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1Analysis of population trajectories is central to assessing risk in populations of conservation concern. In animal studies, researchers realize that probabilities of detection of individuals are often less than one. Plants can also escape detection due to dormancy, herbivory, or observer error, but such information is rarely incorporated into population studies.
2We monitored a population of Asclepias meadii, a rare long-lived prairie perennial. Despite standardized methods, numbers of observed plants fluctuated greatly from 1992 to 2006. Individual plants often had periods of 1–5 years between initial and final sighting when no stems were found. To determine the actual population trajectories, we estimated rates of survival and population growth using mark-recapture models. We also estimated initial and resighting probabilities of detection. In 2007, we repeated surveys to identify reasons for low detection probabilities.
3We estimated 95% annual survival and a population growth rate of 1.023. Probabilities of initial detection were low (typically from 0.120 to 0.311 depending on prairie burn treatment), whereas average probability of detection for marked plants was 0.728.
4Comparisons of survival estimates from 15- and 8-year data sets revealed that survival estimates decline in the final years of a multi-year period, probably due to heterogeneity in encounter histories.
5By conducting three different surveys in 2007, we found that both herbivory over a multiple-week period and observer error contributed substantially to gaps in detection.
6Synthesis. Probabilities of detection that are less than one complicate interpretation of population dynamics, whether of mobile animals or inconspicuous plants. Our work illustrates three general points that could apply to many plant population studies: (i) mark-recapture models may provide insights on vital rates and population trajectories despite the extreme variability in count data that can arise because of low detectability, (ii) probabilities of initial detection can be quantified and can be considerably less than probabilities of resighting, and (iii) repeated surveys can help researchers determine the degree to which dormancy, herbivory, or observer error contribute to low probabilities of detection. Consideration of these points can improve the design and analysis of monitoring programs.
The study of population trajectories is central to conservation biology: loss of biodiversity occurs because individual species decline in numbers, leading to local or global extinction. Superficially, data for studying population dynamics appear simple (i.e. counts of the number of individuals of a species over time). For animals, however, there is a major challenge: individuals are mobile and often not detected. Population sizes and demographic rates such as survival must therefore be estimated using statistical procedures, including ‘mark-recapture’ models (Nichols 1992; Amstrup et al. 2003). In contrast, population biology studies of sessile plants are generally considered to be more straightforward. As noted by Harper (1977), ‘plants stand still to be counted and do not have to be trapped, shot, chased, or estimated’. Despite the general validity of this statement, detection ‘gaps’ can occur, where a plant is not observed in one or more surveys, but seen both before and after. There are at least three reasons for detection gaps: dormancy, removal of a plant from an observable state (e.g. herbivory), or observer error (Shefferson et al. 2001). In the last decade, plant researchers have begun to formally deal with detection probabilities that are less than one by using estimation procedures. In particular, mark-recapture models have provided unbiased estimates of survival rates and other parameters in species with dormant life stages (Shefferson et al. 2001; Shefferson et al. 2003; Kéry & Gregg 2004a,b; Kéry et al. 2005; Shefferson et al. 2005; Gregg & Kéry 2006; Shefferson 2006; Lesica & Crone 2007; Shefferson & Tali 2007; Shefferson & Simms 2007). This in-depth work on dormancy could lead some to conclude that incomplete detection is of primary concern for researchers working with specific taxa (i.e. orchids) or life stages. However, given that that herbivory and observer error can also lead to detection gaps, any plant study is potentially affected by this phenomenon.
The overall goal of our work was to explore population trajectories given the presence of incomplete detection. Some workers may assume that although some plants might be missed, changes in counts of plants over time will mirror actual population dynamics. However, counts are an informative index of population size only if the expected values of detection probabilities are constant over time (Nichols 1992; Kéry & Schmid 2006). Estimates of plant detection probabilities, however, are very rare. As a consequence, we have little knowledge of their magnitude or whether they differ among survey periods. Ignoring low detection probabilities can lead to biases in estimation of survival rates, population size, growth rates and population extinction rates (e.g. Alexander et al. 1997; Shefferson et al. 2001; Slade et al. 2003; Kéry 2004; Kéry et al. 2005).
There are two major types of mark-recapture approaches that estimate detection probabilities as well as vital rates and population size: open models assume that gains and losses can occur over the time period when surveys are being conducted; closed models instead assume there are no additions or losses, and thus population size is fixed over the survey period (Pollock et al. 1990; Amstrup et al. 2003). Variation in detection probabilities is central to the use of mark-recapture models. With plants, it seems apparent that the probability of initial detection (especially for plants existing in a dense vegetation matrix) should often be lower than the probability of finding a plant that has been marked and mapped (Kéry & Gregg 2003).
