Medical and Veterinary Entomology (1990) 4, 167-179

Monitoring tsetse fly populations. I. The intrinsic variability of trap catches of Glossina pallidipes at Nguruman, Kenya

BRIAN WILLIAMS, ROBERT DRANSFIELD and ROBERT BRIGHTWELL International Centre for Insect Physiology and Ecology, Nairobi, Kenya

ABSTRACT. During 1986 the tsetse fly Glossina pallidipes Austen was monitored daily at Nguruman, southwestern Kenya, using three unbaited biconical traps. This was done to investigate the nature and causes of daily variation in trap catches. The variability of the observed catches was com- pared to a model which includes the trapping probability and the stochastic variation in the sex-ratio. By comparing the catches of male and female flies we are able to establish the sampling distribution of the trap catches. In addition to seasonal changes in the trap catches, day-to-day variations are observed and these are considered greater than the variation arising from the stochastic nature of the sampling process. Recommendations are made in relation to sampling tsetse fly populations.

Key words. Glossina pallidipes, tsetse, traps, sampling.

Introduction

Despite the increasingly widespread use of traps for sampling Glossina species, little attention has been paid in recent years to the variability of trap catches over space and time. Whilst such variability was quite high in fly round catches (Ford et al., 1959), several workers noted similar or increased variability when traps were used to catch flies (Glasgow & Duffy, 1961). Spatial variability within vegetation types has mainly been ascribed to various 'site factors' including trap visibility and shading (Morris & Morris, 1949). Daily fluctuations have been ascribed to climatic factors (Hargrove & Vale, 1978; Dransfield, 1984), to the nutritional state of flies (Smith & Rennison, 1961; Glasgow, 1961a), or movement of concentrations of flies in relation to host movements (Morris, 1960; Glasgow, 1961b). Understanding the factors which give rise to such variation is essential for the develop- ment of an optimal sampling strategy. In this

Correspondence: Dr B. Williams, ICIPE, P.O. Box 30772, Nairobi, Kenya.

paper (Part I) we examine the sampling distribu- tion of the trap catches in order to establish the limits of the stochastic variation in the data. In the following paper (Part 11) we will analyse the day-to-day and seasonal trends in the data in re- lation to climatic factors.

Materials and Methods

The data were collected during 1986 at Nguru- man in the Kajiado District of southwestern Kenya (1'50's; 36'05'E). The Nguruman site was chosen for the study of tsetse dynamics and the development of trapping methods because it presents a relatively isolated area of tsetse in- festation within which there are realistic pros- pects for control. At the same time, becabse the study area is connected to other areas of infesta- tion, it provides an opportunity to study the problem of fly movement which is one of the most important factors determining the effec- tiveness of tsetse suppression campaigns.

Both Glossina pallidipes Austen and G.lon- gipennis Corti are present at Nguruman with the

167

168 B. Williams, R. Dransfield and R. Brightwell

J

m N A V I L Y UOOOED

rn -D

pzd uomm MASSLAND

YETLAND

OPEN GRASSLAND

!I SCUE

1 0.- 0,s 1.0

Kli

FIG. 1. Map ot the study area showing the position of the three biconical traps. Trap 1 is to the east, trap 2 is in the middle, trap 3 is to the west.

former more abundant than the latter. Since few G. longipennis were caught (typically less than ten flies per week) the analysis presented here is based on the catches of G.pullidipes only. The main hosts of G.pallidipes at Nguruman are warthog (Potomochoerus aethiopicus) and buf- falo (Syncerus caffer) (Tarimo, unpublished data). The normal seasonal trend is for tsetse numbers to increase during and immediately after the two rainy seasons in April and November and decrease during the dry seasons.

