Estimating Animal Population Size

Using Mark-Recapture Techniques


     Essential to many ecological investigations is a measurement of the density of the organisms being studied. Special techniques may be required to measure animal populations because the individuals move. Investigators are often limited by the amount of time and money needed to measure density. Several models have been developed which enable investigators to estimate numbers of animals without having to census an entire population.

     Sampling methods based on live-capture techniques are frequently used because these allow animals to be released unharmed. By these techniques, animals are captured, marked for identification (banding, tagging, painting, etc.), and released into the population. Because one or more subsequent samples are required before estimates of population density can be determined, the techniques are referred to as capture-recapture or mark-recapture techniques.

Lincoln-Peterson Index

     The simplest of the mark-recapture methods is the Lincoln-Peterson index (Lincoln, 1930). A sample of the population is taken, the animals are marked, released, and a second sample is taken. After their initial release a proportion of the population will be marked. The proportion of marked animals in the second sample should reflect the proportion of animals marked in the entire population. For example, suppose that 10 animals were marked and released in the first sample. If 50% of the animals in the second sample are marked, then you can assume that 50% of the animals in the population are marked. The 10 animals in the first sample therefore represent approximately 50% of the population, so the population = 20. 

     This can be written as follows:

let         N = number of animals in the population. This is to be estimated.

             m = number of animals marked and released

          then      m/N = proportion of marked animals in the population.

If you take a second sample, the proportion of marked animals in the sample should be approximately m/N.

let         r = # of recaptured animals (second sample)

            n = total # individuals in the second sample

then      r/n = proportion of marked animals in the second sample.

If the sampling is unbiased, r/n = m/N. Rearranging the above equation yields:  

N = 




          Equation #1


Several assumptions must be met for this technique to be accurate:

1.  Sampling must be random. Every individual must have an equal probability of capture.

2.  The marked animals must distribute themselves randomly within the population so that the second sample will accurately reflect the population.

3.   This density estimate is based on the ratio of marked to unmarked animals. This ratio must not change. Some factors which could change this ratio and thus increase the amount of error in the estimate are listed below.

a.   In order to estimate the initial population size, there must be no new animals by birth or immigration before the second sample is taken. New animals will change the ratio of marked to unmarked animals (m/N). A second sample will reflect the newer (larger) population, not the original population. The population estimate will be valid for the new population but not for the population which was sampled before births and immigration.

b.  Marked animals must not lose their mark. These animals will appear as unmarked in the second sample.

c.   If the second sample is not taken right away, animals may die or emigrate before the second sample is taken. Mortality and emigration will not affect the density estimate if the proportion of marked and unmarked animals does not change. This would happen when mortality and emigration affect marked and unmarked animals equally. The density estimate provided will be valid for the original population from which the first sample was taken, not the new (smaller) population.


 The accuracy of this estimate can be measured by calculating the standard deviation (S) as follows:

   Equation #2

A larger standard deviation indicates a less accurate estimate.

The standard deviation enables us to calculate a range. For example, if our estimate of population size was 700, we might calculate a range to be plus or minus 100 (or 600 to 800 individuals). We do not know exactly what the population size is but we are fairly confident that it is between 600 and 800. 

This range is called a confidence interval. The standard deviation (equation #2) allows us to calculate a confidence interval. Scientists frequently use a 95% confidence interval. In the above example, if we concluded that the true population size is between 600 and 800 but we would be correct 95% of the time. 5% of the time, it would be outside this range.

The 95% confidence interval of the estimate is calculated using these equations:

N + 1.96S

N - 1.96S

Example Calculation

If our estimate was 85 animals and S was 13 then the 95% confidence interval is 85 plus or minus 1.96 X 13.

85 + 1.96(13) = 110.48

85 - 1.96(13) = 59.52

The confidence interval ranges from 59.52 to 110.48.

Laboratory Exercise

Materials Needed

Dried beans will be used to represent animals. These can be purchased at a grocery store. You will need 1000 white beans (approximately 1 pound) and 500 black (or brown) beans (approximately 1/2 pound). The black beans should be approximately the same size as the white beans. If beans are unavailable, paper squares can be used instead of beans. Click here to print pages with squares. The squares must be cut out using scissors.

A beaker or some other container will be needed to hold the population and for mixing.

Click here to print a blank answer sheet for this exercise. Online students should record the answers to the questions below on the answer sheet but will return to the course to submit the answers. Campus students will turn in the answer sheet.

Lincoln-Peterson Technique

Beans or paper squares will be used to represent a population of animals. Each bean or square will represent one individual. We will know the size of the population (1000 animals), so we can explore the accuracy of the estimation technique.

Experiment 1 - Small Sample Size (5%)

1.   Count 1000 white beans (or squares) and add them to a beaker or other container. They represent your population of animals. 

2.   Remove 5% of the "animals" from the population (50 beans or squares). Replace each white bean with a dark bean. Each dark bean represents a marked animal. If you are using paper squares, mark both sides of the squares with an X.

3.   Return the marked animals to the population and mix them thoroughly. If you are using paper squares, you should take extra care to mix the population thoroughly.

4.   Remove a second sample of 5% of the animals (50 animals).

5.   Count the number of marked individuals (dark beans) in the second sample. You must have at least one recapture each time that you conduct the sampling. If you do not have any recaptures in your second sample, repeat the sample. Enter this information in the data table. Round your calculations to the nearest whole number. For example, 1237.51 should be rounded to 1238 and 495.4 should be rounded to 495.

Experiment 2 - Effect of increased Sample Size (25%)

In this part of the experiment, you will increase the number of marked animals to 250 so that you can see the effect of an increased sample size on your population estimate. 

6.   Replace the sample from step 4 above so that the population contains 50 marked animals and 950 unmarked animals. Mark an additional 200 unmarked animals by removing 200 white beans and replacing them with dark beans. Your population now has 750 unmarked animals and 250 marked animals. 

7.   Mix the beans thoroughly.

8.   Remove a sample of 250 individuals.

9.   Record the number of individuals in the second sample that are marked (dark beans).

Experiment 3 - Effect of Natality and Immigration

10. Replace the sample from step 8 above so that the population contains 250 marked animals and 750 unmarked animals.

11. Add 500 new white beans to the population to represent births or immigration. Mix the beans thoroughly.

12. Take a sample of 250 and record the number of individuals that are marked.