In our work, we have examined detection probabilities in the context of a long-term monitoring program of the Mead's milkweed (Asclepias meadii Torr ex A. Gray). This rare plant is a prime example of a difficult-to-monitor plant because of its low detectability in dense vegetation (Alexander et al. 1997). In addition, it is long-lived and has low annual reproduction (Betz 1989; Kettle et al. 2000; Grman & Alexander 2005). A high probability of survival means adults may persist in a population for years, even if recruitment is not sufficient to maintain a stable population; low reproduction dictates that long-term data sets are needed to estimate recruitment rates, especially if recruitment is episodic. Despite these problems, research on A. meadii is important because it is a federally recognized threatened species in the United States, with greatly restricted distribution and numbers (U.S. Fish and Wildlife Service 2003).
Our long-term records of individual A. meadii plants include multi-year periods when no stems were recorded (= detection gaps); as a consequence, we previously applied mark-recapture methods using 4 (1992–1995, Alexander et al. 1997) and 8 years (1992–1999, Slade et al. 2003) of data. We now present analyses of 15 years of data on individual mapped plants (1992–2006) and of a single year of data (2007) in which multiple surveys within a single season were conducted. This unique long-term data set allows us to address new questions about the role of detectability in studies of population trajectories. We first address whether mark-recapture methodologies can provide insight on rates of survival and population growth for a population despite extreme detection gaps: can the ‘signal’ be seen despite the ‘noise’ (= extreme variability in counts) in the data? Second, we estimate two types of probabilities of detection: the probability that a plant is seen initially and the probability that a labelled plant is seen in subsequent years (= resighting probability). Quantification of these probabilities is important in the careful design of monitoring programs and the valid estimation of population sizes. Third, we describe a field protocol involving multiple surveys for exploring the causes of detection gaps. We believe our approach is the first study to allow identification of the role of herbivory in low detectability, and could be easily adapted by other researchers.
Mead's milkweed is a perennial herbaceous plant restricted to tallgrass prairies and glades (Bowles et al. 1998; U. S. Fish and Wildlife Service 2003). An adult plant consists of a cluster of stems that may or may not flower; in some years no stems are found even though the plant is seen in previous and subsequent years. Flowering occurs in late May/early June; each flowering stem produces one umbel with an average of 12 flowers. Fruits (follicles) containing approximately 60 wind-dispersed seeds mature in mid-August (Betz 1989). Reproductive output can be quite low: at the 4.5 ha field site (see below), 40–100 flowering stems are produced in most years but the total fruit production in the entire site is < 20 fruits, and often less than five (Kettle et al. 2000). Seedlings are projected to take as long as 15 years to flower in field populations (Bowles et al. 2001).
study site and monitoring protocol
The study was conducted at and supported by the University of Kansas Field Station and Ecological Reserves (KSR), a research unit of the Kansas Biological Survey and the University of Kansas. Specifically, the study site consists of the 4.0 hectare Rockefeller Native Prairie and an adjacent 0.5 ha restored prairie in north-eastern Kansas, USA (10 km NE of Lawrence, KS). From 1986 to 2006, the prairie was burned in April of even-numbered years (before the emergence of milkweed plants); the prairie was also burned in 2007 as part of a new management plan. For additional information, see Fitch & Hall (1978) and Kettle et al. (2000).
Starting in 1988, surveys were conducted in late May/early June when flower production peaked. For the first years (1988–1991), the location of individual stems was recorded as their distance and direction from permanent grid stakes. In 1992, a standard protocol was adopted. First, a systematic search was conducted over the entire prairie, looking for plants. Second, we revisited specific locations of known stems from earlier years. Third, for all stems discovered, a numbered nail was placed 18 cm due south from the stem. Data on stem size and flower number were recorded for each stem; individual stems typically appeared within 1–2 cm of their location the previous year. This protocol was repeated from 1992 to 2007. For a long-term survey, there was remarkable consistency in effort each year: the systematic search utilized four to five people working for 2–4 h, while revisiting of past locations and data collection typically involved two people working 2–4 days. Furthermore, there was consistency in personnel: G.L.P. led field crews since 1994 and made all decisions regarding identity of individuals. For details, see Alexander et al. (1997), Kettle et al. (2000) and Slade et al. (2003).