Three unbaited biconical traps (Challier et al., 1977) were placed in a narrow neck of woodland shown in Fig. 1 that joins the main study area at the foot of the Rift Valley escarpment to a more extended area of tsetse infestation in the north. Each trap was set a few metres to the west of thickets along seasonal stream beds running north-south. The thickets close to trap 3 were denser than those close to traps 1 and 2 and trap 3 was set in the flood plain of the seasonal Oloibototo River which runs a few metres to the west of trap 3 . Game animals were more abun- dant near trap 3 than traps 1 and 2 because the

grass on the flood plain remained green through most of the year.

The support poles of the traps were greased to prevent damage to flies by ants. These three traps were put in place to monitor changes in the population with particular reference to day-to- day changes in catches. A larger number of traps operated over the whole study area for 6 days at the beginning of each month in order to study spatial and seasonal changes in the fly popula- tion and the data from these traps will be pre- sented in subsequent publications.

Results

Many factors contribute to the fluctuations in the trap catches. Briefly, the trap catch depends firstly on the density of flies in the vicinity of the trap which in turn depends on the overall rates of change of the population due to mortality and fecundity as well as on fly movement. Of the flies in the vicinity of the trap, only a certain propor- tion will be active at any given time, depending

Monitoring tsetsefty populations. I 169

on the climate and on the physiological state of the flies. The proportion of the active flies which are successfully caught will depend on the effi- ciency of the trap which may depend on the nature of the habitat in the vicinity of the trap and climatic factors.

Fig. 2 shows the numbers of male and female G.pallidipes caught in each of the three traps on each day in 1986. The number of flies caught var- ies slowly and systematically over the year, but there is also a striking variation in the numbers caught on a time scale of about 1 week: from one day to the next the numbers of flies caught are fairly well correlated, but over a period of a week there is very little correlation at all and the numbers typically change by an order of mag- nitude over any 3 or 4 days. These then are the two features of the variations in the trap catch which we need to explain: the seasonal variation from one month to the next and the much more rapid variations which take place over 2 or 3 days. In order to separate systematic changes in the catches from stochastic fluctuations we need to establish the intrinsic variation in the data due to the stochastic nature of the sampling process and this is the subject of this paper.

The sampling distribution of trap catches

Any analysis of the intrinsic sampling variabil- ity should aim to quantify variation in the sam- pled population over space and time. Work on insects has shown that a power law relationship usually holds between the mean number of in- sects sampled and the variance of the number of insects sampled (Taylor, 1984). Two limiting cases are generally considered. In the first case it is assumed that the insects are randomly distributed over the area sampled so that the probability distribution function over the area sampled is constant. The numbers of insects in any area of a given size will then follow a bi- nomial distribution, so that if n is the number of insects sampled from a total population of N insects, and if p is the fraction of the total area which is sampled, the expected value of n, E(n), and the variance of n, V(n) , are given by (Bulmer, 1967)

E(n)=pN V ( n ) = p q N = q E ( n ) (1)

respectively, where q= 1-p. Provided p 4 1 , the binomial distribution approximates closely to the Poisson distribution, the variance is equal to

the expected value and a plot of the logarithm of the variance against the logarithm of the mean will have a slope of 1. Note that in this case the variance of fi is 0.25, so that a square-root transformation will stabilize the variance of the numbers of flies caught (see Appendix).

For the second case, consider an extremely clumped distribution in which the insects all occur in either one or the other of two plots, so that there are N insects in plot 1 and none in plot 2, or N in plot 2 and none in plot 1, each outcome occurring with equal probability. Then the prob- ability distribution function of n is

P(n) =0.56(0) +0.56( N )

(For a discrete distribution the delta-function S(n) is defined so that S(m)=l for n=m and s(m)=O for n f m . ) For the probability density function described by equation ( 2 ) , the expected value and the variance of n are

E(n)=N/2 V(n)=(N/2) ’=E(n)’ (3) so that a plot of the logarithm of the variance against the logarithm of the mean will have a slope of 2. Note that in this case the variance in In(n) is 1 , so that a logarithmic transformation will stabilize the variance in the observed num- bers of counts (see Appendix). The slope of a In- In plot of the variance against the mean there- fore gives a measure of the degree of ‘clumping’, with a slope of 1 indicating no clumping and a slope of 2 indicating ‘perfect’ clumping. Clearly, if two separate effects influence the distribution, one in each of the ways indicated above, then for sufficiently small values of n the slope of such a plot will be 1, and for sufficiently large values of n the slope will be 2 , so that observed slopes are expected to lie between 1 and 2. Taylor (1984) has shown that for many species the variance of the numbers of organisms found in a given area has a power law dependence on the mean with an exponent that is often close to 1.25.

