How can the population size of butterflies be measured which reside in a region of foodplants?

There is a "capture and release" strategy for measuring populations of butterflies which is used to study the sizes of many wild and natural forms of life. It is based on ideas which should be very useful in developing knowledge of mathematical models.

Day 1 - Capture as many butterflies as you can with butterfly nets within a given time. With the butterfly inside the net, use a fine tip felt marker to mark the unside of the wing butterfly with an identifying mark or number. Refer to the number coding system described later in this activity for an explanation of how to label each butterfly with a unique number. After the butterfly has been labeled, it should be released back into the area in which it was caught. This labeling process should take no more than several minutes per butterfly...the butterflies ARE NOT removed from the field in order to accomplish this task!

Day 2 - On this second day (generally the next day), like in day 1, capture as many butterflies as you can during the time period. Carefully record the number of butterflies caught on day 2, and take care to look carefully to see if they were marked with an identifying label the day before. Identify these butterflies which already were labeled as being "Recaptured".

Calculate the size of the population by comparing the total number of butterflies caught on day 2, with the number that had been re-caught from day 1.

The mathematical relationship which has been developed to assist in describing the size of a population in an area is as follows:
n1 = Number of individuals captured and marked on day 1
n2 = Number of individuals captured on day 2
M = Number of individuals captured on day 2 which were captured (and marked) on day 1
N = The estimate of the size of the total population

The ratio exists where:
n1/N  =  M/n2       or       N  =  ((n1)(n2))/M

A correcting factor has been added to the equation producing the following:
N = ((n1)(n2+1))/(M+1)

If 15 butterflies were caught on day 1...and labeled, and on day 2 - 20 butterflies were caught of which 5 had labels indicating that they had also been caught on day 1. Then a calculation can be made which estimates the size of the population of butterflies in the area. That calculation is:

n1 = 15
n2 = 20
M = 5

N = ((15)(20))/(5)
N = 300/5
N = 60

(15 butterflies caught day 1) represent 5/20th of the total population (5 re-caught out of 20 caught on day 2), thus the total population can be estimated at a size of 60 butterflies!

Modeling Population Sampling techniques with more common objects available in the classroom:

There are a number of ways to develop facility with techniques of sampling natural populations with students before the kids actually go "out into the field". These include:

1. Ping Pong balls in a box...where the kids reach into the box and pull out a ping pong ball, label it, return it to the box, mix the box up, and take another ball from the box, observe if the ball has already been marked, record the data...repeating the process in a way which models sampling a number of individuals in the field.

2. Grapes, cherries, beans, or other commonly available food objects are available in a grocery store, and can replace ping pong balls...and might be less expensive!

There are a number of assumptions about sampling which need to be recognized. These include:

• The time of day and weather conditions are the same from one day to the next...
• None of the animals have died...and none were born...
• The behavior of the animals was not changed by the process of being captured and labeled in a way that would influence their being captured again...
• You are sampling the whole area in which the population exists, none of the animals immigrated of migrated into or out of the region being sampled.
• and many more...

Strategies for Labeling Butterflies in a Population Study

Depending on how much information you would like to obtain from your population study, two marking strategies can be used.

If you simply want to get an estimate of the size of a population in an area, and do not want to complicate the issue with a complex system of labeling from which you can extract more information, it is possible to simply mark the butterfly's lower side of either wing with a dot of black ink (from an India Ink pen...very fine or fine tipped is recommended).

However Rudi Mattoni's group in Palos Verdes often uses a more complicated system from which they extract information about the aging process of the butterflies as well as estimating the size of the population. The more complex system involves giving each butterfly it's own unique number when it is labeled. Data is recorded describing not only the number given to each butterfly, but also the butterfly's general condition as apparent from it's appearance (a range of 1 through 5, with 1 being in "very good" condition...and 5 being in "very bad" (almost at the end of their life) condition.

Rudi's system uses the numbering code system of "1, 2, 4, and 7". Other numbers can be made from these four numbers, for example a "3" can be written with a 1 and a 2 label.

The under -surface of the "port-side" (under-side of the left wing as the butterfly is facing forward) wing is labeled with up to four black dots. Refer to the diagram below to see how the number values are labeled onto the wing. Numbers up to "11" can be written on the under-side of the port-side wing.

(drawing of port-side wing labeling system)

For larger number than eleven, the "ten's" column of numbers is labeled under the "starboard-side" wing. For example butterfly number 15 would be labeled with a "one" under the starboard wing, and a "one" and a "four" under the port wing.

There are also other ways to estimate the sizes of populations in nature...

• Walking through an area in which the population resides, counting the individuals which you see... There are a number of assumptions in this sort of sampling also, including:
  1. Individuals are not counted twice...
  2. Your number for "search efficiency" (your estimate of the ratio of the actual population which you are actually observing on your walk through) represents reality...
  3. Your adjustment for sex ratios is real (it may be more likely that you see males on your walk through because of their territorial behavior...yet the ratio of males to females is probably 1:1)...

• Carefully observing what is happening in a very small part of the range of the population...and extrapolating to the larger geographic area. For example, you might carefully count the number of very small butterflies which you observe on a small number of flower heads of one plant. count the number of flower heads on the plant, and the number of plants on a hillside, and estimate the number of hillsides in the geographic range of the butterfly. From this information, and multiplying the data for a small number of the flower heads, it is possible to estimate the size of the population of butterflies!