Mead's milkweed exists in ‘patches’ at the site, with individual patches separated from each other by at least 4 m; a typical patch consists of one to five stems, with stems often 10–60 cm apart (Kettle et al. 2000). We followed Alexander et al. (1997) and Slade et al. (2003) and assigned any two stem locations separated by 1.25 m or less to the same patch. This distance was chosen because 1-m-long rhizomes connecting stems have been occasionally observed (M. L. Bowles, personal communications); a ‘patch’ is thus a reasonable surrogate for a ‘plant’. We have never dug in the soil to look for rhizomes; such digging would be both difficult (rhizomes are not obvious at the soil surface) and destructive to the plants and prairie. Patch identity could be determined in all years, but we have restricted our analyses to 1992 or later when patches were mapped. Nearly all patches were first detected when flowering (see Results); in subsequent years a patch could be in one of three adult stages: non-flowering, flowering (at least one flowering stem present; non-flowering stems could also be present), or not detected (no stems found).
Multiple surveys were conducted in 2007. W.D.K. and H.M.A. visited all patches between 10 and 15 May and put pin flags next to all observed stems. All patches were revisited by W.D.K. and H.M.A. on 29 May to determine the fate of the flagged stems (present vs. removed by herbivory) and to note locations of any newly found stems; comparison of mid and late May surveys allowed evaluation of herbivory effects. All pin flags were then removed. Between 30 May and 1 June, G.L.P. and an assistant conducted the normal yearly surveys using regular procedures. By comparing patch detection between the 29 May and 30 May – 1 June surveys (a short time period when no change in stem numbers was expected), we could determine the extent to which researchers missed plants. The multiple 2007 surveys were done in a spring burn year, when vegetation was short and thus repeated visits led to less trampling than in a year without burning. Trampling of vegetation that did occur did not affect detectability of stems in subsequent surveys because (i) the design of the study meant that all marked patches were observed each survey and (ii) any trampling was generalized around the patch area as opposed to specifically associated with individual stems within a patch. Finally, for the comparisons between 29 May and 30 May – 1 June surveys, it was important to know that stems had not been eaten during this short interval. Hence, W.D.K. and H.M.A. resurveyed all patches between 2 and 3 June for which there were discrepancies between the late May surveys.
Descriptive data on patch numbers, reproduction and gaps in detection
We recorded the presence and absence of flowering and non-flowering patches from 1992 to 2006. We also compiled the number of newly detected patches each year and the percentage of these that were flowering. For an unbiased estimate of the likelihood of a patch flowering in year t, we ignored newly found patches and calculated the proportion of flowering patches among previously detected patches.
We defined a detection gap for a patch as year(s) when no stems were found, preceded and followed by years where we observed stems in the patch. For each gap, we recorded its length (number of years the patch was not detected). Some patches had multiple gaps; to avoid potential dependence among data points, we included a single, randomly selected gap per patch. We examined the frequency distribution of gap length to gain insights on the typical number of years that a patch was not detected. For this analysis, we excluded patches first detected in 2000 or later. We used this ‘cut-off’ point so that we would not bias the data set towards short gap lengths (for example, patches first detected in 2004 could only have 1-year gaps). Given that gap lengths could be 1–5 years but rarely longer (see Results), our choice of 2000 seems appropriate (i.e. 6 years earlier than the last survey year).
Estimation of survival rates and population size when detection probabilities < 1
Before data analysis, we converted our data set into encounter histories (= capture histories), where each data row consisted of a patch identification number followed by 15 0's or 1's representing whether a patch was detected (1) or not detected (0) in the 15 years from 1992 to 2006 (for details, see Alexander et al. 1997, Slade et al. 2003). In contrast to the descriptive data, analyses of encounter histories with mark-recapture models allow us to estimate parameters for all patches, not only those we detected. We analyzed encounter histories with mark-recapture models for open, as opposed to closed, populations because over a 15-year period it is likely that mortality and recruitment occurred. We used the Cormack–Jolly–Seber (C–J–S) procedure within the software package mark (White & Burnham 1999) to compare models with different assumptions about survival (ϕ) and detection (p = probability of detection for an already marked patch = resighting probability). Our goal was to determine whether survival and detection probabilities were constant, varied randomly over years, or depended on the presence or absence of a spring burn at the site. We fit nine different models to explore all combinations of year-specific, burn vs. non-burn, or constant probabilities of survival and detection of marked patches. Because of significant overdispersion (evaluated using the Release goodness of fit test, χ2 = 159.09, d.f. = 47, P < 0.001; ĉ = 3.84) we used quasi Akaike's information criterion (QAICc) values to select the most parsimonious model from this set of candidate models. Following Burnham & Anderson (1998), we chose the best model as that with the lowest QAICc score. Models with similar QAICc values (ΔQAICc ≤ 2) were considered to have significant support, and we used model weights for averaging estimated parameters among those models (Burnham & Anderson 1998). We estimated survival probabilities for 1992–1993 through 2004–2005. We excluded 2007 from our long-term data analysis since it had different management conditions. Note that 2005–2006 survival was not estimated because one needs subsequent years to distinguish between mortality and lack of detection or one would have to assume that resighting probabilities were constant (the latter is inconsistent with our findings; see Results).