In this study we are concerned with changes in the trap catches of tsetse flies over time rather than space. For insects which swarm or, in gen- eral, for which the trapping probability varies systematically with time, the observed catches will appear to be clumped in time, many suc- cessive periods of low catches interspersed with occasional very high catches and the power law dependence of the variance on the mean should be close to 2. For insects which do not swarm but disperse in a random fashion, and for which the

170

trapping probability does not vary systematically with time, catches in successive time periods will follow a Poisson distribution and the power law dependence of the variance on the mean should be close to 1.

B. Williams, R. Dransfield and R. Brightwell

A model for the sampling variance

If the trap catches varied slowly and smoothly with time (apart from the stochastic variation due to the sampling process) the most direct way

100

0

of determining the dependence of the variance on the mean would be to smooth a plot of the catch against time (using splines'or linear filters), use the deviations from the smoothed curve to determine the residual variation and hence the dependence of the variance on the mean. Unfor- tunately, in our data the daily variation in the catches is generally greater than the variation due to the stochastic nature of the sampling pro- cess and we cannot easily remove the underlying trend to isolate the stochastic component of the

J F M A M J J A S O N D

Months FIG. 2(a)

FIG. 2. The numbers of male and female G.paNidipes caught in each trap on each day in 1986. (a) Trap 1 . males. (b) trap 1 . females. (c) trap 2. males. (d) trap 2. females, (e) trap 3. males, (f) trap 3, females.

x rd

v) a,

L L

9 .- -

200

0

J F M A M J J A S O N D

FIG. 2(b) Months

100

~

JF

MA

MJ

JA

SO

ND

FIG

. 2(c

) M

onth

s

FIG

. 2(e

) M

onth

s FI

G. Z

(f)

Mon

ths

173- 6. Williams. R. Dransfield and R. Brightwell

variation. Our approach is instead to take the numbers o f males and females as two measures of the mean population so that differences be- tween them will give a measure of the intrinsic variability of the data.

In the simplest sampling model there are only two factors that contribute to the intrinsic vari- ation in the numbers of flies trapped: the random nature of the sampling and the random nature of the birth process in producing males and females. Other factors, such as changes in the sex-ratio. and the relative efficiency of the traps for males and females, will be considered below and in Part 11.

Suppose that there are N flies of which M are males and F a r e females. Assuming a mean sex- ratio (femalesitotal) at birth of 0.5 and assuming that the number of males and females follows a binomial distribution, the expected number of males and number of females are

while the corresponding variances are

V( M ) = V ( F) = I W l . ( 5 ) In addition, since N=M+F. the covariance of

the number of males and females. C(M,F). i s

C( M , F‘)= - N i l . (6)

These two equations describe the stochastic variation in the sex-ratio. Now suppose that the probability that a particular fly is caught in a given time is p for both male and female flies. Then the expected values of m and f , the num- bers of males and females that are caught, are

E( nr) =E(f)=pN/2 ( 7 )

while the contributions to the variances from the sampling process are

V(m)=V(f )=pqNl2 .

