Estimating Population Growth Rate From Capture Recapture Data in Presence of Capture Heterogeneity

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(1)Supplementary materials for this article are available at 10.1007/s13253-009-0008-8.. Estimating Population Growth Rate From Capture–Recapture Data in Presence of Capture Heterogeneity Roger P RADEL, Rémi C HOQUET, Mauricio A. L IMA, Joseph M ERRITT, and Laurent C RESPIN The direct estimation and modeling of population growth rate from capture– recapture data has now seen a number of applications. However, the original model cannot accommodate heterogeneous capture probabilities. While studying a population of small mammals Peromyscus maniculatus, we became concerned that the peak of population size may be estimated too late in the year because of heterogeneous catchability. Hence, we developed a variation of the original model with a finite number of catchability classes. The results obtained with the new model are more in agreement with the known biology of this population. A bibliographic appendix and computer code are available online. Key Words: Coefficient of proportionality; Losses on capture; Maximum likelihood; Mixture models; Multinomial models; Peromyscus maniculatus.. 1. INTRODUCTION Pradel (1996) developed a model for the estimation of changes in the size of animal populations from capture–mark–recapture (CMR) data. This model has since been used on a variety of species belonging to a wide range of taxa: since 2001, at least 21 times in birds, 16 times on mammals, 11 on reptiles, once on fish, and twice on mollusks; it has also been applied to paleontological data for the estimation of fossil taxa turnover rate (see bibliographic appendix online). Despite its wide applicability, the model makes a number. Roger Pradel () is Research Scientist (E-mail: [email protected]) and Rémi Choquet is Research Engineer, Centre d’Écologie Fonctionnelle et Évolutive, Centre National de la Recherche Scientifique, Equipe Biométrie et Biologie des Populations, 1919 Route de Mende, 34293 Montpellier Cedex 5, France. Mauricio A. Lima is Research Scientist, Center for Advanced Studies in Ecology & Biodiversity, Departamento de Ecología, Pontificia Universidad Católica de Chile, 193 Correo 22, Santiago CP 6513677, Chile. Joseph Merritt is Research Scientist, Illinois Natural History Survey, 1816 S. Oak Street, Champaign, IL 61820, USA. Laurent Crespin is Research Scientist, Centre de Biologie et de Gestion des Populations–IRD, Campus International Agropolis de Baillarguet, CS 30016, 34988 Montferrier-sur-Lez Cedex, France. © 2009 American Statistical Association and the International Biometric Society Journal of Agricultural, Biological, and Environmental Statistics, Volume 15, Number 2, Pages 248–258 DOI: 10.1007/s13253-009-0008-8. 248.

(2) PGR. AND. C APTURE H ETEROGENEITY. 249. of simplifying assumptions and concerns about its reliability have led to the investigation of the more likely causes of departure: study area expansion, permanent trap response (Hines and Nichols 2002), heterogeneous capture probabilities (Hines and Nichols 2002), and tag loss (Rotella and Hines 2005). Capture heterogeneity, in particular, is thought to be widespread in the animal kingdom. Although the bias due to capture heterogeneity in estimates of population growth found by Hines and Nichols (2002) is relatively modest (≈ 5%), these authors also found a spurious negative temporal trend. When studying a population of Peromyscus maniculatus, we became concerned that something similar might be happening as we got seemingly unreasonable estimates of seasonal population growth rates. Additionally, the results of Hines and Nichols (2002) were obtained with the northern spotted owl populations in mind (Franklin et al. 1999) with simulation parameters equal to 0.85 for survival and 1 for population growth rate (PGR hereafter). They may not hold for species with both markedly lower and higher survival rates, species like small mammals that exhibit high variations in population size, or populations exhibiting extreme degrees of capture heterogeneity among individuals (Efford 1998). For all these reasons, a model directly incorporating unequal catchability in the estimation of PGR seemed worth developing. We decided to generalize the model of Pradel (1996) to a population made of a finite mixture of catchability classes; finite mixtures have been used within CR data in other contexts (Pledger 2000; Pledger, Pollock, and Norris 2003). After introducing notation and assumptions (Section 2), we present the general theory (Section 3). We then treat the motivating example of a deer mouse Peromyscus maniculatus population (Section 4).. 