We estimated population size using the open Jolly–Seber model (J–S, in program JOLLY, Hines 1990), which allowed time-specific rates but not grouping into burn and non-burn years. The J–S and C–J–S models both estimate the probability of detection of an already marked patch, and assume that the probability of detecting new patches is equal. The former probability should have been higher than the probability of initial detection because patches were flagged and mapped when found and because initial detection of cryptic patches was difficult. This distinction between the two detection probabilities does not affect survival estimates but will lead the J–S model to underestimate population size (Nichols 1992). Even with this negative bias, we retained the J–S model to determine if there was any overall temporal trend in estimates of population size. Further, given that J–S estimates are known to be negatively biased, the degree to which they exceed count data provides an illustration of the minimum difference between count data and actual population size.
We used the Pradel version of the basic J–S model (Pradel 1996; within mark) to estimate annual rates of recruitment and the population growth rate, λ; in this model, we set the probability of survival to equal 0.95 and allowed the probability of detection to vary among years (as consistent with the selected C–J–S models, Table 1).
Table 1. Comparison of C–J–S models. Survival rates and probability of detection of a patch were modelled as time-dependent (t varies among years), dependent on whether the year was a burn or non-burn year (b/n), or constant over the study period (.). The best fitting model has the lowest QAICc . Model selection was based on ĉ = 3.38
Number of parameters
Survival (.), Detection (t)
Survival (b/n), Detection (t)
Survival (.), Detection (b/n)
Survival (b/n), Detection (b/n)
Survival (t), Detection (t)
Survival (.), Detection (.)
Survival (b/n), Detection (.)
Survival (t), Detection (b/n)
Survival (t), Detection (.)
We also developed a model to estimate probabilities of initial detection, given defined values of probability of annual survival (ϕ) = 0.95 and population growth rate (λ) = 1.023 (from the Pradel model; see Results). This model simultaneously estimated the probabilities of initial detection and initial population size (1991) that minimized the sum of squared differences between observed and predicted newly detected patches. In the simplest case, the model thus had two parameters, N (initial population size) and pf (probability of initial or first detection, Shefferson et al. 2003).
The expected number of newly detected patches in any year (mt) was calculated as:
where mtis the product of the proportion of patches that were initially detected (pf) and the sum of the surviving undetected patches from the previous year (ut–1 ϕ) and the number of newly recruited adult patches, Rt. The expected number of undetected patches in a year (ut) was defined as:
ut= (ut–1ϕ+Rt)(1 –pf)
The number of recruits moving into the adult class, Rt, was defined to be equal to Nt–1( λ – ϕ) even though recruits in any year are actually the offspring of plants that reproduced up to 15 years previously (Bowles et al. 2001). We used this formulation because the population will only grow at a yearly rate of λ if the number of recruits can be defined as the product of the previous year's population size and the part of the population growth rate that is not accounted for by surviving individuals.
We compared the simplest model to three other models: one had three parameters to estimate (N, pf-burn, pf-non-burn) since our previous work (Slade et al. 2003) suggested initial detection was probably higher in burn years than non-burn years. Two other models had four parameters, adding either pf-1994 or pf-1994,2002. These fourth terms were added to evaluate whether initial detection probabilities in 1994 and/or 2004 were distinct from typical burn years. These two years were chosen because they had the two highest probabilities of flowering (see Results) based on our independent data set of previously sighted patches, and nearly all newly detected patches were flowering. We implemented our model using the Solver function in Excel and used AICc values to compare models, following Burnham & Anderson (1998).
We used Model Mt of program capture (within mark) to estimate the number of detectable patches and probability of detection for our late-May 2007 surveys by two independent teams of observers. This closed population model assumes no mortality or recruitment over the 3-day period.
descriptive data on patch numbers, reproduction and gaps in detection
A total of 252 unique patches were identified from 1992 to 2006. However, the occasional appearance of stems between two previously defined patches resulted in seven cases where two patches were merged into one. Thus, our analyses were based on a total of 245 patches. The total number of patches detected, and the number that were flowering and non-flowering, fluctuated over the years (Fig. 1a). New patches (not seen in previous years) were found in every year (Fig. 1b); nearly all (93%) had flowering stems. There was considerable yearly variation in the probability of flowering for previously marked patches (Fig. 1c); probabilities were higher in burn years (mean ± SE, 0.50 ± 0.06) than non-burn years (0.30 ± 0.06; F1,12 = 4.90, P = 0.047). One-year detection gaps were most frequent (53%), whereas 2-, 3-, 4- and 5-year gaps occurred at 23%, 5%, 10% and 6% of the time, respectively.
