L945 ET-225 United States Department of Agriculture Agricultural Research Administration Bureau of Entomology and Plant Quarantine A METHOD OF ESTIMATING BEET LEAFHOPPER POPULATIONS FROM THE PROPORTION OF UNINFESTED PLANTS l/ By M. F. Bowen, 2/ Division of True Is Crop and Garden Insect Investigations In the course of ecological studies of the beet leafhopper ( Eutetttr tenellus (Bak.)), a large number of population counts were taken in beet fields within the range of this inseot. The pur- pose of these counts was to determine the time and magnitude of the spring movement from the breeding areas to the beet fields, and also to follow the trend of populations during the period of the spring movement . Statistical analyses of these data demonstrated that the popu- lations that move Into the beet fields are distributed In a Poisson series • This indicated that the mean number of leafhopper s per plant can be estimated from the proportion of plants not infested, or from the portion of plants infested with a given number of the insects* Since it appeared that this method might be more rapid than the conventional method, a study was made of its reliability and its practical application* Theoretical Considerations. In a Poisson distribution the mean determines the frequency of occurrence of a given number of individuals in the sampling units. In other words, the mean may be calculated from the proportion of the total number of sampling units of a given class, that is, from the proportion of sampling units In which a given number of indi- viduals are found. For example, if beet leafhoppers are distributed in a Poisson distribution and 37 percent of the plants are not 1/ This paper was read at the Joint meetings of the American Association of Economic Entomologists and the Entomological Society of America at San Francisco, Calif., Dec. 29, 19Ul, to Jan. 1, 19^2. 2/ How with the Ninth Service Command, Army Service Forces. -2- infeeted, then it is possible to equate the Bean of the distribution from the equation e-m = 0.37. The mean is 1 leaf hopper per plant. This relation between the mean and the proportion of 0's is shown graphically by curve A in figure 1. Curves B, C, and D in this figure show the relation between the mean and the proportion of l's, 2's, and 3's, respectively. The values shown in the curves may be obtained by direct calculations or from tables prepared by Pearson ^/ The curves for all except the proportion of 0's have two values of the mean corresponding to every relative frequency. For this reason classes greater than must be used with caution. For example], if the percentage of l's is used and this number appears in 20 per- cent of the sampling units, it may indicate a mean of either 0.25 or 2.53 insects per unit area. Similar difficulties in determining mean values will appear when the 2's, 3' a* or higher numbers are used- Therefore, the relative frequency of classes higher than can safely be employed only when the mean is approximately known, and preferably when it is equal to or greater than the size of the class being used., Theoretically, the mean will be estimated from a class fre- quency with the least absolute error by using the olass having the greatest relative frequency. The error of estimation In- creases rapidly with departure from this maximal value. It can be shown by methods of the calculus that for a PolMon distri- bution any class frequency will be at a — ^-t™ when the mean Is equal to the size of the olass. This fact is graphically il- lustrated by the curves In figure 1, which indicate maxima of 0'a, l's, 2's, or 3's when the means are, respectively, 0,0, 1,0, 2,0, or 3.0. Practical Application The application of frequencies of higher-than-0 classes in esti- mating beet leafhopper populations is of doubtful utility, Slnoe the insect is small and very active, the effort required to determine the proportion of samples infested with 1, 2, or a higher number of leafhoppers becomes almost as great as to make a complete census of the sample units. This difficulty becomes increasingly pronounced as the size of the class is increased. On the other hand, the mere presence or absence of the insect may be determined wl ;h speed and 3/ Pearson, K. Tables for statisticians and biometricians Pt. 1, ed. 3, table 51, PP- 113-121. 1930. -3- aocuraoy, and a large number of samples may be taken In a short time. Data obtained In the Grand Valley area of Colorado In the spring and early summer of 1937 are used here to Illustrate the application of the percentage-frequency method to the estimation of populations of the beet leafhopper. The data are summarized In figure 2. Each date mean plotted In figure 2 represents the avecage of the means obtained from 10 fields scattered oyer the area. The mean for each field on each date was based on 50 sampling units. Clearly there Is a striking agreement between the mean popu- lations per beet calculated by the two methods. This agreement sup- ports the conclusion that migrant beet leafhoppers are distributed in a Polsson series, Inasmuch as the calculated values were obtained by a formula derived from this series. The large increases from June 21 to 28 and June 28 to July 15 represent the beginning of the appearance of adults of the first summer generation. The leafhopper population estimated by the two methods agrees very closely but the calculated values tend to be slightly lever than the observed. The correlation coefficient for the 15 pairs of means is r = 0.994, whloh denotes a high degree of association between the two sets of values. Conclusions The foregoing data demonstrate that the proportion of un infested plants may be used to estimate the density of migrant beet leafhopper populations* However, the method has limitations, of which the following should be noted: (1) It is applicable only when the distribution of the insect accords with the Polsson law. As the distribution deviates from this law there will be an Increased tendency for the nethod to underesti- mate the true population. (2) The mean determined from the proportion of 0's will be est!' mated with less precision than that obtained fron the total sample, out the larger number of sampling units that may be taken compen- sates for this deficiency, at least within certain ranges of popu- lation density. (3) The method is adapted primarily to populations of low density, say between and 3.00 insects per unit ar ea. For a Polsson distribution with a mean of 5.00, 0*s will appear only about 7 times in 1,000 samples; and for a mean of 7.00 only once In 1,000 Maple*, Obviously such email probabilities render impracticable the use of the class in estimating dense populations because of the large number of sampling units that would be required. When dense popu- lations are studied, perhaps equal information could be obtained vith lees effort by taking a smaller number of sampling unite and estimating the mean in the usual manner. This is a point that re- quires further study. -5- Pigure 1. — Relation between the ifaean in a Poissoc distri- bution and the relative frequency of • e ( curve A) , of l's (cuma), of 2»s tcurve C) , and of 3's (curve D) . UNIVERSITY OF FLORIDA , urn in; i ill 3 1262 09240 8706 IS 20 25 30 t .JULY Figure 2. — Populations of the beet leafhopper on sugar beets in the Grand Valley area of Colorado in the spring and early summer of 1937. Observed means are shown by the solid line, and the means calculated from the relative frequency of the class by the broken line.