counting the serengeti migratory wildebeest using two-stage sampling

E. Afr. Wildl. J. , 1973, Volume 11, pages 135-149

Counting the Serengeti migratory wildebeest using two-stage sampling*

M . N 0 R T 0 N - G R I F F I T H S Serengeti Ecological Monitoring Programme, Serengeti Research Institute, P.O. Box 3134, Arusha, Tanzania

Summary A method for sample counting the Serengeti migratory wildebeest using vertical aerial photography to sub-sample randomly located transects is described. The population estimate achieved was 754 028 animals, with 95% confidence limits of _+ 8.5% of the population estimate. The application of this method to other wildlife populations is considered.

Introduction One of the objectives of the Serengeti Ecological Monitoring Programme is to collect long-term data on the responses of ungulate populations to fluctuations in the environ- ment. The Serengeti migratory wildebeest (Connochaetes taurinus albojubatus Thomas) are included in this programme. This migratory population has been described by Watson (1 967) who distinguished it from the three resident populations of wildebeest that occupy parts of the migrants’ total range. The migrants spend the rainy season (November-April) on the open grasslands of the Serengeti National Park and the Ngorongoro Conservation Unit. In the early dry season (May-July) they move west into the western corridor of the Serengeti, and in the late dry season (August-October) they move into the Serengeti’s northern extension, usually reaching the Mara Masai Game Reserve in Kenya. These basic migratory movements have been described by Watson (1967) and Bell (1971), while Kruuk (1972) gives an overall description of the area and of the main habitat types.

There is now firm evidence that the size of this migratory population has increased during the last decade, from approximately 200 000 animals in the early 1960s to the present (1972) size of 840 000 animals (Sinclair, 1973, in preparation). Previous estimates of population size have been made from total photographic counts (e.g. Talbot &Stewart 1964; Watson, 1967). Total counts of large numbers of animals over large areas have many disadvantages; they are expensive to carry out and, if a photographic method is used, it takes a very long time to count all the animals on the photographs. It is also difficult to measure the magnitude of biases and errors unless the count

* S.R.I. Publication No. 11 5 .

135

136 M . Norton-Gr@ths

I S very carefully organized, and it is often impossible to ensure that any biases and errors which might occur are kept constant from count to count. This makes it difficult to interpret total count estimates, especially when population changes are being studied.

A new method for estimating the numbers of wildebeest was needed, and a sample count approach was decided on because of the potentially great savings in cost. The sampling method had to meet the following requirements :

( i ) It must be simple and cheap to carry out. (ii) Biases and errors must be minimized and must be held constant from count

to count. (iii) The population estimate must be very precise (i.e. the standard error must be

low). Precision was considered to be more important than accuracy (defined as the actual deviation of the estimate from the true figure) because the main objective was to be able to detect population changes.

There were two main problems to overcome. The first was that the wildebeest were $0 highly mobile that significant numbers could, and on one occasion did, move out of the census area while the census was being carried out. The second was the large numbers of animals which made the counting formidable.

The sampling strategy finally decided on had two main features: (i) The wildebeest herds were sampled by random aerial transects which were so

organized that they could accommodate any movement of the animals during the census.

(ii) The wildebeest were counted from vertical aerial photographs taken systematically along the length of each transect.

This sampling strategy was developed during 1970, and was fully tested in 1971 and 1972. In 1971 a total photographic count was carried out just before the sample count so that the two methods could be compared for costs and efficiency (Sinclair, 1973, Table 4). The following discussion of methods and results refer primarily to the 1971 sample count.

Methods (i) Mappitig the wildebeest Iierd3 Although the annual range of the Serengeti Migratory wildebeest covers at least 25 000 km2, at any one time the animals are concentrated into large herds in a relatively small area. The movements of wildebeest within the Serengeti ecosystem have been studied since June 1969 by a series of monthly survey flights which cover some 35 000 km?. On these flights the density distribution of wildebeest, as well as fourteen other species, is mapped systematically on a 5 X’ 5 km grid square system. A map of the density distribution of wildebeest in February 1970 (Fig. I ) shows how concentrated they can be within the ecosystem.