Of the 123 patches detected from 10 to 15 May 2007, 17 (13.8%) had either no sign of stems on 29 May or stems that were only 2–8 cm tall (i.e. the stem had been clipped to the ground and only tiny resprouts were present). This basal clipping of stems was typical of feeding by small mammals. Of the 131 patches that were observed on either 29 May or 30 May to 1 June, 83.2% (109) were seen in both surveys. Missed patches over the 1–2 day period were nearly always due to observer error, not herbivory.
estimation of survival rates and population growth rate when detection probabilities <1
The best C–J–S model included constant probability of survival and year-specific detection of marked patches; a model with survival varying between burn and non-burn years and year-specific probability of detection fit nearly as well (Table 1). Probabilities of detection of already marked patches ranged from 0.405 to 0.925, with a mean of 0.728 (Fig. 2a). The model-averaged yearly survival rates varied from 0.944 (burn years) to 0.951 (non-burn years), with a mean of 0.948. However, a comparison of the year-specific survival rates from this 15-year data set to the yearly survival rates based on a subset of 8 years (Slade et al. 2003) revealed a clear pattern (Fig. 2b). Specifically, estimated annual survival rates for the earliest years in our data set (1992–1994) were the same regardless of whether analyses were based on 8 years (1992–1999; Slade et al. 2003) or 15 years (1992–2006). Yearly declines in survival rates were, however, apparent at the end of the 8-year data set so that survival estimates for 1995–1998 were considerably higher when based on the 15-year than on the 8-year data set. This decline indicates that survival rates for terminal years in a data set were underestimated.
Using the J–S model, the mean estimated total number of patches per year was 139.6. However, this average includes the very low estimate of 57 in 1993 at the start of our study (Fig. 1a); the average based on all other years, 146.5, was probably more accurate. Estimated numbers of patches always exceeded detected numbers (Fig. 1a). With the exception of 1993, approximately 95% confidence intervals (± 2 SE) for population size overlapped among years. Frequently, the point estimate for 1 year fell within the confidence interval of 1–7 other years, indicating no significant differences in estimated population size among those years.
The estimates of annual recruitment rate and λ from the Pradel model were 0.073 (95% CI: 0.0612–0.0869) and 1.023 (95% CI: 1.010–1.036), respectively. Given that the CI for λ does not include 1, we conclude that the population is not declining (however distinguishing between population growth and stability is not advisable given the variability in the data). Using our model that predicted the number of newly detected patches per year, the model with three probabilities of initial detection, pf-burn, pf-non-burn, pf-1994,2002) fit significantly better than other candidate models (Table 2). With this model, population size was estimated to be 137 patches in 1991, and probability of initial detection was higher in burn (0.311) than non-burn years (0.120), but much higher in 1994 and 2002 (burn years with exceptionally high rates of flowering; 0.663). Because 93% of newly discovered patches were flowering, the probability of finding a new patch of A. meadii can be approximated by the product of the probability of a patch flowering and the probability of detecting the patch given flowering. We estimated the conditional probabilities of initially detecting a flowering patch by dividing the probabilities of initial detection by the probabilities of flowering for each group (using the appropriate average values for burn, non-burn or 1994/2002 years). The probability of first detecting a patch that was flowering was thus estimated to be 0.621 in a burn year (excluding 1994 and 2002), 0.400 in a non-burn year and 0.908 in 1994/2002 years.
Table 2. Estimation of population size in 1991 (N), and initial detection probability (pf) based on modelling new patches per year for a 15-year period, under the assumptions of a fixed survival rate of 0.95 and λ = 1.023. Each column below refers to a different model (1–4): model 1 had two parameters (N and constant pf), model 2 had three parameters (N and pf for burn and non-burn years), and models 3 and 4 had four parameters (N, pffor burn and non-burn years, and pf for unusual years with large amounts of flowering stems). For the latter models, unusual years were defined as either 1994 or as both 1994 and 2002. The best fitting model has the lowest AICc; for this model, estimates of the detection probabilities using the lower (1.010) and upper (1.036) limits of the 95% confidence interval for λ are presented in parentheses (see text)
0.311 (0.267, 0.352)
0.120 (0.094, 0.152)
0.663 (0.554, 0.780)
Number of parameters
In late-May 2007, the estimated number of detectable patches was 132 ± 1.1, with probabilities of detection of 0.92 and 0.89, respectively, for the 29 May and 30 May to 1 June surveys.