Since only a fractionp of the entire population is sampled, the contribution to the overall vari- ance from the variance in the sex-ratio must be scaled by a factor ofp’. For sufficiently large val- ues of p N the spread in the numbers, given by t h e square-root of the variance, is much less than expected value and we may add the variances from the effects of the sex-ratio and the sam- pling. The covariance matrix for the entire pro- cess is then

i pqN12+p2N14 -pzN14

-p’Nl4 pqN12+p2N14 C(m,f) =

(9) from which it follows (see Appendix) that if p = a and +=*, the variance of p-+ is

V ( k-+) =0.5. (10)

If we now consider deviations from the as- sumptions made above we find that this result is quite robust. If only a random sub-set of the population is sampled or, equivalently, if the population is sampled twice, equation (10) holds. If the probability of trapping a fly in the morning differs from the probability of trapping a fly in the evening say, equation (10) holds with p equal to the probability per day of trapping a fly. If the sex-ratio is constant, but not 0.5, or if the probability of trapping a female is not the same as the probability of trapping a male, equa- tion (10) still holds provided the numbers of males and females are scaled appropriately. If we let R be the expected proportion of females caught and we define p* and +* to be

then equation (10) holds provided k* and +* are used in place of p and +. However, equation (10) is strictly only valid in the limit of large fly numbers (so that the standard deviation is al- ways much less than the expected number sam- pled). It is therefore important to establish the range over which equation (10) holds. For small fly numbers it is not difficult to derive analytical expressions for the variance in p,-+ and these are given in the appendix. Furthermore, in the limit p 0 we can catch either one male or no males and one female or no females and writing down the probabilities of all possible outcomes it follows that

V(k-+)-+Np as p-0 (13)

For large values of N it would be possible, but tedious, to work out analytical expressions for the variance and so a Monte Carlo simulation was done for values of N > 3 . The sampling pro- cess was then simulated using a random number

Monitoring tsetsefly populations. I 173

generator to select binomially distributed ran- dom numbers from a population of size N with probability 0.5. This gave the number of males, M, and the number of females, F = N - M . Bino- mially distributed random numbers were then chosen from populations of size M and F with probabilityp to obtain simulated valuesof rn and

f , the numbers of males and females caught in the trap. Values of 6= fl- 6 w e r e then cal- culated for values of N ranging from 2 to 128 and values ofp rangingfrom 0.0125 to 1, the calcula- tion being repeated 4000 times for each value of N and p in order to determine the variance of 6. In all cases the standard deviation of the esti-

0

h

Q) 0

(b) In (Np/2)

FIG. 3. The calculated variance in fi- fl where m and f a re the numbers of male and female G.pal- lidipes trapped with probability p from a population of N flies. The expected number of males and females is Np12. (a) Plotted on linear scales and (b) plotted on logarithmic scales. The slanted solid line in (b) gives the variance in the limit N p O . The dashed lines and the open circles are for calculations in whichfand m are replaced byf+0.5 and m+0.5, respectively. For each value of Np/2 the smaller values of the variance correspond to larger values of N.

174

mate of the variance of 6 was less than 3% of the estimated value.

V(p-+) is plotted against Np12, the expected number of males or females in Fig. 3. For large values of Np the variance converges to 0.5 as predicted by equation (10) while for small values of Np the variance is close to the value given by equation (13), represented by the slanted solid line in Fig. 3(b). The maximum variance occurs when Npi2-1 and decreases from about 1 for small N to about 0.8 for large values of N. For values of Np/2>4 we see that 0.5<V(6)<0.6. In order to increase the range of applicability of equation (10) we can follow the usual practice (Sokal & Rohlf, 1987) of adding 0.5 to the num- bers of males and females before taking the square root. The line corresponding to equation (11) for small values of Np then shifts to the right a5 indicated by the dashed line in Fig. 3(b). Simulating the sampling process for p=0.25 and values of N between 2 and 64 gives the open circles in Fig. 3 and the variance is now within 0.1 of 0.5 for all values of Np/2>1. In the discus- sion that follows 0.5 has been added to each catch before taking the square-root.

