We developed a model for analyzing multi-year demographic data for long-lived animals and used data from a population of Agassiz’s desert tortoise (
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Cover: Desert Tortoise eating along an interpretive trail at the Desert Tortoise Research Natural Area interpretive center, eastern Kern County, California. Photograph taken by Freya Reder, independent contractor, April 2011, used with permission.
We thank Jonathan Rose, Diane Elam, and an anonymous person for reviews. We acknowledge the cooperation and contributions of the Bureau of Land Management, California Department of Fish and Wildlife (Game), and Desert Tortoise Preserve Committee, Inc.
Multiply | By | To obtain |
Length | ||
---|---|---|
millimeter (mm) | 0.03937 | inch (in.) |
centimeter (cm) | 0.3937 | inch (in.) |
meter (m) | 3.281 | foot (ft) |
Area | ||
square kilometer (km2) | 0.3861 | square mile (mi2) |
square kilometer (km2) | 247.1 | acre |
Development of demographic models and calculations of vital rates are desirable for learning more about the potential for population growth, recovery, and survival of species in general and can be especially valuable for providing guidance on recovery efforts for threatened and endangered species (
Agassiz’s desert tortoise (
To demonstrate new methods for modeling demographic attributes, we used data from a long-term study of desert tortoise populations at the Desert Tortoise Research Natural Area (Natural Area) as a case study. In 1979,
Our objectives were to develop demographic models using modern mark-recapture modeling techniques (
Probabilities of a tortoise transitioning from a smaller to a larger size-age state;
Probabilities that a tortoise changes its location with respect to the protective fence (inside-to-outside the fence versus outside-to-inside the fence);
Probabilities of annual and 5-year survival by sex, location (inside or outside the fence), and size-age state;
Probabilities of detecting a tortoise, given it was in the study area, by sex, location (inside or outside the fence), and size-age state, and, as a secondary objective, the potential influence of precipitation on the detectability of tortoises; and
Total population sizes among the sexes, size-age states, and locations (inside and outside the fence) for each sampling year.
In this Open-File Report, we report a model for analyzing multi-year demographic data for long-lived animals and use selected datasets from a population of Agassiz’s desert tortoise at the Desert Tortoise Research Natural Area in the western Mojave Desert of California as a case study to describe the model. We then used the model and datasets along with several other datasets on vegetation, clinical signs of disease and trauma on the tortoises, causes of death and annualized mortality rates, predators, and anthropogenic impacts for a journal article (
The 7.77-km2 study area was located at the southeastern corner of the Desert Tortoise Research Natural Area in eastern Kern County, California, at elevations of 740–790 meters (m). The study area was divided by the Natural Area boundary fence into inside (4.53 km2) and outside (3.24 km2) portions. The fence was made of hog wire and was raised about 25 centimeters (cm) off the ground to allow passage of wildlife and to protect tortoises and habitat inside the fence from sheep grazing and uncontrolled recreational vehicle use, which occurred outside the fence (details in
We conducted censuses of the tortoise population in 7 survey years (1979, 1985, 1989, 1993, 1997, 2002, 2012) spanning 34 years and at intervals ranging from 4 to 10 years. In each survey year, two back-to-back censuses occurred in spring between late March and the first week of June at the height of tortoise activity (
In addition to data collected on the 14 censuses, tortoises were sometimes found during non-census periods, either opportunistically or during other research activities in the study area. These encounters occurred before the first annual survey (before 1979), between the seven annual surveys, and after the last annual survey (after 2012), and provided supplemental information for our demographic model. Three types of non-census encounters were potentially useful—dead encounters of marked individuals, live encounters when a tortoise was first captured, and live encounters when a tortoise was last recaptured. Marked tortoises encountered dead were removed from the study area and, therefore, were unavailable for future recapture. Failure to account for these removals would lead to underestimates of detection probability. Occasionally a tortoise was captured live before being recaptured during a census year. We considered including these opportunistic captures because they represented a more accurate start year of when that tortoise was known to be in the population and available for detection than we would have by relying only on encounters during the survey years. Sixty-six tortoises were captured live opportunistically (that is, not on a survey) and 34 of the 66 tortoises were recaptured during a census year. However, there were relatively few of them (3 percent; 34 of 1,123 modeled tortoises) and they generally occurred shortly before a census recapture. Ultimately, we did not include them in the model because the opportunistic captures would provide relatively little additional information. The same was true for tortoises recaptured live during non-survey years and then not seen again. We also considered including these opportunistic recaptures; however, there were relatively few of them (<3 percent; 29 of 1,123 modeled tortoises) too, and they generally occurred shortly after a census observation. Including them in the model would have provided relatively little information.
We used mark-recapture data from 1,123 tortoises to develop a multistate Jolly-Seber (JS) model to describe size-age, locations relative to the boundary fence, and survival (states) of marked and recaptured tortoises, and to estimate densities in each state category as well as by sex (
We hypothesized that size-age and sex could potentially affect the locations (inside fence versus outside fence) of tortoises, and that size-age, sex, and locations of tortoises could potentially affect their survival and detection rates. These states were subject to change (that is, state transitions) for individual tortoises in consecutive censuses as they grew from smaller to larger size-age, moved locations between inside and outside the fence, and in some cases died. When a marked tortoise was not redetected, then its status was unknown (or latent) and we lacked direct information on its size-age, location, and survival, and thus its inclusion in the population. Our encounter histories for desert tortoises included many unobserved (latent) states. We chose a Bayesian modeling framework to develop our multistate JS model because of its ability to handle unobserved (latent) states (
We summarized observed states of size-age, location, and survival status of tortoises into three categorical variables:
Sex was assigned to adult tortoises based on secondary sex characteristics of the male (that is, longer and often upturned gular horn, concave posterior plastron in the male, longer tail, and larger size;
We applied biological assumptions when constructing hierarchical mathematical models (see section “Mathematical Models” in
Hierarchical relationships between state variables (size-age [gray boxes], location [dark-blue boxes], and survival [mustard boxes]) and state transitions (vertical arrows between state variables going from time
Tortoises in our model were also allowed to move locations freely between inside and outside the fence, and we modeled two probability functions—one for the probability of changing from an inside location to an outside location, and another for the probability of changing from an outside location to an inside location. We allowed location transition probabilities to vary as a function of sex and size-age class, in accordance with differences in home range sizes between these demographic classes. In general, adult males have larger home ranges than adult females and the smaller, younger tortoises have smaller home ranges (
Because surveys were conducted at irregular time intervals, sometimes as short as 1 month and other times as long as 1 decade, we modeled the probability of surviving from one survey to the next survey as the annual survival probability raised to a power equal to the number of years since the previous survey (that is, interval survival = [annual survival] ^ [time since last census]). We modeled annual survival probability as a function of sex, size-age class, and location.
