BIO 412 Principles of Ecology Phil Ganter 302 Harned Hall 963-5782
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Population Estimation

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Introduction:

A fundamental question ecologists must often answer is: "How many are there?" Plants, because they do not move, can be counted by mapping an area or the number can be estimated by quadrat or transect methods (described in the lab on spatial patterns). When estimating population size (or other population parameters, like mortality or migration) for animals with biological characteristics that differ from plants, other techniques must be used. For instance, it may be difficult to distinguish individuals when surveying the population of a soil-dwelling fungus. In such a case, counting individuals may not be as useful as determining the biomass (amount of living tissue) per cubic meter of soil and ignoring the question if it represents many individuals or a single, large individual. Animals present their own difficulties. Some animals are sessile (nonmoving) and colonial, so that individuals are once again difficult to detect even if they don't move. However, most animals are easy to distinguish as individuals but do have the ability to move, which can make counting them difficult to do. Sometimes, counts can be taken so fast that it is impossible for one animal to be counted twice (for instance, flying over a grassland and taking photos in which animals can be counted). In this case, the sample is a snapshot in which no individual is moving and one treats such data like quadrat or transect data on trees in a forest. However, often such an approach is impossible and one must recognize the ability of individuals to move and be counted more than once. Often, ecologists take advantage of this by marking individuals in one sample and taking a second sample in which there is a chance that the same (now marked) individual can be recaptured. There are many sampling designs that are variations on this basic idea and all are collectively called Mark-Recapture Population Estimation Techniques. In this lab, we will simulate an animal population and use several mark-recapture techniques to estimate population number. We will do this with peanuts or beans, as they will allow us to concentrate on the technique without being subject to field error. Any estimation technique will make assumptions about the animals being counted, and we will look at some means of testing their validity.

Just a short note on the methods of capture and marking. Capture methods are as varied as are animals, and some are listed below. Marking is often a very difficult problem, as you want to mark an individual, but you do not want to alter the individual beyond your ability to detect the mark. Specifically, you do not want the capturing and marking to affect individual mortality or fecundity, either through the trauma of marking or by making the animal more visible to predators (if prey) or to prey (if a predator). In addition, you do not want to alter the marked individuals behavior so that it is either more or less easy to capture a second time. The mark should be permanent, or at least last as long as the study. Some capture and marking techniques are listed below with the animals upon which the technique is used.

Animal

• Capture Technique
• Marking Technique

Birds

• netting, nest invasion
• leg banding, recording color patterns, feather clipping

Mammals

• live trapping, sedation
• hair clipping, collars, dyeing hair, recording color pattern, clipping nails, leg banding, subdermal radio transponders

Amphibians

• Pit traps, hand collection
• toe clipping, paint injection, recording color pattern

Insects

• Sweep Netting, Pit trapping, Aspirating, Pheromone Trapping, Light Trapping
• Fluorescent dust, Tags (glued on), Recording Wing Damage Patterns, Radioactive Tracers

Mollusks

• Hand collection
• Paint, Glue-on Tags

Mark-Recapture Models:

The mark-recapture method is perhaps the most common way of estimating population number. The basic idea is to take some animals from the population and mark them for future identification. Then they are returned to the population, where they then mix into the population (so sufficient time must elapse between mark and recapture to allow complete mixing). The simplest mark-recapture techinques assume that the remixing of the population is uniform so that, when you capture a second group from the population, there is an equal chance of capturing marked and unmarked individuals. The population size is estimated from the fraction of captures that are marked individuals.

Single-Recapture Methods:

Closed Populations: the Lincoln/Petersen Index

In a closed population, no mortality, natality, or migration takes place, so that the assumption is that the population does not change between mark and recapture times. The Lincolc-Petersen method is not really an index, rather it is a method of estimating population size, which is usually designated as N. The procedure is close to that described above and you must capture twice. The individuals taken in the first capture are all marked and released into the population (M₁). A second capture (C) is then done and the fraction of the second capture marked (M₂) is noted. The rationale for estimating population size is that the ratio of marked individuals to unmarked in the second capture should be the same ratio of total number of individuals captured the first time to total population size. For example, if you capture 1 % of the population and mark all of them, then you should expect that a second capture to contain 1% marked individuals.

