Estimating Rates of Local Species Extinction, Colonization, and Turnover in Animal Communities

Ecological Applications, 8(4), 1998, pp. 1213±1225 ᭧ 1998 by the Ecological Society of America

ESTIMATING RATES OF LOCAL SPECIES EXTINCTION, COLONIZATION, AND TURNOVER IN ANIMAL COMMUNITIES

JAMES D. NICHOLS,1 THIERRY BOULINIER,2 JAMES E. HINES,1 KENNETH H. POLLOCK,3 AND JOHN R. SAUER1 1U.S. Geological Survey, Biological Resources Division, Patuxent Wildlife Research Center, Laurel, Maryland 20708 USA 2North Carolina Cooperative Fish and Wildlife Research Unit, North Carolina State University, Raleigh, North Carolina 27695 USA 3Institute of Statistics, North Carolina State University, Box 8203, Raleigh, North Carolina 27695-8203 USA

Abstract. Species richness has been identi®ed as a useful state variable for conservation and management purposes. Changes in richness over time provide a basis for predicting and evaluating community responses to management, to natural disturbance, and to changes in factors such as community composition (e.g., the removal of a keystone species). Prob- abilistic capture±recapture models have been used recently to estimate species richness from species count and presence±absence data. These models do not require the common assumption that all species are detected in sampling efforts. We extend this approach to the development of estimators useful for studying the vital rates responsible for changes in animal communities over time: rates of local species extinction, turnover, and colonization. Our approach to estimation is based on capture±recapture models for closed animal populations that permit heterogeneity in detection probabilities among the different species in the sampled community. We have developed a computer program, COMDYN, to compute many of these estimators and associated bootstrap variances. Analyses using data from the North American Breeding Bird Survey (BBS) suggested that the estimators performed reasonably well. We recommend estimators based on probabilistic modeling for future work on community responses to management efforts as well as on basic questions about community dynamics. Key words: animal community ecology; capture±recapture models; community dynamics; conservation and management; local colonization; local extinction probability; North American Breeding Bird Survey; species turnover.

INTRODUCTION pend on the ability to measure or estimate quantities Biological diversity is viewed as a state variable re- that are relevant to theory, management, and associated ¯ecting status of the biosphere (Barbault and Hochberg hypotheses. At the population level of biological or- 1992), and is recognized as ``central to the productivity ganization, ecologists and managers are interested in and sustainability of the earth's ecosystems'' (Chris- such quantities as population size and rates of mortal- tensen et al. 1996:671) and as an important conser- ity, recruitment, and movement. Because animal pop- vation and management objective (e.g., Scott et al. ulation ecologists and managers can seldom enumerate 1993, Mangel et al. 1996). Operationally, species rich- all of the individual animals in a population or a sam- ness is frequently selected as a variable re¯ecting sys- pled area, they must estimate these quantities using tem state, and is often used as such in management and statistical models that explicitly incorporate the prob- conservation efforts and in investigations of the effects ability of detecting an animal, given that the animal is of human development and disturbance on biodiversity in the sampled area (Nichols 1992, Lancia et al. 1994). (e.g., Scott et al. 1993, Conroy and Noon 1996, Keddy Animal population ecologists and managers have and Drummond 1996, Wiens et al. 1996). Estimation placed strong emphasis on the development and use of of rates of change in biodiversity and investigation of appropriate statistical inference procedures that ex- factors responsible for change have been identi®ed as plicitly incorporate unknown animal detection proba- important ecological research topics in the Sustainable bilities (see Otis et al. 1978, Seber 1982, Brownie et Biosphere Initiative (Lubchenko et al. 1991). Here, we al. 1985, Burnham et al. 1987, Pollock et al. 1990, focus on methods for estimating rates of local extinc- Lebreton et al. 1992, Buckland et al. 1993, Lancia et tion, turnover, and colonization, the community-level al. 1994, Wilson et al. 1996). vital rates responsible for changes in species richness Animal community ecology has not seen parallel de- over time. velopment of statistical inference procedures that rec- Advances in ecology and conservation biology de- ognize and explicitly incorporate species detection probabilities. Few animal sampling programs provide Manuscript received 21 July 1997; accepted 10 November community censuses (we de®ne a community census 1997; ®nal version received 15 January 1998. as a complete enumeration of all the species in a com- 1213 1214 JAMES D. NICHOLS ET AL. Ecological Applications Vol. 8, No. 4 munity), leading early ecologists to consider quanti- yet not detected, at j would be erroneously recorded as tative methods to extrapolate from the number of spe- an extinction. These errors in recording species ex- cies observed in samples to the number of species in tinction and colonization events lead to errors in turn- the sampled community (e.g., Fisher et al. 1943, Pres- over metrics as well. ton 1948). This concern for sampling issues was not In the present paper, we introduce some estimators often found in subsequent ecological literature, and that should be useful for studying temporal changes in published research in animal community ecology and animal communities in sampling situations where not management contains numerous studies in which spe- all species are detected. Our personal view is that very cies richness (the number of species in an animal com- few studies can insure species detection probabilities munity) was equated with the number of species enu- of 1. We consider estimation of species extinction prob- merated in a sample. More recently, recognition that ability, species turnover rate, and number of colonizing not all species are detected by sampling efforts has led species. The estimators are based on capture±recapture to the use of probabilistic estimators of species richness models, but the community sampling procedures that (Burnham and Overton 1979, Derleth et al. 1989, Karr produce data for these estimators can be of several et al. 1990, Palmer 1990, Coddington et al. 1991, Bal- possible forms and do not require catching and marking tanas 1992, Bunge and Fitzpatrick 1993, Hodkinson individual animals. We will illustrate some of the es- and Hodkinson 1993, Colwell and Coddington 1994, timators using count data from the North American Solow 1994, Dawson et al. 1995, Thiollay 1995, Wal- Breeding Bird Survey, BBS (Robbins et al. 1986, Pe- ther et al. 1995, Nichols and Conroy 1996, Boulinier terjohn and Sauer 1993). et al. 1998). Explicit recognition of detection probabilities has also led to probabilistic approaches to in- GENERAL APPROACH TO ESTIMATION ferring extinction of a particular species from sighting The estimation approach that we will consider re- data (Solow 1993, Reed 1996). quires sampling an animal community at two different Despite the use of such methods for estimating spe- times. We are interested in estimating quantities as- cies richness, the incorporation of species detection sociated with change in the species composition of the probabilities into estimation methods has not extended community during the interval. Our estimators require to quantities associated with community dynamics. For estimates of the number of species in the entire com- example, recently proposed methods for estimating munity, and/or in speci®ed subsets of the community, species extinction rates and related metrics require at one or both of the time periods. This estimation sampling situations in which species are identi®ed and requires consideration of species detection probability, detected in different samples with probability 1 (e.g., which we de®ne for a particular species as the prob- Pimm et al. 1993, Clark and Rosenzweig 1994, Ro- ability that at least one individual of the species is senzweig and Clark 1994, Burkey 1995, Cook and Han- detected in our sampling efforts. Species detection ski 1995). Discrepancies among published estimates of probability can be written as a function of the number extinction rates (e.g., Budlansky 1994, Heywood et al. of individuals belonging to the species in the area being 1994) emphasize the need for reliable estimation meth- sampled, and the average detection probability for an ods. Recent work on community vital rates other than individual (e.g., the capture probability parameter of extinction (e.g., rates of colonization and turnover) is population-level capture±recapture studies). Species also based on species counts over time, with no effort detection probability thus depends on both population to deal with detection probabilities Ͻ1 (e.g., Hinsley size and the behavioral attributes of individuals relative et al. 1995, Mehlman 1997). to sampling efforts (e.g., activity pattern, vocalization, If certain sampling situations lead to the detection body size, tendency to enter traps or other sampling of all species, then methods based on that assumption devices, etc.) We suspect that species detection prob- are appropriate. However, species detection probabil- abilities are likely to be Ͻ1, and our efforts to deal ities sometimes constitute a nontrivial source of vari- with such detection probabilities distinguish our meth- ation in species count data (Boulinier et al. 1998), and ods from other approaches. analyses that ignore this source of variation can lead to invalid inferences about variation in species richness Sampling animal communities over time and space. For example, local extinction is Several possible approaches can be used to sample generally de®ned as a species detected at time i and an area for the purpose of estimating species richness not detected at some later time j, whereas local colo- and quantities related to changes in richness over time nization is de®ned as a species not detected at time i, and space (Nichols and Conroy 1996). Quadrat sam- but detected at a later time j. Consider species present, pling has been especially popular and involves the sub- yet not detected, at sampling period i. If such a species division of the total area of interest into a number of were not present at j, it would represent an unrecorded quadrats or small sampling units. A random sample of extinction, whereas if it were present and detected at quadrats is selected, and the investigator(s) identi®es j, then it would erroneously be recorded as a coloni- and enumerates species found on each selected quadrat zation. Similarly, a species detected at i and present, using virtually any set of sampling methods (e.g., direct November 1998 ESTIMATORS FOR ANIMAL COMMUNITY DYNAMICS 1215 observation of animals and their sign [tracks, scats, 1984, Chao 1989, Lee and Chao 1994, Norris and Pol- nests, etc.], auditory identi®cation, trapping and netting lock 1996) and richness estimators based on other mod- with different trap and net types, etc.). It is best to use els (e.g., Chao et al. 1992) can also be used in the the same basic set of sampling methods and expend community-dynamic estimators that we present. The similar effort on each of the sampled quadrats, although data required for estimation with the jackknife esti- models permitting variation in species detection prob- mator of model Mh are the frequencies, fh, or numbers abilities among the different quadrats are available (see of species detected on exactly h ϭ 1,2,...,K of the Nichols and Conroy 1996, Boulinier et al. 1998). sampled quadrats, where K is the total number of quadrats sampled (Burnham and Overton 1978, 1979). The Estimating species richness general form of the jackknife estimator for species rich- The result of the sampling efforts is a species list ness is for each quadrat that identi®es the species detected. K Several different estimators have been proposed for use NÃ R f (1) khkhϭ ϩ␣͸ with such data from quadrat species lists (Burnham and hϭ1 Overton 1979, Heltsche and Forrester 1983, Smith and where R is the number of species observed, and the ␣hk van Belle 1984, Chao 1987, Mingoti and Meeden 1992, are constants (see Burnham and Overton 1978, 1979) Bunge and Fitzpatrick 1993). In our work on species corresponding to jackknife estimators of order k (␣hk richness estimation (Boulinier et al. 1998), we have ϭ 0 for h Ͼ k). focused on the use of capture±recapture models ini- tially developed for use with data from closed animal Pollock's robust design populations (see Otis et al. 1978, White et al. 1982, The application of closed models to the estimation Rexstad and Burnham 1991), where ``closed'' popu- of species richness is straightforward (e.g., see Bou- lations refer to those in which no animals enter or leave linier et al. 1998), because it is reasonable to think of the population between sampling periods. a sampled community being ``closed'' to local extinc- The estimation models that we have considered in- tion and colonization for the relatively short periods clude three possible sources of variation in detection over which species presence±absence data are collect- probabilities (Otis et al. 1978, White et al. 1982, Rex- ed. However, in order to estimate quantities concerning stad and Burnham 1991): variation from one quadrat community change between two sampling periods sep- to another (analogous to temporal variation in capture± arated by a long time period (e.g., 10 yr), we require recapture modeling at the population level), hetero- a more general approach. Standard capture±recapture geneity among species (analogous to heterogeneity models for open populations (e.g., Pollock et al. 1990, among individuals), and variation between the ®rst Lebreton et al. 1992) permit such change, but do not quadrat sampled and subsequent quadrats (analogous deal well with heterogeneous detection probabilities to behavioral response). In our initial work using these (see later discussion). Thus, we focused on the robust models to estimate species richness (Boulinier et al. design of Pollock (1982) as a means of developing 1998), we selected a number of BBS routes and ran robust estimators for quantities associated with com- the corresponding data through program CAPTURE munity change. (Rexstad and Burnham 1991), which includes a model Pollock's (1982) robust design involves sampling at selection algorithm to aid the user in selecting the mod- two different temporal scales. Primary sampling peri- el that most closely corresponds to the data. Model Mh ods are separated by times that are suf®ciently large to (with detection probabilities that are heterogeneous expect changes in the population or community from among species) was by far the most frequently selected one primary period to the next. In one of our numerical model in these analyses of BBS data (Boulinier et al. examples, we computed quantities relevant to bird 1998). This ®nding was consistent with our a priori community change occurring between sampling peri- expectation that heterogeneity of detection probabili- ods spaced 20 yr apart (e.g., the primary sampling pe- ties among species is likely to be substantial when sam- riods were 1970 and 1990 and BBS data were obtained pling animal communities, and that the models includ- from a single survey route during both years). Some ing such heterogeneity are likely to be of most use for number of secondary samples or sampling periods oc- estimating species richness. curs within each primary period. The secondary sam- For all of the examples reported here, we use the pling periods should be suf®ciently close together that jackknife estimator for model Mh proposed by Burnham we would not expect the community to change during and Overton (1978, 1979) because it is robust to de- the course of the secondary sampling. In our example viations from underlying assumptions and has per- using data from a single BBS route, secondary samples formed well both in simulation studies (Otis et al. 1978, are represented by the multiple stops made along the Burnham and Overton 1979, Pollock and Otto 1983, route in each of the two years of interest (see later Norris and Pollock 1996) and in ®eld trials with known discussion). species richness (Palmer 1990, 1991). However, other There are multiple approaches to estimating quan- estimators for model Mh (e.g., see Smith and van Belle tities of interest from data collected under the robust 1216 JAMES D. NICHOLS ET AL. Ecological Applications Vol. 8, No. 4 design (e.g., Kendall and Pollock 1992, Nichols et al. served during sampling efforts in period i. Estimators 1992, 1994, Kendall et al. 1995, 1997). In the com- such as Eq. 3, based on the raw count statistics, should munity-dynamic work reported here, all of our esti- have smaller variances than estimators such as Eq. 2, mators are based on use of closed-model estimators for which are based on estimated quantities (e.g., Skalski species richness, computed over the secondary samples and Robson 1992). However, the estimator in Eq. 3 within each primary sampling period. Estimates of pa- will be biased if pÅ i ± pÅ j. A variety of factors could lead rameters relevant to community change are computed to such an inequality in detection probabilities, in- as ratios or other functions of closed model richness cluding temporal changes in observers, habitat, envi- estimates obtained for different primary sampling pe- ronmental conditions relevant to sampling, and species riods. composition.