B. Williams, R. Dransfi’eld and R. Brightwell

Maleifemale variability

If the number of females is plotted against the number of males for any of the traps it is found that the spread in the points increases with the mean number of males and females. The catches were therefore transformed by scaling the square roots of the catches plus 0 .5 using equa- tions (11) and (12) with R, the mean sex-ratio over the year, equal to 0.571,0.560 and 0.563 for traps 1. 2 and 3, respectively. This transforma- tion stabilizes the variance considerably and gives the result shown in Fig. 4. The standard de- viations of I**-+*. estimated from the data, are 1.4 and 1.3 for traps 1 and 2 , respectively, while fortrap 3 the standard deviation of p*-+* is 2.9.

In order to determine the power law depend- ence of thc variance on the mean we used the fits in Fig. 4 to estimate the number of female flies on each day which we will call f. We then took the difference between these estimates and the observed values, squared the result and took the logarithm. This gives an estimate of the logarithm on the variance on each day. We then took the logarithm of the values o f f for each day, sorted the data using these values and then averaged the data points in groups of ten. The

resulting plots of the variance against the mean values are shown in Fig. 5. For traps 1 and 2 the slopes of the plots shown in Fig. 5 (with 95% confidence limits) are 1.2050.28 and 1.02k0.24, respectively, and do not differ significantly from 1 while for trap 3 the slope is 1.6350.26 which is significantly greater than 1.

So far we have assumed that the sex ratio is constant over time but Fig. 6 shows that there are significant changes in the sex ratio over time. This is particularly apparent for trap 3 which catches an order of magnitude more flies than traps 1 and 2. In order to allow for the variation of the sex-ratio with time the plots shown in Fig. 6 were smoothed using a five-point smoothing al- gorithm and used to correct the transformed catches for the sex-ratio as indicated in equa- tions (11) and (12). After making this correc- tion, none of the exponents in the power law relationship between the variance and the mean differ significantly from 1 and the standard de- viations of p*-+* are 0.7, 0.8 and 1.2 for traps 1.2 and 3 respectively, all much closer to the pre- dicted value of 0.7 than before.

Having established that the sampling distribu- tion of the trap catches follows a Poisson dis- tribution we can consider the actual catches of males and females. From equation (9) it follows that for large values of rn andf

V(V%)=V(fl)=(l-p/2)/4 (14) and

V( Mf ) = 1 14

while for small values of rn and f

V( V%) = V( q) = Np12

V ( m f l = N p

For large values of Np and small values of p the variance in the number of flies caught is 114. In monitoring changes in tsetse fly populations over time one typically records the catch in a number of traps, distributed over the study area, for a number of successive days in each month. Since the Poisson distribution is additive, the total catch of all of the traps over the sampling period will also follow a Poisson distribution and the arguments developed above apply to the total catch.

We have also seen that at Nguruman there is a systematic variation from day to day that is generally greater than the stochastic sampling

cn a, - E a, Y-

-0 a,

0 cn C

E LC

z! I-

U a,

0 cn C

E -4-

z

(4 Transformed males FIG. 4. Transformed female G.pallidipes catch plotted against the transfarmed male G.pal?idipes catch. The chained lines give 95% confidence limits for the points. (a) Trap 1 , (b) trap 2, ( c ) trap 3 .

> c -

> C -

' E * / m I- /

a

(c) 2 4 6 In M

FIG. 5. Natural logarithm of the variance, V , of the difference between the transformed male catches and the transformed female catches plotted against the natural logarithm of the mean, M, of the transformed catches. (a ) Trap 1 , (h) trap 2 . (c) trap 3 . The calculation of the variances and means are descrihed in the text.

h

Proportion of female flies S A

A

3 c,

Proportion of female flies A

I

2 Proportion of female flies A

A

178

variation. Since trapping is a time-consuming process we have also estimated the minimum number of days over which sampling should be done to achieve a given level of accuracy in the mean monthly catch. To determine the effect of the day-to-day variation in trap catches on esti- mates of the mean monthly catch we have calcu- lated 1 , 2 . 1 , 8. 16 and 32 day running means of the total catch in all three traps for both sexes and then taken the ratio of the 1 to 16 day running means to the 32 day running mean. The median ratios together with the 0.95 percentile ranges are given in Table 1.