This proposition is easily made into an equation. If M₁ is the number of individuals marked (the number taken in the first capture, M₂ is the number of marked individuals, C is the total second catch (including previously marked individuals), and N is the population size (the only unknown that you are estimating) then:

or

.

In an analysis of this method, Bailey (1952) found that the above formulation did not give as accurate an estimate as when he added one to the number of recaptured marked individuals and the total caught in the second capture:

.

The hat over the N means that this is an estimator of N and is read "N-hat". Now that we have the estimator, we need to know just how good the estimate is. We need a confidence interval. The idea of a confidence interval is always linked to an arbitrarily chosen probability, like 95%. We will use this number, and so we will construct a 95% confidence interval for our estimate of population size. This confidence interval should be interpreted thus: "given the data and our calculated estimate of N, in 95 out of 100 cases, the true population size is within the boundaries of the confidence interval." There is more that one way to calculate this interval, but we will do it as follows.

Calculate the standard error of N (the expected spread of multiple estimates done from this population) for our estimate as:

.

Then the 95% confidence interval is:

.

This method assumes that:

1. the marked individuals (those taken during the first capture) have had time to mix into the population so that each marked individual is "equally catchable" as any unmarked individual (we will examine this assumption later)
2. marked and unmarked animals have an equal probability of being caught. This is sometimes violated if animals learn to avoid traps or if they become "trap happy" because they know they can get a good meal there
3. all marking must be done at the same time and the population size must remain the same between captures
4. the number of marked individuals does not change between the captures (no loss of marking, no mortality of marked individuals, no migration of marked individuals).

Multiple-Recapture Methods:

Multiple capture methods can not always be done, but they improve the accuracy of the population estimate when they can be done.

Closed Populations: the Schnabel Method

This works well in closed populations such as the fish in a lake, the spiders in an old field (where the surrounding woods are hostile territory for the spiders), etc. Beans in a bucket work well too! To do this method, capture on multiple occasions, mark, and make a chart with the columns below:

In the table, the # marked is just that, the number marked at each sampling. The recapture column is also easy to understand. It is simply the number recaptured. The marks here are the same for all individuals and for all samplings. This complicates the third column, the number of marked individuals in the population. For each sampling, it should be the sum of all marked individuals from the beginning. However, if you catch 8 and two already have marks, you are really only increasing the number of marked individuals by six. This is why the F_t entry for March 18 is 71. If you simply add 6 to the previous total number of marked individuals (67 for March 16), you would be counting the two recaptures on March 18 twice (once when they were first marked and once when they were recaptured). So you should add 4 (= 6 caught - 2 already marked) for a total of 71. The next two columns are for calculating the estimate of N. The M_tF_t² column is the product of M_t and the square of F_t. The last column is the product of R_t and F_t. The estimate of N is:

The standard deviation of this value is difficult to calculate and so we will merely mention that it exists and that one can calculate a confidence interval for the Schnabel estimate as well.

This method assumes that the marked individuals have had time to mix into the population so that each marked individual is "equally catchable" as any unmarked individual (we will examine this assumption later). The population size must remain the same during the course of the sampling. Also, it is assumed that the number of marked individuals does not change between the captures (no loss of marking, no mortality, no migration). Actually, there can be mortality, but the free marked individuals must be reduce by the number that died.

Open Populations:

the Jolly Stochastic Method:

This method allows individuals to die or migrate, so the population size can fluctuate. This method requires that more than two captures are done and that the marks made on one date are different from those made at another time. Note that no organism gets more than one mark and that that mark identifies not just that it has been captured before but also identifies when it was first captured.