In the final component of our hierarchical model, we modeled the process of capturing and recapturing tortoises by allowing detectability (that is, the probability of capture or recapturing a tortoise given it is there to capture) to vary as a function of sex, size-age class, and location. Sex differences could occur because males have greater movement activity and are more likely to spend time above ground where they are easier to find. Size-age differences could occur because large mature tortoises are easier to detect than small young tortoises. We expected tortoises outside the fence to be easier to detect than tortoises inside the fence because substantial parts of the area outside the fence gradually became denuded and the ground compacted due to off-road vehicle activity and sheep grazing. Therefore, our models assume that detectability outside the fence was equal to or greater than detectability inside the fence.
We used a Bayesian implementation of a multistate JS model as outlined by
Censuses were conducted at unequal intervals ranging from 1 month (between consecutive censuses in the same year) to 10 years (from 2002 through 2012); therefore, we allowed the probability of growth and survival between intervals to depend on the length of time of the preceding interval.
We expanded the state-space in our multistate model to accommodate not just multiple states of a variable but also multiple state-space variables (size-age states JV, IM, and AD; location states Inside and Outside; and entry/survival states Not-yet-entered, Alive and present, and Dead or removed or permanently emigrated; see
The standard multistate JS model assumes that dead animals are not observable, whereas in our study the carcasses of marked tortoises were often found during and between censuses and then removed. We incorporated death information into our model whenever available to account for these deaths and to better estimate survival probability (see section “Death Records” in
We modified the JS model in a manner that could accommodate data from non-census detections occurring at variable times midway during intervals (see section “Non-Census Data Modeling” in
Age transitions in our model were one-way (for example, juveniles could become adults, but adults could not become juveniles); therefore, we non-randomly initialized the Markov chains for the Bayesian model simulations to avoid conflicts between randomized initial age and realized age after later intervals in the data. To ensure that chains were not converging to solutions that could be biased by their initial values, we applied highly contrasting rules when initializing different chains (see section “Initializing Markov Chains” in
We drew on monthly PRISM records for rainfall data for each of the fall-winter periods (October 1–March 31) preceding the annual surveys (
[The precipitation figures are from PRISM records (
Winter years | Spring years | Winter rain |
Recapture ratio |
1978–79 | 1979 | 172.32 | 0.9275 |
1984–85 | 1985 | 117.51 | 2.112 |
1988–89 | 1989 | 46.89 | 1.064 |
1992–93 | 1993 | 316.81 | 0.713 |
1996–97 | 1997 | 82.02 | 1.050 |
2001–02 | 2002 | 41.73 | 1.240 |
2011–12 | 2012 | 97.31 | 1.556 |
The initial results from the preliminary models suggested a potentially non-significant or negative correlation between precipitation and detectability. Therefore, we re-evaluated the potential for meaningful precipitation effects and ultimately decided to exclude precipitation from our final models. In that evaluation, we defined, for each year, a raw index of detectability based on the ratio of recaptures (number of times a tortoise was recaptured in that year, ignoring any history from previous years) divided by first-time captures (the time a tortoise was captured for the first time that year, ignoring previous years). In
Tortoise recapture ratio in relationship to winter precipitation (in millimeters [mm]). Recapture ratio is a raw index of detectability calculated as the number of times a tortoise was recaptured in a given year divided by the number of times a tortoise was captured for the first time that year, ignoring any capture history from previous years.
We used Markov Chain Monte Carlo (MCMC) simulations to fit and evaluate our JS model. We used the `jagsUI` package in R software as an interface for using Just Another Gibbs Sampler (JAGS) software to conduct these MCMC simulations (
Over the course of the period from 1979 to 2012, 1,120 desert tortoises were live captured and marked on surveys. An additional 3 desert tortoises were found dead and included with the 1,120 tortoises for modeling. Among the live captures, there were 420 females, 373 males, and 327 of unknown sex. Most (734) were AD, and 129 were IM and 257 were JV captures. Most (751) captures occurred inside, although 369 occurred outside. There were 1,119 live recapture events, with 606 female recaptures, 418 male recaptures, and 95 of unknown sex.
There were 58 JV recaptures, 60 IM recaptures, and 1,001 AD recaptures. Most (825) recaptures occurred inside the fence compared with outside (294). Of these desert tortoises, 248 were recovered dead. Slightly more than one-half of the dead recoveries were female (136) compared to 87 males and 25 of unknown sex. Most (223) were AD, 10 were IM, and 15 were JV. Most dead recoveries occurred inside the fence (215) compared with outside (32).
Following recommendations by
Among three size-age classes (JV, IM, AD), three growth transitions were possible across the wide range of intervals between censuses (1 month to 10 years). We modeled the transitions of JVs growing to IMs, JVs growing to ADs, and IMs growing to ADs. The probability of growth to a larger size-age is highest for the IMs growing to the AD transition, with this probability increasing rapidly between years 1 and 5 (
Estimated probabilities of a desert tortoise growing to a larger size based on intervals between years for the 34-year study at the Desert Tortoise Research Natural Area, western Mojave Desert, California. Bands of color refer to 90-percent credible intervals. AD, small to large-sized adult tortoises; IM, large immature tortoises; JV, juvenile and small immature tortoises; No., Number o.
Overall, the probability for tortoises to change locations regarding the boundary fence was low: from inside to outside the fence, the estimated probability ranged from a low of 0.019 to a high of 0.057, and from outside to inside the fence, the estimated probability ranged from 0.068 to 0.145. The probability of a tortoise moving from outside the boundary fence to inside was 184-percent greater (90-percent CI = 106 to 300 percent) than vice versa (
[The range of figures within the parentheses is 90-percent credible intervals]
Size class | Females | Males | ||
Inside to outside | Outside to inside | Inside to outside | Outside to inside | |
JV | 0.0190 |
0.0685 |
0.0374 |
0.0859 |
IM | 0.0238 |
0.0906 |
0.0458 |
0.1129 |
AD | 0.0293 |
0.1179 |
0.0568 |
0.1338 |
The estimated probability of annual survival (survival of >1 year) did not differ much between sexes, size-age classes, and locations of tortoises: the median values for survival ranged from a low of 0.791 to a high of 0.862 (
Estimated median probabilities (points) with 90-percent credible intervals (error bars) of annual survival by sex and size-age class state for desert tortoises developed from mark-recapture data collected over a 34-year period at the Desert Tortoise Research Natural Area, western Mojave Desert, California. AD, small to large-sized adult tortoises; IM, large immature tortoises; JV, juvenile and small immature tortoises.