A primary goal of any population dynamics study or monitoring program is to determine if the numbers of individuals increase, decrease or remain constant through time. Given real-world time constraints, however, achieving a complete census of plants at a site is often challenging (Lesica & Steele 1994; Menges & Gordon 1996; Elzinga et al. 1998; Philippi et al. 2001). With A. meadii, for example, counts of observed patches (our surrogate for plants) increased and decreased over time, implying more variability in population size than was consistent with a long-lived plant with low reproduction. However, using methods that take into account incomplete detection of patches, we estimated population growth rates that were consistent with a stable to increasing population size. Mark-recapture models thus allowed us to interpret data on population trajectories even for a plant with clear challenges for detectability. It is also noteworthy that we were successful using these models despite a relatively small number of individual patches and resightings.
All models are oversimplications of real-world populations. In our work, the C–J–S model was overdispersed, indicating that the variation in the data exceeds the variation expected from the model. Overdispersion is typically caused either by a lack of independence of individuals or when estimated parameters (e.g. probabilities of detection and survival) vary among individuals (Amstrup et al. 2003). An example of lack of independence would be if survival and/or resighting was dependent on whether the patch was detected in the previous year or if there were spatial patterns to herbivory, leading patches near each other to have similar probabilities of detection. Although such scenarios might occur, we expect that heterogeneity in encounter histories was the principal cause of the overdispersion, with some individual patches detected year after year, while other patches have sizeable gaps in detection. For example, stage-dependence could occur in probabilities of survival or resighting if, for example, eaten patches differ in parameter values compared to patches that did not experience herbivory. Unfortunately, this heterogeneity is not obviously linked to plant traits that we have data on for the entire time period of the study and thus it is hard to define groups for stratified mark-recapture analyses. However, even when substantial individual heterogeneity in detection probability is present, simulation studies have shown that the effect on survival estimates is primarily limited to the final years of a study (see below, and Carothers 1979; Buckland 1982).
For the Pradel model, heterogeneity in resighting probabilities would affect estimates of recruitment in an analogous way to their effect on survival in C–J–S models. This similarity exists because Pradel's (1996) approach to estimating recruitment is identical to the C–J–S method for estimating survival, but with the encounter histories ‘reversed’ so that one goes backwards in time to estimate the rate of ‘entry’ into the population. An additional assumption of Pradel models is that there is no difference between probabilities of initial detection and resighting (White et al. 2002). Clearly, we have violated this assumption with our data. Violation of this assumption could lead to biases if one estimated year-specific recruitment and population growth rate in models with time-dependence in survival or if one was modelling organisms with low survival rates. In the latter case, for example, significant numbers of recruits may be missed because of low initial detection probabilities coupled with high mortality. However, in our work with A. meadii, we are studying a long-lived species, lack time-dependency in survival and are estimating a single value of λ. Therefore, our estimate of λ should not be affected by the violation of the assumption of equal initial and resighting probabilities. One of us (A.W.R.), for example, found that running the Pradel model on simulated data with high survival rates but with a low initial probability of detection and several different probabilities of resightings yielded the same estimates of λ (see Appendix S1 in Supporting Information).
The λ derived from these ‘temporal symmetry’ models is a ‘realized’ population growth rate (Sandercock 2006; Pradel & Henry 2007), and thus a very desirable rate to estimate for a plant of conservation concern. There are four potential drawbacks to this approach, however, as summarized by Sandercock (2006). Two of these (the need for a long time series of data and for defined boundaries to the study site) are not a problem for our work. A third, heterogeneity in capture, has already been addressed. The fourth issue, that of age-structure effects, is worthy of further consideration. Sandercock (2006) states that these models need to be applied with caution if there is age-specific variation in fecundity or survival; Pradel & Henry (2007) further note that marked individuals must be representative of the population as a whole. With plants, there is probably age- (or, more likely, stage or size) specific demography (e.g. Shefferson 2006). By restricting our analyses to adult patches of A. meadii, we should have reduced the impact of this problem. Furthermore, since we began our yearly surveys in 1988, but our analyses start with 1992, the patches in our data set should be representative of the population (i.e. if there were attributes that distinguish initially detected patches from the rest of the patches (Pradel & Henry 2007), this ‘pre-analysis’ survey period should have reduced this problem). Overall, although caution is always wise, we expect the estimated λ provides useful information on the population trajectory and is superior to interpretations based only on the extremely variable count data.
Estimation of population size in open models is widely known to be challenging (Link 2003). We know, for example, that the J–S population sizes are underestimates since probabilities of initial detection differed substantially from probabilities of subsequent detection. Given that the J–S estimates are negatively biased, however, it is intriguing how much higher these estimates are than the observed counts (Fig. 1a). Furthermore, the variability in the estimated population sizes is less than that of the observed counts. Variable counts can arise for many reasons, including the simple binomial sampling process. The overlap in confidence intervals of population size estimates (Fig. 1a) is consistent with our interpretation of a stable presence of the species at the site. However, it also must be noted that small deviations from stability would be difficult to detect for a long-lived plant with low recruitment.