We shall use an example taken from the inventor of the method (Jolly, 1965). In the table below, the diagonal table represents the actual number of marked insects collected over 13 days of marking. If you look at the fourth column you can see that that night 209 insects were trapped, that 56 were carrying marks (153 had no marks and had never been captured before) and that 202 of the 209 captured were subsequently marked and released. Presumably the 7 that were not released died either during capture or marking. Of the 56 that were captured with pre-existing marks, 5 were last captured on the first night, 18 on the second night, and 33 on the third night (notice that 56 = 5 + 18 + 33

Also, in our case, s_i and n_i will usually be the same. They differ when an organism is captured but is not released (maybe marking or capturing it harmed it or killed it). This will not happen to you unless you lose a bean in the process of counting. However, the data below reflects lots of mortality due to this. The first capture, none were killed and all were released, so n_i = s_i. The second time, three died, so n_i (146) is three greater than s_i (143. The number killed is entered in the row above s_i.

We need to define some parameters to calculate the size of the population from this sort of data. The first is the proportion of the capture that was marked:

.

For example, this is 56/209 (=0.268) for the fourth capture, 53/220 (0.241) for the fifth capture. Next, we calculate the size of the marked population, M_i (notice that it is a capital letter, and is not the same thing as m_i, the total number of marked individuals in a capture):

.

In this expression, s_i is the total of released individuals (202 for the fourth capture, 214 for the fifth), z_i is the sum of all individuals marked in earlier captures that are ever caught after the capture. Thus, z_i is

z₄ = 2+2+1+0+0+0+1+0+0+8+4+6+4+2+0+2+1+1+13+5+8+0+4+1+3+3+0 = 71

Note that this sum is from the table above. The terms "2+2+1+0+0+0+1+0+0" that begin it are from the first row (time of last capture 1) from the fourth column (time of capture) to the end of the row (the cells are in yellow). Thus, these are all of the recaptures of individuals marked in the first capture (past the fourth recapture). We also need the same data for those individuals marked in the second and third captures (remember we are calculating the population size for the fourth capture date) and if you look at the last 18 terms of the summation, you will see the next two rows of recaptures there.

z₅ = 2+1+0+0+0+1+0+0+4+6+4+2+0+2+1+1+5+8+0+4+1+3+3+0+20+10

+3+2+2+1+1+2 = 89

Here, you can see that z₅ starts out identical to the z4 case, but one cell to the left. This is so because we are now looking at all of the recaptures after time 5, not time 4. Also, since it is time 5, we look at the recaptures marked at time 4 in addition to those at times 1, 2, and 3 (as it was for z₄).

R_i is the sum of times individuals marked at time i are ever caught. . BE AWARE THAT R₄ AND z₄ BOTH EQUAL 71, BUT THIS IS BY CHANCE, NOT BECAUSE THEY HAVE TO BE THE SAME.

R₄ = 30+20+10+3+2+2+1+1+2 = 71 (the cells are in light green)

R₅ = 43+34+14+11+33+0+1+3 = 129

So, M₄ = (202*71)/71 + 56 = 258 and M₅ = (214*89)/139 + 53 = 190.02. Make sure you know where the numbers come from in the table and examples above! You will not be able to complete this lab if you do not.

With these parameters, we can estimate population size, N for each capture date (N_i):

.

N₄ = 258/0.268 = 963 and N₅ = 190/0.241 = 789.

We can also estimate the probability of surviving from time i to time i+1:

Notice that this value can be greater than 1 if the population grows between i and i+1. F ₄ = 190/(258 - 56 + 202) = 0.47 and F ₅ = 386/(190 - 53 + 214) = 1.10. Finally, we can estimate the number of animals added to the population between time i and time i+1:

With this method, one can get an accurate estimation about the dynamics of a population, not just a static estimate of its size at one time.

Testing assumptions of the multiple capture models:

Equal Catchability:

There are several methods for testing the assumption that all individuals are equally catchable. We will use a simple index that requires that the population is sampled at least three times (Cormack, 1966). For this test, we need:

• n₁ = number of animals marked at time 1.
• m₁₀ = number of animals caught at time 1, marked, and never seen again.
• m₁₂ = number of animals caught at times 1 and 2.
• m₁₃ = number of animals caught at times 1 and 3.
• m₁₂₃ = number of animals caught at times 1, 2 and 3.

