Estimating Population Sizes by Mark-recapture and Removal Sampling Methods

Overview
        In this lab, we will conduct two studies of population size.  One study involves estimating population size of natural populations of harvester ants (Pogonomyrmex) at BFL.  The other study is an examination of population estimation sampling techniques using a population of known size of a laboratory-reared beetle (Tenebrio).
        Directly counting the exact number of individuals of organisms is difficult if not impossible, hence sampling of populations to estimate their size has become an extremely important part of studies in population dynamics and structure in ecological and evolutionary studies.  Historically the field implementation for estimation of population size in animals has led to a wide variety of indices proposed by numerous authors (reviewed by Southwood 1978 and Krebs 1989) all which attempt to correct potential biases and peculiarities of the taxa being sampled.  Two general techniques have been used in population estimates the permanent removal method and the mark and recapture method.

Mark and recapture studies of Pogonomyrmex barbatus (F. Smith) populations

Introduction
      

 Mark and recapture, also known as the Lincoln-Pearson method in honor of two early contributors to its development, involves capturing individuals from a population of interest, marking them, releasing them for a relatively short period of time, and then later recapturing individuals from the same population and counting the number of marked and unmarked individuals. The size of the entire population can be accurately estimated from the proportion of marked and unmarked animals given that certain assumptions are met (as far as practicable).
        These assumptions, taken from Southwood (1978) are:
1. The marked animals are not affected (neither in behavior nor life expectancy).
2. The marked animals are completely mixed in the population.
3. The probability of capturing a marked animal is the same as that of capturing any member of the population.
4. Sampling must be at discrete time intervals and the actual time involved in taking the samples must be small in relation to the total time.
When using the simple Lincoln Index it is also assumed that:
5. The population is a closed one, or, if not, immigration and emigration can be measured or calculated.
6. The are no births or deaths in the period between sampling or, if there are, allowance must be made for them.
        In the particular case of estimation of colony size of P. barbatus, we meet, in a general but not stringent way, these assumptions and can use sampling methodology outlined by Porter and Jorgensen (1980, 1981) in their similar, but more detailed studies of P. owyheei Cole.  We will tailor this methodology to our particular goals.

Statistical Background
        Assume the total population size to be estimated contains N individuals. From this population, take a sample of M individuals, mark and return them to the population.  Later, take a second sample of n individuals from the population.  This sample contains R recaptured animals. The Lincoln-Pearson equation for estimating population size, N, is:

This equation overestimates the actual population size. This bias can be reduced by using Bailey's modification of the Lincoln-Pearson equation:

        Bailey's modification is appropriate if the second sample of animals is obtained by collecting the animals one at a time, returning each to the population before taking another. Therefore, individual organisms could b e recaptured more than once. His equation is thought to yield a better estimate when sample size is small (less than circa 20). A more typical situation is where the animals are captured all at once so an individual can only be counted once as a member of that sample. The preferred unbiased population estimate is then:

(Chapman 1951, cited in Brower et al. 1998).

        As with all population estimates made from samples, there is an uncertainty caused by the error associated with examining a sample rather than the entire population.  A measure of this error that expresses the uncertainty of a capture-recapture population estimate is the standard error (SE).  for N_B, this is computed as

        N_B is a nearly unbiased estimate of population size if the number of recaptured individuals, R, is at least 7 and N_C is also nearly unbiased if R is at least 8 (Krebs 1989: p. 17, cited in Brower et al. 1998). The precision of all of the population size estimates for mark-recapture techniques is dependent on the number of marked animals recaptured so attempt to obtain a reasonably large R by making n, the second sample size, large.

        Confidence intervals for mark-recapture estimates may be approximated from the standard error.  It is computed as

N ± (t) (SE),

where t is Student's t (see table under readings/handouts on the course web page) for DF = infinity.

Materials and Methods
        We will sample the populations of two mounds of P barbatus at the Brackenridge Field Laboratoy of the University of Texas at 2907 Lake Austin Boulevard in Austin, Texas at the beginning of lab. We will mark fifty exterior workers with a nontoxic paint (Liquid Paper©, Liquid Paper Corp., Boston, MA) at the beginning of the afternoon for potential recapture later in the afternoon. During this time interval the ants will be allowed to freely mix with other exterior workers of the mound. In preparation for ant resampling a 30 cm line tangential to the primary mound opening at a distance of 15 cm from the opening will be gently marked and divided into two 15 cm intervals (Fig 1). The placement of the 30 cm line will be such that it crosses an apparent trunk foraging trail. All harvester ants crossing this tangential line will be noted and recorded as either marked or unmarked during a measured time interval. Two observers will watch the ants, each will be responsible for one 15 cm segment of the 30 cm interval. Two recorders will tally the observers calls. Sampling will continue for 10 minutes or until at least 10 marked individuals have crossed the tangential line. Estimate the population size in the mound using the Lincoln Index and Bailey's modification.