B. Williams, R . Dransfield and R . Brightwell

TABLE 1 The median value, M. of the ratio of the N day running means of the trap catches to the 32 day running means of the trap catches together with lower bounds, L, and upper bounds, U giving the 0.95 per- centile ranges

Y 1. M U

1 0,27 0.92 3.10 2 0.40 0.97 2.35 4 0.51 0.98 1.88 8 0.65 0.99 1.51

16 0.81 1 .oo 1.22

From Table 1 it is seen that the catch on a single day gives an estimate of the monthly mean catch which is only accurate to within a factor of 3 (95% confidence limits) and that if 50% ac- curacy, say, in the estimate of the mean monthly catch is required, with 95% confidence, trapping at Nguruman should be carried out over a period of at least 8 days.

Discussion

For the data presented above the daily variation in trap catches of G.pallidipes is generally greater than the variation due to the stochastic nature of the sampling process even though the number of flies caught is generally small. Un- fortunately few other studies of the day-to-day variation in catches of G.pallidipes have been published. Glasgow (1961a) examined the varia- bility of catches of G.pallipides on three fly rounds over 24 days. The ratios of the biggest to the smallest catches in each of the fly rounds were 2.8, 3.1 and 3.9. Glasgow & Duffy (1961) also published data for fourteen Morris traps

over 18 days. The greatest number of flies caught in one trap on one day was fifty and in each of the traps there was at least one day on which no flies were caught. The ratio of the maximum catch in each trap on any one day to the mean catch for that trap, ranged from 3.1 to 7.5 with a mean value of 4.3. In both of these studies the varia- tion is of the same order of magnitude as the var- iation given in Table 1 for the 1 day catches.

The only study of tsetse flies in which variance and mean were related was carried out on G.pal- palis palpalis (Robineau-Desvoidy) . Randolph et a l . (1984) determined the power law depen- dence of the variance on the mean trap catches at ten sites and obtained values ranging from 0.78 to 3.50. They point out that if the variances within and between sets of data vary greatly, direct statistical comparisons will be meaning- less and they therefore transform each set of data in order to ensure that the variance does not depend on the mean. Unfortunately, this may not always be desirable. Since we are interested in identifying systematic variations which un- derlie the stochastic variation, transforming the data to minimize the dependence of the variance on the mean may well mask the very effects we seek to investigate.

From the analysis presented here it appears that the intrinsic variation in the numbers of flies caught in each of the traps is consistent with a model which includes only the stochastic vari- ation in the sex-ratio and the variation arising from the random nature of the sampling. For trap 1 and trap 2 the variance is proportional to the mean, although the residual variance is rather greater than the predicted value of 0.5. In the case of trap 3, which caught an order of mag- nitude more flies than either trap 1 or trap 2 , the variance is proportional to the mean raised to the power of 1.6, but the deviation from 1 can be accounted for largely as a result of systematic changes in the sex-ratio with time.

It is usual to use trap catches taken over a few days to estimate populations of tsetse flies but the day-to-day variability in trap catches of G.pallidipes is such that the traps should be op- erated over the longest possible period if reliable population estimates are to be obtained. If the reasons for the day-to-day variability can be un- derstood it may be possible to allow for the var- iability or to design trapping systems in such a way as to minimize this variability and obtain more consistent population estimates. In Part I1

Monitoring tsetsefly populations. I 179

we examine the effect of various climatic factors on the day-to-day variability in trap catches,

Acknowledgments

We would like to thank the International Centre for Insect Physiology and Ecology for the provi- sion of research facilities, Dr D. Rogers for a number of important comments and suggestions, Mr R. Kruska and Mr J . Akiwumi for preparing Fig. 1, and our field assistants.

References

Bulmer, M.G. (1967) Principles of Statistics, p. 87. Oliver and Boyd, London.