It is only practical to census wildebeest during the rainy season (December-May) when they are on the Serengeti Plains (Fig. 1). At this time of year the migratory animals are clearly separated from the resident animals to the west (Kirawirra), to the north-east (Loliondo) and to the north (Mara Masai), and the open treeless plains habitat allows photographic methods to be used.

Distribution maps like that shown in Fig. 1, which are based on a systematic coverage of the whole ecosystem, clearly show that there is a real boundary to the distribution. When the wildebeest were first censused in February 1970 an area within which to sample was demarcated from a distribution map by drawing in a boundary

Counting nzigratovy wildebeest 137

D A T E F E R U A R Y - 1 9 7 0 W I L D E R E E S T E D l S T R l R U T l O h p F C C E F L I G H T N O : 9

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7 8 . . . . 1 3 . . . . 1 9 . . . . . ?I!. . . . . 2 9 . . . . . . . . . 1 . 1 . 3 2 . 3 3 . 2 3 * 2 . . . . . . . L 4 . 3 2 . 1 . .

2 z . . . . . . 2 3 . . . . . . 2 1 . . . . . . . . . . 24. . . . . . . . . . 24. . . . . . . . . . . 25. . . . . . . . . . . 2 5 . . . . . . 2 6 . . . . . . 2 b . . . . . . 2 7 . . . . . . . . . . . . . . 2 7 . 21). . . . . . . . . . . . * . 2 8 . . . . . . 2 9 . . . . . . . * . . . . . . . . . . . . . . . .

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x x * y y * z z * . . * . . * . . * . . * . . * . . + . . + . . * . . * . . * . . * . . * . . * . . * . . + . . * . . * . . * . . + , . + . . * . . + . . * . . * . . * . . * . . * . . * * . * . . * . . * . . * . . * . . . . * . . * . . * . . * . . * . . * . . * * . * . . * . . * . . * * . * . . + . . * . . * . . + . . * . . * . . * . . + . . + . . + . . *

Fig. 1. Computer generated density distribution map for wildebeest in February 1970. The outer line marks the boundary of the systematic sample, the inner line the boundary of the Serengeti National Park. Each figure represents the density within a 5x 5 km grid square. Approximate density values in animals/km2: 1=10, 2=80, 3=200, 4 ~ 5 0 0 .

138 M . Norton-Grlfitks

line around the occupied grid squares. However, in further tests in 1970 considerable problems were encountered when significant numbers of animals moved out of the area during the census. Sampling within a previously demarcated area was therefore abandoned.

Instead, a more fluid sampling strategy was used which could accommodate any movement of the animals. Since the boundary of the distribution was so clear cut, the transects were flown along until there were no more wildebeest ahead of the aircraft, and similarly they were started when the wildebeest were reached (see (iii) below).

A detailed distribution map was therefore no longer necessary. All that was required was information on: (a) the broad pattern of distribution; (b) the general direction of movement; (c) the direction of any gradient of density within the herds. This allowed much more flexibility in choosing when to carry out the census. Tn May 1971 this information was known from the total count which was carried out 8 days before the sample count and from an additional reconnaissance flight, while in June 1972 the distribution was mapped froni a 16-h flight across the Serengeti plains.

(ii) Locating the transects The sampling strategy was to fly random transects across the herds with the transects oriented so that they ran roughly parallel to any contours of density within the herds. This was achieved by drawing on a map a base-line which followed the density gradient. The transects were then located at random points along the base-line, and were flown at right angles to the base-line through the points.

Fig. 2 shows the broad distribution of the herds in May 1971. The herds were moving in a north-westerly direction, and there was a north-west/south-east density gradient. The base line follows this gradient, and the transects therefore run parallel to the density contours.

(iii) Movements of’ the herds The movements of the herds were accommodated in two ways. Firstly, the base line was made long enough so that any movement off it at either end was extremely un- likely to occur during the time elapsing between drawing in the base-line and carrying out the census. Secondly. the transects were variable in length so that they could accommodate any movement away from the base line. The transects were ‘started’ when the wildebeest were reached, and they were ‘stopped’ when the wildebeest had been passed over, i.e. when no wildebeest could be seen ahead of the aircraft or to either side of it. The beginning and end points of each transect were then marked on a 1 : 250 000 map. Errors from inaccurate mapping were minimized by starting and stopping the transects at easily recognizable landmarks e.g. roads, drainage lines, kopjes. In the 1971 census the three most easterly transects were not flown at all because it was clearly seen at the time that there were no animals in the area.

