Opportunities to study animals and plants in their natural settings arise in school biology and in the process pupils use a number of classical ecological techniques based on sampling methods. Being classical, each technique tends to be used without it first being explored, with the result that pupils are not always aware of the underlying principles of a technique nor of the conditions necessary for its proper use. An ecological technique can be explored by using it on a model of the ecological system to be studied, and one that the author has used successfully for some years is presented here.

The capture/recapture technique (also known as the Lincoln Index) is used to arrive at estimates of the size of populations of mobile animals, like ground beetles and woodlice. An initial sample of the population in question is caught, its individuals marked and then released back into the wild, and a note taken of the number released. These marked individuals are allowed to become randomly dispersed throughout the population and then a second sample is taken. Its size and the number in it of marked, and hence recaptured, individuals is noted.

The size of the population estimated on the principle that the proportion marked in the second sample equals the proportion of marked individuals in the population as a whole, so that

where a is the number marked and released into the population, b is the size of the second catch,

c is the number recaptured in the second catch, and d is the size of the population as a whole, such that

Estimate of population size (d) = a.b/c

On this occasion a quarter pound (113 g) box of Smarties serves as a suitable model not least because the size of the population of the Smarties in the box is unknown both to pupils and to teachers for the box is new and unopened. The following has been found to be a suitable procedure:

Discussion

The count of Smarties in the box serves as an empirical check on the estimated population size and hence on the reliability of the method. Pupils are impressed by the close agreement between the estimate based on the mean of the samples and the count, and are thereby convinced of the technique and of the estimates that arise from it. They find it interesting that it is the estimate from the mean of the samples rather than the individual estimates from each of the samples that is convincing, for these vary considerably—in the results given in Table 1 from 67 to 176 (to the nearest whole number), and can explore the merit of increasing the number of samples particularly after the fourth has been taken.
 

TABLE 1. Estimation of population size using the capture/recapture technique

Number marked and released into the population (a) 16
 

Data sample number	Size of second catch	Number recaptured in second catch	Population estimate from each sample	Running combined sample estimate	Running 95% confidence interval
1	17	3	90.7	90.7	44.7 to infinity
2	32	2	170.7	130.7	74.7 to 521.8
3	27	3	144.0	135.1	83.7 to 349.5
4	28	3	149.3	138.7	90.5 to 296.4)
5	25	4	100.0	130.9	88.4 to 238.3)
6	21	5	67.2	120.3	81.8 to 189.4
7	25	4	100.0	117.4	82.2 to 175.8
8	22	2	176.0	124.7	86.4 to 179.7
9	28	4	112.0	123.3	87.5 to 172.5
10	24	5	76.8	118.7	85.0 to 158.6

 

So the same population needs to be sampled repeatedly for the estimate to be highly reliable. In ecology, however, the population is such that it is often invalid to make more than one attempt at a second sampling from a single marking, for many characteristics essential to the technique, and which are fulfilled in the physical model of the box of Smarties, do not hold in nature.

The number marked and released into the population must not change, yet in a biological population the number must be expected to decrease in time for, unlike the Smarties, marked individuals in the wild may move out of the population or die, and it is quite likely that some will lose their mark. Pupils can devise tests with the box of Smarties to find out what effect the (unknown) loss of marked individuals will have on the final estimates. They find it results in an overestimate of the size of the population, with the overestimate itself related to the proportion of the original number of marked individuals remaining in the population. Table 2 shows a set of results obtained by reducing the number of marked individuals to twelve while supposing it had remained at sixteen.
 
 

TABLE 2. Effect of lost marked individuals on estimates of population size derived from the cap ture~ recapture technique

Number marked and released into the population (a) taken as 16, but deliberately reduced to 12 before sampling.
 

Data sample number	Size of second catch	Number recaptured in second catch	Population estimate from each sample	Running combined sample estimate	Running 95% confidence interval
1	22	2	1760	176.0	75.80 to infinity
2	22	3	1173	146.7	77.1 - 805.6
3	23	2	1840	159.1	90.0 - 512.3
4	29	5	928	142.5	83.7 - 271.9
5	28	3	1493	143.9	89.7 - 251.7
6	19	2	1520	145.2	93.1 - 243.0
7	31	2	2480	159.9	102.9 - 254.6
8	23	3	1227	155.3	102.8 - 236.4
9	28	2	2240	162.9	108.8 - 241.2
10	29	2	1547	162.1	111.0 - 233.9

The number of individuals in the population as a whole must also remain constant throughout the sampling. In the wild there is always the risk of a movement into the population and of births such that the number of ‘not marked’ individuals increases. Whenever this occurs the population size is liable to be over-estimated, as pupils find out for themselves when they devise and run suitable tests on the Smarties. It follows that an overall decrease in the number 'not marked’ leads to an underestimate of population size.

These effects show when the Smarties are set aside deliberately, or when they are eaten secretly. From the outset pupils are asked not to spoil the results by eating the Smarties during the class and are warned that they are liable to be found out if they do, for this has happened. A class noticed one pupil’s figures consistently overestimated the size of the population and, on finding out who was responsible, knowingly decided she had eaten some. She promptly blushed bright red, embarrassed to find her secret made public, and admitted to eating two red ones. The pupils were entertained by the incident. They were also impressed at the detection.

Whenever the population changes in the ways described, the second catch is no longer representative of the population as a whole, and gives rise to estimates that are misleading.

The model system is carefully set up so that the marked individuals in it make up more than a tenth of the population and the population is sampled ten times. In this way one can anticipate an average estimate close enough to the count for pupils to be impressed by the technique, and they could go on to explore the effects of taking fewer samples and smaller samples. But even now pupils will have found the estimate, particularly from a single recapture attempt, is accurate only within broad limits. They have cause not to agonise over the first decimal place and the spurious accuracy arising from the arithmetic. The final figure for population size is no more, but also no less than, an estimate.

University of Keele, Staffs.

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