Under model Mh, the hypothesis of equal detection Variance estimation probabilities for two samples can be tested using the

Here, we present only point estimators for quantities raw frequency data, fh, with a 2 ϫ K contingency table of interest in community dynamics. Because estimates ␹2 test of the null hypothesis that the proportions of of parameters relevant to community change are com- species found in h ϭ 1,2,...,K secondary samples puted as ratios or other functions of closed-model rich- are similar for the two primary sampling periods, i and ness estimates, it is possible to use the delta method j. Finally, note that the estimated quantity, rate of (e.g., Seber 1982) and other approximations to obtain change in species richness, does not provide infor- estimators of associated variances and covariances. mation about possible changes in species composition However, these approximations would be based nec- (the identities of the species). The estimators dealing essarily on the variance estimates for the closed-model with local extinction, turnover, and colonization do per- richness estimators; in our examples, they would be mit inferences about changes in community composi- based on the variance estimator for the Mh jackknife tion. estimates. These variance estimators themselves do not Local extinction probability always perform well, as the coverage of approximate 95% con®dence intervals based on the jackknife esti- Survival and reproductive rates are the so-called vital mator and its variance has sometimes been poor in rates responsible for population changes, and the anal- simulation studies (e.g., Otis et al. 1978, Burnham and ogous community vital rates are local extinction and Overton 1979). Therefore, we would not expect ap- colonization rates. We de®ne local extinction proba- proximations that are functions of these variance es- bility as the probability that a species present in the timates to necessarily perform well. For this reason, community during primary sampling period i is not we have chosen to compute bootstrap variance esti- present at some later period j. We can employ the robust mates for all of the estimators presented. An outline of design approach of Kendall and Pollock (1992) and our bootstrap computational methods is presented in Nichols et al. (1992) to estimate this quantity using the the Appendix. reasoning that underlies open capture±recapture survival estimators (Jolly 1965, Seber 1965). We ®rst con- ESTIMATORS OF COMMUNITY DYNAMICS dition on the number of species observed in period i