Estimated median probabilities (points) with 90-percent credible intervals (error bars) of desert tortoises surviving >5 years by sex and size-age at the Desert Tortoise Research Natural Area, western Mojave Desert, California. AD, small to large-sized adult tortoises; IM, large immature tortoises; JV, juvenile and small immature tortoises.
For a given sex and size-age state, no significant differences existed between detections for tortoises occurring either inside or outside the boundary fence (
Median probabilities (points) with 90-percent credible intervals (error bars) of detecting desert tortoises at the census level, when a tortoise is available to be detected (that is, alive and present) at the Desert Tortoise Research Natural Area, western Mojave Desert, California. AD, small to large-sized adult tortoises; IM, large immature tortoises; JV, juvenile and small immature tortoises.
Estimates of densities varied between censuses in a given survey year. Estimates were higher for the second censuses of years 1979, 1985, and 2012 inside the fence, and for 1979 and 1985 outside the fence (
When densities were arrayed by sex, size-age class, and location (
Density estimates of all sizes of desert tortoises per square kilometer (km2) for the 14 censuses conducted between 1979 and 2012 at the Desert Tortoise Research Natural Area, western Mojave Desert, California. Error bars are 90-percent credible intervals.
Density estimates of all sizes of desert tortoises per square kilometer (km2) occurring inside the fence (top) and outside the fence (bottom) by census at the Desert Tortoise Research Natural Area, western Mojave Desert, California, 1979–2012. Error bars are 90-percent credible intervals.
Density estimates of desert tortoises per square kilometer (km2) by survey census, sex, size-age class, and location with regard to the fence at the Desert Tortoise Research Natural Area, western Mojave Desert, California. Error bars are 90-percent credible intervals for the sum total densities of females and males. AD, small to large-sized adult tortoises; IM, large immature tortoises; JV, juvenile and small immature tortoises.
Our JS model satisfied the Gelman-Rubin test for convergence for some but not all model parameters. Our model included large matrices (14 survey censuses for 1,123 marked tortoises), which we augmented with several hundred more possible uncaptured tortoise records (
Although some of our Markov chains did not converge to the same solution, they appeared to converge to solutions that were near one another. We initialized different chains with very different values based on biologically opposing guesses about demographic and movement patterns (normal-growth/high-survival versus slow-growth/low-survival, and minimal movement versus random movement;
The differences among chains might partly be explained by possible artifacts due to differences in the way we initialized them. The chains differed only for parameters involving size-age classification (that is, the growth model and effects of size-age on movement, survival, and detection;
Models can produce different results depending on assumptions and techniques employed.
[Assumptions and abilities are described both generally and for the specific applications of SLI implemented by
Assumptions/abilities | Stratified Lincoln Index (SLI) | Jolly-Seber (JS) | ||
Can do? | Did |
Can do? | Does this study do? | |
Population closure | ||||
---|---|---|---|---|
Assume open population within year (i.e., recruitment and death between same-year censuses allowed) | No | No | Yes | Yes |
Assume open population between years (i.e., recruitment and death between years allowed) | Yes | Yes | Yes | Yes |
Detection probability | ||||
Assume detection probability varies by year | Yes | Yes | Yes | No |
Assume detection probability varies between locations | Yes | Yes | Yes | Yes |
Assume detection probability varies between sexes | Yes | No | Yes | Yes |
Assume detection probability varies between size-age groups | Yes | Yes | Yes | Yes |
Assume detection probability varies in relation to annual covariates (e.g., precipitation) | No | No | Yes | No |
Movement | ||||
Assume movements occur between location states/strata | Yes | Yes | Yes | Yes |
Ability to estimate movement | Yes | Yes | Yes | Yes |
Demographic processes | ||||
Ability to estimate size-age growth | No | No | Yes | Yes |
Ability to estimate survival | No | No | Yes | Yes |
Population size | ||||
Ability to estimate population size | Yes | Yes | Yes | Yes |
Use of information | ||||
Ability to use full encounter history (e.g., if tortoise is observed in previous and later year, but not current year, then can it contribute to current year estimate?) | No | No | Yes | Yes |
Ability to include covariates (e.g., annual precipitation) | No | No | Yes | No |
Computing issues | ||||
Minimum required sample size | Recommended >50 | None, but smaller sizes are less precise. | ||
Other computing notes | Requires calculating the inverse of a |
Bayesian Monte Carlo models can take long time or be difficult to converge, as this one was. |
The density estimates for 1979 and 1985, reported by
[Density estimates for the Stratified Lincoln Index for adult and all sizes are from
Size-age class, carapace length at the midline, size (mm [MCL]) | Year | Stratified Lincoln Index: midpoint of density estimate, numbers per km2 (95-percent confidence interval) | Jolly-Seber: Median density estimate (numbers per km2, 90-percent credible interval) | ||
Inside fence | Outside fence | Inside fence | Outside fence | ||
All sizes | 1979 | 131 (111–155) | 114 (90–146) | 91 (86–100) | 69 (65–77) |
<140 mm MCL (JV) | 1979 | 44 (18–89)* | 42 (9–154)* | 17 (15–22) | 13 (11–17) |
≥180 mm MCL (AD) | 1979 | 70 (58–84) | 53 (41–69) | 61 (59–64) | 46 (42–49) |
All sizes | 1985 | 89 (77–101) | 52 (42–65) | 80 (78–86) | 47 (45–51) |
<140 mm MCL (JV) | 1985 | 14 (8–25)* | 9 (5–16)* | 12 (11–16) | 9 (8–11) |
≥180 mm MCL | 1985 | 69 (60–79) | 40 (32–52) | 62 (61–64) | 35 (33–37) |
Differences in modeling assumptions may have also contributed to differences in results. The JS model made fewer assumptions than the SLI method with regard to population closure, with the difference being that the SLI method assumes that no entries or death/removals occurred between two censuses in the same year, whereas such changes were allowed in the JS model. Differences in closure assumptions might not have greatly affected the results because incidents of death or growth to a larger size-age during the short interval between consecutive censuses were infrequent, and we structured the JS model to estimate smaller transition probabilities in correspondence to short intervals. There may have been more significant differences between the JS and SLI methods when it came to detection probability. In our JS model, we assumed that detection probability could vary with certain factors such as sex, size-age, and location, but not with year. In contrast, the SLI model, which is calculated independently by year, is naturally structured to allow for annual variations in detection probability, but ignores information about sex, size-age, and tortoise capture information from other years. It is difficult to assess how these differences might have affected each model’s estimates or contributed to their differences, but the differences in JV tortoise detection probability for 1979 between the SLI and JS models are striking and led to correspondingly large differences in population density for this size-age group (
Models can only be as reliable as the data on which they are based. For example, estimates for probability of growth, survival, and mortality rely on capture-recapture data. In accumulating long-term, accurate data for analyses of these topics, shortening the interval times for capture, recapture, and collection of remains of tortoises is highly desirable, shorter than the four- to ten-year intervals of this study. With lengthy intervals, critical information is lost on whether a tortoise was increasing in size, when a tortoise was last alive, and when and where deaths of individuals occurred. The investigators do not know if the animals walked away from the plot, were illegally collected, or died. Furthermore shell-skeletal remains deteriorate (
The variables used in models need to be appropriate for a given situation. As mentioned in section, “Methods” and described in more detail in section, “Models,” we explored preliminary models that included effects of precipitation on the detectability component of the model and ultimately discontinued that model development in favor of a model without precipitation. In general, an unnecessarily complex model can slow the convergence of chains, requiring many days for models to run with a possibility of non-convergence.
