Mark Recapture Technique Assumptions And Critical Thinking
BIO 412 Principles of Ecology Phil Ganter 302 Harned Hall 963-5782 | |
Australian Sealions basking on shore |
Population Estimation
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Introduction:
A fundamentalquestion ecologists must often answer is: "How many arethere?" Plants, because they do not move, can be counted bymapping an area or the number can be estimated by quadrat ortransect methods (described in the lab on spatial patterns). Whenestimating population size (or other population parameters, likemortality or migration) for animals with biologicalcharacteristics that differ from plants, other techniques must beused. For instance, it may be difficult to distinguishindividuals when surveying the population of a soil-dwellingfungus. In such a case, counting individuals may not be as usefulas determining the biomass (amount of living tissue) per cubicmeter of soil and ignoring the question if it represents manyindividuals or a single, large individual. Animals present theirown difficulties. Some animals are sessile (nonmoving) andcolonial, so that individuals are once again difficult to detecteven if they don't move. However, most animals are easy todistinguish as individuals but do have the ability to move, whichcan make counting them difficult to do. Sometimes, counts can betaken so fast that it is impossible for one animal to be countedtwice (for instance, flying over a grassland and taking photos inwhich animals can be counted). In this case, the sample is asnapshot in which no individual is moving and one treats suchdata like quadrat or transect data on trees in a forest. However,often such an approach is impossible and one must recognize theability of individuals to move and be counted more than once.Often, ecologists take advantage of this by marking individualsin one sample and taking a second sample in which there is achance that the same (now marked) individual can be recaptured.There are many sampling designs that are variations on this basicidea and all are collectively called Mark-RecapturePopulation Estimation Techniques. In this lab, we willsimulate an animal population and use several mark-recapturetechniques to estimate population number. We will do this withpeanuts or beans, as they will allow us to concentrate on thetechnique without being subject to field error. Any estimationtechnique will make assumptions about the animals being counted,and we will look at some means of testing their validity.
Just a short noteon the methods of capture and marking. Capture methods are asvaried as are animals, and some are listed below. Marking isoften a very difficult problem, as you want to mark anindividual, but you do not want to alter the individual beyondyour ability to detect the mark. Specifically, you do not wantthe capturing and marking to affect individual mortality orfecundity, either through the trauma of marking or by making theanimal more visible to predators (if prey) or to prey (if apredator). In addition, you do not want to alter the markedindividuals behavior so that it is either more or less easy tocapture a second time. The mark should be permanent, or at leastlast as long as the study. Some capture and marking techniquesare listed below with the animals upon which the technique isused.
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-recapturemethod is perhaps the most common way of estimating populationnumber. The basic idea is to take some animals from thepopulation and mark them for future identification. Then they arereturned to the population, where they then mix into thepopulation (so sufficient time must elapse between mark andrecapture to allow complete mixing). The simplest mark-recapturetechinques assume that the remixing of the population is uniformso that, when you capture a second group from the population,there is an equal chance of capturing marked and unmarkedindividuals. The population size is estimated from the fractionof captures that are marked individuals.
Single-RecaptureMethods:
ClosedPopulations: the Lincoln/Petersen Index
In a closedpopulation, no mortality, natality, or migration takes place, sothat the assumption is that the population does not changebetween mark and recapture times. The Lincolc-Petersen method isnot really an index, rather it is a method of estimatingpopulation size, which is usually designated as N.The procedure is close to that described above and you mustcapture twice. The individuals taken in the first capture are allmarked and released into the population (M_{1}).A second capture (C) is then done and the fraction of thesecond capture marked (M_{2}) is noted. Therationale for estimating population size is that the ratio ofmarked individuals to unmarked in the second capture should bethe same ratio of total number of individuals captured the firsttime to total population size. For example, if you capture 1 % ofthe population and mark all of them, then you should expect thata second capture to contain 1% marked individuals.
