Closed population – a population in which there is no recruitment (birth or immigration) or losses (death or emigration) during the period of study.
a. Geographic:
b. Demographic:
Closedpopulation models
a. two samples – LincolnPetersen model (and Chapman modification)
b. several samples (k>2) – Schnabel (SchumacherEschmeyer) model and models in program CAPTURE.
Openpopulation models (note: geographic closure is still a critical assumption)
a. CormackJollySeber models (based on k>2) in Program MARK.
A sample of C_{1} animals is Caught, marked, and released (M_{2}). Later a sample of C_{2} animals is Captured, of which R_{2} animals are Recaptures that were previously marked. If capture probability (p) is independent of marking status, then the proportion of marked animals in the second sample should be equivalent to the proportion of marked animals in the total population so that
R/C = M/N
where N is the total catchable population size. Solving for N yields the estimator:
If sample size is small, the above estimator is biased. For example, what happens if the number of recaptures is zero? A modified version with less bias was originally developed by Chapman (1951) and is commonly called the modified Petersen estimate in fisheries:
Note that Ricker (1975) and many other texts simply drop the 1 as negligible. Chapman showed that the variance of N^{^} can be estimated as:
var (N_{c}^{^} ) = [(M+1) (C+1) (M R) (CR)] / [(R+1)^{2} (R+2)]
An approximate 95% CI (normality for N^{^}_{c} is assumed) can be estimated as
N_{c}^{^} ± 1.965 * [var (N_{c}^{^} )]^{0.5}
The population is closed (geographically and demographically).
All animals are equally likely to be captured in each sample.
Capture and marking do not affect catchability.
Each sample is random.
Marks are not lost between sampling occasions.
All marks are recorded correctly and reported on recovery in the second sample.
Note: capture probability ( p) is often defined as the probability of an animal being caught in any trap. Possible sources of variation in p include: (1) heterogeneity (e.g., sex, age, social status, size of fish), (2) behavior (e.g., trap happy or trap shy), and (3) time (e.g., effects of weather or sampling effort on p). 
This method extends the LincolnPeterson method to a series of samples in which there are 2, 3, 4,..., n samples. Individuals caught at each sample are first examined for marks, then marked and released. Only a single type of mark need be used because we just need to distinguish 2 types of individuals: marked, caught in one or more prior samples; and unmarked, never caught before. For each sample t, the following is determined:
C_{t} = Total number of individuals caught in sample t
R_{t} = Number of individuals already marked (Recaptures) when caught in sample t
M_{t} = Number of marked animals in the pop'n just before the tth sample is taken.
Schnabel treated the multiple samples as a series of LincolnPeterson (LP) samples and obtained a population estimate as a weighted average of the LP estimates which is an approximation to the maximum likelihood estimate of N:
N^ = SUM (M_{t} C_{t}) / ((SUM R_{t}) + 1)
Note: Var (N^{^} ) is a
function of recaptures, therefore precision of the estimate is increased by
increasing total recaptures (SUM R_{t}). It
can be calculated by inverting the following: Var^(1/N) = SUM (R_{t}) / ( SUM (M_{t} C_{t}))^2 
Same as LincolnPetersen estimator but assumptions apply to all sampling periods. In other words, every individual in the population is assumed to have the same capture probability for a given sampling occasion (although capture probabilities can vary among sampling periods).
The major advantage of multiple sampling is that it is possible to evaluate the data for violations of assumptions, such as unequal capture probabilities.
Violations of the assumptions of equal capture probabilities in capturerecapture studies can lead to unreliable estimates of population size. Program CAPTURE was developed to deal with unequal capture probabilities. It uses different estimators (models) to account for different sources of variation in capture probabilities:
M_{0} = Equal Catchability Model (null model)  Assumes every animal in the population has the same p^{^} for each sampling period in the study.
M_{h} = The Heterogeneity Model  Assumes that each animal has a unique p^{^} that remains constant over all trapping occasions. Furthermore, capture probabilities are assumed to be a random sample of all individuals in the population.
M_{b} = The Trap Response Model  Adjusts for a change in capture probabilities caused by a response to trapping. An assumption of the M_{b} model is that the initial p^{^} for all animals is the same (equal catchability).
M_{bh} = The Heterogeneity and Trap Response Model Based on the assumption that each animal has its own unique pair of potential capture probabilities, p_{j} and c_{j} (j = 1, ..., N animals in the population), where p_{j} is the initial capture probability and c_{j} is the recapture probability.
M_{t} = The Time Variation (Schnabel) Model  Based on the assumption that every individual in the population has the same p^{^ }for a given sampling occasion, but capture probabilities can vary at each sampling time.