Rate of change in richness (denote this number as Ri), and then estimate the number of these species still present in period j (denote this Rate of change in species richness between two sam- number asMRi ). Local extinction probability is then pling periods i and j can be estimated as j estimated as Ã Nj Ã Ri ˆ Mj ␭ij ϭ Ã (2) 1 Ϫ␾ˆ ϭ 1 Ϫ (4) Ni ij Ri where Ni denotes species richness at time i and where, where ␾ij is the complement of extinction probability typically, i Ͻ j ( j is the more recent of the two time and denotes the probability that a species present in i periods). The estimates of species richness are obtained is still present in j. using species occurrence data in conjunction with Ri The estimation ofMj is based on closed models with closed-model capture±recapture estimators such as species occurrence data from period j, and can be ac- those for Mh (e.g., the jackknife estimator of Eq. 1; complished using either of two approaches. The most also see Burnham and Overton 1978, 1979). conservative approach is to use only occurrence data If pÅ i ϭ pÅ j, that is if the average species detection for period j (i.e., species presence±absence data over probabilities are the same for the two periods, then an the secondary samples within primary sampling period alternative estimator for rate of change is j) from species also observed at primary period i (i.e., use onlymRi ; members of R that are also seen at j). Rj j i ␭ˆ ij ϭ (3) These data are used directly with a closed-population R i Ri estimator such as Eq. 1 to estimateMj . where Ri denotes the number of species actually ob- The second approach requires the additional as- November 1998 ESTIMATORS FOR ANIMAL COMMUNITY DYNAMICS 1217 sumption that average species detection probabilities the species present in period j are new (not present at Ri at time j are the same for theMj species that are present i) and its minimum value (0) when all of the species Ri in both i and j, and the Nj Ϫ Mj species present at j present at j are survivors from period i. but not at i. If this assumption holds, then we can Estimation of this turnover parameter is based on an estimate the number of Ri still present at j as observation by Pollock et al. (1974) about the temporal symmetry of capture±recapture data for open popula- Ri mj Ã Ri tions. Speci®cally, they observed that if capture history Mj ϭ (5) pÅˆ j data are viewed in reverse time order (treating the ®nal sampling period, K, as the initial period and treating where pÅ j is the average detection probability of all species present at j. This pÅ is estimated by using occur- sampling period 1 as the ®nal period), then the standard j Cormack-Jolly-Seber estimator for survival between rence data from all species observed at period j (Rj)in conjunction with program CAPTURE to estimate total two periods estimates the fraction of species in the more recent period that are ``old,'' or survivors from species richness at j (Nj), and then using the following estimator: the previous period. The complement of this estimator estimates the proportion of species that are new and

Rj were not present in the previous period (Pollock et al. pÅˆ j ϭ Ã . (6) 1974, Nichols et al. 1986, Pradel 1996). Nj Our estimation of community turnover will thus be Ri The ®rst approach for estimatingMj should have accomplished by using the extinction probability es- smaller bias, but larger variance. The second approach timator of Eq. 4 with data placed in reverse time order. Ã Ri should yieldMj with larger bias (perhaps), but smaller The notation is the same as that for the survival esti- variance. A decision about the appropriateness of the mator (Eq. 4), except for a change in the subscripting second approach can be based on a test for equality of that denotes the change in temporal ordering. We es- average detection probabilities for species present in timate turnover as primary period j that were and were not detected in MÃ Rj period i. This test can be conducted as a ␹2 test for a 1 Ϫ␾ˆ ϭ 1 Ϫ i (i Ͻ j). (7) ji R 2 ϫ K contingency table using the fh for the two groups j of species detected in primary period j: those detected Ã Rj The key to the estimation involvesMi , the estimated Ri in primary period i (mj ) and those not detected in pri- number of species observed in j that were also present Ri 2 mary period i (Rj Ϫ mj ). The resulting ␹ statistic pro- in i. Estimation is accomplished, as with extinction vides a test of the null hypothesis of equal proportions probability, by conditioning on the subset of species of species detected from the two groups. Whenever the actually observed at j and then estimating the number Ri group, Rj Ϫ mj , is relatively small, the contingency of these that were also present in i. As was the case table test for equal detection probabilities is not likely Ri with extinction, the estimation ofMj can be accom- to be very powerful; in such cases, we recommend the plished in either of two ways. The most conservative Ri ®rst approach for estimatingMj for use in the extinc- approach, requiring fewest assumptions, involves use tion probability estimator of Eq. 4. of a species richness estimator with species occurrence data for the members of R that were also observed in Local species turnover j Rj period i(mi ). The other approach involves estimating Ã The literature of community ecology contains a num- average detection probability for period i, pÅ i, from all ber of de®nitions of turnover. Most of these de®nitions species observed at i, and then applying this estimate are based on statistics (functions of data) rather than to the number of species observed at i that were also on a speci®ed underlying parameter of interest. Thus, later observed at j (see Eqs. 5 and 6). As was noted in we are faced with different turnover metrics that re¯ect the discussion of extinction probability estimators, the the different intuitions of their respective devisers ®rst approach should have smaller bias and larger vari- about what turnover actually is. Here, we follow an ance than the second approach. At this point, we rec- approach that we adopted in paleobiological work with ommend that the estimation of turnover be based on fossil data (Nichols et al. 1986), and de®ne turnover the ®rst approach. between two times, i and j, where j is the more recent period (i Ͻ j), as the probability that a species selected Number of local colonizing species at random from the community at time j is a ``new'' Denote as Bij the number of species not present in species (i.e., it was not present in the community at the local area at time i that colonize the area between time i). This turnover parameter arises naturally in the times i and j and are still present at time j. The approach modeling of capture±recapture data (Pollock et al. to estimation is similar to that used under the robust 1974, Pradel 1996). It is a function of rates of extinction design to estimate number of new recruits entering the and colonization and re¯ects dissimilarity between studied population between two sampling periods (Pol- communities at two different points in time. The pa- lock 1982, Pollock et al. 1990). This approach simply rameter achieves its maximum value (1) when all of subtracts from the estimated species richness at time j, 1218 JAMES D. NICHOLS ET AL. Ecological Applications Vol. 8, No. 4 the estimated number of surviving species from a pre- We note that Eq. 9 represents the kind of parame- vious time, i: terization that will probably be needed for open-model Ã Ã Ã Ã approaches to the estimation of community-level quan- Bij ϭ Nj Ϫ ␾ijN i (8) tities. Open models are based on capture history data, where denotes the number of species not present at Bij with row vectors of 0's and 1's representing no capture/ time i that entered the community between times i and detection and capture/detection, respectively. In the j and are still present at time j. usual capture±recapture framework, a ``0'' appearing Annual extinction and recolonization probabilities between 1's (e.g., the 0 in capture history 101) indicates The extinction probability estimator of Eq. 4 esti- an animal that is present but not caught in the sampling mates the probability that a species present at time i is period. However, in community studies, an interior 0 absent at some later time j, but speci®es nothing about in a detection history can indicate either a species that the detailed process leading to this event. In the case is present but not detected (sometimes referred to as a of previously published turnover and extinction esti- ``sampling 0''), or a species that is not present (locally mators based on census data (pi ϭ 1), this concern for extinct) yet recolonized at a later time (sometimes re- underlying process has been linked to interest in meth- ferred to as a ``structural 0''). In this respect, the mod- ods for scaling estimators for the length of the interval eling of species detection history data in the presence between census/sampling periods (e.g., Diamond and of local extinction and recolonization is similar to the May 1977, Clark and Rosenzweig 1994, Russell et al. modeling of capture±recapture data in the presence of 1995). First-order Markov process models of the gen- temporary emigration. The robust design (Pollock eral type considered by Simberloff (1969; also see Di- 1982) provides the information needed to estimate amond and May 1977, Clark and Rosenzweig 1994, quantities of interest in the presence of temporary em- Russell et al. 1995) provide a reasonable underlying igration (Kendall et al. 1997), and may provide a basis model for community dynamics. This type of model for estimating community parameters from detection his- has led to appropriate estimators for local, annual ex- tory data using parameterizations such as that of Eq. 9. tinction and recolonization probabilities based on cen- In the formulation of Eq. 9, our parameters corre- sus data collected at both annual (Diamond and May spond directly to events that are potentially observable 1977, Rosenzweig and Clark 1994) and longer (Clark for the time scale of our sampling. If assumptions are and Rosenzweig 1994) intervals. Here, we consider made about the number of events that can occur be- estimation under Markovian models in the situation tween sampling periods, then it is sometimes possible where species detection probabilities are Ͻ1. to estimate parameters under alternative models. For In order to survive (species survival probability is the complement of extinction probability) between two example, Clark and Rosenzweig (1994) and Rosen- primary periods, i and j, a species present at i may zweig and Clark (1994) consider a Markov model sim- survive every year between i and j, or it may go locally ilar to Eq. 9, except that a species present at times i extinct and then recolonize during this period. Assume and i ϩ 1 can re¯ect either a species that survived the a simple situation in which we are interested in changes interval or a species that went locally extinct and then within a bird community between years i and i ϩ 2. recolonized. Under their formulation, a species absent We can write at i and then present at i ϩ 1 re¯ects only recolonization (e.g., it cannot re¯ect recolonization, followed by ex- ␾ ϭ␾ ␾ ϩ (1 Ϫ␾ )␥ (9) i,iϩ2 i,iϩ1 iϩ1,iϩ2 i,iϩ1 iϩ1,iϩ2 tinction, followed by recolonization), and estimation where ␥iϩ1,iϩ2 is the probability that a species present of both extinction and recolonization probabilities is in the community at some earlier time (in this case, possible (assuming species detection probabilities of 1 time i), but not at time i ϩ 1, recolonizes the area during and assuming a stationary Markov process [extinction the interval (i ϩ 1, i ϩ 2) and is present in the com- and recolonization probabilities do not vary over munity at i ϩ 2. time]). By considering stationary processes and by We can estimate each of the three species survival carefully de®ning the number of events that can occur probabilities in Eq. 9 using the general estimator pre- during the interval between sampling periods, it is thus sented in Eq. 4. We can then solve Eq. 9 for ␥ to iϩ1,iϩ2 sometimes possible to estimate parameters for such obtain models, although this becomes more dif®cult when p