This proposition iseasily made into an equation. If M_{1} isthe number of individuals marked (the number taken in the firstcapture, M_{2} is the number of markedindividuals, C is the total second catch (includingpreviously marked individuals), and N is the populationsize (the only unknown that you are estimating) then:
or
.
In an analysis ofthis method, Bailey (1952) found that the above formulation didnot give as accurate an estimate as when he added one to thenumber of recaptured marked individuals and the total caught inthe second capture:
.
The hat over the Nmeans that this is an estimator of N and is read"N-hat". Now that we have the estimator, we need toknow just how good the estimate is. We need a confidenceinterval. The idea of a confidence interval is always linked toan arbitrarily chosen probability, like 95%. We will use thisnumber, and so we will construct a 95% confidence interval forour estimate of population size. This confidence interval shouldbe interpreted thus: "given the data and our calculatedestimate of N, in 95 out of 100 cases, the true population sizeis within the boundaries of the confidence interval." Thereis more that one way to calculate this interval, but we will doit as follows.
Calculate thestandard error of N (the expected spread of multiple estimatesdone from this population) for our estimate as:
.
Then the 95%confidence interval is:
.
This method assumesthat:
- 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)
- 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
- all marking must be done at the same time and the population size must remain the same between captures
- 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-RecaptureMethods:
Multiple capturemethods can not always be done, but they improve the accuracy ofthe population estimate when they can be done.
ClosedPopulations: the Schnabel Method
This works well inclosed populations such as the fish in a lake, the spiders in anold field (where the surrounding woods are hostile territory forthe spiders), etc. Beans in a bucket work well too! To do thismethod, capture on multiple occasions, mark, and make a chartwith the columns below:
Capture | # Captured | Recaptures | Free Marks | ||
Date | Mt | Rt | Ft | MtFt^2 | Rt*Ft |
4-Mar | 10 | 0 | 0 | 0 | 0 |
6-Mar | 27 | 0 | 10 | 2700 | 0 |
8-Mar | 17 | 0 | 37 | 23273 | 0 |
10-Mar | 7 | 0 | 54 | 20412 | 0 |
12-Mar | 1 | 0 | 61 | 3721 | 0 |
14-Mar | 5 | 0 | 62 | 19220 | 0 |
16-Mar | 6 | 2 | 67 | 26934 | 134 |
18-Mar | 15 | 1 | 71 | 75615 | 71 |
20-Mar | 9 | 5 | 85 | 65025 | 425 |
22-Mar | 18 | 5 | 89 | 142578 | 445 |
24-Mar | 16 | 4 | 102 | 166464 | 408 |
26-Mar | 5 | 2 | 114 | 64980 | 228 |
28-Mar | 7 | 2 | 117 | 95823 | 234 |
30-Mar | 19 | 3 | 122 | 282796 | 366 |
sum 1 | sum 2 | ||||
989541 | 2311 | ||||
N= | sum1/sum2= | 428 |
In the table, the #marked is just that, the number marked at each sampling. Therecapture column is also easy to understand. It is simply thenumber recaptured. The marks here are the same for allindividuals and for all samplings. This complicates the thirdcolumn, the number of marked individuals in the population. Foreach sampling, it should be the sum of all marked individualsfrom the beginning. However, if you catch 8 and two already havemarks, you are really only increasing the number of markedindividuals by six. This is why the F_{t} entry for March18 is 71. If you simply add 6 to the previous total number ofmarked individuals (67 for March 16), you would be counting thetwo recaptures on March 18 twice (once when they were firstmarked and once when they were recaptured). So you should add 4(= 6 caught - 2 already marked) for a total of 71. The next twocolumns are for calculating the estimate of N. The M_{t}F_{t}^{2}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 standarddeviation of this value is difficult to calculate and so we willmerely mention that it exists and that one can calculate aconfidence interval for the Schnabel estimate as well.