Finally, although models provide valuable information about several demographic attributes (for example, density, sex, size-ages, spatial distributions, and survival) of imperiled animal species, they also have limitations. For example, they may not be able to provide critical information about such topics as causes of population increases or decreases or death unless the appropriate variables are known and added to the models. The appearance of a new and emerging infectious disease is one example. The drivers of change are essential to understand for recovery efforts for federally and State-listed threatened and endangered species. When land managers are provided with such data on causes of population declines or mortality, they can institute adaptive management procedures to reduce human-caused deaths and thereby aid in recovery of the species.
Detecting early life stages of reptiles and amphibians can be very difficult for some species and can lead to assumptions of poor survival (
Our models show significant declines in the population at the southeastern part of the Desert Tortoise Research Natural Area over a 34-year period and concomitant low survival after 5 years, regardless of tortoise size or whether inside or outside the fence. Population declines were probably underway at the time our study area was first sampled. Similar declines were documented elsewhere in some populations in the Mojave and western Sonoran (Colorado) deserts at other sites first sampled in the late 1970s (
We compared our results from the JS model with results from other studies using different methods and report similarities in results. In 2004, the USFWS initiated distance sampling throughout designated critical habitat for adult desert tortoises (
The low survival rates observed at our study site were associated with the steep declines in population density observed over a relatively short time. Study populations of desert tortoises have experienced high mortality rates (or low survival) over short periods in other desert regions, and the authors have associated the high death rates of adults with drought (for example, 18.4 percent of adults in 1 year in the northeastern Mojave Desert in Ivanpah Valley, California;
High mortality rates of adults in long-lived freshwater and terrestrial chelonians can have catastrophic consequences, because many species of turtles and tortoises require several years to reach reproductive maturity, produce relatively few eggs annually, and have low survival rates of young turtles (
We developed a Bayesian multistate or state-space, hierarchical JS model that can be used for analyzing demographic data for animal or plant populations. We used selected long-term datasets from Agassiz’s desert tortoise at the Desert Tortoise Research Natural Area as a case study. For future work with desert tortoises and other long-lived species in long-term studies, our models can be modified by altering the underlying assumptions. Some modifications might include different numbers of size-age states, study areas, and intervals between surveys; use of opportunistic captures; and allowing for year-specific effects in probabilities of detection and (or) survival models. Environmental variables (precipitation, drought) and probability of hyperpredation events are other possibilities to add to the models. Models drawn from a dataset with brief intervals between surveys may be easier to work with than those with long intervals between surveys.
We used a combination of modeling examples presented by
We made several modifications or accommodations to this model to address several circumstances unique to our study design. These circumstances included (1) uneven interval spacing between census occasions; (2) state processes and detection processes likely influenced by the variable interval spacing, other state variables, or covariates; (3) removal of tortoise carcasses from the study area whenever found; (4) extraneous information on capture, recapture, or recovery of tortoises during intervals between censuses (non-census data); (5) large numbers of census occasions and multiple state processes (size-age states, location states, and survival states) generated a very large number of latent states (unobserved states) for tortoises at census occasions when they were not captured and, although the model is capable of handling latent states if time and memory are adequate, we took additional measures to eliminate some whenever possible; and (6) our state processes necessitated special consideration when setting the initial values of chains in the Markov-Chain Monte Carlo (MCMC) process, and we developed a customized set of initialization rules. We discuss our handling of these circumstances separately in the next six sections.
Throughout this report, unless otherwise noted, we use the term “capture” to refer to the first-time capture of an unmarked tortoise or recapture of a marked tortoise. We also use the term “detection” to refer to the capture of a tortoise or recovery of a dead tortoise. In our model, we assume that the first-time capture probability, recapture probability, and recovery probability of a given tortoise are equal, and can be referred to equivalently as detection probability.
Two month-long censuses were conducted in each spring of 7 years (1979, 1985, 1989, 1993, 1997, 2002, and 2012), with each pair of censuses occurring consecutively (that is, the second census in the pair beginning immediately after the first census concluded, so that the two censuses were approximately 1 month apart). Because of the short time separating both censuses per given year, we weighed the advantages and disadvantages of collapsing the information from each pair of censuses and modeling as a single census, for a 7-occasion model, versus modeling each census separately, for a 14-occasion model. We chose the 14-occasion model because of its similarity to Pollock’s Robust Design Model (
We did not assume complete closure between consecutive censuses in the same year. A small number of tortoises suffered mortality between the two censuses (for example, captured or recaptured alive in the first census, then dead or moribund in the following month’s census). Rather than assuming population closure (that is, forcing probability of survival to equal 1), we modeled the probability of survival between a pair of censuses by taking annual survival probability and raising it to the power of 1/12 to represent monthly survival probability (see section “Mathematical Models”). We also allowed new tortoises to enter the population at any census.
To estimate population size as well as allow new tortoises to enter the population, the Bayesian implementation of the JS model includes a strategy called parameter-expanded data augmentation (
The indexing of occasions was important to our modeling. Each tortoise contributes information about the mark-recapture population model at every census occasion (whether or not the tortoise is observed at that occasion), except a tortoise does not contribute information to census occasions after it is recovered dead and removed. We calculated our model by iterating the likelihood of each tortoise’s data from occasions 1–15, except for recovered tortoises, which were iterated over a narrower range of occasions (see also sections “Death Records” and “Non-Census Data Modeling”).