(␾ˆ i,iϩ2 Ϫ (␾ˆ i,iϩ1␾ˆ iϩ1,iϩ2)) Ͻ 1. At present, our estimates obtained using the robust ␥Ã iϩ1,iϩ2 ϭ ,or (1 Ϫ␾ˆ i,iϩ1) design correspond to parameters de®ned at the same temporal scale at which extinction and recolonization MMMÃ RRi Ã i Ã Riϩ1 events are observable. However, we require no as- iϩ2 Ϫ iϩ1 iϩ2 RRR sumptions about the stationary nature of the underlying iii[]΂΃΂΃ϩ1 ␥Ã iϩ1,iϩ2 ϭ . (10) Markov process, and permit local extinction and re- MÃ Ri 1 Ϫ iϩ1 colonization probabilities to be time speci®c, as indi- ΂΃Ri cated by the subscripting in Eqs. 9 and 10. November 1998 ESTIMATORS FOR ANIMAL COMMUNITY DYNAMICS 1219

TABLE 1. Species detection statistics for Maryland BBS (Breeding Bird Survey) route 25, 1970 and 1990.

No. No. species detected on exactly h of the ®ve groups of stops (f ) species h Species group detected f1 f2 f3 f4 f5

Total no. species detected (1970), R70 65 15 8 10 12 20 Total no. species detected (1990), R90 55 15 8 13 12 7 R70 No. members of R70 detected in 1990, m 90 48 10 6 13 12 7 R90 No. members of R90 detected in 1970, m 70 48 7 6 6 9 20

EXAMPLE ANALYSES Wisconsin route 1, and computed estimates relevant to We illustrate some of our estimators using data col- community dynamics between 1970 and 1990. The raw lected as part of the BBS. This survey is carried out data (Tables 1 and 2) on which all of these estimators every spring on permanent survey routes randomly lo- are based include simply the number of species de- cated along secondary roads throughout the United tected on 1, 2, . . . , 5 of the route segments, for all States and southern Canada. Each route is 39.4 km long species observed in 1970 (R70) and 1990 (R90), for the and consists of 50 stops spaced at 0.8-km intervals. number of species observed in 1970 that were also R70 The observer drives along the route, exiting the vehicle detected in 1990 (m90 ), and for the number of species at each stop to record all birds seen and heard within observed in 1990 that were also detected in 1970 R90 0.4 km of the stop during a 3-min observation period ().m70 (Robbins et al. 1986, Peterjohn and Sauer 1993). In the Model Mh adequately (P Ͼ 0.10) ®t all four sets of BBS ®les, data are summarized by groups of 10 stops. detection frequency data in Table 1 for Maryland BBS Hence, there are ®ve such summary records for each route 25, 1970 and 1990. The richness estimate for survey route, and for each summary record (each group 1990 was smaller than that for 1970, although the 95% of 10 stops), we have a species list and the number of CIs for the two estimates overlapped substantially (Ta- individuals counted for each detected species. For our ble 3). The estimated rate of change based on the rich- examples, the raw data used to compute our estimates ness estimates was 0.88, but the con®dence interval for on each survey route are simply the species lists for rate of change included values Ͼ1. However, the test each of the ®ve groups of stops. We thus treat each for equal average detection probability in 1970 (esti- group of 10 stops along a survey route as a ``quadrat'' mate, 0.83) and 1990 (estimate, 0.79) provided little 2 sampling the area covered by the entire survey route. evidence of a difference (␹4 ϭ 5.86, P ϭ 0.11), justi- These ®ve quadrats are the secondary samples of our fying the use of Eq. 3 for estimating rate of change. robust design approach. It is possible to apply our The resulting estimate of 0.85 was similar to that based methods to any subset of total species (e.g., de®ned by on Eq. 2, but was substantially more precise. The 95% taxonomy, foraging habit, etc.), but in our examples, CI for the Eq. 3 estimate of rate of change was 0.73± we include all avian species. 1.00 (Table 3), providing evidence that avian richness All computations were conducted using program declined on the route between 1970 and 1990. COMDYN, developed by Hines et al. (in press). Based The complement of extinction probability was esti- on the general applicability of model Mh to BBS data mated at 0.84, and the 95% CI included 1.00 (Table 3). (Boulinier et al. 1998), all of the estimators in program The complement of turnover indicated that an estimated COMDYN are based on the jackknife estimators of 93% of the species present in 1990 were also present Burnham and Overton (1978, 1979). Variance esti- in 1970, re¯ecting an estimated species turnover of 7% mation in program COMDYN is accomplished using a (Table 3). Consistent with this fairly low turnover, the bootstrap approach (Appendix). COMDYN also in- estimated number of new species colonizing between cludes goodness-of-®t tests of the detection frequency 1970 and 1990 and present in 1990 was small (Ͻ5 Ã data to model Mh and tests of the null hypothesis that species; see B70,90, Table 3). The overall conclusion for two sets of detection frequency data were produced by this route was weak evidence of a decline in species the same detection probabilities. richness, with the number of colonizing species not We selected two BBS routes, Maryland route 25 and quite balancing the number of local extinctions.

TABLE 2. Species detection statistics for Wisconsin BBS route 1, 1970 and 1990.

No. No. species detected on exactly h of the ®ve groups of stops (f ) species h Species group detected f1 f2 f3 f4 f5

Total no. species detected (1970), R70 66 17 19 9 10 11 Total no. species detected (1990), R90 80 23 15 9 12 21 R70 No. members of R70 detected in 1990, m 90 57 9 10 6 11 21 R90 No. members of R90 detected in 1970, m 70 57 10 17 9 10 11 1220 JAMES D. NICHOLS ET AL. Ecological Applications Vol. 8, No. 4

TABLE 3. Estimates of quantities associated with community dynamics based on avian species seen on Maryland BBS route 25 in 1970 and 1990.

Naive Quantity (␪) ``estimates''² Estimator ␪Ã SE(␪Ã) 95% CI Ã Species richness (1970) 65.0 N70 78.5 10.4 66.9±104.1 Ã c Species richness (1990) 55.0 N90 69.4 11.1 55.6±94.3 Ã R70 No. members of R70 present in 1990³ 48.0 M 90 54.7 13.2 35.0±86.0 Ã R90 No. members of R90 present in 1970³ 48.0 M 70 51.1 6.9 38.5±67.1 Ã Complement of extinction probability 0.74 ␾70,90 0.84 0.15 0.54±1.00 Ã Complement of turnover 0.87 ␾90,70 0.93 0.09 0.70±1.00 Ã Rate of change in richness (Eq. 2) 0.85 ␭70,90 0.88 0.18 0.61±1.28 Ã Rate of change in richness (Eq. 3) 0.85 ␭70,90 0.85 0.07 0.73±1.00 Ã Number of colonizing species 7.0 B70,90 3.3 12.2 0.0±40.9

Average detection probability (1970) 1.0§ pÅˆ 70 0.83 0.10 0.62±0.97

Average detection probability (1990) 1.0§ pÅˆ 90 0.79 0.11 0.58±0.99 ² ``Estimates'' are based on the assumption that all species are detected. Ã RRii ³ Con®dence intervals forMmj can include values Ͻ j , because we consider variation associated with the extinction Ri Ã process (M j ϳ bin[Ri, ␾ij]). § By assumption.