This method assumesthat the marked individuals have had time to mix into thepopulation so that each marked individual is "equallycatchable" as any unmarked individual (we will examine thisassumption later). The population size must remain the sameduring the course of the sampling. Also, it is assumed that thenumber of marked individuals does not change between the captures(no loss of marking, no mortality, no migration). Actually, therecan be mortality, but the free marked individuals must be reduceby the number that died.
Open Populations:
the JollyStochastic Method:
This method allowsindividuals to die or migrate, so the population size canfluctuate. This method requires that more than two captures aredone and that the marks made on one date are different from thosemade at another time. Note that no organism gets more than onemark and that that mark identifies not just that it has beencaptured before but also identifies when it was first captured.
We shall use anexample taken from the inventor of the method (Jolly, 1965). Inthe table below, the diagonal table represents the actual numberof marked insects collected over 13 days of marking. If you lookat the fourth column you can see that that night 209 insects weretrapped, that 56 were carrying marks (153 had no marks and hadnever been captured before) and that 202 of the 209 captured weresubsequently marked and released. Presumably the 7 that were notreleased died either during capture or marking. Of the 56 thatwere captured with pre-existing marks, 5 were last captured onthe first night, 18 on the second night, and 33 on the thirdnight (notice that 56 = 5 + 18 + 33
Also, in our case,s_{i} and n_{i} will usually be the same. Theydiffer when an organism is captured but is not released (maybemarking or capturing it harmed it or killed it). This will nothappen 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 killedis entered in the row above s_{i}.
Time of capture | |||||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
Total Caught ni | 54 | 146 | 169 | 209 | 220 | 209 | 250 | 176 | 172 | 127 | 123 | 120 | 142 |
Total Marked mi | 0 | 10 | 37 | 56 | 53 | 77 | 112 | 86 | 110 | 114 | 77 | 72 | 95 |
Total Unmarked | 54 | 136 | 132 | 153 | 167 | 132 | 138 | 90 | 62 | 13 | 46 | 48 | 47 |
Number that died | 0 | 3 | 5 | 7 | 6 | 2 | 7 | 1 | 3 | 1 | 3 | 0 | 0 |
Total Released si | 54 | 143 | 164 | 202 | 214 | 207 | 243 | 175 | 169 | 126 | 120 | 120 | 142 |
Time of | |||||||||||||
Last Capture | |||||||||||||
1 | 10 | 3 | 5 | 2 | 2 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | |
2 | 34 | 18 | 8 | 4 | 6 | 4 | 2 | 0 | 2 | 1 | 1 | ||
3 | 33 | 13 | 8 | 5 | 0 | 4 | 1 | 3 | 3 | 0 | |||
4 | 30 | 20 | 10 | 3 | 2 | 2 | 1 | 1 | 2 | ||||
5 | 43 | 34 | 14 | 11 | 33 | 0 | 1 | 3 | |||||
6 | 56 | 19 | 12 | 5 | 4 | 2 | 3 | ||||||
7 | 46 | 28 | 17 | 8 | 7 | 2 | |||||||
8 | 51 | 22 | 12 | 4 | 10 | ||||||||
9 | 34 | 16 | 11 | 9 | |||||||||
10 | 30 | 16 | 12 | ||||||||||
11 | 26 | 18 | |||||||||||
12 | 35 |
We need to definesome parameters to calculate the size of the population from thissort of data. The first is the proportion of the capture that wasmarked:
.
For example, thisis 56/209 (=0.268) for the fourth capture, 53/220 (0.241) for thefifth capture. Next, we calculate the size of the markedpopulation, M_{i} (notice that it is a capital letter,and is not the same thing as m_{i}, the total number ofmarked individuals in a capture):
.
In this expression,s_{i} is the total of released individuals (202 for thefourth capture, 214 for the fifth), z_{i} is the sum ofall individuals marked in earlier captures that are ever caughtafter the capture. Thus, z_{i }is
Here, you can see that z_{5} 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 z_{4}).