2. ASSUMPTIONS AND NOTATION We consider s successive random samples taken at time intervals, which need not be equal, from a population of independent individuals. The individuals are identical except in terms of catchability where they fall into C classes. It is assumed that each individual belongs to the same catchability class during the whole duration of the study. Unless they carry natural marks, the animals are individually marked when first captured with marks that cannot be lost nor misread. When recaptured, the mark is read and recorded. All individuals are immediately released or removed after capture or recapture. There is no temporary emigration (except possibly random). Finally, for convenience, parameters are considered in the proofs as time-specific although the approach proposed is valid more broadly, e.g., for constant survival and/or capture probabilities. The notation is that of Pradel (1996), extended to accommodate the catchability classes. • h = capture history, • i = event at time i (0 = nonobservation, 1 = observation), • η = indicator of right censorship (η = 1 if the animal was removed after last capture, 0 if it was released), • e = index of the earliest observation in h, • l = index of the last observation in h,.

(3) 250. R. P RADEL. ET AL .. • i + (i − ), instant immediately following (preceding) sampling occasion i. “i − ” and “i + ” used in what follows for the sake of generality can be replaced with “i” when there are no removals. • γi , seniority probability, probability that an animal present at i − was already present at (i − 1)+ . • φi , survival probability, probability that an animal present at i + will still be present at (i + 1)− . (j ). (j ). • pi (ri ), capture probability, probability of being captured at time i for an individ(j ) (j ) ual of class j present at i − (i + ) (pi and ri differ if some animals are removed). • ρi , population growth rate between i + and (i + 1)− . • μi , probability of being released for an individual captured at time i (we will denote μ•i , the complement probability not to be released). • ξi , probability of not being seen before time i for an animal of class j present at i − . (j ). • χi , probability of not being seen after time i for an animal of class j present at i + . (j ). • Ni , expected population size at time i. We will distinguish Ni− , the population size at i − , from Ni+ , the population size at i + . (j ). (j )−. • Ni , expected size of class j at time i. We will distinguish Ni (j )+ i − , from Ni , the class size at i + .. , the class size at. (j ). • πi , proportion of class j individuals in the population at time i. We will distinguish (j )− (j )+ πi , the proportion at i − , from πi , the proportion at i + .. 3. THE CAPTURE–RECAPTURE POPULATION GROWTH RATE MODEL WITH HETEROGENEITY The change in population size between two successive sampling occasions, more precisely from i + to (i + 1)− , i.e., excluding the sampling occasions themselves, results from demography alone and is unaffected by capture heterogeneity. The formula given by Pradel (1996, see for details) that expressed PGR as a function of survival and seniority probabilities thus still holds: ρi =. − Ni+1. Ni+. =. φi . γi+1. (3.1). Now because each class is sampled with a different intensity, removals during sampling may affect the composition of the population. After sampling, each class is depleted by the number of individuals captured and not released (j )+ (j )− (j ) 1 − pi μ•i . (3.2) = Ni Ni The population size after sampling is thus: (j )− (j )− (j ) (j ) 1 − pi μ•i = Ni− 1 − pi μ•i = Ni− (1 − pi μ•i ). Ni πi Ni+ = j. j. (3.3).