Year | Census | Occasion |
Dummy | Dummy | 1 |
1979 | 1 | 2 |
1979 | 2 | 3 |
1985 | 3 | 4 |
1985 | 4 | 5 |
1989 | 5 | 6 |
1989 | 6 | 7 |
1993 | 7 | 8 |
1993 | 8 | 9 |
1997 | 9 | 10 |
1997 | 10 | 11 |
2002 | 11 | 12 |
2002 | 12 | 13 |
2012 | 13 | 14 |
2012 | 14 | 15 |
We used recapture data across all 14 survey occasions to model population demographic patterns (rates of transition to larger size-age groups, and survival rates) and movement (rates of transition between inside and outside the fence) in a Bayesian JS modeling framework (
In these models, we represented the status of size-age (JV, IM, AD), location (IN, OUT), and live-dead (not-yet-entered, entered/alive, or dead/removed) as states; and growth, movement, and survival processes as transitions between states. We used logit functions to express the probabilities of state transition in relationships to predictor variables such as sex, length of time between surveys, and other states. Specifically, we used multinomial logistic regression for transitions between the three size-age states as a function of time between surveys, and binomial logistic regression for transitions between location states as a function of sex and size-age and for transitions between live-dead states as a function of sex, size-age, and location. We accounted for imperfect detection by incorporating another binomial logistic regression for detection probability as a function of sex, size-age, and location.
The transition probabilities for a JV tortoise follow a multinomial logistic function:
The gamma parameters are not uniquely identified, that is, an arbitrary constant can be added to all parameters without affecting the model; therefore, we set ξ
No further changes to size-age occur after a tortoise reaches AD size, and transition probabilities are set to 0.
We also used logistic functions to express transition probabilities between the location states, inside and outside the fenced area, as a function of sex and size-age.
ψ
ψ
We used a logistic function to represent annual survival:
We raised the annual survival probability to the power of
We defined detection probability, the probability of detecting a tortoise conditional on it being alive and present:
We applied the data augmentation technique to estimate the number of undetected tortoises living according to each sex, size-class state, and location state (
Out of 1,123 tortoises analyzed, 141 were recovered dead on a census and 107 others were recovered dead on non-census detections. As carcasses were recovered, they were removed from the study area and unavailable for detection. Tortoises that were never found dead had the possibility of being dead or alive throughout the remainder of the study and were modeled at all occasions from censuses 1–14 (that is, occasions 1–15, including the dummy occasion; see section “Intervals”). Conversely, recovered tortoises were modeled over a truncated range of occasions from census 1 through the last census up to its removal. For example, a tortoise (call it “Tortoise A”) that is recovered in census 3 was modeled from censuses 1 through 3 (that is, occasions 1–4). Every recovered tortoise was assigned a “last occasion” index ranging from 2 to 15 depending on when its recovery occurred, and tortoises that were never recovered were assigned a “last occasion” index of 15.
Occasionally, between census occasions and during the course of other non-census tortoise research activities at the Natural Area, tortoises were sometimes encountered. When this occurred, they were processed in a similar manner as tortoises captured or recaptured during a census occasion—that is, new captures were assigned an ID; their ID was recorded along with their location, size, and sex (when MCL ≥180 only); and dead tortoises were removed. In cases of non-census recoveries (carcass removals) of marked tortoises, then those individuals became unavailable for detection in subsequent censuses. We incorporated non-census recovery data into our model because they were too valuable to ignore. All recoveries, whether census or non-census related, provide certain information of a tortoise’s death and contribute directly to estimates of survival probability. If we had restricted our analysis to tortoise capture and recapture data collected only during censuses, then any tortoise that was removed during a non-census recovery would have only live encounters recorded for it; in other words, the model would be unable to separate these dead removed tortoises from surviving tortoises that simply evaded further recapture.
The standard mark-recapture model is structured around an interval-based framework, with captures and recaptures occurring in relatively short capture periods (in our case census occasions, which divide those intervals). Because non-census recoveries did not occur during census occasions, we considered several approaches: (1) assign the non-census recovery to the nearest census occasion, thus treating it as if it had occurred on a census; (2) assign the non-census recovery to the next census occasion, thus treating it as if had occurred on a later census; or (3) modifying the definition of an occasion so that it could represent non-census detection events. We rejected the first approach because it introduced potential data conflicts for tortoises actually captured or recaptured alive in the census immediately before being recovered dead. We rejected the second approach because the gap between censuses (except for consecutive censuses from the same year) ranged from 4 to 10 years, and we were uncertain of the potential biases in misattributing the year of recovery. We chose the third approach; for tortoises with non-census recoveries, we modified its next occasion by assigning it to the time of the non-census recovery.
For example, a tortoise (call it “Tortoise B”) that is recovered in 1981 is modeled for two census occasions (censuses 1 and 2 in 1979) and an additional occasion for 1981. The data for Tortoise B resemble that of Tortoise A (from section, “Death Records” example) in the sense that (1) they are both modeled the same number of occasions with respect to demographic processes such as growth, movement, and survival, and (2) they are both dead on the last occasion. There are also two important differences. First, the timing of the third occasion differs (1985 for Tortoise A, and 1981 for Tortoise B), and consequently the “time since last census” variable, which is used to model growth and survival, also differs (for example, since the last census would be 1979, then 6 years for Tortoise A, and 2 years for Tortoise B). Second, the recovery of Tortoise A occurred as part of a census and is admissible for modeling the detection process, but the recovery of Tortoise B is not. Therefore, whereas Tortoise A is modeled through census 3 (that is, occasion 4) for both detection as well as demographic processes (growth, movement, and survival), Tortoise B is handled slightly differently and modeled through census 2 (occasion 3) for the detection process and through a modified version of occasion 4 (modified for the non-census recovery timing) for the demographic processes.
To manage the different occasions that were admissible for the different processes (demographic or detection), for non-census recovered tortoises, we defined a “last census” index equal to the last census occasion prior to the non-census recovery (for example, for Tortoise B, occasion 3). For non-census recovered tortoises, the “last census” index was also one occasion less than the “last occasion” index (see section “Death Records”; for Tortoise B, occasion 4). Demographic processes were modeled through the “last occasion” index, whereas detection processes were modeled through the “last census” index. For tortoises that were recovered on a census, the “last census” index was equal to the “last occasion” index. For tortoises that were never recovered nor confirmed dead, the “last census” index and “last occasion” index were both equal to 15.
Year | Timing in relation to censuses | Assigned occasion |
1978 | Before census 1 | 1 |
1981 | Between censuses 2 and 3 | 4 |
1982 | Between censuses 2 and 3 | 4 |
1986 | Between censuses 4 and 5 | 6 |
1987 | Between censuses 4 and 5 | 6 |
1988 | Between censuses 4 and 5 | 6 |
1989 | Between censuses 6 and 7 | 8 |
1990 | Between censuses 6 and 7 | 8 |
1991 | Between censuses 6 and 7 | 8 |
1992 | Between censuses 6 and 7 | 8 |
1997 | Between censuses 9 and 10 | 11 |
1998 | Between censuses 10 and 11 | 12 |
1999 | Between censuses 10 and 11 | 12 |
2000 | Between censuses 10 and 11 | 12 |
2001 | Between censuses 10 and 11 | 12 |
2014 | After census 14 | NA |
In addition to non-census recoveries, we considered incorporating records of non-census live captures or recaptures into the model. These live records, when they occur prior to the first detection on a census or after the last detection on a census, can provide some additional information about the period of years when a tortoise was present. We visually inspected first and last non-census live records and found that they generally occurred close in time to the nearest detection on a census. We decided that there was little to gain from incorporating live non-census records into the model.
