Research Article  Open Access
L. L. Eberhardt, J. M. Breiwick, "Trend of the Yellowstone Grizzly Bear Population", International Journal of Ecology, vol. 2010, Article ID 924197, 5 pages, 2010. https://doi.org/10.1155/2010/924197
Trend of the Yellowstone Grizzly Bear Population
Abstract
Yellowstone's grizzlies (Ursus arctos) have been studied for more than 40 years. Radiotelemetry has been used to obtain estimates of the rate of increase of the population, with results reported by Schwartz et al. (2006). Counts of females with cubsoftheyear “unduplicated” also provide an index of abundance and are the primary subject of this report. An exponential model was fitted to such counts, using nonlinear leastsquares. Estimates of the rate of increase, r, were about 0.053. 95% confidence intervals, were obtained by several different methods, and all had lower limits substantially above zero, indicating that the population has been increasing steadily, in contrast to the results of Schwartz et al. (2006), which could not exclude a decreasing population. The grizzly data have been repeatedly misused in current literature for reasons explained here.
1. Introduction
Yellowstone’s grizzlies have been studied extensively for over 40 years. A recent monograph [1] provided a detailed analysis of much of the accumulated data. Two methods have been used to estimate trends of the population. One, the “unduplicated” counts of females with cubsoftheyear, was developed by Dr. R. R. Knight and has been maintained since 1973. The second approach uses radiotelemetry data and was pioneered by Craighead [2]. The “unduplicated” counts along with some earlier observations were converted to an index described as a “minimum population” estimate [3]. That index used a threeyear moving average to smooth the annual numbers of adult female bears and a calculated proportion of adult females in the population of 0.274. A long series of “study committees” [4] continued use of the index because a “minimum population” estimate appeared to be something readily understood by administrators and the public.
A weakness in the “minimum population” index is the presence of a strong serial correlation, induced by the fact that each annual estimate contains the counts from 2 other years. Difficulties caused by serial correlation have been assessed by Watt [5, 6], Chapman[7], and Freckleton et al. [8]. Eberhardt [9] indicated the sizable degree of correlation thus induced when the underlying data are constructed only from random numbers and showed that a correlation of −0.707 could be induced in analysis of two independent sets of random numbers. Royama [10] reported the same result. The “minimum population” index is not used here, and the DurbinWatson test [11] does not show evidence of serial correlation in the data used here.
An alternative approach using radiotelemetry was applied by Eberhardt, et al. [12] using an approximation to Lotka’s equation proposed by Eberhardt [13]. That analysis yielded an estimate of of 1.046 with bootstrap 95% confidence limits of 1.00 to 1.09. That study depended on bears caught and marked in a set of “backcountry” traps, located away from centers of human activity. The major, and essentially the only, cause of adult grizzly mortality comes from conflicts with humans. When such conflicts arise, efforts are made to capture and move bears. Much experience has shown that bears involved in such a situation have a low survival rate. Consequently, bears first captured in “conflict” situations were not used in the analysis, but bears firstcaptured in the socalled “research” trapping continued to be used in the analysis after they were involved in conflicts with humans. Subsequently, Pease and Mattson [14] proposed that there are “wary” and “unwary” bears so that the “unwary” bears firstcaptured in “conflict” situations should be included in the telemetry sample. This, of course, reduced the lower confidence limit on population trend below 1.0. The issues thus involved have been extensively explored in the monograph by Schwartz et al. [1] and need not be further considered here. However, probability distributions generated by stochastic simulations of Yellowstone grizzly numbers [15] have lower limits below =1.00, thus not excluding the prospect that the population has not increased. Again, these results have been amply examined in the monograph of Schwartz et al. [1]. The purpose of the present study is to utilize the original “unduplicated” counts as an independent measure of population trend and to correct some erroneous uses of the data.
2. Study Area
The study area constitutes Yellowstone and Grand Teton National Parks, 6 adjacent National Forests plus some state and private lands, and totals about 34,500 km^{2}. It is known as “The Greater Yellowstone Ecosystem” (GYE). The GYE constitutes the Yellowstone Plateau, and surrounding mountain ranges above 1500 m. Long cold winters and short summers characterize the climate. Low elevations are covered by grasslands or shrub steppes. Douglas fir (Pseudotsuga menziesii) and lodgepole pine (Pinus contorta) are the dominant tree species. A detailed description of the study area is available in Schwartz et al. [1].