(4) PGR. AND. C APTURE H ETEROGENEITY. 251. (j )− (j ) It depends only on the mean capture probability p i = j πi pi , not on the individual pi ’s. From Equations (3.2) and (3.3), we can also derive the proportion of each class in the population after sampling: (j ) • (j )− (1 − pi μi ) . (1 − p i μ•i ). (j )+. (j )+. πi. =. Ni. Ni+. = πi. We note that the composition of the population would remain unchanged if there was no (j ) heterogeneity (pi = p i , ∀j ) or no removal (μ•i = 0). From Equations (3.1) and (3.3), we get the PGR from i − to (i + 1)− , i.e., including this time sampling at time i: − Ni+1. Ni−. = ρi. Ni+. = ρi (1 − p i μ•i ).. Ni−. (3.4). Again, it depends only on the mean capture probability in the population. The probability of a capture history will be obtained as in Pradel (1996) as the ratio of the expected number of individuals with this particular capture history, mh , to the expected total number of individuals observed at least once during the entire duration of the study, M. We start by calculating M, which is easier. M is the total of individuals observed for the first time at the different occasions in the different catchability classes: M=. s C . (j )− (j ) (j ) ξ e pe. =. Ne. e=1 j =1. s . Ne−. C . (j )− (j ) (j ) ξ e pe .. πe. j =1. e=1. Similarly, in calculating mh , we go over the catchability classes. We must distinguish three cases: • The animal is immediately removed upon first encounter (l = e and η = 1) mh = Ne− μ•e. C . (j )− (j ) (j ) ξ e pe .. πe. j =1. • The animal is released upon first encounter but never seen again (l = e and η = 0) mh = Ne− μe. C . (j )− (j ) (j ) (j ) ξe pe χe .. πe. j =1. • Otherwise (l > e), mh = Ne− μe. C . (j )− (j ) (j ) (j ) ξe pe Pcond (h),. πe. j =1. where (j ) Pcond (h) =. l−1 i=e. φi. l−1 (j ) i (j ) 1−i (j ) (j ) 1−η pi μ i 1 − pi , pl (μ•l )η μl χl i=e+1. is the probability of h conditional on the time of first capture at e for an animal of class j ..

(5) 252. R. P RADEL. ET AL .. The class-specific probabilities of not being seen, respectively, before and after sample i, (j ) (j ) (j ) ξi , and χi , are functions of the φi ’s and pi ’s with the following recurrence formulas: (j ) (j ) (j ) (j ) i = 1, . . . , s − 1 χs = 1 , χi = (1 − φi ) + φi 1 − pi+1 χi+1 , (j ) (j ) (j ) (j ) i = 2, . . . , s ξ1 = 1 . ξi = (1 − γi ) + γi 1 − ri−1 ξi−1 , (j ). In the last formula, ri , is the capture probability at occasion i of an animal belonging (j )+ (j ) (j )− to Ni . It differs from pi , the capture probability of an animal of Ni , because some captured animals may have been removed during sampling occasion i. However, (j )− (j )+ because all the animals missed at occasion i belong both to Ni and Ni , the change in the probability to be missed results only from the change in the class size given by (j ). Equation (3.2). Thus, 1 − ri. =. (j ). 1−pi. . If there are no removals (μ•i = 0), ri. (j ). (j ) 1−pi μ•i. (j ). = pi .. Now, we can come back to the calculation of P (h) as mMh . We first iterate Equation (3.4) to get: e−1 − − ρi (1 − p i μ•i ) . Ne = N1 i=1. Now, we can factor out N1− in mh and M, so that it eventually cancels out in P (h). After some algebra, we reach: l−1 l−1 s e−1 • i φi γi (1 − pi μi ) μi (μ•l )η P (h) = i=1. i=e+1. × s. j. (j )− (j ) ξe {. πe. i=1 {. i=e. i=1. (j )i (j ) (j ) l (1 − pi )1−i }(μl χl )1−η i=e pi (j )− (j ) (j ) . i−1 • )}{ s φ (1 − p μ γ } ξ i pi k k k j πi k=i+1 k=1 k. (3.5). (j ). In case of homogeneity (C = 1), the π terms disappear and p i = pi = pi . Then, it can be seen that Equation (3.5) reduces to the formula for Pg (h) in Pradel (1996, p. 706). The likelihood L for a set of observed animals ω is eventually obtained as the product of the probabilities of all individual capture histories: L= P (h(ω)) = P (h)#{h} . ω. h. Just as in the original article (Pradel 1996), the likelihood can be expressed as a function of the φ and γ parameters as from Equation (3.5), but it can also be regarded as a function of the φ’s and ρ’s or of the γ ’s and ρ’s, the link between the various expressions being Equation (3.1); a further possibility is the use of the fecundity parameter ft (Pradel 1996) equal t+1 to 1−γ γt+1 in place of γt ’s. If the parameters were to remain unconstrained, the choice of the parameterization would be immaterial. However, when constraints are used to maintain the parameters within their natural range ([0, 1] for probabilities, [0, +∞] for ρ and f ), some parameterizations are insufficiently restrictive as regards other parameters: the (γ , ρ) and the (f , ρ) parameterizations let φ take any positive value, and the (φ, ρ) parameterization lets γ take any value in [0, +∞] and f take any value in [−∞, +∞]. Regarding the encounter probabilities, there is no guarantee in what precedes that the different classes will remain ranked from highest catchable to lowest catchable in the same.

