Prepared by: Dr. Steve Amstrup, PBSG member, and Trent McDonald
Most of what we know about polar bears we know from capturing them and then releasing them alive at the site of capture. Capturing bears allows the collection of biological samples (e.g. blood or fat samples) and measurements of physical stature and condition. Perhaps most importantly, capture efforts that are repeated regularly over multi-year periods allow the estimation of vital rates like reproduction and survival and they allow estimation of population size. These parameters have been estimated by what are called capture-recapture analyses, and it is capture-recapture that has provided most of the population assessments we currently have for polar bears worldwide.
Polar bear marked with a lip tattooImage: Andrew Derocher
So what is capture-recapture? Capture-recapture or mark-recapture is a method, for estimating population size and other parameters, that is based on ratios of marked to unmarked individuals. Perhaps the most famous early application of capture-recapture was estimation, by Pierre Simon Laplace, of the population size of France in 1802 (Cochran, 1978; Stigler, 1986). At that time, live births were recorded for all of France on an annual basis. In the year prior to September 1802, Laplace estimated the number of such births to be approximately X = 1,000,000 nation wide. These newly born individuals constituted his marked population. Laplace then obtained census and live birth data from several communities scattered across all of France. At that time, census data were not available for most communities, but Laplace chose these communities for his estimate because their “zealous and intelligent mayors” made sure they did have annual census data. Recognizing temporal variation in annual birth rates, Laplace summed the number of births reported in these sample communities for the three years leading up to the time of his estimate, and divided by three to determine that there were x = 71,866 births per year (marked individuals) in those communities. The total number of individuals in these sampled communities was determined by the mayors to be y = 2,037,615. The ratio of marked to unmarked was then p = x/y = 71,866/2,037,615 = 0.0353. Assuming that the ratio of marked to unmarked in the sample was the same as the ratio of marked to unmarked in the population meant that the total estimated population of France in 1802 was N = X/p ≈ 1,000,000/0.0353 = 28,328,612. In its simplest form, a similar estimate for a polar bear population could be derived if you went out one year and captured and marked 100 bears. If the following year you went into the same area and captured another 100 bears and 10 of them were marked, you might conclude, ignoring details such as births, deaths, emigration and immigration, that one in ten bears in the population was marked or p = 0.10. Because the total sample size was 100 bears the total population size could be estimated as N = 100/0.10 = 1000.
So, this is the concept of capture-recapture in its simplest form. Of course, in this simple form, you are limited to a simple estimate of population size. Even more importantly, with capture-recapture as with lots of other things, the devil is in those details that were ignored in these examples. In these simple examples, some rather important assumptions were made. For example, the total number of live births in France was treated as a known entity. Of course in fact, it was an estimate. Similarly, the simple polar bear estimate assumed that there was no effect of births or deaths or migration by animals in or out of the area sampled. That is, it was assumed that the total number of marks in the population during the second year was still 100 and that they were all equally available for capture in that second year. These things and many more create complications in capture-recapture estimates. To address these complications, the theories and applications of capture recapture have moved far beyond the simple models just described. In an effort to make these increasingly complicated developments available to biologists working with real world problems, we developed the Handbook of Capture Recapture Analysis (Amstrup et al. 2005). That volume begins with the very earliest applications of capture-recapture and brings the reader, with easy to understand language and numerous examples, through some of the most up to date methods available. Here, we frequently refer to appropriate sections in that volume.
Closed population models
A major bifurcation in capture-recapture analyses is the division between estimates for populations that are considered open and those that are considered closed. A closed population is one in which the total number of individuals is not changing through births, deaths, immigration or emigration. The first applications of capture-recapture methods were with populations that were assumed to be closed for the period of estimation. In practice, animal populations are not closed. If the interval among sampling events is short enough, however, the changes over the time period of interest may be small enough that the assumption of closure is reasonable. There are many cases in practice where this is a reasonable assumption. However, even in populations assumed to be closed many complications arise. The analysis of capture-recapture data from closed populations, therefore, continues to be a topic of interest to biologists and managers. This topic is introduced and discussed in detail in Chapters 2 and 4 of (Amstrup et al. 2005).