Latent states (unobserved states) of tortoises occurred on census occasions when they were not captured or recaptured and hence were unavailable for us to observe their states. As part of the Bayesian analysis, the MCMC follows an iterative process that simulates the distributions of likely values for all latent variables. Because the MCMC process can be extremely time-consuming and memory-intensive, especially when large numbers of parameters or latent variables are involved, we reduced the number of latent variables by applying constraints and filling data gaps whenever possible. Specifically, we filled data gaps in size-age or survival states when a comparison of observed states in previous and successive occasions indicated that the tortoise must have remained in the same state during any intervening occasions. We also reduced the number of occasions that we input into the model whenever a tortoise was recovered and removed and unavailable for detection in subsequent occasions.
For example, Tortoise 1023 was captured alive on occasions 2, 4, and 5 (that is, censuses 1, 3, and 4), and recovered dead on occasion 8 (census 7). Survival state variables are coded as 1=not-yet-entered, 2=live, 3=dead, or NA=not captured or recaptured. Occasion 1 is the dummy census (see section “Intervals”), and all tortoises are set to the “not-yet-entered” state (
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Survival state | 1 | 2 | NA | 2 | 2 | NA | NA | 3 | NA | NA | NA | NA | NA | NA | NA |
Although the tortoise was not observed at occasion 3, we know that it must have been alive in that occasion because it was recaptured alive in two later occasions. We filled the gap in occasion 3 with a live code (2); however, we could not fill the gaps for occasions 6 and 7. Because the tortoise was found dead in occasion 8, we only needed to model this tortoise’s data through occasion 8, thus leaving 2 latent survival state variables for the MCMC to model (
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Survival state | 1 | 2 | 2 | 2 | 2 | NA | NA | 3 |
As a second example, Tortoise 907 was captured alive on occasions 4 and 5 (that is, censuses 3 and 4) and never encountered again (
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Survival state | 1 | NA | NA | 2 | 2 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA |
Because the survival status of Tortoise 907 was unknown throughout the remainder of the study, the state variables for occasions 6 through 15 are latent. The state variables for occasions 2 and 3 are also latent because we do not have data indicating whether this tortoise had entered the population at those occasions. We modeled this tortoise’s data through occasion 15 and had 12 latent survival state variables to model.
We iterated (MCMC) simulations to generate statistical distributions for the probable values of latent state variables (see section “Latent States”) and, thus jointly, of model coefficients (see section “Mathematical Models”). A list of these parameters is shown in
[Parameter coefficients and sex were initialized at random from
Parameter | Description | Initialization method |
JV to IM or AD growth coefficients | ||
IM to AD growth coefficients | ||
IN to OUT movement coefficients | ||
OUT to IN movement coefficients | ||
Annual survival coefficients | ||
Detection coefficients | ||
Detection coefficient, location effect | ||
Sex (1=F, 0=M) of tortoise i, if latent | ||
Size-age (1=JV, 2=IM, 3=AD) of tortoise i at occasion t, if latent | Chain 1: normal-growth rule |
|
Location (1=inside, 2=outside) of tortoise i at occasion t, if latent | Chain 1: minimal-move rule |
|
Survival state (1=not-yet-entered, 2=live, 3=dead) of tortoise i at occasion t, if latent | Chain 1: high-survival rule |
The latent states included survival state (1=not-yet-entered, 2=live, or 3=dead), size-age state (1=JV, 2=IM, or 3=AD), and location state (1=Inside or 2=Outside). We did not simply apply random generation to initialize all of these states because of the high chance of generating contradictory sequences (for example, AD size-age in one occasion and IM in the next, or dead in one occasion and live in the next). It was impossible for the model to process a chain when it was initialized with a contradictory state sequence.
We considered two alternative approaches to initializing state variables in the chains: (1) initialize the chains randomly but constrained by a set of rules that ensured allowable state sequences, or (2) initialize the chains deterministically using a set of rules that ensured allowable state sequences but assign different chains with different rules based on very different assumptions about survival and growth. We chose the latter approach because it had two advantages over the former approach: (1) it was easier to implement and (2) it allowed us to choose rules that produce very different sets of initial values (that is, greater over-dispersion) between the chains to better assess model convergence. When multiple chains converge to the same solutions, after being initialized with widely varying values, then those solutions are considered to be robust (
We initialized chains using biologically contrasting scenarios about the population demographics. For the best-case scenario, we initialized the first two chains by assigning the latent size-age and survival state variables to initial values that were consistent with normal-growth and high-survival rates. We varied the two “best-case scenario” chains according to different movement scenarios. Specifically, for the first chain we initialized location state variables according to a minimal-movement rule by initializing latent locations to last known previous location or next soonest location, whichever was more immediate. We initialized the second chain using a random-movement rule by initializing latent locations randomly as inside or outside. For the worst-case scenario, we initialized a third chain by initializing the latent size-age and survival state variables with values consistent with slow-growth and low-survival rates. For example, under low survival, we initialized tortoises as dead for latent survival states after the last live observation, and we initialized size-age states to the slowest growth possible within the intervals that tortoises were detected or initialized live (growth could not occur after death). We describe our rules for initializing survival and size-age states in greater depth in the following section. We initialized movement states in the worst-case scenario chain according to minimal-movement rules, also respecting the rule that tortoises would not move after death.
We applied two different initialization rules for latent variables of survival state based on worst-case (low-survival) and best-case (high-survival) scenarios for survival rates. Under the high-survival rule, we initialized all latent survival state variables to the “live” state. Under the low-survival rule, we initialized all latent survival state variables to the “dead” state, except in cases where a tortoise was later observed alive. To illustrate, we return to the examples of Tortoises 1023 and 907 (see section “Latent States”). Because Tortoise 1023 was last captured alive in occasion 5 and recovered dead in occasion 8, we assigned two different sets of initial values for the latent survival state in occasions 6 and 7.
The other occasions have data; therefore, they do not require initial values. Tortoise 907 was known to be alive in the population during occasions 4 through 5. The latent variables after the last occasion on which it was seen alive were all initialized to a live status under the high-survival rule, and to a dead status under the low-survival rule. We initialized the occasions prior to the first capture (occasions 2 and 3) to a not-yet-entered state under both rules.