(6) PGR. AND. C APTURE H ETEROGENEITY. 253. way over time. The model can be fitted as is but, in order to fit a model more in agreement with what is expected in a heterogeneous population, capture probabilities can also (j ) be reparameterized by introducing coefficients of proportionality αt maintained in the (j ) (j ) (j −1) , ∀j ∈ [2, C]. This trick ensures that the caprange [0, 1] such that pt = αt × pt ture probabilities will decrease with the class index, making class 1 consistently the class of highest catchability and class C the class of lowest catchability. This is the option retained in the example. A computer program implementing this model has been written in MATLAB (see supplemental materials).. 4. EXAMPLE OF PEROMYSCUS MANICULATUS The deer mouse Peromyscus maniculatus is a small mammal whose populations can exhibit high variations in size. It is commonly live-trapped and the data we use here come from a study conducted in the Appalachian Mountains (U.S.A.) from 1979 to 2003 on a trapping grid of 1 ha (details are given in Merritt 1984). It is known that the abundance of P. maniculatus on the study site increases during summer, reaching peak numbers in late summer and early autumn and gradually decreasing through winter (Merritt 1987). The analysis we present is restricted to the first 10 years, from the autumn of 1979 until the autumn of 1989, and corresponds to 52 trapping sessions (12 in autumn, 12 in winter, 15 in spring, and 13 in summer) for a total of 448 individuals, all males. The intervals between sessions being of varying lengths, the parameters used were all expressed per time unit, here the week; in this way only is it sensible to compare PGRs pertaining to different years or to different seasons. The goodness-of-fit test (Pollock, Hines, and Nichols 1985; Pradel 1993) of the Jolly–Seber model (Jolly 1965; Seber 1965) implemented in 2 = 82.40, program U-CARE (Choquet et al. 2009), although overall not significant (χ86 P = 0.59), revealed in its details an excess of never seen again individuals among the newly marked (z = 3.18, P = 0.0007, one-sided test) (Pradel et al. 1997) as well as an excess of individuals recaptured at the occasion immediately following a previous release (z = −3.39, P = 0.0007, two-sided test) (Pradel 1993). The first feature could be due to the presence of transients on the plot and the second to trap-dependence (here traphappiness), a commonly found behavior of this species, e.g., see Hammond and Anthony (2006). Both could be true. However, the simultaneous significance of both component tests to a similar degree is more parsimoniously explained by heterogeneity of catchability among individuals in the population. Thus, the assumption of homogeneous catchability did not seem to hold for this population. This was confirmed when we compared models with and without heterogeneity (Table 1). The estimates actually revealed a very strong degree of heterogeneity (Table 2): in model (φs , pw , ρs , α, π ) with two classes of catchability, the ratio α of the low to the high capture probability is estimated at 0.140 (SE 0.017). This model has seasonal variation (index s) in ρ, the PGR, and φ, the weekly survival probability; p, the detection probability of the higher detectable class is much lower in winter (index w) than during the other seasons (Table 2); π , the proportion of individuals with high detectability is constant. For the sake of clarity, we present the PGR estimates from this model (Figure 1) although.