(iv) Stratifjcation Stratifying the sample in advance was abandoned because the animals moved between strata during the census (tests during 1970). A technique of stratifying after the sample had been carried out was used instead (see post-sampling stratification below). The transects were oriented parallel to the density contours in order to make this technique more effective.

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I40 M. Norton-Grifliths

( v ) Counting thr rc,iltlebeest Each transect was sub-sampled by taking vertical aerial photographs every 10 sec of flying time. Continuous strip photography was not used because of the difficulties of ensuring complete coverage (given variations in the ground-speed of the aircraft), and because of the labour involved in demarcating the areas of overlap on successive photographs. Sub-sampling made everything much more simple, and each photograph is treated as one sub-sample unit .

The sub-sampling within each transect can be systematic rather than random because the sub-sample variance need not be enumerated. It has a negligible effect on the population variance (see Results). This is discussed by Cochran (1963, Chapters 10 and 1 1 ) and by Yates (1960, Chapter 7).

(v i ) Aircraft and p/iorographjs A Cessna 182 aircraft was used with the starboard door removed so that a 35 mm Nikon camera (fitted with motor drive and 250 exposure cassettes) could be mounted on the door sill. The aircraft was flown at full cruising speed and the height above ground was controlled by reference to a radar altimeter (which was calibrated against the pressure altimeter before the census).

A three man crew was used. The pilot was responsible for navigation and height control; the recorder mapped the start and end points of each transect, and recorded at intervals unknown to the pilot the height indicated by the radar altimeter; and the photographer in the back seat was responsible for operating the camera.

The height at which the photographs were taken (see Table 1 ) had been chosen to give a large photo-area whilst still being able to distinguish clearly between wildebeest

Table 1. Heights used for photographing wildebeest using 35mm Nikon eauiument

May 1971 June 1972

Focal length of lens (mrn) 55 35 Height a.g.1.

ft I800 1141 metres 545.5 345.8

ft 785 782 metres 237.9 237

ft 1178 1173 metre3 357 355.5

Scale 1 : 9974 1 : 9933

Photo uidth

Photo length

and other species when the negatives were printed. No image motion was apparent with a shutter speed of 1 1000 sec. In the 1971 census a 55-mm micro-nikkor lens was used. This gave evcellent photographs but the maximum f. stop of 3.5 was a dis- tinct disadtantage in cloudy conditions. In the June 1972 census a 35-mm lens was used instead, but this produced noticeable distortions around the edges of the negatives (see Biases and Errors, below) which made counting more difficult.

21 cm. The wildebeest, including calves, were counted using a hand tally counter, and each animal was pinpricked when it was counted. This prevents both double counting and missing animals. Power magnifica- tion

Each negative was printed 14

8 was used when necessary.

Counting migratory wildebeest 141

Results The results presented here refer to the sample count of wildebeest in May 1971. Thirty-four transects were chosen with equal probability and without replacement along a 68-9-km base-line (Fig. 2) and each transect, with the exception of numbers 32-34, were systematically sub-sampled.

The basic approach to analysing these data is to estimate the total number of animals in the whole of each transect (the transect estimate) from the average number of animals counted on the photographs taken in each transect. These transect estimates are then used as if they are transect total counts.

The results from each transect are shown in Table 2. Subscript refers to a transect, superscript - indicates an average value, and superscript A indicates an estimate. Two measurements are made from each transect :

zt the number of photographs taken along the length of the transect y , the total number of wildebeest counted on all those photographs.

Table 2. Transect data from sample count of wildebeest in May 1971

Photos Animals Transect No. taken counted

=t Yf

Transect area

2,

Transect estimate

j f

Density per photograph

J t

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

22 24 35 43 47 64 64 62 63 53 54 49 46 41 42 41 31 34 37 40 38 42 30 34 16 14 11 15 16 14 15

Not photographed Not photographed Not photographed

4311 4933 3506 2433 3140 3434 3469 2919 2703 2287 3342 1926 2001 1947 2714 706 795