Model Mh adequately ®t (P Ͼ 0.10) all of the data DISCUSSION sets in Table 2 for Wisconsin route 1, 1970 and 1990. The most dif®cult methodological problem faced by Estimated species richness was greater in 1990 than biologists studying animal populations or communities 1970, and there was little overlap between the respec- in the ®eld is the inability to detect every animal or tive con®dence intervals (Table 4). Both estimates for species present in a sampled location. Recognition of rate of change in species richness were Ͼ1.2, and nei- this problem has led to recent efforts to use capture± ther con®dence interval included 1.00 (Table 4), pro- recapture models to estimate species richness in studies viding evidence for an increase in richness between of animal community ecology (Derleth et al. 1989, Karr 1970 and 1990. The test for similar distribution of de- et al. 1990, Palmer 1990, Coddington et al. 1991, Bal- tection frequencies ( 2 3.37, P 0.50) provided no ␹4 ϭ ϭ tanas 1992, Bunge and Fitzpatrick 1993, Hodkinson evidence for different detection probabilities between and Hodkinson 1993, Colwell and Coddington 1994, 1970 (estimate, 0.85) and 1990 (estimate, 0.80). Dawson et al. 1995, Thiollay 1995, Walther et al. 1995, The estimated ␾70,90 re¯ected only a 7% local ex- Nichols and Conroy 1996). We agree that these models tinction probability between 1970 and 1990, but the are well-suited for estimating species richness, and we estimated ␾90,70 indicated that 22% of the species pres- have found them to be very useful for estimating avian ent in 1990 were new (not present in 1970). Similarly, species richness from BBS data (Boulinier et al. 1998). the estimator for new species (Eq. 8) indicated Ͼ25 Our intent here has been to provide suggestions about species as local colonists between 1970 and 1990 (Ta- extending the capture±recapture estimation and mod- ble 4). Thus, the data for this route indicated an increase eling framework to the study of animal community in the number of species, with the number of local dynamics. We have presented several estimators, and colonists exceeding the number of local extinctions. estimates computed using some of these in our example Tables 3 and 4 also include the naive ``estimates'' analyses seemed reasonable. Precision was not always of the quantities of interest obtained, based on the as- good, but was consistent with expectations based on sumption that all species are detected. Many of the other work with capture±recapture models. We note naive estimates are biased low The bias of the naive that our examples used all avian species detected on estimate for number of colonizing species may be either the BBS routes, whereas it would be possible to restrict positive or negative, depending on the relative detec- interest to speci®c subsets of species (guilds or taxo- tion probabilities and the actual numbers of local ex- nomic subgroups). However, precision of estimates tinctions and colonizations. The naive estimate for rate would be lower with the smaller sample sizes that of change in species richness is nearly unbiased when would necessarily accompany subsetting of the data. detection probabilities for the two time periods are Burnham (1981:325) noted that ``Using just the equal, as in these two examples. Finally, when consid- count of birds detected . . . as an index [of] abundance ering the magnitudes of the difference between the na- is neither scienti®cally sound nor reliable.'' This crit- ive and probability-based estimates, it should be noted icism can be applied to any analysis of counts in which that higher detection probabilities than those observed the proportion of individuals (or species) detected is will lead to smaller differences, whereas lower detec- Ͻ1. Even extensive monitoring programs such as the tion probabilities will produce even larger differences. BBS have great constraints on any analysis, because November 1998 ESTIMATORS FOR ANIMAL COMMUNITY DYNAMICS 1221

TABLE 4. Estimates of quantities associated with community dynamics based on avian species seen on Wisconsin BBS route 1 in 1970 and 1990.

Naive Quantity (␪) ``estimates''² Estimator ␪Ã SE(␪Ã) 95% CI Ã Species richness (1970) 66.0 N70 77.2 5.5 68.6±89.8 Ã c Species richness (1990) 80.0 N90 99.9 11.4 85.5±129.9 Ã R70 No. members of R70 present in 1990³ 57.0 M 90 61.6 7.7 47.2±74.4 Ã R90 No. members of R90 present in 1970³ 57.0 M 70 62.5 13.8 39.4±84.7 Ã Complement of extinction probability 0.86 ␾70,90 0.93 0.09 0.72±1.00 Ã Complement of turnover 0.71 ␾90,70 0.78 0.16 0.49±1.00 Ã Rate of change in richness (Eq. 2) 1.21 ␭70,90 1.29 0.17 1.06±1.70 Ã Rate of change in richness (Eq. 3) 1.21 ␭70,90 1.21 0.08 1.05±1.37 Ã Number of colonizing species 13.0 B70,90 27.9 14.1 8.4±62.7