R_{i} isthe sum of times individuals marked at time i are ever caught. .BE AWARE THAT R_{4} AND z_{4} BOTH EQUAL 71, BUTTHIS IS BY CHANCE, NOT BECAUSE THEY HAVE TO BE THE SAME.
So, M_{4} =(202*71)/71 + 56 = 258 and M_{5} = (214*89)/139 + 53 =190.02. Make sure you know where the numbers come from in thetable and examples above! You will not be able to complete thislab if you do not.
With theseparameters, we can estimate population size, N for each capturedate (N_{i}):
.
N_{4} =258/0.268 = 963 and N_{5} = 190/0.241 = 789.
We can alsoestimate the probability of surviving from time i to time i+1:
Notice that thisvalue can be greater than 1 if the population grows between i andi+1. F _{4} = 190/(258 - 56 + 202) = 0.47 and F _{5}= 386/(190 - 53 + 214) = 1.10. Finally, we can estimate thenumber of animals added to the population between time i and timei+1:
With this method,one can get an accurate estimation about the dynamics of apopulation, not just a static estimate of its size at one time.
Testingassumptions of the multiple capture models:
EqualCatchability:
There are severalmethods for testing the assumption that all individuals areequally catchable. We will use a simple index that requires thatthe population is sampled at least three times (Cormack, 1966).For this test, we need:
- n_{1} = number of animals marked at time 1.
- m_{10} = number of animals caught at time 1, marked, and never seen again.
- m_{12} = number of animals caught at times 1 and 2.
- m_{13} = number of animals caught at times 1 and 3.
- m_{123} = 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 willcalculate a statistic that, if less than 1.96, allows us to makethe statement above about equal catchability. If the value is1.96 or above, the we can not make the statement above becausesome animals were more difficult to catch than others. We have tosay that it is likely that not all animals are equally catchable.The statistic is calculated as:
This is long, butall of the terms are simple and no complicated summations areneeded. I will not go into the details of how this is derived,but I will say that it has to do with a probability distributioncalled the multinomial.
An example: Supposethat you capture 25 individuals, mark all of them an releasethem. The next time, you recapture 8. These 8 keep their oldmarks and have a new mark added to them. The third captureproduces 9 recaptures, 7 of them with only the first mark on themand two with both marks. Then n1 is 25, m12 is 8, m13 is 7, m123is 2. The last term, m10 is the difference between n1 (25) andall those ever caught. This total is m12 + m13 (exclude m123, asthey 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 yourjolly data if you record the number of beans captured and addmarks to those beans you have caught before. Image that you havejust used blue and yellow marks (in that order) The nextrecapture, you get 9 recaptures with blue marks. Three also haveyellow marks. Thus, you know that the three with the blue andyellow marks are the m123 types and the other six (which haveblue but not yellow marks) are m13 types.
Gathering the Data:
ClosedPopulations &endash; Lincoln/Petersen Index:
- 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.
- 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.
- 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.
- Place the marked beans in the container and thoroughly mix the marked beans with the unmarked beans. Recapture with a second handful.
- 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:
- mark all of the individuals captured the second time in the same fashion as the first capture
- add the number of marked individuals from the first and second capture and treat them as the first capture.
- capture a third time and treat it as the second capture (if you have at least one marked individual
- if you still don't get a marked indivudual, then repeat this process from step 1 until you do.
ClosedPopulations - Schnabel method:
- The data from the Lincoln/Peterson index can be used here. After the second capture, mark all beans that have not been previously marked.
- Replace all marked beans in the container and thoroughly mix the marked beans with the unmarked beans.
- Recapture with a third handful and repeat the procedure.
- Recapture with a fourth handful and repeat the procedure.
- Recapture with a fifth handful and repeat the procedure.
- 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 thispoint, you will do either Jolley A or one of the four experimentsfrom Jolley B. DO NOT DO BOTH. Talk to your instructor about thetwo variations. It is your choice, but be sure you understand thechoices before you do anything.
Jolley A -population fluctuations
- 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.