In animal studies, Pollock's robust design (Pollock et al. 1990) is advocated for studies of population dynamics; this design combines the advantage of open models for estimation of survival probabilities between years and closed models for estimation of population size within years. Furthermore, such an approach is the best way to deal with Markovian temporary emigration (i.e. situations where the lack of detection of an individual in a single survey depends on whether it was not detected in a previous survey, a situation that could occur with plants; Schaub et al. 2004). To use the robust design with A. meadii, we would need to repeat the late-May/early-June survey at least twice, but preferably four or five times, to allow estimation of population size with a closed model. In contrast to our multiple surveys in 2007, these proposed within-year surveys would be of the entire prairie, examining not only previously observed patches but also looking for previously undiscovered patches. Unfortunately, the trampling of the vegetation that would occur with four to five closely spaced within-year surveys would likely introduce dependency among the surveys (i.e. patches detected in the first survey may be more likely to be found in subsequent surveys), as well as be detrimental to the population or environment. Shefferson et al. (2001) and Kéry et al. (2005) have made similar observations. To reduce trampling effects, population sizes within a year could be estimated with a removal model; for such analyses, one does not need to resample patches once they have been observed.
probabilities of detection
We focused on two detection probabilities: the probability of initially detecting an unmarked patch, and the probability of detecting a patch once it is marked. Estimation of the latter resighting probability is central to models that estimate survival rates. However, standard models for open populations assume no difference between probabilities of initial detection and resighting probabilities, even though such a distinction is central to closed mark-recapture models for estimation of population size (Williams et al. 2002, p. 151). For many plants, initial detection probabilities are likely to be low relative to detection of permanently marked plants: plants can be inconspicuous, the matrix which plants live in can obscure detection and surveyors with time limitations cannot search intensively for every individual. Furthermore, initial detection probabilities may vary depending on plant size (Kéry & Gregg 2003; Shefferson 2006). Our modelling approach suggests one way to estimate such initial detection probabilities from long-term survey records. It is noteworthy that our estimates of initial detection were typically 50% lower than estimates of probability of detection of patches that were already marked, although initial detection of flowering plants was higher (see Results). Quantification of these detection probabilities is important for any monitoring program. For example, if probabilities of initial detection are much lower than resighting probabilities, it is clear that population estimates from open models like J–S will be underestimates. Low initial detection probabilities also suggest that field protocols should be reconsidered: for example, one might survey smaller areas, or increase the amount of effort per survey.
The probabilities of resighting were also less than one. In the past, we had assumed that dormancy accounted for many of the detection gaps, as reported in many long-lived perennial plants (e.g. Lesica & Steele 1994; Kéry et al. 2005; Shefferson 2006), including another milkweed species (Breaden 1999). However, if 2007 is representative, herbivory and observer error are responsible for many missed patches. Although others have conducted two surveys over a short period of time (= double-observer survey) to study detectability with plants (e.g. Shefferson et al. 2001; Kéry & Gregg 2003; Lesica & Crone 2007), we believe ours is the first study that included multiple surveys over several weeks to distinguish between herbivory and dormancy. In retrospect, the prevalence of early season herbivory was not unexpected given the high herbivory on A. meadii stems in mid-summer (Grman & Alexander 2005). The incidence of observer error was initially surprising to us, but is consistent with similar studies (i.e. 75–92% detectability, Kéry & Gregg 2003; 86–100% detectability, Lesica & Crone 2007). Several factors likely contribute to observer error: the inconspicuous nature of vegetative stems, the complex vegetation matrix, variation in plant size, and the difficulty of finding the relatively small number of stems (1–5) that occur per year in a patch. Given that observable patches were sometimes missed, it is likely that increased effort (number of people surveying, amount of time invested) could increase the probability of detection. Finally, another possible issue is that plants could vary in the timing of emergence and thus a survey at an early date could result in missed plants. We believe this issue is unlikely to be relevant for our study (our early-June survey occurred weeks after stems emerge based on our 2007 census) but could be a factor for other species.
Some detection gaps exceeded 4 years and were longer than reported for many species (e.g. Lesica & Steele 1994; Kéry et al. 2005; Lesica & Crone 2007). Repeated herbivory could potentially be responsible. Alternatively, rhizomes may die out and be replaced by a new seedling at the same location. However, since fruit production at the site is very low, seeds are wind dispersed and seedlings may take up to 15 years to flower, this scenario seems unlikely. We also lack data on whether seeds can remain dormant at a site for extended periods.