Average detection probability (1970) 1.0§ pÅˆ 70 0.85 0.06 0.73±0.96

Average detection probability (1990) 1.0§ pÅˆ 90 0.80 0.08 0.61±0.94 ²``Estimates'' are based on the assumption that all species are detected. Ã RRii ³ Con®dence intervals forMmj can include values Ͻ j , because we consider variation associated with the extinction Ri Ã process (M j ϳ bin[Ri, ␾ij]). § By assumption. of the need to model ``nuisance'' variables that affect performance in the face of strong heterogeneity. In par- the proportion of animals detected and that can bias ticular, it will be useful to base such an evaluation on estimation (e.g., Sauer et al. 1994). The procedures a criterion such as mean squared error that includes described here have the potential to expand the appro- both bias and precision. priate use of count data in hypothesis tests, and can be applied to many existing data sets that were formerly Potential problems considered to be inappropriate for statistical analysis. Although the estimators we have presented should For example, data gathered in the BBS and other mon- be superior to ad hoc approaches based on species ac- itoring programs such as the Christmas Bird Count tually detected, our estimators are not without prob- (Butcher 1990) and atlas programs (Robbins 1990) can lems. A recurring problem in all statistical modeling be used to test hypotheses about changes in species involves what to do when the available models do not richness and associated community-level vital rates. ®t the data. Model Mh was found to be the most frequently selected model, by a wide margin, in analyses Open vs. closed models of numerous BBS data sets (Boulinier et al. 1998), so All of the community-dynamic estimators presented we developed program COMDYN basing all estimators here are based on capture±recapture models developed on this model (Hines et al., in press). In cases where for closed animal populations. In our earlier paleo- this model does not ®t the data well, the data could be biological work (Nichols and Pollock 1983a, Conroy run through program CAPTURE (Otis et al. 1978, and Nichols 1984, Nichols et al. 1986), we used pri- Rexstad and Burnham 1991). If the selected model has marily open-model estimators, relying on the relative available estimators and ®ts the data reasonably, then robustness of estimators of turnover and extinction resulting estimates can be used with the estimators pre- probability to heterogeneity of detection probabilities sented here. If no model ®ts the data well, then we (variation among different taxa in fossil encounter would recommend relying on the general robustness of probabilities). We selected closed models as the basis the jackknife estimator for model Mh and using this for the estimators reported here, because of the a priori estimator with caution. It may be reasonable to use a likelihood, and the strong evidence from BBS data quasi-likelihood (see Burnham et al. 1987, Lebreton et (Boulinier et al. 1998), that detection probabilities are al. 1992) approach, computing variance in¯ation fac- strongly heterogeneous among species. Only capture± tors based on the goodness-of-®t test results (this is recapture models for closed populations permit unbi- another topic meriting investigation). We emphasize ased estimation in the presence of heterogeneous de- that lack of model ®t is not an adequate reason for tection probabilities. However, some open-population abandoning an estimation approach and resorting to the capture±recapture estimators are quite robust to het- use of ad hoc estimators, because model-based esti- erogeneous capture probabilities (Carothers 1973, mates will probably perform much better (i.e., exhibit 1979). We suspect that sampling variances are likely less bias) than ad hoc approaches, even when model to be smaller for open-model estimators than for the assumptions are not met (e.g., see Nichols and Pollock estimators presented here. The variances associated 1983b). with many of the estimates computed in our examples Potentially the most serious problem with the com- were large. Thus, an important topic for investigation munity-dynamic estimators we have presented involves will be comparative closed- and open-model estimator the effects of heterogeneity on the ␾ estimators (local 1222 JAMES D. NICHOLS ET AL. Ecological Applications Vol. 8, No. 4 extinction probability, local turnover). The methods re¯ecting the proportion of the species present at time that we propose were developed speci®cally to deal j that were not present at a previous time i, is a natural with heterogeneity of detection probabilities. However, turnover statistic and is preferable to other turnover detection probability for a species will be a function metrics. However, for those who do not share this pref- of the morphological and behavioral characteristics of erence, other statistics such as the widely used turnover individuals in the species and the number of individuals index of Diamond (1969) can readily be computed us- in the sampled species population. Unfortunately, local ing the estimates of species richness, the number of extinction probability and probability of having been local colonists, and the local extinction probability pre- present at some previous time period (1 Ϫ turnover) sented. Use of capture±recapture estimates to compute will probably also depend on the number of individuals such indices avoids the typical assumption of species in the species. Extinction probability for a population detection probabilities of 1. In addition, our bootstrap should generally decrease as the number of individuals approach can be used to compute associated variances, increases (e.g., Bailey 1964, MacArthur and Wilson permitting statistical inference on these indices. 1967, Goel and Richter-Dyn 1974, Gilpin and Soule 1986, Boyce 1992, Burgman et al. 1993), although em- Conclusions pirical results are not always consistent with this pre- For ®eld situations in which all species present in diction (e.g., Karr 1990). an area are detected at each sampling occasion, we Our estimators for ␾ condition on the number of recommend methods such as those developed by Clark species actually observed at a speci®c period (Eqs. 4 and Rosenzweig (1994) and Rosenzweig and Clark and 7) and then estimate how many species in that (1994) for estimating local extinction and colonization subset are present at a different time. If detection prob- probabilities. In situations where species detection ability for an individual species is closely tied to the probabilities are Ͻ1, we believe that the estimation number of individuals in that species, then the species methods described here will be useful to community on which we condition our estimates (e.g., the members ecologists, managers, and conservation biologists. We of Ri) will tend to have more individuals, on average, recommend use of these estimators for estimating than species present but not observed (e.g., Ni Ϫ Ri). changes in species richness and for investigating po- Thus, the species on which we condition our estimates tential in¯uences of such factors as environmental will also tend to have greater probabilities, on average, change and habitat management on the vital rates that of being present in some other sampling period. If we determine such changes. want our inferences to apply to the entire community, ACKNOWLEDGMENTS then this positive covariance between p and ␾ within We thank Jeff Brawn and Curt Flather for constructive species will tend to result in a positive bias in our reviews. This work was partially funded and supported by estimates,␾ˆ . Heterogeneous survival probabilities also the U.S. Forest Service. present problems in models for estimating quantities LITERATURE CITED at the population level (Nichols et al. 1982, Pollock and Raveling 1982, Johnson et al. 1986, Rexstad and Bailey, N. T. J. 1964. The elements of stochastic processes with applications to the natural sciences. Wiley, New York, Anderson 1992, Burnham and Rexstad 1993), and the New York, USA. covariance between survival and recapture/recovery Baltanas, A. 1992. On the use of some methods for the es- probability is an important determinant of estimator timation of species richness. Oikos 65:484±492. performance (e.g., Nichols et al. 1982). Our initial work Barbault, R., and M. E. Hochberg. 1992. Population and community level approaches to studying biodiversity in on this problem in community analyses suggests that international research programs. Acta Oecologica 13:137± bias is not large, but we will continue to investigate 146. possible bias and methods for bias reduction. Most im- Boulinier, T., J. D. Nichols, J. R. Sauer, J. E. Hines, and K. portantly, we note that ad hoc approaches to estimation H. Pollock. 1998. Estimating species richness: the impor- tance of heterogeneity in species detectability. Ecology 79: share this problem, and the dif®culties are magni®ed 1018±1028. because of the absence of an ability to draw inferences Boyce, M. S. 1992. Population viability analysis. Annual about species present, yet not detected. Thus, the ap- Review of Ecology and Systematics 23:481±506. proaches presented here are still far preferable to meth- Brownie, C., D. R. Anderson, K. P. Burnham, and D. S. Rob- son. 1985. Statistical inference from band recovery data: ods that incorrectly assume detection probabilities of 1. a handbook. Second edition. United States Fish and Wild- life Service, Resource Publication 156. Other quantities Buckland, S. T., K. P. Burnham, D. R. Anderson, and J. L. Laake. 1993. Density estimation using distance sampling. We believe that the estimators presented here will Chapman and Hall, London, UK. be useful for studying local species extinction, colo- Budlansky, S. 1994. Extinction or miscalculation. Nature nization, and turnover. However, we note that our cap- 370:105. Bunge, J., and M. Fitzpatrick. 1993. Estimating the number ture±recapture approach can be used to estimate other of species: a review. Journal of the American Statistical metrics relevant to these community processes. For ex- Association 88:364±373. ample, we believe that our turnover metric (Eq. 7), Burgman, M. A., S. Ferson, and H. R. AkcËakaya. 1993. Risk November 1998 ESTIMATORS FOR ANIMAL COMMUNITY DYNAMICS 1223