- If you have not counted the beans, DO SO NOW.
- 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.
- 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.
- After the fourth capture, add about 33% more beans to your population (record how many beans you add).
- Continue for another two captures.
- 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.
- Continue for another two captures.
- 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 themarking procedure outlined above, but manipulate the populationin the following manner. Each group will do ONLY ONE ofthe four procedures outlined below. Each procedure is designed toexplore different kinds of pupulation changes on the results ofthe jolly method. We will do this by using set sample sized andmanipulating 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 procedureshould go like this
- Sample the population
- Count the number marked (for each of the previous markings) and the number not marked in the sample
- Mark all of the unmarked individuals with the symbol for the current capture.
- Record the data
- Remove the required number of beans to simulate deaths
- Add the required number of beans to simulate births
- 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
Experiment Number | Starting Population Size | Capture | Sample Size | Removals (Deaths) | Add Unmarked (Births) |
1 | 400 | 1 | 60 | 40 | 40 |
Stationary | 2 | 60 | 40 | 40 | |
Population | 3 | 60 | 40 | 40 | |
4 | 60 | 40 | 40 | ||
5 | 60 | 40 | 40 | ||
6 | 60 |
Experiment Number | Starting Population Size | Capture | Sample Size | Removals (Deaths) | Add Unmarked (Births) |
2 | 400 | 1 | 60 | 0 | 26 |
Growing | 2 | 60 | 0 | 36 | |
Population | 3 | 60 | 0 | 52 | |
4 | 60 | 0 | 64 | ||
5 | 60 | 0 | 80 | ||
6 | 60 |
Experiment Number | Starting Population Size | Capture | Sample Size | Removals (Deaths) | Add Unmarked (Births) |
3 | 400 | 1 | 60 | 80 | 0 |
Declining | 2 | 60 | 64 | 0 | |
Population | 3 | 60 | 52 | 0 | |
4 | 60 | 36 | 0 | ||
5 | 60 | 26 | 0 | ||
6 |
FW (ZO) 353
Wildlife Management
To establish and to appraise management practices, wildlife managers must estimate the sizes of wildlife populations. For game species, such inventories are ideally taken 3 times a year: during the breeding season, after the young are born or hatched and before the start of the hunting or trapping season, and after the hunting or trapping season. In practice, population estimates are usually done only once a year, at best, because of manpower and funding shortages.
Wildlife managers use 4 general approaches to estimate population sizes of wildlife: total counts, incomplete counts, indirect counts, and mark-recapture methods. We shall examine each of these methods and detail some of their advantages and disadvantages.
COMPLETE COUNTS OR TOTAL COUNTS
A complete count, or total count, counts every member of a population. Where populations of large species occur in open areas, such as waterfowl on lakes, seals on breeding beaches, or pronghorns on shortgrass prairie, aerial counts of most individuals are possible, especially with the aid of photography. Sometimes, wildlife managers can count deer in enclosed populations using a drive approach: a large group of people crosses the enclosure in a line, counting all deer that pass in each direction. Distances between the members of the drive crew are critical for success because all deer must be counted, even those hiding. Nonetheless, wildlife managers seldom use this approach because lack of funds or personnel usually make censussing an entire population impractical or impossible and, in addition, such an undertaking disturbs, and can even destroy, the population or its habitat. Even when used, this approach is usually expensive.
INCOMPLETE COUNTS
An incomplete count involves counting part of a population and then extrapolating to the entire population. Quadrats may be established in a sample area and an attempt made to count all the individuals in each quadrat. A "deer drive" census, using large sized quadrats, can be an effective way to estimate deer populations on wooded areas. Stationary observers stand along 3 sides of a quadrat and count all deer leaving and entering the area in front of a drive crew walking across the quadrat from the 4^{th} side. The total number of animals is then calculated as the sum of the animals leaving the area ahead of the drive crews plus the animals passing back through the drive line minus the animals entering the quadrat through one of the sides or through the drive line. As with complete counts, distances between observers and between members of the drive crew are critical for success.