1284 1766 1968 1004 670 250 335 327

0 1

243 315 230 111

44.0 46.4 55.4 71.0 70.7 96.8 91.9 95.4 87.4 87.4 85.3 83.9 73.5 67-5 64.1 64.4 47.4 51.5 54.0 52.5 52.9 54.0 47.7 48.7 24.4 36.6 39.0 34.1 26.1 25.8 23.7 0.0 0.0 0.0

8622 9545 5545 401 8 4721 5193 4982 4491 3749 3771 5279 3691 3196 3207 4139 1109 1214 1946 2576 2586 1398 861 398 481 498

0 3

553 514 423 175

0 0 0

196.0 205.5 100.2 56.6 66.8 53.7 54.2 47.1 42.9 43.2 61.9 39.3 43.5 47.5 64.6 17.2 25.7 37.8 47.7 49.2 26.4 16.0 8.3 9.9

20.4 0.0 0.1

16.2 19.7 16.4 7.4 0 .o 0.0 0.0

142 M. Norton-Grifiths

From these are calculated:

The length of each transect is measured from the map. Dividing this length by the length of a photograph (from Table 1) gives:

Multiplying each j, by its 2, gives

whole length of the transect). The final two parameters needed are:

j I the average number of wildebeest per photograph.

2, the total number of possible photographs in each transect (transect area).

j , the transect estimate (i.e. the estimated total number of wildebeest along the

n the number of transects which were sub-sampled N the number of possible transects along the length of the base-line.

N is calculated by dividing the length of the base-line by the width of a photograph (from Table I ) . Here, N

The estimated population total (P) and its variance (Var ( Y)) are calculated using Jolly's (1969a) Method 1 for equal-sized units (Table 3). Jolly points out that although his Method 1 is strictly for use with equal-sized sampling units (in this case the transects) it can be validly used for unequal-sized units although the variance of P will be unduly large if the units vary considerably in size. It can be seen from Fig. 2 and Table 2 that the transects do vary considerably in size. However, since this sampling strategy was specifically designed to be independent of a predetermined area, neither of Jolly's alternative Methods are applicable.

288.

Table 3. Results of the May 1971 wildebeest sample count

Unstratified Stratified

Population total P 752 900 754 028 Population variance Var( El 13 400 914 253 1 068 930 619 Standard error SE( P) 115 762 32 694 95 confidence limits of ( P) z 226 894 5 6 4 081 95 "/, confidence limits

as '4 of P +30% + 8.5 %

The formulae for Jolly's Method 1, using my notation, are:

Population total: Y - N . j , .

Z h where 5, is the average of the transect estimates (j,= -). n

N( N-n) n

Population variance: Var ( P) = - . $25 , .

Population standard error: SE ( P) - 4 V a r ( E) 9536 confidence limits of Y= YL- 1.96. SE ( P)


Since each of the transect estimates must have a variance attached to it, there is a second component to the population variance caused by these transect variances. This component is given by Cochran (1963, 11.14) to be:

t N I-fit 2; -.x-.- n 2, 2,-1

where the

where

This component can be calculated from the raw data to be 192 495 769. This represents 1.4% of the variance calculated from formula (2) above, and it is small enough to be ignored in the calculations.

The sub-sampling could be systematic because this variance component caused by the transect estimates can be ignored (although Cochran, 1963, points out that this is only so when the primary sample fraction n/N is small). The systematic sub-sample is advantageous because systematic samples tend to give more accurate estimates than do random samples of the same size.

y indicate the actual counts on the individual photographs in each transect and

fit is the sub-sample fraction in each transect (i.e. z, /f ,) .

Post sampling stratijication In many wildlife censuses the area to be sampled is often divided up into strata before the sample takes place and samples are drawn independently within each stratum. The objective of this prestratification is usually to reduce the variance of the population total by delineating the strata and the sampling intensity on the basis of animal density (optimal stratification, e.g. Siniff & Skoog, 1964).

With these highly mobile wildebeest it was not possible to stratify into high and low density areas, or even into high and low density portions along the base-line, because experience showed that the wildebeest were quite capable of obliterating the effect of the stratification by moving between strata during the census.

However, once the sample has been taken, information on the density of animals in each sampled unit is available from which the sample may then be stratified. The objective here is to reduce the variance of the population total by delineating theunits in the sample into strata of similar density and therefore of low variance. The method described below for doing this follows the principles outlined by Cochran (1963) and by Yates (1960).