Assignment
        The results will be your estimates of the population size of the ants in each of the mounds and an average for the two mounds. Warning, the population you are sampling may be different from the actual population of the ants in the mound, see Porter and Jorgensen (1980, 1981) for clarification.  Compare the population sizes you found to other estimates of colony size and biological attributes in Pogonomyrmex as reviewed in Hölldobler & Wilson (1990) and Cole (1968).  Discuss any problems, biases, or inconsistencies which might need attention in future studies.  Consider any other questions of population size in these ants which you encounter during your literature search.  Of course this lab experience should be written as a scientific paper and handed in at the appropriate time which I (Dr. Abbott) will give you.  A special requirement for this paper is that at least five good bibliographic references should be cited.

Literature Cited
Bailey, N. T. J. 1951. On estimating the size of mobile populations from recapture data. Biometrika 38: 293-306.

Brower, J. E., J. H. Zar, and C. N. von Ende. 1998. Field and laboratory methods for general ecology. 4th ed. Boston: McGraw-Hill. 273 pp.

Cole, A. C., Jr. 1968. Pogonomyrmex harvester ants. Univ. Tennessee Press, Knoxville, TN, 222 pp.

Hölldobler, B. and E. O. Wilson. 1990. The ants. Harvard Univ. Press, Cambridge, MA, 732 pp.

Krebs, C. J. 1989. Ecological methodology. Harper Collins Publishers, New York, 654 pp.

Le Cren, E. D. 1965. A note on the history of mark-recapture population estimates. J. Anim. Ecol. 34: 453-545.

Lincoln, F. C. 1930. Calculating waterfowl abundance on the basis of banding returns. U. S. D. A. Circ. 118: 1-4.

Porter, S. D. and C. D. Jorgensen. 1980. Recapture studies of the harvester ant, Pogonomyrmex owyheei Cole, using a 
fluorescent marking technique. Ecol. Entomol. 5: 263-269.

Porter, S. D. and C. D. Jorgensen. 1981. Foragers of the harvester ant, Pogonomyrmex owyheei: a disposable caste? Behav. Ecol. Sociobiol. 9: 247-256.

Southwood, T. R. E. 1978. Ecological methods. Halsted Press, Chapman and Hall. London. 524 pp.

Population size estimates of an isolated laboratory colony of Tenebrio molitor Linnaeus (Insecta: Coleoptera: Tenebrionidae)

Introduction
        The sampling of mobile and secretive animal populations is more difficult than sampling of plants or other sedentary organisms.  Not only are the statistical problems of sampling anything encountered but biological attributes of the animals themselves make locating them in space and time problematic (Southwood 1978).  The mark and recapture method allow estimates of population size to be made with a minimum of harm to the populations of interest.  However, if the assumptions are not met, these methods may not be of value in estimating population size.  If the population of interest can be sampled without replacment, removal methods can be used.

        The removal method has been used frequently in the past to estimate population size in small animals including arthropods and rodents.  Sampling entails permanent removal of the study organisms when encountered in randomized but consistent procedures.  A prerequisite for this method is that the number of individuals encountered become successively smaller as sampling ensues.  For this sampling regime the natality, metamorphosis, mortality, and migration are taken to be insignificant during the sampling period.  Migratory populations do not satisfy these basic assumptions so a different sampling regime is necessary to estimate their populations.  The population must be reasonably stationary, be large enough to assure a significant catch in each proceeding sample, and small enough to note a reduction in catch as sampling proceeds.  Three common modes of analysis are used in removal sampling routines: the hand graphing method, the two sample method (Moran-Zippin Method), and linear regression plots.  In this experiment we will make use of these three removal methods for estimating the population size of a discrete colony of the darkling beetle, Tenebrio molitor Linnaeus.

Statistical Background
The data analyzed below to illustrate these two methods are shown in Table 1.

Table 1. Number of darkling beetles encountered and used to illustrate various methods of estimating population size using removal.