Other TimeDependent Models: M_{th}, M_{tb}, and M_{tbh} (Note: there was no estimator available for M_{tbh} until recently and it is only calculated within MARK).
Tests between models
Goodnessoffit tests
Mathematical model selection procedure (i.e., Selection Criteria)
Note:

Program CAPTURE now consists of 2 executable components: an analytical component (still named CAPTURE) that performs abundance and density estimation from information provided by the user, and a new (1991) component named 2CAPTURE, which provides a userfriendly interface to CAPTURE.
Note: Program 2CAPTURE is used to build the input file, which then must be submitted to program CAPTURE for analysis. You will likely run into "memory" problems if you try to run program CAPTURE from within 2CAPTURE. Instead, use 2CAPTURE to create your input file, then exit 2CAPTURE and run CAPTURE in standalone mode. We will show you how to do this in lab!
*.CAP = input file (an ASCII file containing raw data, without command lines)  used by 2CAPTURE to build an *.INP file.
*.INP = input file (an ASCII file containing command lines and, possibly, raw data)  created by using 2CAPTURE or some type of editor. This file tells CAPTURE what to do with the data. Note: older versions of program CAPTURE expect this file to be named CAPTIN.
*.OUT = output file (summary and results of analyses) from CAPTURE. If you do not designate an output file, the program will name the output files CAPTUR1, CAPTUR2, etc. If there is a runtime error, you may also end up with a FORT1 in your directory (results of analyses that were successfully executed). Note: program CAPTURE will automatically save output to a file named CAPTLP if you don't specify an output file.
TITLE=' a title of your choice ' TASK READ CAPTURES (MATRIX) OCCASIONS=(#) SUMMARY FILE=optional DATA='a name of your choice that describes the data, source, etc.' Note:optional FORMAT='(Fortran code)' Note: describes data location by column number. READ INPUT DATA Data are entered here or stored in the file referenced above. TASK MODEL SELECTION OCCASIONS=## (occasions command is optional) TASK CLOSURE TEST OCCASIONS=## (occasions command is optional) TASK POPULATION ESTIMATE APPROPRIATE
List of task statements, commands, warning messages, etc. Note: ignore warnings if you got a "Capture Normal End" statement on your screen.
Definitions of symbols/variables used in the output (you may not get this page in your output  see next section)
Summary of capture history for each animal (if applicable and if SUMMARY is included in the TASK READ CAPTURES line).
Results of modelselection tests (7 tests possible). Note: see Selection Criteria
Population estimates and confidence intervals
Test for closure (if requested and if applicable to the model selected)
Notation used in output from program CAPTURE
t  number of trapping occasions.
n(j)  number of animals captured in the jth sample, j=1,...,t. This is the total number of captures in the experiment.
u(j)  number of new (unmarked) animals captured in the jth sample, j=1,...,t.
f(j)  the capture frequencies. The number of individuals captured exactly j times in t days of trapping j=1,...,t.
M(j)  the number of marked animals in the population at time of the jth sample, j=2,...,t. Note that M(1) equals zero.
M(t+1)  the number of distinct individuals caught during the experiment.
N^{^}  the estimated population.
p^  estimated probability of capture.
p^(j)  the estimated probability of capture by occasion.
pbar  average probability of capture for the generalized removal model.
c^  estimated probability of recapture.
Krebs, C. J. 1989. Ecological methodology. Harper and Row, Publ., New York. 654pp.
Lancia, R. A., J. D. Nichols, and K. H. Pollock. 1994. Estimating the number of animals in wildlife populations. Pages 215253 in T. A. Bookhout, ed. Research and management techniques for wildlife and habitats. Fifth ed. The Wildlife Society, Bethesda, Md.
Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson. 1978. Statistical inference from capture data on closed animal populations. Wildl. Monogr. 62. 135pp.
Pollock, K. H., J. D. Nichols, C. Brownie, and J. E. Hines. 1990. Statistical inference for capturerecapture experiments. Wildl. Monogr. 107. 97pp.
Rexstad, E., and K. Burnham. 1991. User's guide for interactive program CAPTURE. Colorado Coop. Fish and Wildl. Res. Unit, Colorado State University, Fort Collins. 29pp. Note: TA has a copy of this manual if you have a question about program 2CAPTURE.
Ricker, W. E. 1975. Computation and Interpretation of Biological Statistics of Fish Populations. Department of the Environment, Fisheries and Marine Service, Ottawa.
White, G. C., D. R. Anderson, K. P. Burnham, and D. L. Otis. 1982. Capturerecapture and removal methods for sampling closed populations. Los Alamos Nat. Lab., Los Alamos, New Mexico. 235pp. Note: there is a copy of this manual on reserve at the UI library.
Revised: August 25, 2011