Strip censuses, roadside counts, flushing counts and booming or drumming ground counts are all incomplete count methods. A strip census can be used to estimate grouse population sizes. An observer walks a transect through a representative section of habitat and records the distances at which birds flush to either side. The population size, P, is estimated to be
where A is the area of the habitat censussed, Z is the total number of grouse flushed, X is the total distance walked and Y is twice the average distance from the observer to the bird when flushed. The fundamental assumptions of this method are 1) birds vary randomly in distances at which they flush, 2) birds are scattered randomly across the study area and 3) the average flushing distance is a good estimate of the "true" average. Which of these assumptions are likely to be met? What if some birds will not flush? A Wildlife Monograph has dealt extensively with these types of population size estimates (Burnham et al. 1980).
INDIRECT COUNTS
As it is often impossible to obtain accurate, visual or auditory counts of the animals in a population, wildlife managers use indirect signs of the animals present as indices of relative abundance. An index of population indicates relative size of a population and shows population trends (up, down, stable) but does not provide an actual estimate of the number of animals. Examples of indirect counts include counting numbers of muskrat houses, counting scats (fecal pellets) of deer and rabbits, and counting numbers of nests or den sites in a given area. Sometimes counting the number of birds heard singing is considered an incomplete count and sometimes it is considered an indirect count. Which makes more sense?
One can count fecal pellets of deer or rabbits along transects or in delineated study plots. In either case, the first thing to do after establishing the transects or plots is to remove all old pellets. Then, at a predetermined interval, count all new piles of fecal pellets. This is an index of the number of deer or rabbits in the area: the more animals, the more pellets produced. What assumptions does this index make?
In those areas where muskrats build houses of vegetation in marshes, the number of active, maintained houses in a marsh year to year is an index of the number of muskrats: more muskrats make more houses. If, for a given area, one knows the average number of muskrats living in each house, then the number of houses can be used to estimate the population size. It should be remembered, however, that indirect counts are only indices of population sizes unless other information is known, such as the average number of muskrats living in each house.
MARK-RECAPTURE METHODS
These methods are used extensively to estimate populations of fish, game animals, and many non-game animals. The approach was first used by Petersen (1896) to study European plaice in the Baltic Sea and later proposed by Lincoln (1930) to estimate numbers of ducks. Petersen's and Lincoln's method is often referred to as the Lincoln-Petersen Index, even though it is not an index but a method to estimate actual population sizes. (Should it not be the Petersen-Lincoln Estimate?) Their method involves capturing a number of animals, marking them, releasing them back into the population, and then determining the ratio of marked to unmarked animals in the population. The population (P) is estimated by the formula:
where M is the number of animals marked in the first trapping session, C is the number of animals captured in a second trapping session, and R is the number of marked animals recaptured in the second trapping session. This is derived from the equation:
To express your confidence in this estimate, you calculate the 95% confidence limits for your estimate. The upper and lower 95% confidence limits are
upper: 59
lower: 5.5
This means that if you trap muskrats in this way many, many times, 95% of the time that you obtained an estimate of 20 muskrats, the true population size would be somewhere between 6 and 59 animals. Since you actually captured 14 muskrats, you know that the population size is at least 14.
Otis et al. (1978) developed sophisticated modifications of the L-P Estimator that attempt to insure that data are consistent with the assumptions. Several modifications construct stratified indices whereby data are collected separately for specific sub-groups of the population, such as age and sex categories or trap-happy and trap-shy animals. Thus, researchers must uniquely mark each individual captured and record information about that individual, such as sex and age. These modifications also insure an order of magnitude increase in the complexity of the mathematics and are available in computer software, such as Capture.
