant height. Huggins’ model (1989; 1991) could also be used to directly model the relationship between detectability and plant height without taking account of plant life-states. However, most plant demographic work is based on models for discrete life-states rather than models for continuous size. Therefore, we retained the life-state distinction in all models. The animal ecology literature is rich in references to capture–recapture models, many of which are useful for plant ecologists (e.g. Otis et al. 1978; Nichols et al. 1986; Williams et al. 2002).

When nothing is known about possible factors related to detectability, models that allow for unspecified heterogeneity of detectability between individuals may be used, such as the estimators for model Mh implemented in program CAPTURE (Otis et al. 1978), or the finite mixture estimators of Pledger (2000) that can be fitted using program MARK (White & Burnham 1999).

In conclusion, although plants stand still and wait to be counted (Harper 1977), they may sometimes hide. The problem will be greater for smaller and less conspicuous species or life-states and for larger plots. However, even in small-scale plot studies of conspicuous plants, detectability should not be assumed to equal 1. If this assumption is not met, estimates of demographic rates are bound to be distorted. Capture–recapture models offer a flexible framework to estimate detectability and to test what factors affect it. We suggest that plant demographers should estimate detection probabilities at least once during their study. Then, if detection probabilities are indeed < 1 for some states, it is possible to adjust demographic models of survival and recruitment processes.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

We greatly appreciate valuable discussions with Jim Nichols at all stages of the study. Jim Hines prepared Fig. 1 and helped with the implementation of the models in program MARK. We thank Michael Schaub, Markus Fischer, Gerard Oostermeijer, Norman Slade, Mike Runge, Michael Hutchings, Pieter Zuidema, and two anonymous reviewers for valuable comments on earlier versions of the manuscript. Mrs Jack Beavers allowed us to work on her land. Our sincere thanks go to them all. MK was supported by a grant of the Swiss National Science Foundation (81ZH-64044).

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  • Alexander, H.M., Slade, N.A. & Kettle, W.D. (1997) Application of mark-recapture models to the estimation of the population size of plants. Ecology, 78, 12301237.
  • Anderson, D.R. & Burnham, K.P. (1999) Understanding information criteria for selection among capture-recapture or ring recovery models. Bird Study, 46 (Suppl.), S14– S21.
  • Buckland, S.T., Anderson, D.R. & Augustin, N.H. (1997) Model selection: an integral part of inference. Biometrics, 53, 603618.
  • Burnham, K.P. & Anderson, D.R. (1998) Model Selection and Inference. Springer, New York.
  • Burnham, K.P. & Anderson, D.R. (2001) Kullback-Leibler information as a basis for strong inference in ecological studies. Wildlife Research, 28, 111119.
  • Cahill, J.F. Jr, Castelli, J.P. & Casper, B.B. (2002) Separate effects of human visitation and touch on plant growth and herbivory in an old-field community. American Journal of Botany, 89, 14011409.
  • Caswell, H. (2001) Matrix Population Models, 2nd edn. Sinauer Associates, Sunderland MA.
  • Catling, P.M. & Gregg, K.B. (1992) Systematics of the genus Cleistes in North America. Lindleyana, 7, 5773.
  • Gregg, K.B. (1989) Reproductive biology of the orchid Cleistes divaricata (L.) Ames var. bifaria Fernald growing in a West Virginia meadow. Castanea, 54, 5778.
  • Gregg, K.B. (1991) Variation in behaviour of four populations of the orchid Cleistes Divaricata, an assessment using transition matrix models. Population ecology of terrestrial orchids (eds T.C.E.Wells & J.H.Willems), pp. 139159. SPB Academic Publishing, The Hague.
  • Harper, J.L. (1977) Population Biology of Plants. Academic Press, London.
  • Hegland, S.J., Van Leeuwen, M. & Oostermeijer, J.G.B. (2001) Population structure of Salvia pratensis in relation to vegetation and management of Dutch dry floodplain grasslands. Journal of Applied Ecology, 38, 12771289.
  • Huggins, R.M. (1989) On the statistical analysis of capture experiments. Biometrika, 76, 133140.
  • Huggins, R.M. (1991) Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics, 47, 725732.
  • Hutchings, M.J. (1987) The population biology of the early spider orchid, Ophrys sphegodes Mill. I. A demographic study from 1975 to 1984. Journal of Ecology, 75, 711727.
  • Kéry, M. (2000) Population dynamics of the rare plant Gentianella germanica, Ecology of Small Populations, PhD thesis, pp. 125160, University of Zurich, Switzerland.
  • Luer, C.A. (1975) The native orchids of the United States and Canada, excluding Florida. New York Botanical Garden, New York.
  • Nichols, J.D. & Kendall, W.L. (1995) The use of multistate capture-recapture models to address questions in evolutionary ecology. Journal of Applied Statistics, 22, 835846.
  • Nichols, J.D. & Pollock, K.H. (1983) Estimation methodology in contemporary small mammal capture-recapture studies. Journal of Mammologyalogy, 64, 253260.
  • Nichols, J.D., Tomlinson, R.E. & Waggerman, G. (1986) Estimating nest detection probabilities for white-winged dove nest transects in Tamaulipas, Mexico. Auk, 103, 825828.
  • Oostermeijer, J.G.B., Brugman, M.L., De Boer, E.R. & Den Nijs, H.C.M. (1996) Temporal and spatial variation in the demography of Gentiana pneumomanthe, a rare perennial herb. Journal of Ecology, 84, 152166.
  • Oostermeijer, J.G.B., Van’t Veer, R. & Den Nijs, J.C.M. (1994) Population structure of the rare, long-lived perennial Gentiana pneumomanthe in relation to vegetation and management in the Netherlands. Journal of Applied Ecology, 31, 428438.
  • Otis, D.L., Burnham, K.P., White, G.C. & Anderson, D.R. (1978) Statistical Inference from Capture Data on Closed Animal Populations. Wildlife Monograph 62.
  • Pascarella, J.B. & Horvitz, C.C. (1998) Hurricane disturbance and the population dynamics of a tropical understory shrub: Megamatrix elasticity analysis. Ecology, 79, 547563.
  • Pledger, S. (2000) Unified maximum likelihood estimates for closed capture-recapture models using mixtures. Biometrics, 56, 434442.
  • Shefferson, R.P., Sandercock, B.K., Proper, J. & Beissinger, S.R. (2001) Estimating dormancy and survival of a rare herbaceous perennial using mark-recapture models. Ecology, 82, 145156.
  • Tamm, C.O. (1972) Survival and flowering of perennial herbs. III. The behaviour of Primula veris on permanent plots. Oikos, 23, 159166.
  • White, G.C. & Burnham, K.P. (1999) Program MARK: survival estimation from populations of marked animals. Bird Study, 46, 120139.
  • Williams, B.K., Nichols, J.D. & Conroy, M.J. (2002) Analysis and Management of Animal Populations. Academic Press, San Diego.