Table 2 shows the density of animals in each sample unit (the 7,). Not only do these densities vary considerably, but also it is apparent that neighbouring transects tend to fall into groups of similar density. The following two assumptions, based on the principles of simple random sampling, can therefore be made:

(i) since the transects were chosen with equal probability and without replacement they truly represent the full extent of variation present in all transects

(ii) since neighbouring transects fall into groups which are of similar density, these groups truly represent strata of similar density along the length of the base-line.

I44 M . Norton-Grlfiths

The sampled transects can therefore be grouped into strata on the basis of density. Formulae ( I ) and (2) are then applied to each of the strata in turn. Using subscript h

to indicate a stratum:

Population total %Z F/, . . (3)

Population variance Var (F)-=zVar( PI,) . ' (4) Two new parameters need to be known before applying these formulae to each stratum:

nh the number of sampled transects in each delineated stratum N l , the total number of transects which could have been sampled in each delineated

stratum. nit and N,, are used in place of 17 and N in formulae (1) and ( 2 ) . / l i t is the number of transects in each of the delineated strata. The only problem is deciding on the size of each N/,. Care must be taken to d o this in a way which is not completely dependent upon the sample itself. For example, if the Nh were estimated from the ratio of nl, : n, then formulae ( 3 ) and (4) above would reduce, by substitution, to formulae (1) and (2), and there would be no gain from the stratification. The best method to use is to take the stratum boundaries as being exactly half way between neighbouring transects in different strata. The Nh are then calculated by measuring each segment of the base line and dividing this by the width of a photograph.

The procedure for this post-sampling stratification was therefore as follows : ( i ) Neighbouring transects were delineated into strata on the basis of density. ( i i ) The boundaries between each stratum were marked in along the base-line and

each N/ , was calculated. ( i i i ) Formulae (1) and (2) were applied to each stratum in turn, substituting nh and

Nil for n and N, and the results summated as shown by (3) and (4). This procedure was repeated until the minimum Var ( 9 ) was achieved. The results

of this are shown in Table 3. The lowest Var (p) was obtained by using six strata made up from transects 1-2, 3-15, 16-25, 26-27, 28-31 and 32-34. The stratum boundaries therefore fell mid-way between transects 2-3, 15-1 6, 25-26, 27-28 and 31-32. The population total was hardly changed at all, while the population variance was greatly reduced.

The post-sampling stratification was made considerably more effective by the smooth gradient of density along the sampled transects (Table 2). The base-line was specifically oriented so that this should occur.

Biasrs and errors There are two main sources of bias inherent in this sampling strategy. The first is caused by the pilot not flying at the correct height. Although height above ground level was controlled by reference to a radar altimeter in both the 1971 and 1972 census, only in the latter census were records kept of the height readings. The height aimed for was 1140 ft agl. (345.5 m), and the average height flown, calculated from 581 readings from the radar altimeter, was 1141 ft (345.8 m), with a standard deviation of 45 ft (13.6 m). It is the average height flown which gives the mean length and width of a photograph. and in this census this average height of I141 ft (345.8 m) was used to calculate the dimensions of the photographs.

This shows the real value of a radar altimeter, Tt should be used not only t o control the height flown, but also to find out afterwards exactly how high the aircraft was in


fact flying. Bias due to incorrect height was therefore greatly reduced in the 1972 census. Since the same aircraft and pilot was used for both censuses the bias due to height error can be assumed constant and negligible.

The second source of bias will result from aircraft bank, for any bank will increase the dimensions of a photograph. This will result in a consistent positive bias. Short of photographing the artificial horizontal at intervals and subsequently calculating the magnitude of the bias it is best to assume that this bias will be constant from count to count. An estimate of the magnitude of the bias can be made from the formulae given by Pennycuick & Western (1972). Their data were collected from a Piper Cruiser (PA-12-130) flying at 300-400 ft agl., so their figures are not exactly comparable with a Cessna 182 flying at 1400-1 800 ft agl. By applying their formulae a bias of approximately + 1 % for ‘smooth air’ is calculated. Since the Cessna 182 is a more stable aircraft than the Piper Cruiser the bias may well be less than this.