When wildlife managers or researchers establish long-term population studies with frequent samplings, they can estimate not just the population size but the numbers of animals entering and leaving the population (Jolly 1963, 1965; Seber 1973). The Jolly-Seber Method relaxes the assumption that a population is closed. That is, the population can be open and have ingress (births and immigration) and egress (deaths and emigration). By keeping track of capture histories for individual over many capture sessions, ingress and egress can be estimated. Jolly-Seber Estimates can be calculated by hand but the exercise is complicated. Several software packages provide Jolly-Sever Estimates. The Wildlife Management Techniques manual shows how to make Jolly-Seber Estimates.
Pollock and his colleagues (Kendall & Pollock 1992, Nichols, et al. 1984, Pollock1991, Pollock & Otto. 1983) developed the Robust Design for estimating animal populations, which incorporates capture-recapture methods for both closed and open populations. In its simplest form, the Robust Design uses an L-P Estimate for total population size during each of several, regularly scheduled trapping sessions and uses of the Jolly-Seber approach to estimate ingress and egress between trapping sessions.
Krebs (1966) formally introduced the Minimum Number Alive (MNA) method, though it had been used by many researchers for years. The MNA method avoids the use of estimators, using instead the minimum number of animals known to be alive during a sampling period as a biased estimator of the population size. Hilborn et al. (1976) tested the sensitivity of this method to five important population parameters in mouse populations. They used simulation models and actual data to estimate the expected error on the MNA in actual studies. Their results showed that the MNA method, though clearly a biased estimator of population size, is an unbiased estimator of critically important population characteristics such as age distribution, pregnancy rate and lactation rate. In addition, in most cases an MNA population estimate is as good as or better than a Jolly-Seber Estimate.
The Frequency of Capture Method (Eberhardt 1969) can be used when capture data are available over several trapping days. Plot the number of times that individuals are captured against the numbers of animals captured each number of times. For example, imagine that you live trap gray squirrels on campus over the course of 2 weeks and trap 15 squirrels only once, trap 10 twice, 5 3-times, 3 4-times, 3 5-times, 2 6-times, and 1 7-times. You plot these captures like so:
These data are then fit to a statistical distribution to determine how many squirrels were never trapped at all even though they were present. In this case, if we assume that squirrels were captured at random, the estimate of the number never trapped is between 16 and 17. The number never trapped at all but present is added to the MNA to give the Frequency of Capture estimate. In this example, the population size estimate is 55.
Lastly the DeLury Method, first worked out for fish populations, uses kill data to estimate game populations. The critical assumption is that the number of animals killed per unit of hunting time is proportional to the population density; if this assumption is true, then each unit of hunting effort takes a constant proportion of the population. By plotting the kill rate (number of animals killed per unit hunting effort) against the total kill, it is possible to estimate the total population by extending the line to the X-axis. The value at the point of intersection is the estimate of the original population, P_{o}. The validity of this method rests heavily on the assumption of each unit of hunting effort taking a constant proportion of the population. This DeLury Method also assumes that: 1) the population is closed; 2) animal vulnerability remains constant; 3) variable hunter skills average out; and 4) hunting is done individually. Of these assumptions, the one most likely to be violated is constant vulnerability. This can be affected by factors both intrinsic and extrinsic to the hunted population.
Example of the DeLury Method: Imagine that you are a wildlife biologists monitoring the game populations on a designated Wildlife Management Area. Assume further that your Area allows deer hunting for 7 successive days each year and that hunters must apply for a permit to hunt on the Area. Hunters must check in before and after hunting and must report their kill. On each of the 7 successive days, hunters hunt for a total of about 400 hours each day. You record the hours hunted each day, record the number of deer killed each day, and calculate the cumulative kill, producing a table like the following.
Day | Animals Killed | Hours Hunted | Kill/Gun-Hr | Cumulative Kill |
1 | 100 | 400 | .250 | 100 |
2 | 90 | 375 | .240 | 190 |
3 | 81 | 410 | .200 | 271 |
4 | 73 | 405 | .180 | 344 |
5 | 66 | 390 | .170 | 410 |
6 | 59 | 385 | .153 | 469 |
7 | 53 | 395 | .134 | 522 |
You then graph kill/gun-hr against cumulative kill to estimate of the initial population size before hunting began. This it the graph you get.