Challier, A,, Eyraud, M., Lafaye, A. & Laveissitre, C. (1977) Ameloration du rendement du pitge biconique pour glossines (Diptera, Glossinidae) par l’emploi d’un cBne infereur bleu. Cahiers ORSTOM, SCrie Entomotogie Me‘dicale et de Parasitologie, 15, 283-286.

Dransfield, R.D. (1984) The range of attraction of the biconical trap for Glossina pallidipes and Glossina brevipalpis. Insect Science and its Application, 5,

Ford, J., Glasgow, J.P., Johns, D.L. & Welch, J.R. (1959) Transect fly-rounds in field studies of Glos- sina. Bulletin of Entomological Research, 50,275- 285.

Glasgow, J.P. (1961a) The variability of fly-round catches in field studies of Glossina. Bulletin of En- tomological Research, 51,781-788.

Glasgow, J.P. (1961b) Seasonal variations in size and colour and daily changes in the distribution of Glossina pallidipes Aust. in the South Busoga Forest, Uganda. Bulletin of Entomological Re- search, 52,647-666.

Glasgow, J.P. & Duffy, B.J. (1961) Traps in field studies of Glossina pallipides Austen. Bulletin of Entomological Research, 52,795-814.

Hargrove, J.W. & Vale, G.A. (1978) The effect of host odour concentration on catches of tsetse flies (Glossinidae) and other Diptera in the field. Bul- letin of Entomological Research, 68,607-612.

Morris, K.R.S. & Morris, M.G. (1919) The use of ‘traps against tsetse in West Africa. Bulletin ofEn- tomological Research, 39,491-528.

Morris, K.R.S. (1960)Trapping as a means of studying the game tsetse Glossina pallidipes Aust. Bulletin of Entomological Research, 51,533-537.

Randolph, S. , Rogers, D. & Kuzoe, F.A.S. (1984) Local variation in the population dynamics of Glossina palpalis palpalis (Robineau-Desvoidy) (Diptera: Glossinidae). 11. The effect of insect- icidal spray programmes. Bulletin of Entomolo- gical Research, 74,425-438.

Seber, A.F. (1982) The Estimation of Animal Abund- ance. Charles Griffin, London.

Smith, l.M. & Rennison, B.D. (1961) Studies of the sampling of Glossina pallidipes Austen. I. The numbers caught daily on cattle in Morris traps on a

363-368.

fly round. Bulletin of Entomological Research, 52,

Sokal, R.R. & Rohlf, F.H. (1987) Introduction to Biostatistics, 2nd edn. W.H. Freeman & Co. , New York.

Taylor, L.R. (1984) Assessing and interpreting the spatial distributions of insect populations. Annual Review of Entomology, 29,321-357.

1 65-1 82.

Accepted 16 August 1989

Appendix

Variance stabilizing transforms

Let 0 be a random variable with variance V(0) and expected value e . Let 4=g(0) be a function of 0. Then

V(+)-(d0/d+)2V(0) (Al l

(Seber, 1982). If V(0)=6 and it follows from equation (Al) that V(+)=1/4 while if V(0)=6* and +=ln0, it follows from equation (Al) that V(+)=l.

Variance of p-+ for largepN

C(rn,fl is (Seber, 1982) I f p=g(m) and + = g o the covariance matrix

C(l.4) = (dp/dm)(d+/df)C(m,fl (‘42)

so that if C(m,f) is given by equation (10) with E(m)=E(f)=pN/2, then

and

Variance of p--+ for small N

For small values of N the variance of S = q - 6 may be calculated analytically and is given by the following expressions:

N = l : V(S)=p (A51

N=2: V(S)=2p-p2 ( 4

N = 3 : V ( 6 ) = 3 ( p - p z + ( 1 - 1 / ~ ) p ’ ) (A7)

N=4, p = l : V(S)=(5-2VT)/2 (AS)

N=5, p=l: V(6)=(15-5VC)/4 (A9)