        The Hayne method involves taking a series of samples from a population.  Each sample must involve the same amount of effort.  The number of individuals from each collection is plotted against the previous number captured. If the probability of capture remains constant, the points on the graph should fall along a straight line.  The estimated total population size is derived from extrapolating the line to the X axis.  This extrapolation represents the theoretical condition in which all animals have been removed by sampling.  Using the manual graphing technique the first data point had an x value of 0 (since no beetles had been removed previously) and a y value of 84; the second data point was 84 versus 49, the third 133 versus 29, and so on.  Visual inspection of this scatter plot (Fig. 2) and hand fitting a line to these data produced a line which crossed the x axis at 200, an indication that 200 animals were in the population before sampling began.  The accuracy of this method depends on counting a very large proportion of the population and obtaining enough samples to reliably draw a straight line.

Figure 2. Plot of catch per effort and cumulative prior catch for Tenebrio molitor population.

        The Hayne method can be combined with linear regression analysis.  This type of analysis is valid if one of the variables is dependent on another.  This method is easily used in conjunction with many graphics and analysis software packages and of course can be calculated manually as well.  This is generally perceived to be the most accurate of the methods for locating and projecting a straight line based on a series of data points.  The formula for a straight line is the familiar:

Where y is the cumulative catch; m is the slope of the line; x is the catch per unit effort; and b is the point intercept on the y axis. The population size estimate is given by the y intercept, b.  The manual equations follow this reasoning: if and when all the animals had been removed from the population, x would equal zero and thus the projected estimate of population size would be the y intercept (Fig. 3).

Fig. 3. Regression analysis of the number of Tenebrio molitor collected in successive samples from 3 l of wheat bran. Y intercept, 210 individuals, estimates the population size.

For your own enrichment, think about what the slope of the line means. If we obtained different slopes using different sample sizes (in this example, size of scoop) what might we learn?

Moran-Zippin Method
        The Moran-Zippin method requires fewer samples (just two) than the Hayne method.  The equation used to estimate population size is as follows:

Where N is the estimated population size; Y1 is the number of individuals in the first sample; and Y2 is the number of individuals in the second sample. Please note that the second sample must be smaller than the first or the procedure cannot be used. Why?

Applying this equation to the data in Table 1:

The standard error of this population estimate is

For the above example, SE = (84)(49)(133)^½ / (84-49)² = (4116)(11.53) / 1225 = 38.75

Confidence intervals may be calculated as N ± (t) (SE)

where t is Student's t for DF = infinity.  So, for a 95% confidence interval, t = 1.96 (available on course web site, under handouts & readings) and the confidence interval would be calculated as

201.6 ± (1.96) (38.75) = 201.6 ± 75.95

Methods & Materials
        Colonies of the darling beetle, Tenebrio molitor, are maintained at the Brackenridge Field Laboratory of the University of Texas as food source for other laboratory animals.  A known population of beetles was prepared by counting the various life stages such that the true population consisted of 100 larvae, 50 pupae, and 50 adults.  The total true population was thus 200 individuals.  These individuals were then placed in three liters of wheat bran and mixed thoroughly.  Sampling consists of removing a 500 ml sample of bran with associated beetles and counting the number of each stages.  This is repeated for a total of 5 sets of 5 samples each.

Assignment
        For each of the three methods, calculate population size appropriately, either using extrapolation by hand graphing, computer-aided graphing with linear regression (Hayne method) or by using the Moran-Zippin method for the first two samples. Also calculate the sampling error and confidence intervals for the Moran-Zippin method.

Consider the following in your analyses:
1. How did your extrapolations compare with the actual population size of 200?
2. How did each of the three extrapolation methods compare with each other?

Additional information regarding age structure for the colony can be noted. Note that the number of larvae is roughly two times that of either pupae or adults. Examine these data and find a violation of basic assumptions. In science apparent violations of initial assumptions require an explanation. These explanations do not necessarily mean that the assumptions are wrong or that the experiment was totally faulty. It is in minimizing the number of ad hoc, or specific violations, of the hypothesis that science can progress and models become more reflective of reality.

Literature Cited
Brower, J. E., J. H. Zar, and C. N. von Ende. 1998. Field and laboratory methods for general ecology. 4th ed. Boston: McGraw-Hill. 273 pp.

Southwood, T. R. E. 1978. Ecological methods. Halsted Press, Chapman and Hall. London. 524 pp. Table 1. Number of darkling beetles encountered during sampling.

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