3. Material and Methods
The model used here is the exponential is the annual rate of increase and is time in years, while a denotes an initial value. This equation was fit using nonlinear leastsquares [16], as implemented in the Rlanguage [17], and ordinary leastsquares regression after a logtransformation of the trend index. Utility of the logtransformation was demonstrated by Eberhardt [13]. The data were also examined using a smoothing technique, “lowess” [18, 19]. Confidence intervals on were obtained by bootstrapping [19, 20] for the nonlinear model and from the usual linear regression model for the logtransformed data. Several different confidence intervals were calculated in the Rlanguage program boot.ci, as expounded by [20, chapter 5] A generalized linear model (GLM) [21] with Poisson errors was also fit to the data using program glm in the Rlanguage. All bootstraps used 5,000 calculations.
The DurbinWatson test [11] was used to test for serial correlation. This test depends on the fact that the squared difference between successive deviations can be used to approximate the variance of the deviations if the pattern of deviations is random. The test is where the observations are deviations from a fitted model. Values of d lie between 0 and 4, with zero correlation at . Bias can be estimated in bootstrapping, using the following:
where is the mean of the bootstraps and tis the original estimate [19].
Data used here on the “unduplicated” females with cubs in the GYE are those given in of the 2006 Annual Report of the Interagency Grizzly Bear Study Team, C. C. Schwartz, M. A. Haroldson, and K. West, Editors (available at http://www.nrmsc.usgs.gov/research/igbsthome.htm).
4. Results
The unduplicated data set was initially assessed by the lowess program, which is essentially a data smoothing program that calculates values from a weighted linear regression of a subset of points adjacent to the current location. It was used to produce Figure 1.
The Jshaped nature of the plot is in consequence that measures to protect and enhance grizzly populations in Yellowstone did not take effect until about 1983 [1, 3]. All subsequent analyses given here use the data from 1983 onwards. Fits of the data from 1983 onwards using nonlinear leastsquares with (1) appear in Figure 2.
The estimate of rate of increase for the exponential fit (1) was 0.052 with a standard error of 0.0071. The DurbinWatson test when applied to the exponential fit yielded a Dstatistic of 2.32, a nonsignificant result, indicating no evidence of serial correlation. Bias calculated from (3) was negligible in all instances. Results of the various analyses appear in Table 1.

5. Discussion
The fact that the difference between from the exponential fit and 0 is over 7 times the standard error provides strong evidence that it is significantly different from zero. This result is clearly supported by the several confidence interval calculations (Table 1), so it seems clear that the “unduplicated” counts show a significant upwards trend. Two different models are considered, one the linear model with additive errors resulting from the logarithmic transformation of the data and the second a multiplicative error structure for the nonlinear leastsquares fit to the exponential model. Confidence intervals from both models demonstrate that the rate of increase, , is substantially greater than zero. We use , the “intrinsic rate of increase” here, rather than = exp as calculated from reproductive and survival rates, but, for small , the difference is trivial.
The data indicate that the GYE bear population does not yet show signs of approaching an asymptote. This is a bit surprising, inasmuch as Schwartz et al. [1]. detected signs of decreasing survival in the population surrounding the primary inhabited area. One possible explanation is that the expansion zone, being crudely an annulus of substantial area, can still provide a substantial increment to the population in spite of having lower overall survival rates. Clearly, the population will ultimately level off, and its numbers may then be fit with a logistic curve. We have favored a “generalized logistic” model in which the rate of increase remains high until the asymptote (K) is approached [22], but believe the present data set is too variable for that model to be useful until the population begins to approach an asymptotic value.