(7) 254 Table 1.. R. P RADEL. ET AL .. Some CR models fitted to the male compartment of a Peromyscus maniculatus population inhabiting the Appalachian Mountains (U.S.A.). Parameters are: φ, survival probability; p, detection probability; ρ, realized population growth rate; α, proportionality constant for p; π , proportions in different classes. Effects considered are: t , time variation; s, seasonal variation; y, yearly variation; w, winter vs other seasons. AICc is the difference in Akaike information criterion adjusted for small sample size between the current and the best model.. Model. Deviance. Number of parameters. φs∗y , pw , ρs∗y , α, π φs , pw , ρs , α, π φs∗y , ps∗y , ρs∗y , α, π φs∗y , ps∗y , ρs∗y φs , pw , ρs φt , pt , ρt , αt , πt. 4384.43 4593.42 4328.84 4364.85 4649.67 4200.39. 86 12 126 124 10 235. AICc 0 36.99 55.91 85.98 89.11 329.08. strong differences also exist among years (Table 1). These results are in agreement with what is known of the biology of the species in the area with a peak of population size in early autumn (September to mid-October) followed by a sharp decrease during winter (December–January–February). By contrast, the corresponding model without heterogeneity (φs , pw , ρs ) yields a higher estimate of PGR for autumn, and thus predicts a later peak of population size. On the other hand, it yields lower estimates for summer (June–July– August) and winter. The spring (March–April–May) PGR is estimated very close to 1 by both models.. Table 2.. Results from two CR models for the estimation of population growth rate applied to the male compartment of a Peromyscus maniculatus population inhabiting the Appalachian Mountains (U.S.A.). Model (φs , pw , ρs , α, π ) is a model with seasonal variation (index s) in ρ, the population growth rate, and φ, the weekly survival probability. It has two classes of catchability. p, the detection probability of the higher detectable class differs in winter (index w) from the other seasons; α is the proportionality constant of the low to the high detection probability; π is the proportion of individuals with high detectability. Model (φs , pw , ρs ) is the corresponding model without heterogeneity whose estimates are given for comparison. Standard errors are between parentheses. Parameter. (φs , pw , ρs , α, π ). (φs , pw , ρs ). φautumn φwinter φspring φsummer ρautumn ρwinter ρspring ρsummer pwinter pothers α π. 0.946 (0.012) 0.947 (0.010) 0.950 (0.011) 0.937 (0.013) 1.001 (0.013) 0.952 (0.011) 1.000 (0.011) 1.047 (0.011) 0.680 (0.077) 0.918 (0.045) 0.140 (0.017) 0.167 (0.045). 0.935 (0.013) 0.935 (0.010) 0.935 (0.011) 0.918 (0.013) 1.008 (0.013) 0.947 (0.011) 1.000 (0.011) 1.043 (0.012) 0.468 (0.050) 0.678 (0.040) NA NA.

(8) PGR. AND. C APTURE H ETEROGENEITY. 255. Figure 1. Weekly growth rate by season of a population of male Peromyscus maniculatus. The estimates are from the model (φs , pw , ρs , α, π : full dots) where weekly survival (φ) and PGR (ρ) are season specific, encounter probability (p) differs in winter, the proportionality constant for p (α) and the probability to belong in one of two classes of catchability (π ) are constant over time. They are compared to estimates from the corresponding model without heterogeneity (φs , pw , ρs : empty dots). 95% confidence intervals are represented around each estimate.. In spring, the snow cover, thought to provide to small mammals a shelter both against predators and low temperatures (Merritt and Merritt 1980; Marchand 1996), is melting, but breeding can compensate mortality. Merritt and Merritt (1980) also emphasized that for Peromyscus maniculatus another critical period regarding survival is the autumn freeze, but there again some reproduction is occurring at the onset of the season. At last, winter is known to be a season of high survival but no reproduction for small mammals (Ostfeld and Canham 1995). Although reproduction does occur in early autumn, it becomes limited and can hardly compensate durably losses from mortality, contrary to what the model without heterogeneity predicts. A possible explanation for the autumn overestimation of PGR by the model without heterogeneity is that the individuals born earlier during the year become only progressively more trappable. As a result, because their earlier inobservation is incompatible with the average capture probability estimated under the homogeneity model, these individuals are wrongly seen by this model as entering the population with a delay, mainly in autumn.. 5. DISCUSSION Despite a huge improvement in the quality of fit as measured by the reduction in AICc (by 52.12 points from model φs , pw , ρs to model φs , pw , ρs , α, π ), consistent with the detection of a very strong degree of heterogeneity (low capture probability is only 14% of high capture probability), the relative bias in Peromyscus maniculatus weekly PGR estimates between the models with homogeneous and heterogeneous capture probabilities.