There are two main sources of error. The first is that sub-sampling may not give a correct estimate of Y t , the average number of wildebeest per photograph in each transect. This was tested in the May 1971 census by flying one transect twice at an interval of approximately 70 min from different directions. The two estimates of J , were 37-8 (n = 34, SE = 9.9) and 37.7 (n = 33, SE = 11.0). Although more tests like this should in practice be carried out, this one result indicates that this source of error is likely to be small.

The second source of error is associated with counting the animals on the photographs. Wildebeest show up very clearly on the scale of photograph used in these censuses, and they appear as little white blobs (the silvery grey patch on their back- sides) surrounded by a dark border. They are easily distinguished from zebra (Equus burchelli Gray) which look like fat white sausages, and from Thomson’s gazelle (Gazella thomsoni Gunther) which appear as tiny white dots with no black border. Other species have different shapes to wildebeest and do not have the light patch. Wildebeest calves are also easily distinguished from gazelle. This however only applies to photographs taken in strong sunlight and inevitably some photographs appear as a meaningless blurr and have to be discarded.

The small number of photographs enabled one person, myself, to count them all. A random sample of forty and seventy photographs were recounted from the 1971 and 1972 censuses respectively and no bias was found using Jolly’s (1969b) method. For the 1971 census: l? = 0.997; SE (I?) = 0.006; t = -0.58. For the 1972 census: l? =

0.995; SE (1) = 0.008; t = -0.63. The standard errors of the B’s show that the counting errors are both small and remarkably consistent between the two censuses. The effect of this counting error will be on the variances of the transect estimates and not directly on the population variance. We have already seen that this component of the variance is small enough to ignore, so it seems safe to assume that the counting errors are having a negligible effect on the population variance.

Discussion (i) Costs Costs should be considered in comparison with alternative methods of estimation. The total count, carried out a few days before the sample count, gave essentially the same estimate of numbers with the same precision (Table 4). The sample count was very much cheaper, especially in terms of the man hours necessary for counting the animals on the photographs and for working out the results.

146 M . Norton-Grijiths

Table 4. Comparison between sample counts and total count of wildebeest, 1971-72

May 1971 May 1971 June 1972 sample count total count sample count

Population estimate Standard error 95 :,( confidence limits

754 028 720 769 841 359 32 694 28 825 78 260

&8.57; %8.0% *18% Flying costs (E.A.shs) 9301- 3 1 20/- 1800/-

Time taken to count photographs (5 h) (17 h) (10 h)

and process data 3 weeks 10 weeks 3 weeks

The only practical alternative method of sample counting these wildebeest would be to take photographs at random locations across the herds. If the 1143 photographs taken in the 1971 sub-sampling census had been spread randomly across the herds, the 95% confidence limits of the population estimate would have been approximately k 7”,. However, the costs would have been considerably higher. An order of magnitude estimate of these costs, based on the calculation of the average nearest neighbour distance between the randomly placed photographs, shows that at least 12 h flying would have been needed. This would still be cheaper than the total count, but more expensive than the sub-sampling. This is in agreement with the general rule that for fixed costs, and for a fixed amount of material to be sampled, sub-sampling will give a more precise estimate than will simple random sampling.

( i i ) Precision and accuracy The precision of an estimate is judged by its standard error, and in a sample count the standard error is made up from two components. The first is due to random sample error and the second is due to errors associated with counting. The latter may be biased errors, or they may be random errors. In the analysis of biases and errors (above) it appeared that the errors from counting were both acceptably small and unbiased. The precision of this sample estimate is therefore based entirely on sample error. By using the post-sampling stratification technique a reasonably precise estimate was achieved in 1971.

The accuracy of an estimate refers to the actual deviation of the estimate from the true value. It is usually difficult to determine accuracy apart from demonstrating a lack of bias but in this case we can determine the accuracy of the sample estimate by comparing it with the estimate achieved by the total count. These two estimates are very close, and they were obtained independently using two totally different techniques each with very different sources of bias and error (Table 4). The close agreement between the two estimates indicates a high degree of accuracy.

( i i i ) Measuring population changes Population changes can be measured either by analysing the trend between a number of population estimates, or by measuring the difference between two estimates separated in time. Whichever method is used it is essential that the biases and errors present in the estimates remain the same, otherwise successive estimates will not be comparable.