[States for occasions 1, 4, and 5 do not require initialization because they were not latent states. The observed survival state is copied from
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Observed survival state | 1 | NA | NA | 2 | 2 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA |
Initialized latent survival state (High- survival rule) | NA | 1 | 1 | NA | NA | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
Initialized latent survival state (Low-survival rule) | NA | 1 | 1 | NA | NA | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
We also applied two different initialization rules for latent variables of size-age state based on worst-case (slow-growth) and best-case (normal-growth) scenarios for growth rates. We initialized latent size-age states using a combination of information based on size-age and number of years since the last previous detection, size-age and number of years until the next soonest detection, and the survival status at the next soonest detection.
Under the slow-growth rule, we initialized all latent size-age state variables to the last known size-age state, except in cases where the next detection of the tortoise was as a dead recovery. Because we only used the slow-growth rule in combination with the low-survival rule, for initializing the worst-case scenario chain, we were constrained to initializing latent size-age states in a manner that was consistent with initialized latent survival states under the low-survival rule. For any latent states preceding a dead recovery, we initialized the latent size-age to the size-age at the time of recovery because the latent survival state would have been initialized to the dead state and tortoises cannot grow when dead.
Under the normal-growth rule, we initialized size-age based on a combination of information about the observed size-age of that tortoise during any previous or successive detections, and the timing or occasion numbers of those detections. For any latent states that preceded the very first detection of a tortoise, we initialized the latent size-age to the first detected size-age. For any latent states after the first detection, we developed a hierarchy of rules for choosing which detection event (the last previous or the next successive) would provide more useful information from a biological perspective for initializing the latent size-age under normal growth conditions:
If <5 years since last previous detection, then initialize latent size-age to the size-age observed at the last previous detection; or
If ≥5 years since last previous detection but <5 years until the next successive detection, then initialize size-age to the size-age observed at the next successive detection; or
If ≥5 years since last previous detection, and either ≥5 years until the next successive detection or no successive detections, then initialize size-age based on the size-age state at the time of the last previous detection and the number of years since the last previous detection; and
If the tortoise was in the JV size-age class at the time of the last previous detection, and there were ≥5 years and <15 years since the last previous detection, and ≥5 years since the next detection, then initialize size-age to the IM size-age class; or
If the tortoise was in the JV size-age class at the time of the last previous detection, and there were ≥15 years since the last previous detection, and ≥5 years until the next detection, then initialize size-age to the AD size-age class; or
If the tortoise was in the IM size-age class at the time of the last previous detection, and there were <10 years since the last previous detection, and ≥5 years since the next detection, then initialize size-age to the IM size-age class.
For example, the size-age states for Tortoises 1023 and 907, respectively, are presented in
[Under the slow-growth rule for the worst-case scenario Markov chain, the latent size-age states in occasions 6 and 7 are initialized to the same size-age (AD) of the tortoise when it was recovered dead in occasion 8, for consistency with latent survival states having been set to the dead state under the low-survival rule (see
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Year | — | 1979 | 1979 | 1985 | 1985 | 1989 | 1989 | 1993 |
Observed size-age | NA | 1 | NA | 2 | 2 | NA | NA | 3 |
Initial values (slow-growth) | 1 | NA | 1 | NA | NA | 3 | 3 | NA |
Initial values (normal-growth) | 1 | NA | 1 | NA | NA | 2 | 2 | NA |
[States for occasions 1–5 and 8 do not require initialization because they were not latent states. The observed survival state is copied from
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Observed survival state | 1 | 2 | 2 | 2 | 2 | NA | NA | 3 |
Initialized latent survival state (High-survival rule) | NA | NA | NA | NA | NA | 2 | 2 | NA |
Initialized latent survival state (Low-survival rule) | NA | NA | NA | NA | NA | 3 | 3 | NA |
Occasion | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
Year | — | 79 | 79 | 85 | 85 | 89 | 89 | 93 | 93 | 97 | 97 | 02 | 02 | 12 | 12 |
Obs. size-age | NA | NA | NA | 1 | 1 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA |
Slow-growth | 1 | 1 | 1 | NA | NA | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Normal-growth | 1 | 1 | 1 | NA | NA | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
We used the “jagsUI” package in R software as an interface for using Just Another Gibbs Sampler (JAGS) software to conduct MCMC simulations with Gibbs sampling algorithms (
When chains had still not reached convergence or failed to mix, we adapted our model to address the possibility that our initial values could be too overdispersed or that our prior distributions could be too diffuse to converge to the correct solution. Gibbs sampling methods, and more generally Metropolis-Hastings algorithms, have been noted to have problems converging in high-dimensional models (
More specifically, we simplified our model by omitting population estimation, which contributes to the high dimensionality of our model with parameter-extended data-augmentation, the dummy occasion, and the “not-yet-entered” survival state. This essentially reduced our model to a CJS model, but we retained the same model equations with the same diffuse priors and specifications as our JS model. Another difference between the CJS model and the JS model is that the CJS model only uses data from a tortoise conditional upon its first capture. We ran simulations based on the CJS model and assessed convergence in the same manner as for the JS model.