assessment in conservation biology. Chapman and Hall, richness from capture and count data. Journal of Applied London, UK. Statistics 22:1063±1068. Burkey, T. V. 1995. Extinction rates in archipelagoes: im- Derleth, E. L., D. G. McAuley, and T. J. Dwyer. 1989. Avian plications for populations in fragmented habitats. Conser- community response to small-scale habitat disturbance in vation Biology 9:527±541. Maine. Canadian Journal of Zoology 67:385±390. Burnham, K. P. 1981. Summarizing remarks: environmental Diamond, J. M. 1969. Avifaunal equilibria and species turn- in¯uences. Studies in Avian Biology 6:324±325. over rates on the Channel Islands of California. Proceed- Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, ings of the National Academy of Sciences (USA) 64:57± and K. H. Pollock. 1987. Design and analysis methods for 63. ®sh survival experiments based on release±recapture. Diamond, J. M., and R. M. May. 1977. Species turnover American Fisheries Society Monograph 5. rates on islands: dependence on census interval. Science Burnham, K. P., and W. S. Overton. 1978. Estimation of the 197:266±270. size of a closed population when capture probabilities vary Fisher, R. A., A. S. Corbet, and C. B. Williams. 1943. The among animals. Biometrika 65:625±633. relation between the number of species and the number of Burnham, K. P., and W. S. Overton. 1979. Robust estimation individuals in a random sample of an animal population. of population size when capture probabilities vary among Journal of Animal Ecology 12:42±58. animals. Ecology 60:927±936. Gilpin, M. E., and M. E. SouleÂ. 1986. Minimum viable pop- Burnham, K. P., and E. A. Rexstad. 1993. Modeling hetero- ulations: processes of species extinctions. Pages 19±34 in geneity in survival rates of banded waterfowl. Biometrics M. E. SouleÂ, editor. Conservation biology: the science of 49:1194±1208. scarcity and diversity. Sinauer, Sunderland, Massachusetts, Butcher, G. S. 1990. Audubon Christmas Bird Counts. Pages USA. 5±13 in J. R. Sauer and S. Droege, editors. Survey designs Goel, N. S., and N. Richter-Dyn. 1974. Stochastic models and statistical methods for the estimation of avian popu- in biology. Academic Press, New York, New York, USA. lation trends. United States Fish and Wildlife Service, Bi- Heltshe, J. F., and N. E. Forrester. 1983. Estimating species ological Report 90(1). richness using the jackknife procedure. Biometrics 39:1± Carothers, A. D. 1973. The effect of unequal catchability on 11. Jolly-Seber estimates. Biometrics 29:79±100. Heywood, V. H., G. M. Mace, R. M. May, and S. N. Stuart. . 1979. Quantifying unequal catchability and its effect 1994. Uncertainties in extinction rates. Nature 368:105. on survival estimates in an actual population. Journal of Hines, J. E., T. Boulinier, J. D. Nichols, J. R. Sauer, and K. Animal Ecology 48:863±869. H. Pollock. COMDYN: software to study the dynamics of Chao, A. 1987. Estimating the population size for capture± animal communities using a capture±recapture approach. recapture data with unequal catchability. Biometrics 43: Bird Study, in press. 783±791. Hinsley, S. A., P. E. Bellamy, and I. Newton. 1995. Bird . 1989. Estimating population size for sparse data in species turnover and stochastic extinction in woodland capture±recapture experiments. Biometrics 45:427±438. fragments. Ecography 18:41±50. Chao, A., S. M. Lee, and S. L. Jeng. 1992. Estimation of Hodkinson, I. D., and E. Hodkinson. 1993. Pondering the population size for capture±recapture data when capture imponderable: a probability-based approach to estimating probabilities vary by time and individual animal. Biomet- insect diversity from repeat faunal samples. Ecological En- rics 48:201±216. tomology 18:91±92. Christensen, N. L., et al. 1996. The report of the Ecological Johnson, D. H., K. P. Burnham, and J. D. Nichols. 1986. The Society of America committee on the scienti®c basis for role of heterogeneity in animal population dynamics. Pages ecosystem management. Ecological Applications 6:665± 1±15 in Proceedings of the 13th International Biometrics 691. Conference, Session 5. The Biometric Society, Alexandria, Clark, C. W., and M. L. Rosenzweig. 1994. Extinction and Virginia, USA. colonization processes: parameter estimates from sporadic Jolly, G. M. 1965. Explicit estimates from capture±recapture surveys. American Naturalist 143:583±596. data with both death and immigration-stochastic model. Coddington, J. A., C. E. Griswold, D. Silva Davila, E. Pen- Biometrika 52:225±247. aranda, and S. F. Larcher. 1991. Designing and testing Karr, J. R. 1990. Avian survival rates and the extinction sampling protocols to estimate biodiversity in tropical eco- process on Barro Colorado Island, Panama. Conservation systems. Pages 44±60 in E. C. Dudley, editor. The unity Biology 4:391±397. of evolutionary biology. Proceedings of the Fourth Inter- Karr, J. R., S. K. Robinson., J. G. Blake, and R. O. Bierre- national Congress of Systematic and Evolutionary Ecology. gaard, Jr. 1990. Birds of four neotropical forests. Pages Dioscorides Press, Portland, Oregon, USA. 237±269 in A. H. Gentry, editor. Four neotropical rainfor- Colwell, R. K., and J. A. Coddington. 1994. Estimating ter- ests. Yale University Press, New Haven, Connecticut, USA. restrial biodiversity through extrapolation. Philosophical Keddy, P. A., and C. G. Drummond. 1996. Ecological prop- Transactions of the Royal Society of London B 345:101± erties for the evaluation, management, and restoration of 118. temperate deciduous forest ecosystems. Ecological Appli- Conroy, M. J., and J. D. Nichols. 1984. Testing for variation cations 6:748±762. in taxonomic extinction probabilities: a suggested meth- Kendall, W. L., J. D. Nichols, and J. E. Hines. 1997. Esti- odology and some results. Paleobiology 10:328±337. mating temporary emigration and breeding proportions us- Conroy, M. J., and B. R. Noon. 1996. Mapping of species ing capture±recapture data with Pollock's robust design. richness for conservation of biological diversity: concep- Ecology 78:563±578. tual and methodological issues. Ecological Applications 6: Kendall, W. L., and K. H. Pollock. 1992. The robust design 763±773. in capture±recapture studies: a review and evaluation by Cook, R. R., and I. Hanski. 1995. On expected lifetimes of Monte Carlo simulation. Pages 31±43 in D. R. McCullough small-bodied and large-bodied species of birds on islands. and R. H. Barrett, editors. Wildlife 2001: populations. El- American Naturalist 145:307±315. sevier, London, UK. Dawson, D. K., J. R. Sauer, P. A. Wood, M. Berlanga, M. H. Kendall, W. L., K. H. Pollock, and C. Brownie. 1995. A Wilson, and C. S. Robbins. 1995. Estimating bird species likelihood-based approach to capture±recapture estimation 1224 JAMES D. NICHOLS ET AL. Ecological Applications Vol. 8, No. 4 of demographic parameters under the robust design. Bio- Breeding Bird Survey annual summary 1990±1991. Bird metrics 51:293±308. Populations 1:1±15. Lancia, R. A., J. D. Nichols, and K. H. Pollock. 1994. Es- Pimm, S. L., J. Diamond, T. M. Reed, G. J. Russell, and J. timating the number of animals in wildlife populations. Verner. 1993. Times to extinction for small populations of Pages 215±253 in T. Bookhout, editor. Research and man- large birds. Proceedings of the National Academy of Sci- agement techniques for wildlife and habitats. The Wildlife ences (USA) 90:10871±10875. Society, Bethesda, Maryland, USA. Pollock, K. H. 1982. A capture±recapture sampling design Lebreton, J.-D., K. P. Burnham, J. Clobert, and D. R. An- robust to unequal catchability. Journal of Wildlife Man- derson. 1992. Modeling survival and testing biological agement 46:752±757. hypotheses using marked animals: a uni®ed approach with Pollock, K. H., J. D. Nichols, C. Brownie, and J. E. Hines. case studies. Ecological Monographs 62:67±118. 1990. Statistical inference for capture±recapture experi- Lee, S.-M., and A. Chao. 1994. Estimating population size ments. Wildlife Monographs 107. via sample coverage for closed capture±recapture models. Pollock, K. H., and M. C. Otto. 1983. Robust estimation of Biometrics 50:88±97. population size in closed animal populations from capture± Lubchenco, J., et al. 1991. The sustainable biosphere initia- recapture experiments. Biometrics 39:1035±1049. tive: an ecological research agenda. Ecology 72:371±412. Pollock, K. H., and D. G. Raveling. 1982. Assumptions of MacArthur, R. H., and E. O. Wilson. 1967. The theory of modern band-recovery models, with emphasis on hetero- island biogeography. Princeton University Press, Princeton, geneous survival rates. Journal of Wildlife Management New Jersey, USA. 46:88±98. Mangel, M., et al. 1996. Principles for the conservation of Pollock, K. H., D. L. Solomon, and D. S. Robson. 1974. wild living resources. Ecological Applications 6:338±362. Tests for mortality and recruitment in a K-sample tag±re- Mehlman, D. W. 1997. Change in avian abundance across capture experiment. Biometrics 30:77±87. the geographic range in response to environmental change. Pradel, R. 1996. Utilization of capture±mark±recapture for Ecological Applications 7:614±624. the study of recruitment and population growth rate. Bio- Mingoti, S. A., and G. Meeden. 1992. Estimating the total metrics 52:703±709. number of distinct species using presence and absence data. Preston, F. W. 1948. The commonness, and rarity, of species. Biometrics 48:863±875. Ecology 29:254±283. Nichols, J. D. 1992. Capture±recapture models: using Reed, J. M. 1996. Using statistical probability to increase marked animals to study population dynamics. BioScience con®dence of inferring species extinction. Conservation 42:94±102. Biology 10:1283±1285. Nichols, J. D., and M. J. Conroy. 1996. Estimation of species Rexstad, E. A., and D. R. Anderson. 1992. Heterogeneous richness. Pages 226±234 in D. E. Wilson, F. R. Cole, J. D. survival rates of mallards (Anas platyrhynchos). Canadian Nichols, R. Rudran, and M. Foster, editors. Measuring and Journal of Zoology 70:1878±1885. monitoring biological diversity. Standard methods for Rexstad, E., and K. P. Burnham. 1991. User's guide for in- mammals. Smithsonian Institution Press, Washington, teractive program CAPTURE. Abundance estimation of D.C., USA. closed animal populations. Colorado State University, Fort Nichols, J. D., J. E. Hines, K. H. Pollock, R. L. Hinz, and Collins, Colorado, USA. W. A. Link. 1994. Estimating breeding proportions and Robbins, C. S. 1990. Use of breeding bird atlases to monitor testing hypotheses about costs of reproduction with cap- population change. Pages 18±22 in J. R. Sauer and S. Droe- ture±recapture data. Ecology 75:2052±2065. ge, editors. Survey designs and statistical methods for the Nichols, J. D., R. W. Morris, C. Brownie, and K. H. Pollock. estimation of avian population trends. United States Fish 1986. Sources of variation in extinction rates, turnover, and Wildlife Service, Biological Report 90(1). and diversity of marine invertebrate families during the Robbins, C. S., D. Bystrak, and P. H. Geissler. 1986. The Paleozoic. Paleobiology 12:421±432. Nichols, J. D., and K. H. Pollock. 1983a. Estimating taxo- breeding bird survey: its ®rst ®fteen years, 1965±1979. U.S. nomic diversity, extinction rates, and speciation rates from Fish and Wildlife Service, Resource Publication 157. fossil data using capture±recapture models. Paleobiology Rosenzweig, M. L., and C. W. Clark. 1994. Island extinction 9:150±163. rates from regular censuses. Conservation Biology 8:491± Nichols, J. D., and K. H. Pollock. 1983b. Estimation meth- 494. odology in contemporary small mammal capture±recapture Russell, G. J., J. M. Diamond, S. L. Pimm, and T. M. Reed. studies. Journal of Mammalogy 64:253±260. 1995. A century of turnover: community dynamics at three Nichols, J. D., J. R. Sauer, K. H. Pollock, and J. B. Hestbeck. timescales. Journal of Animal Ecology 64:628±641. 1992. Estimating transition probabilities for stage-based Sauer, J. R., B. G. Peterjohn, and W. A. Link. 1994. Observer population projection matrices using capture±recapture differences in the North American Breeding Bird Survey. data. Ecology 73:306±312. Auk 111:50±62. Nichols, J. D., S. L. Stokes, J. E. Hines, and M. J. Conroy. Scott, J. M., F. Davis, B. Csuti, R. Noss, B. Butter®eld, C. 1982. Additional comments on the assumption of homo- Groves, H. Anderson, S. Caicco, F. D'Erchia, T. C. Ed- geneous survival rates in modern bird banding estimation wards, Jr., J. Ulliman, and R. G. Wright. 1993. Gap anal- models. Journal of Wildlife Management 46:953±962. ysis: a geographic approach to protection of biological di- Norris, J. L. III, and K. H. Pollock. 1996. Nonparametric versity. Wildlife Monographs 123. MLE under two closed capture±recapture models with het- Seber, G. A. F. 1965. A note on the multiple-recapture census. erogeneity. Biometrics 52:639±649. Biometrika 52:249±259. Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson. . 1982. Estimation of animal abundance and related 1978. Statistical inference from capture data on closed an- parameters. Macmillan, New York, New York, USA. imal populations. Wildlife Monographs 62. Simberloff, D. S. 1969. Experimental zoogeography of is- Palmer, M. W. 1990. The estimation of species richness by lands: a model for insular colonization. Ecology 50:296± extrapolation. Ecology 71:1195±1198. 314. . 1991. Estimating species richness: the second-order Skalski, J. R., and D. S. Robson. 1992. Techniques for wild- jackknife reconsidered. Ecology 72:1512±1513. life investigations: design and analysis of capture data. Ac- Peterjohn, B. G., and J. R. Sauer. 1993. North American ademic Press, San Diego, California, USA. November 1998 ESTIMATORS FOR ANIMAL COMMUNITY DYNAMICS 1225