Harris et al. [15]. proposed use of the Chao2 estimator [23] to “estimate the total number of females with cubs present from the estimated number observed” [15, page 17]. Using their estimates [24, Table ] we fitted (1) by nonlinear leastsquares giving the results shown in Figure 3. This fit had a residual standard error of 8.36 as contrasted to a value of 6.11 from the data of Figure 2, suggesting that the Chao2 estimator gives substantially more variable results than using the unadjusted estimates of females with cubs of the year.
Conducting livetrapping, marking and radiotelemetry operations on grizzlies are very expensive operations. Hence, the main lesson from the present study is that index studies should also be attempted as a check on results. Transforming observed index data with the Chao2 estimator produced slightly more variable results, so the method may not be useful unless some improvements are possible. The radiotelemetry results provide essential data for management so should also be conducted whenever possible.
The time span of the reported data (approaching 40 years) has led to its use in a long list of applications [25–32]. All of these ignore the fact that the initial data in the series were constructed from different data sources not compatible with the 1973 development of the “unduplicated” index. As noted above, the population was evidently decreasing until about 1983. This lead to various forecasts of ultimate extinction [27, 28] a somewhat decreased risk [31,Table ], a lower limit on lambda of 0.99 [26], and a very small probability of extinction [29]. Staudenmayer and Buonaccorsi [31] indicate that some of their estimates indicate a rate of increase less than zero. They also used the data to estimate extinction rates. Brook and Bradshaw [25] used the data to study densitydependence and did not consider extinction of the population.
Acknowledgments
The senior author is grateful to Dr. Richard Knight for the opportunity to work on data for the Yellowstone grizzly bears for over 20 years and to Bonnie M. Blanchard for her continual and cheerful efforts at tabulation and analysis of the data. Dave Stradley of the Gallatin Flying Service provided expert piloting for many hours of “looking for bears”. Support from the Marine Mammal Commission and the National Marine Mammal Laboratory is acknowledged.
References
 C. C. Schwartz, M. A. Haroldson, G. C. White et al., Temporal, Spatial and Environmental Influences on the Demographics of Grizzly Bears in the Greater Yellowstone Ecosystem, Wildlife Monographs, Washington, DC, USA, 1992.
 J. J. Craighead, J. Varney, and F. C. Craighead, A Population Analysis of the Yellowstone Grizzly Bears, Montana Forest and Conservation Experiment Station, Bulletin 40, School of Forestry, University of Montana, Missoula, Mont, USA, 1974.
 L. L. Eberhardt and R. R. Knight, “How many grizzlies in Yellowstone?” Journal of Wildlife Management, vol. 60, no. 2, pp. 416–421, 1996. View at: Google Scholar
 C. Servheen and R. Schoemaker, “Delisting the Yellowstone grizzly bear,” Yellowstone Science, vol. 16, pp. 25–29, 2008. View at: Google Scholar
 K. E. F. Watt, “Density dependence in population fluctuations,” Canadian Entomologist, vol. 96, pp. 1147–1148, 1964. View at: Google Scholar
 K. E. F. Watt, Ecology and Resource Management, McGrawHill, New York, NY, USA, 1968.
 D. G. Chapman, “Statistical problems in dynamics of exploited fisheries populations,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkley, Calif, USA, 1961. View at: Google Scholar
 R. P. Freckleton, A. R. Watkinson, R. E. Green, and W. J. Sutherland, “Census error and the detection of density dependence,” Journal of Animal Ecology, vol. 75, no. 4, pp. 832–851, 2006. View at: Publisher Site  Google Scholar
 L. L. Eberhardt, “Correlation, regression, and density dependence,” Ecology, vol. 51, pp. 306–310, 1970. View at: Google Scholar