(9) 256. R. P RADEL. ET AL .. remains low, being at most 0.75% in autumn (10.2% for the seasonal PGR). Our study thus confirms the findings of Hines and Nichols (2002) that, contrary to population size estimators (Link 2003), the PGR estimators of CR models are robust to capture heterogeneity. However, if the bias remains low, it is not systematic in its direction: depending on the season, the highest estimate comes from the heterogeneity or from the homogeneity model. Hence, qualitatively, the biological message is altered. The treatment of heterogeneity is thus especially required when investigating patterns. The treatment of capture heterogeneity by a mixture of discrete hidden classes each with its own capture parameter is a very classical approach (Carothers 1973, 1979; Hines and Nichols 2002; Pledger, Pollock, and Norris 2003). As commented by Pledger, Pollock, and Norris (2003), there is no need for the classes to correspond to actual categories in the population. Nonetheless, it may happen that the population is effectively divided into classes with intrinsically different catchabilities like stayers and movers, dominants and subordinates, or young and adult (Crespin et al. 2008). In this latter instance, a more thorough investigation of capture heterogeneity can be carried out using recently developed CR survival models that allow for movement among hidden classes (Pradel 2005, 2009). For the Peromyscus maniculatus dataset, we did not find any strong evidence of movements among the classes, nor could we ascribe the individuals with a high catchability to an identifiable category in the population. Thus, the nature of capture heterogeneity in this study remains unknown. To decide whether a restoration plan is being effective and to take management decisions, PGR is often the relevant parameter; its study can be carried out by different means (Pradel and Henry 2007). In particular, the use of matrix population models (Caswell 2001) is a well established approach that predates the direct approach from CR data that we extended in this article. The two methods were used concurrently for six years in the course of a study on the assessment of the status of the northern spotted owl populations of the western United States (Franklin et al. 1999, 2004; Courtney et al. 2004). In 2004, Courtney et al. concluded that the direct method was superior and they used it exclusively afterwards. The reasons for this choice are detailed in Franklin et al. (2004, pp. 12–13). To summarize, the matrix population model approach is analytical and hence very demanding as it requires the detailed knowledge of the population dynamics, such as the juvenile survival rate, which is often very poorly known, or the emigration rate, which is even more challenging to estimate (Anthony et al. 2006; Sandercock 2006). A common practice in this context is to model gains based only on reproduction—hence leaving immigration out— while losses are modeled with the local survival estimate derived from CR data—hence taking emigration into account. This asymmetry of treatment between gains and losses is particularly problematic (Nichols, pers. comm.). Conversely, the direct approach deals with the net gains and losses, be they from death, birth, emigration, or immigration, and is applicable even for animals with totally unobservable larval stages like fish (Pradel and Henry 2007). The current study gives a further incentive to using the CR direct approach by showing that capture heterogeneity can be accommodated and treated for its qualitative as well as quantitative effects in a relatively simple way. By contrast, the effects of capture heterogeneity on the performance of the matrix population model approach remain largely unknown..