Open population models
Polar bear populations clearly are not closed. Just as importantly, many other factors can contribute to uncertainties and heterogeneity in the data collected in polar bear studies. For example, polar bears can live 30 years in the wild. Reproduction is protracted and the period of caring for young is long. Thus, at any one time there is a variety of sex and age groups in any population. There is variation among individuals and among sex and age groups in movements and habitat use patterns. Some bears may run when they hear a helicopter engine others may hide. Bears are probably more individualistic than innate in their behaviors. The bottom line is that all individual bears in a population may not be uniformly available for capture. Further, the interest in real populations monitored over time is not just “how many animals are there.” Rather, recognizing populations as dynamic, people also want to know reproduction and survival rates and other characteristics that may indicate population health or status.
Polar bears may be among the most challenging species to which capture-recapture methods are applied, but they are not alone in these challenges. Hence, great advances have been made in methodologies of estimating population parameters from capture-recapture methods. A major advancement was the development of maximum likelihood estimation for the analysis of open population capture-recapture data by Cormack (1964), Jolly (1965) and Seber (1965). Early methods for analyzing capture-recapture and tag-recovery data relied upon ad-hoc models for their justification. As people became ever more aware of the many complications in capture-recapture modeling of real populations, the inadequacies of these approaches was increasingly obvious. This led to the development of what are now called the Cormack-Jolly-Seber (CJS) and the Jolly-Seber (JS) models. These and all of the more recent developments in capture-recapture are maximum likelihood approaches to estimation.
The fundamental difference between JS and CJS models is that the JS model incorporates the assumption that all animals are randomly sampled from the population and that captures of marked and unmarked animals are equally probable. That means that the JS model can provide direct estimates of population size as illustrated above. The CJS model is based solely on recaptures of marked animals and provides estimates of birth rates, survival rates, and capture probabilities only. Recent studies have shown that population estimates can be derived from CJS type models by application of Horvitz-Thompson (Horvitz and Thompson 1952) type estimators. These derived estimators effectively overcome what at first seems a limitation of CJS type modeling (McDonald and Amstrup 2001, Amstrup et al. 2005), and are now the pretty standard practice for estimation of both size and vital rates of animal populations.
Of course most studies of wild populations are not simply capture-recapture studies. Biologists increasingly utilize radio-telemetry to learn about movements. They monitor the harvests, and they may have varying additional observational methods. Recent advances in capture-recapture methodology have been designed to take advantage of these different kinds of observations. New methods allow covariates, supplemental characteristics of the data, to be incorporated to explain variation among data points. For example, radio-collared animals probably have higher recapture probabilities than non collared animals. Incorporation of such covariates can result in complicated model structures which can be difficult to handle even with available software and powerful computers. A major recent advance that facilitates building of these complicated models is the direct regression approach to capture recapture (McDonald and Amstrup 2001, Amstrup et al. 2005, Chapter 9). Routines to accomplish this approach are available at http://cran.cnr.berkeley.edu/web/packages/mra/index.html. Because they dramatically simplify construction of complex models, serious practitioners may want to become familiar with these routines.
Similarly, modern methods allow joint modeling of harvest returns, resightings, and live recaptures to take simultaneous advantage of independent data streams. In this approach, animals can be recorded after their initial tagging by (i) live recaptures, (ii) live resightings at any time between tagging operations, and (iii) tags recovered from animals killed or found dead between tagging occasions (Amstrup et al. 2005: Chapter 7) This may be particularly valuable for polar bears because most populations are hunted and the hunter harvest offers a good opportunity to “recapture” tags that may have been deployed by researchers but which have not shown up in the standard recapture samples.