Smith, E. P., and G. van Belle. 1984. Nonparametric esti- 1982. Capture±recapture and removal methods for sam- mation of species richness. Biometrics 40:119±129. pling closed populations. Los Alamos National Laboratory Solow, A. R. 1993. Inferring extinction from sighting data. Publication LA-8787-NERP. Ecology 74:962±964. Wiens, J. A., T. O. Crist, R. H. Day, S. M. Murphy, and G. . 1994. On the Bayesian estimation of number of spe- D. Hayward. 1996. Effects of the Exxon Valdez oil spill cies in a community. Ecology 75:2139±2142. on marine bird communities in Prince William Sound, Alas- Thiollay, J.-M. 1995. The role of traditional agroforests in ka. Ecological Applications 6:828±841. the conservation of rain forest bird diversity in Sumatra. Conservation Biology 9:335±353. Wilson, D. E., F. R. Cole, J. D. Nichols, R. Rudran, and M. Walther, B. A., P. Cotgreave, R. D. Price, R. D. Gregory, and S. Foster, editors. 1996. Measuring and monitoring bio- D. H. Clayton. 1995. Sampling effort and parasite species logical diversity. Standard methods for mammals. Smith- richness. Parasitology Today 11:306±310. sonian Institution Press, Washington, D.C., USA. White, G. C., D. R. Anderson, K. P. Burnham, and D. L. Otis.

APPENDIX In order to facilitate the understanding of our variance es- cluded only the ``error of estimation'' associated with a de- timation methods, we will describe the speci®c case of BBS tection probability Ͻ1, and not any variation associated with data, but we have encoded general versions of this approach the stochasticity of population- or community-dynamic pro- in program COMDYN (Hines et al., in press). The ®rst step cesses that produced the true Ni. Our reasoning for omitting in computing bootstrap variance estimates is to use the actual such variation from our variance estimates for species rich- data (e.g., from the BBS) to compute the estimate of interest. ness is that we are interested in the species actually present All such estimates are based on one or more estimates of at a particular time i, not in a statistic characterizing the species richness, sometimes for all of the members of the distribution of the possible realizations of that random vari- community and sometimes for speci®c subsets of species. able. This estimation of the number of species in the face of un- Our approach to variance estimation is different for turn- known species detection probabilities is the source of all of over and extinction probability, because we are more inter- the variation considered in the variance estimates for some ested in the underlying probability that a species present at Ã Ã of the estimators computed by COMDYN (e.g., Ni,.␭ˆ ij, pÅ i) i is still (or was, in the case of turnover) present at j than in For estimating variance of these estimators, each estimate of the actual proportions that survived (or were survivors). We Ri richness based on the actual data is treated as the true number introduce this additional variation by treatingMj as a random Ã Ri of species. The observed frequencies of species seen on 1, variable, rather than conditioning on the actualMj estimated 2, . . . , 5 of the BBS route segments (denoted as f1, f2,..., from the data. f5, where fh denotes the number of species observed on h The ®rst step in the bootstrap procedure is the same as for Ã Ri segments) are then used to estimate multinomial cell prob- species richness, as described. We treat the estimate,Mj , as abilities as the true number of species present in period j that were survivors from the species observed in period (members of ). f i Ri Ã h We then use the observed frequencies (f , f , ..., f ) with ␲h ϭ Ã h ϭ 1,2,...,5 and 1 2 5 Ni which members of Ri present at j were observed at period j 5 on 1, 2, . . . , 5 of the BBS route segments, to estimate mul- f tinomial cell probabilities as ͸ h ␲Ã ϭ 1 Ϫ hϭ1 . (A.1) 0 NÃ fh i ␲Ã h ϭ h ϭ 1,2,...,5 Ã Ri Mj These estimated cell probabilities,␲Ã h , correspond to the probability that a species in the community is seen on h ϭ 0, 1, 5 f . . . , 5 route segments. ͸ h Ã hϭ1 For each bootstrap replicate, we then use the Ni and the ␲Ã 0 ϭ 1 Ϫ . (A.3) MÃ Ri ␲Ã h to generate a set of multinomial random variables,f *h , and j Ã use them to compute a new richness estimate,Ni** . This new Ri* For each bootstrap replicate, we ®rst generateMj as a richness estimate is then used in the estimator of interest, and Ri* binomial random variable (i.e., Mjijϳ bin[Ri,␾ˆ ]). We then resulting ``estimates'' are the replicates that are used to com- * use thisMRi , with the␲Ã from Eq. A.3, to generate a set of pute empirical variance estimates jh multinomial random variables,f hh* . These frequencies,f * , are n * then used to compute a new estimate,MÃ Ri , which is used in (ˆ* ˆ¯*)2 j ͸ ␪j Ϫ ␪ conjunction with the estimators of Eqs. 4 and 7 to compute jϭ1 var(␪ˆ) ϭ (A.2) the bootstrap replicates of the␾ˆ ij . The bootstrap variance n(n Ϫ 1) estimates are then computed as in Eq. A.2. where␪ˆ j* is the jthd bootstrap replicate of the estimated pa- It is possible for some of our estimators to assume values rameter of interest, n is the number of replicates, and␪¯Ã * is that we judge to be inadmissable, and we had to decide on a the arithmetic mean of the n replicates. In addition to com- way to deal with these in the bootstrap replications. Estimates puting variances, we used the empirical distribution of the of ␾ˆ ij Ͼ 1 were logically impossible, so when this condition bootstrap replicate estimates,␪ˆ j* , to compute our 95% CIs. occurred in a bootstrap replicate, we set the estimate to 1. Ã The bootstrap variance estimates computed as described Similarly, estimates of Bij Ͻ 0 were set equal to 0. Finally, can be viewed as conditional on the true value of the estimated the process of generating bootstrap replicates led to the pos- quantity. For example, the bootstrap estimate for variance of sibility that a richness estimate was smaller than the number Ã species richness could be written (using the notation of Jolly of species observed (Ni** Ͻ Ri). In such cases, the bootstrap Ã 1965) asvar (Niiͦ N ) , indicating that the variance estimate in- estimate was set equal to the number of species observed.