 T. Royama, Analytical Population Dynamics, Chapman and Hall, London, UK, 1992.
 N. R. Draper and H. Smith, Applied Regression Analysis, John Wiley & Sons, New York, NY, USA, 1998.
 L. L. Eberhardt, B. M. Blanchard, and R. R. Knight, “Population trend of the Yellowstone grizzly bear as estimated from reproductive and survival rates,” Canadian Journal of Zoology, vol. 72, no. 2, pp. 360–363, 1994. View at: Google Scholar
 L. L. Eberhardt, “Assessing the dynamics of wild populations,” Journal of Wildlife Management, vol. 49, pp. 997–1012, 1985. View at: Google Scholar
 C. M. Pease and D. J. Mattson, “Demography of the Yellowstone grizzly bears,” Ecology, vol. 80, no. 3, pp. 957–975, 1999. View at: Google Scholar
 R. B. Harris, C. C. Schwartz, M. A. Haroldson, and G. C. White, “Trajectory of the yellowstone grizzly bear population under alternative survival rates,” in Temporal, Spatial and Environmental Influences on the Demographics of Grizzly Bears in the Greater Yellowstone Ecosystem, C. C. Schwartz, M. A. Haroldson, G. C. White et al., Eds., pp. 44–55, Wildlife Monographs, Washington, DC, USA, 2006. View at: Google Scholar
 D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications, John Wiley & Sons, New York, NY, USA, 1988.
 R Development Core Team, “R: A language and environment for statistical computing,” R Foundation for Statistical Computing, Vienna, Austria, 2004, http://www.Rproject.org. View at: Google Scholar
 W. S. Cleveland, “Robust locallyweighted regression and smoothing scatterplots,” Journal of the American Statistical Association, vol. 74, no. 2, pp. 829–836, 1979. View at: Google Scholar
 B. Efron and R. J. Tibishirani, An Introduction to the Bootstrap, Chapman and Hall, New York, NY, USA, 1993.
 A. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Application, Cambridge University Press, New York, NY, USA, 1997.
 P. McCullagh and J. A. Nelder, Generalized Linear Models, Chapman and Hall, New York, NY, USA, 2nd edition, 1989.
 L. L. Eberhardt, J. M. Breiwick, and D. P. DeMaster, “Analyzing population growth curves,” Oikos, vol. 117, no. 8, pp. 1240–1246, 2008. View at: Publisher Site  Google Scholar
 R. M. Wilson and M. F. Collins, “Capturerecapture estimation with samples of one using frequency data,” Biometrika, vol. 79, pp. 543–553, 1992. View at: Google Scholar
 R. B. Harris, G. C. White, C. C. Schwartz, and M. A. Haroldson, “Population growth of Yellowstone grizzly bears: uncertainty and future monitoring,” Ursus, vol. 18, no. 2, pp. 168–178, 2007. View at: Google Scholar
 B. W. Brook and C. J. A. Bradshaw, “Strength of evidence for density dependence in abundance time series of 1198 species,” Ecology, vol. 87, no. 6, pp. 1445–1451, 2006. View at: Google Scholar
 J. P. Buonaccorsi and J. Staudenmayer, “Statistical methods to correct for observation error in a densityindependent population model,” Ecological Monographs, vol. 79, no. 2, pp. 299–324, 2009. View at: Publisher Site  Google Scholar
 B. Dennis, P. L. Munholland, and J. M. Scott, “Estimation of growth and extinction parameters for endangered species,” Ecological Monographs, vol. 61, no. 2, pp. 115–143, 1991. View at: Google Scholar
 B. Dennis and M. L. Taper, “Density dependence in time series observations of natural populations: estimation and testing,” Ecological Monographs, vol. 64, no. 2, pp. 205–224, 1994. View at: Google Scholar
 S. T. Lindley, “Estimation of population growth and extinction parameters from noisy data,” Ecological Applications, vol. 13, no. 3, pp. 806–813, 2003. View at: Google Scholar
 W. F. Morris and D. F. Doak, Quantitative Conservation Biology: Theory and Practice of Population Variability Analysis, Sinauer Associates, Sunderland, Mass, USA, 2002.
 J. Staudenmayer and J. P. Buonaccorsi, “Measurement error in a random walk model with applications to population dynamics,” Biometrics, vol. 62, no. 4, pp. 1178–1189, 2006. View at: Publisher Site  Google Scholar
 D. F. Staples, M. L. Taper, and B. Dennis, “Estimating population trend and process variation for PVA in the presence of sampling error,” Ecology, vol. 85, no. 4, pp. 923–929, 2004. View at: Google Scholar
Copyright
Copyright © 2010 L. L. Eberhardt and J. M. Breiwick. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.