This sampling strategy was designed to enable population changes to be measured, and in June 1972 a second census of the wildebeest was made to test the repeatability,


of the method. Abnormal rainfall in May made the wildebeest disperse off the plains, but they returned for one week in mid June. It was decided to carry out the census even though the distribution of the animals was far from favourable (the animals were widely dispersed and there was no smooth density gradient across the herds). The precision of the estimate was very disappointing; 841 359 animals with 95% confidence limits of k 18%.

Even though the magnitude of the difference between the estimates made in 1971 and 1972 was the same as that predicted from independent studies of recruitment and mortality, the two estimates are not statistically different. This underlines the problems of measuring changes, for obviously no change can be detected if it is less than the confidence limits of successive counts. Although it would be ideal to be able to detect annual changes it appears that the level of precision required is unattainable.

However, in 1971 two independent and very precise estimates were made, and these can be combined to give a population estimate of great precision and accuracy against which both past and future changes may be measured. The two estimates can be averaged by weighting them inversely to their variance (G. M. Jolly, personal communication). From the total count we have: PI = 720769; Var (pl) =

830 880625; w1 = l/Var (fJ. From the sample count we have 2', = 754028; Var ( P2) = 1 068 897 636 and w2 = 1 /Var ( F2), then

* w1. Yl + w2. F2 combined estimate P= w1 +w2

with a variance Var ( f') = 1 /( 1 /Var ( fl) + 1 /Var ( Y2)) This gives a weighted average estimate of 735 307 with 95% confidence limits of f 6%.

It was demonstrated above that the biases and errors were constant between 1971 and 1972 censuses, and there should be little difficulty in holding them constant in future censuses. The main problem will lie with counting errors for the analysis of these errors was based on recounts by the same person (myself) rather than on a number of different counters. If different people ever use this sampling strategy on the Serengeti wildebeest it will be essential to carry out recounts against my counts to establish correspondence. Another important point with these wildebeest is to ensure that the censuses are carried out at the same time of year so that the proportion of calves in the population will be roughly the same. Censuses carried out at different times of year can always be corrected so long as completely independent estimates of the proportion of calves are made at the same time.

(iv) Other applications This sampling strategy has been used successfully on two other occasions in 1971. The first was when estimating the number of wildebeest in a study area of 61 km2 in the western corridor of the Serengeti National Park (P. Duncan, personal communication). The data were analysed in the way described except that Jolly's (1969a) formulae for unequal-sized units were applied (Jolly's Method 2). Twenty transects were flown giving an estimate of 99 258 animals, with 95% confidence limits of & 7%. A visual transect count was carried out from the aircraft at the same time. This produced an estimate of 62 080 wildebeest (95% limits of f 14%), which clearly demonstrates the advantages of a photographic method when large numbers of animals are involved. The second successful application was by Mr R. M. Bradley of the S.R.I. (personal

I48 M . Norton-Grifitlis

communication, and thesis in preparation) to estimate the number of Thomson’s gazelle.

Although this sampling strategy was developed for estimating large numbers of animals, there are two features which have general application. The first is the variable length transect, which is useful under certain circumstances ; for example, when accurate maps are not available, or when the boundary of a census area can not be well defined-or not defined at all. In the present case when sub-sampling was used the length of each transect must be known. If however each transect is total counted then the length need not be known and all numbers are expressed in terms of sample units. Dr R. Bell (personal communication) has used this to count lechwe (Kobus leche Gray) in Zambia, in a situation where the animals are distributed in bands along a river. The transects were run at right angles to the river at random points along it. Counting started when the lechwe were reached, and stopped when the lechwe were passed. Each transect was thus counted over its entire length so the only parameter needed was N . This was found by dividing the length of the river line by the width of a transect.

The only drawback to this is that the data have to be analysed by the equal-sized unit method (Jolly’s Method I ) , and this may lead to a very high variance if the units differ widely in size. Jolly’s Method 2 for unequal sized units may only be used if a completely independent way is found of estimating the area under survey (2 in Jolly’s terminology). I t is not permissible to use the mean area of the transects to estimate the total area Z because, by substitution, the Method 1 and Method 2 population estimates are then equivalent. This means that the population variance calculated by Method 2 is not applicable to the population estimate.