We want to be 95% confident that all animals are equally catchable. (I chose 95% arbitrarily and it can be anything between 0 and 1.) We will calculate a statistic that, if less than 1.96, allows us to make the statement above about equal catchability. If the value is 1.96 or above, the we can not make the statement above because some animals were more difficult to catch than others. We have to say that it is likely that not all animals are equally catchable. The statistic is calculated as:

This is long, but all of the terms are simple and no complicated summations are needed. I will not go into the details of how this is derived, but I will say that it has to do with a probability distribution called the multinomial.

An example: Suppose that you capture 25 individuals, mark all of them an release them. The next time, you recapture 8. These 8 keep their old marks and have a new mark added to them. The third capture produces 9 recaptures, 7 of them with only the first mark on them and two with both marks. Then n1 is 25, m12 is 8, m13 is 7, m123 is 2. The last term, m10 is the difference between n1 (25) and all those ever caught. This total is m12 + m13 (exclude m123, as they were counted in the first 8 caught). So, z is:

And so, z equals ~180/147 or ~1.22. This z is less than the critical value and so these organisms were equally catchable and a key assumption of the Jolly (and other methods) has been tested and found to be acceptable.

You can use your jolly data if you record the number of beans captured and add marks to those beans you have caught before. Image that you have just used blue and yellow marks (in that order) The next recapture, you get 9 recaptures with blue marks. Three also have yellow marks. Thus, you know that the three with the blue and yellow marks are the m123 types and the other six (which have blue but not yellow marks) are m13 types.

Gathering the Data:

Closed Populations &endash; Lincoln/Petersen Index:

1. We will simulate captures of animals from real populations with captures of beans from buckets. Get a bag of beans, a container, and a marker.
2. Put the beans in the container. Then, without looking, grab a handful of beans from the container. For practical reasons, a sample size between 50 and 80 is fine. If the sample is larger, then it will take too long, and ,if smaller, then the accuracy of the estimate is very weak.
3. Count, record and mark the beans. Allow the mark to dry and mark both sides of the bean so that you won't have to turn the beans over to see the marks. Make the marks small enough that at least two marks can be made on each side of one bean.
4. Place the marked beans in the container and thoroughly mix the marked beans with the unmarked beans. Recapture with a second handful.
5. Count and record the second capture and note how many are marked. If you do not get a marked bean in the second capture go to Oops! section below. IF YOU GET SOME MARKED BEANS DO NOT PUT THIS SECOND CAPTURE BACK INTO THE GENERAL POPULATION AS OF YET -- YOU WILL NEED TO MARK THEM TO BEGIN THE SCHNABEL METHOD.

   OOPS! The second time you capture, you must get at least one marked individual or this technique will not work, as the number of marked individuals is a divisor, and division by zero is undefined. If you get at least one marked individual, skip the four steps below. If you get no marked beans in the first capture:

   1. mark all of the individuals captured the second time in the same fashion as the first capture
   2. add the number of marked individuals from the first and second capture and treat them as the first capture.
   3. capture a third time and treat it as the second capture (if you have at least one marked individual
   4. if you still don't get a marked indivudual, then repeat this process from step 1 until you do.

Closed Populations - Schnabel method:

1. The data from the Lincoln/Peterson index can be used here. After the second capture, mark all beans that have not been previously marked.
2. Replace all marked beans in the container and thoroughly mix the marked beans with the unmarked beans.
3. Recapture with a third handful and repeat the procedure.
4. Recapture with a fourth handful and repeat the procedure.
5. Recapture with a fifth handful and repeat the procedure.
6.  At the end of the Schnabel sampling, count all of the beans so that we can see how accurate your estimates are as the number of beans is the real N.