(10) PGR. AND. C APTURE H ETEROGENEITY. 257. SUPPLEMENTAL MATERIALS Bibliographic Appendix and MATLAB code: The supplemental materials contain some applications of the direct estimation of population growth rate from capture–recapture data (MS08-064R code.pdf file) and function computing the likelihood of the heterogeneity model (MS08-064R additional references.pdf file). (13253_2009_8_MOESM1_ESM. pdf, 13253_2009_8_MOESM2_ESM.pdf). ACKNOWLEDGEMENTS The authors thank Jim Nichols for his very useful comments on the biological implications of our model and an anonymous referee for pointing out the incorrect formulation of an equation. This article is the result of a French–Chilean collaboration supported by the program ECOS, action n◦ C05B02. It was partly done while RP was in a sabbatical with the Spanish Ministry of Science (SAB-2006-0014/Roger Pradel). [Received July 2008. Revised December 2008. Published Online January 2010.]. REFERENCES Anthony, R. G., Forsman, E. D., Franklin, A. B., Anderson, D. R., Burnham, K. P., White, G. C., Schwarz, C. J., Nichols, J. D., Hines, J. E., Olson, G. S., Ackers, S. H., Andrews, L. S., Biswell, B. L., Carlson, P. C., Diller, L. V., Dugger, K. M., Fehring, K. E., Fleming, T. L., Gerhardt, R. P., Gremel, S. A., Gutierrez, R. J., Happe, P. J., Herter, D. R., Higley, J. M., Horn, R. B., Irwin, L. L., Loschl, P. J., Reid, J. A., and Sovern, S. G. (2006), “Status and Trends in Demography of Northern Spotted Owls, 1985–2003,” Wildlife Monographs, 163, 1–48. Carothers, A. (1973), “The Effects of Unequal Catchability on Jolly–Seber Estimates,” Biometrics, 29, 79–100. (1979), “Quantifying Unequal Catchability and Its Effects on Survival Estimates in an Actual Population,” Journal of Animal Ecology, 48, 863–869. Caswell, H. (2001), Matrix Population Models: Construction, Analysis and Interpretation (2nd ed.), Sunderland, MA: Sinauer Associates. Choquet, R., Lebreton, J.-D., Gimenez, O., Reboulet, A.-M., and Pradel, R. (2009), “U-CARE: Utilities for Performing Goodness of Fit Tests and Manipulating CApture–REcapture Data,” Ecography, in press. Courtney, S., Blakesley, J., Bigley, R., Cody, M., Dumbacher, J., Fleischer, R., Franklin, A. B., Franklin, J., Gutiérrez, R., Marzluff, J., and Sztukowski, L. (2004), “Scientific Evaluation of the Status of the Northern Spotted Owl, ” scientific report, Sustainable Ecosystems Institute, Portland, OR. Crespin, L., Choquet, R., Lima, M. A., Merritt, J. F., and Pradel, R. (2008), “Is Heterogeneity of Catchability in Capture–Recapture Studies a Mere Sampling Artifact or a Biologically Relevant Feature of the Population?” Population Ecology, 50, 247–256. Efford, M. (1998), “Demographic Consequences of Sex-Biased Dispersal in a Population of Brushtail Possums,” Journal of Animal Ecology, 67, 503–517. Franklin, A. B., Burnham, K. P., White, G. C., Anthony, R. G., Forsman, E. D., Schwarz, C., Nichols, J. D., and Hines, J. E. (1999), Range-Wide Status and Trends in Northern Spotted Owl Populations, Portland, OR: U.S. Fish and Wildlife Service. Franklin, A. B., Gutiérrez, R. J., Nichols, J. D., Seamans, M. E., White, G. C., Zimmerman, G. S., Hines, J. E., Munton, T. E., LaHaye, W. S., Blakesley, J. A., Steger, G. N., Noon, B. R., Shaw, D. W. H., Keane, J. J., McDonald, T. L., and Susan, B. (2004), “Population Dynamics of the California Spotted Owl (Strix occidentalis occidentalis): A Meta-Analysis,” Ornithological Monographs, 54, 1–110..

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