You draw a line through your data points and extend the line to the X-axis. Your estimate of P_{o }, the estimated population size before the hunting season started, is about 1350 on the graph. You also calculate a linear regression through the data points and calculate the X-intercept. Here you find that your best estimate of P_{o} is actually 1335. Because 522 deer were killed, the population after the hunting season is estimated to be 813.
COMPARISONS
Many researchers have used more than method of estimating populations on the same population at the same time. Let us look at 3 of these comparisons.
Morgan & Bourn (1981) compared an Incomplete Count and an L-P Index of the giant tortoise population on Aldabra atoll in the Indian Ocean. To make the incomplete count, the atoll was divided into quadrats 100 m square. All tortoises were counted and marked in 5% of the quadrats and the total number counted was multiplied by 20. The L-P Index was made by counting marked tortoises on transect lines. Morgan & Bourn believed that almost all assumptions for each technique were satisfied, yet the estimates of the population size differed significantly: 87,300 for the incomplete count and 68,100 for the L-P Index. Evidently the assumptions for one or both methods were not met as well as believed. Morgan & Bourn had more confidence in their incomplete count estimate than in their L-P Estimate and cautioned readers about using elaborations on the L-P Index unless all assumptions are completely met.
Mares et al. (1981) compared the L-P Estimate, the Schnabel estimate (a variation on the L-P Estimate that tends to underestimate population sizes slightly), and a removal estimate on a population of known size of eastern chipmunks in Pennsylvania. The chipmunks, they found, fell into 2 categories: those that readily entered traps and those that were hesitant to enter traps. Thus, all methods tended to estimate the population as being composed mostly of the former group and, thus, all methods tended to underestimate the total population size. The 95% confidence limits for the L-P Estimates on successive days always included the known population size, whereas this was not the case with the Schnabel method. They concluded that, for populations with unequal catchability, the L-P Estimate was the best.
Boufard & Hein (1978) used 7 different methods concurrently for 6 months during 1976 to estimate the size of a gray squirrel population in Pennsylvania. Four of their methods have been discussed (at least briefly) in this handout: Schnabel, Frequency of Capture, Jolly, and MNA. Their results are as follows:
Month | Schnabel Estimate | Frequency of Capture Estimate | Jolly-Seber Estimate | MNA Estimate | ||||
June | 115 � 200 | 392 | 54 | 27 | ||||
July | 76 � 19 | 96 | 38 | 52 | ||||
August | 85 � 24 | 82 | 51 | 48 | ||||
September | 118 � 84 | 35 | 31 | |||||
October | 159 � 97 | 115 | 111 | 32 | ||||
November | 179 � 59 | 195 | 20 | 34 |
The Schnabel estimates were the most consistent, despite their variability, and the Frequency of Capture estimates were similar to the Schnabel. Twice (July and November), the Jolly-Seber Estimates were less than the MNA.
It is obvious from these comparisons that estimating populations is an exercise fraught with inperfections. The best we can do is to choose the method or methods whose assumptions are best met by the population we wish to study. When possible, one should always collect data in such a way that more than one population estimate can be made. Often, the estimate made using the method whose assumptions are best met turns out not to work as well as anticipated. Having other estimates to augment that "best" one can save the day.
Also note from these comparisons that Lincoln-Petersen Estimates are often reasonably accurate despite violation of their assumptions. I have noted this pattern and, therefore, tend to use L-P Estimates (or variants available through Capture software) over other methods when no other method is an obvious choice (for example, using the DeLury Method to estimate the size of a harvested population). Pollock's Robust Design is consistent with my informal observation.
Up to now, this handout may appear very straightforward. There is a problem, however. When you determine a wildlife population size, you automatically determine density. What I mean is, you determine the population size in a particular area - and that means you have determined
population
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