Open Populations - Jolley Method:

At this point, you will do either Jolley A or one of the four experiments from Jolley B. DO NOT DO BOTH. Talk to your instructor about the two variations. It is your choice, but be sure you understand the choices before you do anything.

Jolley A - population fluctuations

1. We will sample the beans again, but we need to be able to mark them with a mark that is recognizable as a mark from a particular capture. For this we will use different colored markers and position the marks on the bean to make the marks unique at each capture. The instructor will show you how to do this.
2. If you have not counted the beans, DO SO NOW.
3. Use the same capture method as above, but place unique marks on unmarked beans each time you capture so you can know when a bean was first captured.
4. Make four captures. After each capture, count and record the number of marked individuals from each of the previous captures and mark all unmarked beans. NO BEAN SHOULD RECEIVE MORE THAN ONE JOLLY MARK. If the bean already has a mark, simply record that it was captured and return it to the population.
5. After the fourth capture, add about 33% more beans to your population (record how many beans you add).
6. Continue for another two captures.
7. Finally , remove about 2/3 of the population (count the number of beans removed). The beans removed should be randomly chosen and should contain marked and unmarked individuals.
8. Continue for another two captures.
9. This last manipulation represents a population fluctuation due to a catastrophic event. Count the actual number of beans in the population (including those you removed). Record all data.

Jolley B - Births and deaths

Following the marking procedure outlined above, but manipulate the population in the following manner. Each group will do ONLY ONE of the four procedures outlined below. Each procedure is designed to explore different kinds of pupulation changes on the results of the jolly method. We will do this by using set sample sized and manipulating the population size.

• Notice that the starting population size is set for each and that some beans are added and taken away each time.
• Notice that the sample sizes are set. Do this by taking small samples until you get to the required sample size.

The procedure should go like this

1. Sample the population
2. Count the number marked (for each of the previous markings) and the number not marked in the sample
3. Mark all of the unmarked individuals with the symbol for the current capture.
4. Record the data
5. Remove the required number of beans to simulate deaths
6. Add the required number of beans to simulate births
7. Take the next capture and repeat as many times until you have done two more captures than the number of captures listed under the capture column

Data Analysis, Presentation, and Interpretation:

First - report the number of beans that were in your population.

Closed Populations - Lincoln/Petersen Index:

• Make a table of you results
• Estimate the population size
• Calculate the 95% confidence interval for your estimate.
• Present all calculations and clearly label them so that a grader can see which number is the estimated population size

Closed Populations - Schnabel method:

• Make a table of you results
• Estimate the population size
• Present all calculations and clearly label them so that a grader can see which number is the estimated population size

Open Populations - Jolley Method:

Jolley A

• Make a table of you results
• Estimate the population size at each capture
• Calculate the number added to (or lost from) the population since the last capture, and the probability of surviving from one capture to the next.
• Present all calculations and clearly label them so that a grader can see which number is the estimated population size, difference between successive estimates, and probabilities of surviva

Jolley B

• Make a table of you results
• Estimate the population size at each capture
• Calculate the number added to (or lost from) the population since the last capture, and the probability of surviving from one capture to the next.
• Present all calculations and clearly label them so that a grader can see which number is the estimated population sizes, difference between successive estimates, and probabilities of survival

Equal Catchability:

Use your Jolly results. Test for equal catchability at each date (up to three before the last capture). Conclude about the catchability of the beans.

Thinking about the experiment:

1. Was the actual number of beans within your 95% confidence limit for the Lincoln/Peterson estimate?
2. Was the Schnabel estimate any better than the Lincoln/Peterson?
3. Was the Jolley method able to track the population changes in the first experiment (if you did Jolley A)?
4. Was the Jolley method able to track the changes in population size, new recruits, and survivorship in the second experiment (if you did Jolley B)?
5. Explain the conclusions you reached about catchability.
6. Do the "bean animal" populations and our procedures violate any assumptions of the methods used?

References:

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

Cormack, R. M. 1966. A test for equal catchability. Biometrics 22:330-342.

Last updated September 6, 2006