Our findings lead us to concur with Kéry & Gregg (2003), who were sceptical about the common assumption that all plants at a site will be found. Interestingly, research supporting the need for consideration of ‘detectability’ comes not only from studies conducted in dense vegetation matrices (our work and Lahoreau et al. 2004), but also from work in relatively open vegetation (Kéry & Gregg 2003). To provide information on the extent of detectability problems and possible causes, within-year double-observer surveys should be carried out early in a monitoring study. Had we performed our double-observer surveys earlier, for example, we could have conducted intensive subsampling to improve detectability and established experiments to explore the effect of herbivory.
Information from double-observer surveys also can be considered in light of multi-state mark-recapture models, which have been effectively used to model dormancy (e.g. Kéry et al. 2005; Shefferson 2006; Lesica & Crone 2007). In models of dormancy, researchers often assume perfect redetection of above-ground plants and a zero probability of detection of dormant individuals. For A. meadii, such an assumption is incorrect; we would need much more complex models to incorporate probabilities of patches being dormant, eaten or present but not observed. Further, herbivory (and possibly dormancy) may occur at both the individual rhizome/stem level or at the entire patch level; these factors affect whether probabilities of detectability are zero or simply reduced, and complicate model development. Finally, we do not know if the 2007 data are representative of our 15-year data span, especially since the presence or absence of spring burns may alter dormancy, herbivory, or observer error. Thus, we chose to not develop multi-state models at this time.
Additional estimates of detection probability come from the closed population model using data from adjacent days in 2007. These detection probabilities (0.92 and 0.89) are not comparable to the average value of 0.73 estimated from the C–J–S model because the former values reflects the repeatability of finding observable patches, whereas detection probabilities from open models are affected by dormancy, herbivory and observer error. As a consequence, the estimated population size of 132 from the closed model in 2007 is likely an underestimate, since it only incorporated the role of observer error in detection.
value of long-term monitoring programs
Long-term studies in ecology are frequently lauded (Likens 1989). However, we expect few plant ecologists realize that repeated surveys can also provide insights on incomplete detection and thus refine estimates of population trajectories. For example, our long-term monitoring, by chance, included the high flowering year of 1994 (which followed a very wet year; Kettle et al. 2000). The chance occurrence of this unusual year early in our study allowed us to detect many non-flowering patches in subsequent years. Furthermore, our ability to compare two long-term data sets (8 and 15 years) led us to infer that lower estimated survival rates at the end of the 8-year data set (and reported in Slade et al. 2003) were artifactual. Such a decline in survival estimates at the end of a survey period has been observed in other empirical studies (e.g. Kéry et al. 2006); theoretical work has shown that such a negative bias in survival rates can be due to heterogeneity in probability of detection among individuals (Carothers 1979; Buckland 1982; Lebreton 1995). A decline in estimated survival rates at the end of the 15-year data set might also be expected, but the best fitting C–J–S (MARK) model did not include year-specific survivorship, and thus is not consistent with a decline. Such terminal declines in survival probability are perhaps less influential in very long data sets since the final years are a smaller proportion of the total data set. Alternatively, actual survival rates could increase at the end of the 15-year period but not be detected due to heterogeneity-induced bias. Regardless of these details, our work illustrates that one must be cautious when interpreting estimates of survival at the end of any time series as heterogeneity in detection and incomplete capture histories may make these estimates unreliable.
Long-term studies are also essential for exploring recruitment in long-lived species. For A. meadii, our work suggests that recruitment is occurring throughout the 15-year period (Fig. 1b). For many prairie species, the appearance of new plants through seed is a rare event in comparison to the major role of vegetative reproduction (Benson & Hartnett 2006). Given the impossibility of finding naturally occurring seedlings in dense prairie vegetation, the continued appearance of new patches of A. meadii through time provides important insights that successful reproduction is occurring.
In conclusion, this study reveals that even highly variable long-term data can yield information on population trajectories if analyses take into account incomplete detection of plants. We illustrate this point with our work on a ‘hard-to-study’ plant but the issues raised in this study are very general in nature. Specifically, given that dormancy, herbivory, observer error, and variation in timing of emergence can all contribute to plants being missed in surveys, incomplete detection is likely to occur in many plant ecology studies and potentially affect data collection and interpretation. Given that understanding of population dynamics is impossible without unbiased estimates of vital rates and population growth rates, we need increased attention to both field methods and statistical approaches to cope with this source of uncertainty.
The University of Kansas Biological Survey provided financial support and staff for this project. Authors thank all the people who have participated in surveys over the last 20 years. S. Bodbyl-Roels, L. Habibi and B. Kuhn assisted in 2007, C. Moore encouraged authors to perform multiple surveys, and S. Bodbyl-Roels, M. Kéry, R. Shefferson, S. Roels and anonymous referees provided useful comments on previous manuscripts.