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.