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NOV 4; 1945ET2e5 United States Department of Agriculture Agricultural Research Administration Bureau of Entomology and Plant Quarantine A MM0D OF ESTIMATING BUT LZAFlPPE POPULATION FROM MM PROPORTICM OF IJNIIFTED PLANTS _/ By M. F. Bowen, 2/ Division of Track Crop and Garden Insect Investigations In the course of ecological studies of the beet leafhopper (Eutettix tenellus (Bak.)), a large number of population counts vere taken in beet fields within the range of this insect. 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 popup. lations that move Into the beet fields are distributed in a Poisson series. This indicated that the mean number of leafhoppers per plant can be estimated from the proportion of plants not infested, or from the portion of plants infested with a given mvnber 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 It 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 numer 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 Entological Society of America at San Francisco, Calif., Dec. 29, 1941, to Jan. 1, 1942. 2/ Now with the Ninth Service Command, Army Service Forces. 2 infested, then it is possible to equate the mean of the distribution from the equation em Z 0.37. The mean is I leafhopper per plant. This relation between the mean and the proportion of O's is shown graphically by curve A in figure 1. Curves B and D in this figure show the relation between the man 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 3/ The curves for all except the proportion of O's have two values of the mean corresponding to every relative frequency. For tks reason classes greater than 0 must be used with caution. For ezauplq, if the percentage of l's is used and this number appears in 20 per cent of the sampling units, it my indicate a mean of either 0.25 or 2.53 insects per unit area. Similar difficulties in determnng mean values will appear when the 2's, 3's, or higher numbers are use4. Therefore, the relative frequency of classes higher than 0 can Wafell be employed only when the mean is approximately knovn, and preferablS when it is equal to or greater than the size of the class being used., Theoretically, the mea will be estimated fron a Oas fre quency with the least absolute error by using the *se having thei atest i~tire f uen, The error of eet1 tIon _ eaes raid j tii departre from_ this ixum value. Itca be ahuwn iuie of 't calculus that for a Poseon datri bution auz class frequency will be at a mxrl when the man is equ o te z sIze of the class. This fact is gaphially I lustrated by the curves in figure l, vhich indicate mxim of Ole* l's, 2's, or 3'6 whea the means are, respeotively, 0.0, 1.0, 2.O, or 3.0. Practical Application The application of fquencies of bigherthwa0 classes in est mating beet leafhopper populations is of doubtful utitTi. Sine the insect is small and very active, the effort required to detern the papoxtion ot e eple ic eated with I, 2, or a higher number of leafnahpers becoes almost as great as to uake a complete census of the ssmple unite, This difficulty becomes increasingly pronounced as the 8ize ,e cas i increased. On the other hand, the ue~e pesee x &bsenee of the ise~t iy be determined with speed and _/ 2 ub+ or ,tatisticiane and biometrioians. i i, ed. 3, Vble i, pp. 113121. 1930. 3 aaoiwaoyp and a large number of samples my be taken in a short time. Data obtained in the Grand Valley area of Colorado in the spring ad early sg r of 1937 are used here to illustrate the application of the peroentagefrequenoy method to the estimation of populations of the beet leafhopper. The data are eummarized in figure 2. ach date mean plotted in figure 2 represents the average of the means obtained f 10 fields scattered over 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 latioms per beet calculated by the two methods. This agreement sup ports the conclusion that migrant beet leafhoppers are distributed in a Polson series, Inasmuch as the calculated values were obtained by a formla 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 seer generation. The leafhopper population estimated by the two methods agrees very closely but the calculated values tend to be slightly lower than the observed. The oorrelaticn coefficient for the 15 pairs of means is r : 0.991, which denotes a high degree of association between the two sets of values. Conclusions The foregoing data demonstrate that the proportion of uninfeeted plants my 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 Poisson law. As the distributing deviates from this law there will be an increased tendenci for the method to underesti mate the true population. (2) The mean determined from the proportion of O's will be esti umted with less precision than that obtained froi the total sample, 'but the larger number of sampling units that my be taken compen sates for this defiolency, at least within certain ranges of popu. lation density. (3) The method is adapted primarily to population of low density, say between 0 and 3.00 insects per unit area. For a Poidon distribution with a mean of 5.00, 00s will appear only about 7 times In 1,000 samples; and for a mean of 7.00 only once in ,O00 amplas. Obviously such wall probabilities render imp tIcable the us. of the 0 class in estimting dense populations because of the lag number of sampling units that vould be required. When dens. popu lations are studied, perhaps equal inform tion could be obtained vith les effort by taking a smaller number of sampling uaits =A estimting the mean in the usual manner. This In a point that re quires further study. 5 z be _j 0 hi  ME AN Figure I.Relatiou between the lean in a Poisson distri bution and the relative frequency of O's (curve A), of l's (w 1),, of 2's (curVe C), and of 3's (curve D). UNIVERSITY OF FLORIDA 3 1262 09240 8706 I 2.00O 1.80 OBSERVED MEANS 1.0 CALCULATED MEANS I40 I ,1.20 w z/ 0 < o '/ o0 / JI 0 0. so1 z 60 40 20 0 p p I p p p p p p p p 15 20 25 30 4 9 14 19 24 29 4 I 14 I9 MAY JUNE 4ULY Figure 2.Populations of the beet leafhopper on sugar 1 eets 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 0 class by the broken line. 