The second useful feature is the post-sampling stratification. This gives great flexibility when designing a sample count, especially in cases like the present one where it was not possible to stratify the sample in advance, and it allows reasonably precise estimates to be achieved without having to take very large samples. Even in cases when a sample has been stratified in advance it allows strata to be subdivided again should the variance of any stratum be too high.

I do not know whether the principles outlined above for post-sampling stratification apply to other sampling strategies. I f a sample was drawn with probability pro- portional to size then I think that this procedure would give highly biased results. A frequently used sampling method is to choose randomly located quadrats or transects with a census area. Whether these can be later stratified in the same sort of way I do not know, although Yates (1960) seems to suggest that they can.

Perhaps the most useful application of post-sampling stratification is when more than one species is being counted. for under these circumstances a different optimal stratification can be worked out for each species in turn. A theoretical example is given in Table 5, where it can be seen that the optimal stratification for species A is

Table 5(a). Theoretical example of post-sampling stratification of a two species transect count.

Numbers of each species counted in each transect

Transect no. I 2 3 4 5 6 7 8 9 10 NumterofspeciesA 10 11 14 11 10 15 105 104 100 116 Number of species B 4 6 8 49 54 56 52 10 1 1 12

ILet I I - 10 and R; - 50


Table 5(b). Population estimate and confidence limits (expressed as a % of the populstion estimate) for each species

Species A Species B Population total: 2480 1310 95 % confidence limits

Unstratified 56 % 50 % Optimal stratification for Species A 5 % 52 % Optimal stratification for Species B 34 % 5 %

to have two strata made up from transects 1-6 and 7-10, while for species B it is to have three strata made up from transects 1-3, 4-7 and 8-10. Sinclair (1972, in press) has made extensive use of this procedure in the analysis of a multi-species sample count in the Serengeti National Park.

Acknowledgments My thanks are due to the Trustees and Director of Tanzania National Parks, and to the Director of the Serengeti Research Institute, for permission to reside and work in the Serengeti National Park.

The Serengeti Ecological Monitoring Programme is funded by the African Wild- life Leadership Foundation. Flying costs were met from a grant from the Royal Society of London.

Dr G. Jolly of the A.R.C. Unit of Statistics, Edinburgh, Scotland assisted in the design and analysis of this sample count. His help is gratefully acknowledged. My thanks are also due to members of the research and technical staff of the S.R.I. for their help in carrying out the censuses and in criticizing the manuscript, in particular Dr H. F. Lamprey, Dr A. R. E. Sinclair, and Dr C . J. Pennycuick.

References BELL, H.V. (1971) A grazing ecosystem in the Serengeti. Scient. Amer. 224(1), 86-93. COCHRAN, W.G. (1963) Sampling Techniques. John Wiley, New York. JOLLY, G.M. (1969a) Sampling methods for aerial censuses of wildlife populations. E. Afr. agric.

JOLLY, G.M. (1969b) The treatment of errors in aerial counts of wildlife populations. E. Afr. agric.

KRUUK, H. (1972) The Spotted Hyena. Univ. Chicago Press. PENNYCUICK, C.J. & WESTERN, D. (1972) An investigation of some sources of bias in aerial transect

SINCLAIR, A.R.E. (1972) Long term monitoring of mammal populations in the Serengeti: census of

SINCLAIR, A.R.E. (1973) Population increases in buffalo and wildebeest in the Serengeti. E. Afr.

SINIFF, D.B. & SKOOG, R.O. (1964) Aerial censusing of caribou using stratified random sampling.

TALBOT, L.M. & STEWART, D.R.M. (1964) First wildlife census of the entire Serengeti-Mara Region,

WATSON, R.M. (1967) The ecology of wildebeest in the Serengeti. Ph.D. thesis, Univ. of Cambridge.

YATES, F. (1960) Sampling Methods for Censuses and Surveys. Charles Griffin, London.

For. J . 34, 4649.

For. J. 34, 50-55.

sampling of large mammal populations. E. Afr. Wildl. J . in press.

non-migratory ungulates, 1971. E. Afr. Wildl. J . in press.

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J. Wild/. Mgrnt, 28(2), 391-401.

East Africa. J. Wildl. Mgmt, 28(4), 815-827.

Unpublished.

(Manuscript received 5 November 1972)

counting the serengeti migratory wildebeest using two-stage sampling

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