When the CJS model reached convergence, we took the posterior distributions of model coefficients and used them to generate new initial values and new prior distributions for model coefficients in our JS model (
[Parameters were initialized at random from
Parameter | Description | Revised initial values | Revised prior distribution |
JV to IM growth intercept | |||
JV to AD growth intercept | |||
Effect of years on JV to IM growth | |||
Effect of years on JV to AD growth | |||
IM to AD growth intercept | |||
Effect of years on IM to AD growth | |||
IN to OUT movement intercept | |||
Movement of females vs males | |||
Movement of larger vs smaller sizes | |||
OUT to IN movement intercept | |||
Movement of females vs males | |||
Movement of larger vs smaller sizes | |||
ψ |
Annual survival intercept | ||
ψ |
Survival of females vs males | ||
ψ |
Survival of larger vs smaller sizes | ||
ψ |
Survival outside vs inside fence | ||
θ |
Detection intercept | ||
θ |
Detection of females vs males | ||
θ |
Detection of larger vs smaller sizes | ||
θ |
Detection outside vs inside fence |
We ran Markov chain Monte Carlo (MCMC) simulations on our model as described in
Following recommendations by
[Model coefficients are listed in terms of their mathematical symbol in model equations (see
Model coefficient | parameters | Posterior distribution | 90-percent credible interval | PSRF | Effective sample size | |||
Symbol | Code name | Mean | SD | Median | 5% | 95% | ||
xi.intercept[1,2] | –3.507 | 0.457 | –3.474 | –4.293 | –2.790 | 1.165 | 16 | |
xi.intercept[1,3] | –4.552 | 0.951 | –4.286 | –6.387 | –3.347 | 2.738 | 4 | |
xi.intercept[2,3] | –2.258 | 0.397 | –2.266 | –2.915 | –1.607 | 1.471 | 8 | |
xi.slopeYrs[1,2] | 0.546 | 0.294 | 0.655 | 0.071 | 0.913 | 4.583 | 3 | |
xi.slopeYrs[1,3] | 0.988 | 0.368 | 1.149 | 0.374 | 1.414 | 5.326 | 3 | |
xi.slopeYrs[2,3] | 0.733 | 0.482 | 0.979 | –0.009 | 1.246 | 7.756 | 3 | |
psiIO.intercept | –3.500 | 0.841 | –3.469 | –4.976 | –2.181 | 1.067 | 39 | |
psiIO.slopeSex | –0.694 | 0.297 | –0.690 | –1.183 | –0.205 | 1.006 | 354 | |
psiIO.slopeSize | 0.227 | 0.288 | 0.223 | –0.216 | 0.734 | 1.069 | 39 | |
psiOI.intercept | –2.630 | 0.590 | –2.669 | –3.562 | –1.650 | 1.054 | 43 | |
psiOI.slopeSex | –0.241 | 0.264 | –0.246 | –0.677 | 0.204 | 1.006 | 285 | |
psiOI.slopeSize | 0.285 | 0.204 | 0.298 | –0.050 | 0.600 | 1.050 | 47 | |
phi.intercept | 1.687 | 0.491 | 1.854 | 0.854 | 2.299 | 4.054 | 3 | |
phi.slopeSex | 0.200 | 0.074 | 0.200 | 0.076 | 0.320 | 1.037 | 59 | |
phi.slopeSize | –0.026 | 0.181 | –0.104 | –0.239 | 0.270 | 5.378 | 3 | |
phi.slopeLoc | –0.131 | 0.087 | –0.129 | –0.277 | 0.010 | 1.028 | 77 | |
po.intercept | –0.167 | 0.374 | –0.053 | –0.827 | 0.313 | 3.363 | 4 | |
po.slopeSex | –0.013 | 0.105 | –0.013 | –0.189 | 0.157 | 1.001 | 1,800 | |
po.slopeSize | 0.370 | 0.126 | 0.332 | 0.205 | 0.593 | 3.349 | 4 | |
po.slopeLoc | 0.050 | 0.045 | 0.036 | 0.003 | 0.141 | 1.000 | 1,800 |
Our model did not reach convergence for any parameters associated with size-age variables, including all parameter coefficients for growth processes and the slope parameter coefficients for the effects of size-age on movement, survival, and detection probabilities (
When examining individual chains for the non-converged parameters, for example, the effect of time (in years) on growth rates, there were consistent patterns (
Trace plots and posterior distribution density plots of three Markov chains for four parameters the intercepts of growth models from JV to IM size-age (xi.intercept [1,2]) (top row), JV to AD size-age (xi.intercept [1,3]) (second from top row), IM to AD size-age (xi.intercept [2,3]) (third from top row), and the effect of time (in years) since previous survey on growth from JV to IM size-age (xi.slopeYrs [1,2] (bottom row). Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial values.
Trace plots and posterior distribution density plots of three Markov chains for four parameters—the effect of time (in years) since previous survey on growth from JV to AD size-age (xi.slopeYrs [1,3]) (top row), IM to AD size-age (xi.slopeYrs [2,3]) (second from top row), and the intercept and effect of sex (being female) in the movement model from inside to outside (psiIO.intercept [third from top row], psiIO.slopeSex [bottom row]). Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial values.
When differences between chains occurred, the median values of model parameters never differed by more than approximately 2 (
Having converged chains is important when making statistical inferences. When chains are unconverged or when they converge to different posterior distributions, they cannot all be correct. This calls into question the validity of results from any one chain, and we are unable to determine the true posterior distribution or even which chain is closest to it. We present posterior estimates based on the combined results of unconverged chains because, although less ideal than those of converged chains, they can still be useful when they provide similar inferences to one another. Differences among chains reveal some sensitivities to their initial values, which we had systematically set to opposite ends of the biological spectrum for growth (normal or slow), survival (high or low), and movement (minimal or random). We believe that the truth falls somewhere between the ends of the spectrum, and that our 90-percent credible intervals from the posterior distribution based on unconverged chains likely encapsulate most of the true posterior distribution. Therefore, unconverged chains can still be useful when they indicate similar population inferences (
Trace plots and posterior distribution density plots of three Markov chains for four parameters—the effect of size-age on inside-to-outside movements (psiIO.slopeSize) (top row), intercept in the outside-to-inside movement model (psiOI.intercept) (second from top row), and effects of sex (being female) and size-age on outside-to-inside movements (psiOI.slopeSex [third from top row], psiOI.slopeSize [bottom row]). Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial values.
Trace plots and posterior distribution density plots of three Markov chains for four parameters—the intercept in the survival model (phi.intercept) (top row), and effects of sex (being female; phi.slopeSex) (second from top row), size-age (phi.slopeSize) (third from top row), and location (phi.slopeLoc) (bottom row) on survival. Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial values.
Trace plots and posterior distribution density plots of three Markov chains for four parameters—the intercept in the detection model (po.intercept) (top row), and effects of sex (being female; po.slopeSex) (second from top row), size-age (po.slopeSize) (third from top row), and location (po.slopeLoc) (bottom row) on survival. Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial values.
Medians (dots) and 90-percent credible intervals (bars) from posterior distributions based on three separate Markov chains for estimating the probability of transitioning from JV to AD size-age in relationship to increasing time since the previous survey. Variables V1–V19 correspond to time intervals (time since previous survey) ranging from 1 to 10 years in increments of one-half-years. Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial value.
Medians (dots) and 90-percent credible intervals (bars) from posterior distributions based on three separate Markov chains for estimating the probability of transitioning from IM to AD size-age in relationship to increasing time since the previous survey. Variables V1–V19 correspond to time intervals (time since previous survey) ranging from 1 to 10 years in increments of one-half-years. Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial value.
Medians (dots) and 90-percent credible intervals (bars) from posterior distributions based on three separate Markov chains for estimating the population size of desert tortoises across 14 surveys in 1979 (N[1] and N[2]), 1985 (N[3] and N[4]), 1989 (N[5] and N[6]), 1993 (N[7] and N[8]), 1997 (N[9] and N[10]), 2002 (N[11] and N[12]), and 2012 (N[13] and N[14]). Two chains (red and black) were started with normal-growth and high-survival initial values, and the third chain (green) was started with slow-growth and low-survival initial value.
For more information concerning the research in this report, contact the
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U.S. Geological Survey
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