The original maximum likelihood models (CJS, JS) for capture-recapture data assumed that the animals in the population being considered were homogeneous in the sense that every one has the same probability of being captured when a sample was taken, and the same probability of surviving between two sample times. Later developments illustrated that the homogeneity assumption could be relaxed, with covariates being used to describe different capture and survival probabilities. Under some circumstances, spatial or demographic separation of animals into different groups, with random movement between these groups, is difficult to quantify and covariates may not be efficient at accounting for the variation that such separation introduces. For example, if members of a population move among different geographic locales (e.g. feeding, breeding or wintering areas), and the probability of movement among these locales is unknown, movement among locales has the ability to affect the estimates derived. Covariates associated with the individual animals or sample times may not be well suited to model the situation because the state of the animal and hence the covariate necessary to represent it may not be known when animals are not seen. In that case the movement between locations or states is best modeled directly. In addition to geographic locations, states also could be different stages of maturation of animals, or animals that are with young versus those that are not. To address these kinds of challenges, “multi-state” capture-recapture models have been developed (Amstrup et al. 2005: Chapter 8). If sufficient sample sizes are available, such methods can directly resolve many issues that can lead to uncertainty in estimates.
Most of what we know about polar bear populations is known from capture-recapture kinds of studies. Methods have improved, and we have better estimates of parameters of polar bear populations than ever before. Many challenges remain, however. For example, polar bears are the most mobile of all non-aquatic mammals and practical study areas often sample only a small portion of the area in which members of each population live. What portion, then, of the total population is represented by derived estimates. And, what about the ability of temporary emigration to bias estimates. Clearly, actual capture probabilities may be very different among individuals and among capture seasons and these differences must be accounted for in models. Further, because capture-recapture estimates of survival are “apparent” survival, an animal that has temporarily left the study area cannot be distinguished from one that has died. Therefore, the polar bears’ great mobility and the researchers’ limited sampling ability are likely to be persistent issues. The polar bears’ multi-year reproductive cycle and the dependence of reproductive success on things that vary greatly among years, like sea ice development and persistence, make assessment of reproductive rates very difficult and add to the length of studies necessary to derive good estimates. Yet, understanding reproduction is an important part of projecting what may happen to polar bear populations in the future. Sample sizes obtainable for most populations are small enough to make the confidence intervals around many of our estimates a matter for concern. Despite these and other issues, however, we largely understand polar bear population dynamics because of capture-recapture efforts.
The body of capture-recapture literature is now so large that this type of estimation method almost comprises its own branch of estimation. Regardless of their complexity, however, all of the approaches described above share a common feature with the simplest applications of over 200 years ago. That is, they are all based at some level upon the ratios of marked to unmarked animals in some kind of sample. Recognition of the “devil” in the details of such estimators has resulted in many novel developments that have resulted in more accurate and precise parameter estimates. Recent developments summarized in Amstrup et al. (2005) illustrate what those developments mean for estimates of the parameters in which polar bear biologists and managers are interested.
Amstrup, S. C., T. L. McDonald, and B. F. J. Manly (eds.). 2005. Handbook of Capture-Recapture Analysis. Princeton University Press: Princeton, New Jersey, 296 pp.
Cochran, W. G. 1978. Laplace’s ratio estimators. Pages 3-10 in H. A. David, editor. Contributions to survey sampling and applied statistics. Academic Press, New York.
Cormack, R. M. 1964. Estimates of survival from the sightings of marked animals. Biometrika 51:429-438.
Horvitz, D. G., and D. J. Thompson. 1952. A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47: 663-685.
Jolly, G. M. 1965. Explicit estimates from capture-recapture data with both death and immigration - Stochastic model. Biometrika 52:225-247.
McDonald, T. L., and S. C. Amstrup. 2001. Estimation of population size using open capture-recapture models. Journal of Agricultural, Biological, and Environmental Statistics 6:206-220.
Seber, G. A. F. 1965. A note on the multiple recapture census. Biometrika 52:249-259.
Stigler S. M. 1986. The history of statistics: the measurement of uncertainty before 1900. The Belknap Press of Harvard University Press. Cambridge.