An interpretation of aphid population development by statistical model

Shiyomi (1967a and b) proposed two models which describe the reproduction and the plant-to-plant movement of aphids. For the explanation of the whole process of development of population of aphids, the above two models were incorporated into a new model (called Model C). This model is superior in the following points to the negative binomial model:

1. Model C has 7 parameters and gives a fuller explanation compared with the negative binomial model which has 2 parameters.
2. Model C describes the structure of population at any stage of its development, while the negative binomial model describes that of a well developed stage of population.

     

Experimental studies on the distribution of the aphid counts

The present paper dealt with the sequential changes of the distribution pattern of apterous females aphid populations, that were artificially settled at the beginning on the experimental barley ‘field'. The aphids were settled at random or even with a fixed denisty per plant. For five or six days after the settling, the number of individuals followed the negative binomial distributions in all cases while the parameters k and p were varying. The estimated values of k were rather small for the first one week after the settling, which may suggest that the number of moving aphids between plants was relatively small and the degree of concentration expressing the intrinsic increase was high. After that, as the number of individuals increased, the number of moving aphids between plants would be considered to be increased. It was found that with the lapse of time the degree of concentration decreased or k became larger. The distribution of aphids per blade in a plant was also described briefly.     

Influence of parental density on the distribution pattern of eggs in the common cabbage butterfly,Pieris rapae crucivora

To investigate the relation between the distribution pattern of eggs and the parental density in the common cabbage butterfly, Pieris rapae crucivora, the countings of egg number per plant were made on both cabbage plants cultivated in the farm and planted in the net house in which the female butterflies were released at various densities. The frequency distribution of eggs fits well to the negative binomial excepting the cases where they agree withPoisson series, and the degree of aggregation expressed as the reciprocal of the parameter, 1/k, tends to decrease as the egg or parental density increases. At the same parental density, however, the distribution of eggs can be described by the negative binomial with a common parameter, k_c, regardless of the difference in the density of laid eggs. In the case where a single female butterfly lays eggs, the spatial pattern of egg distribution is always lean, while its frequencies conform toPoisson or the negative binomial series. This lean changes toward patchy with increasing the parental density. From these results, it is concluded that the degree of aggregation in the distribution of eggs decreases with the increase of the parental density.     

A new method of sequential sampling to classify populations relative to a critical density

A method of sequential sampling for grading population level in relation to a critical density is proposed. The method is based on the relationship and can be used without restrictions on the distribution patterns. The formulae for simple random sampling as well as for two-stage sampling are given.     

Spatial distribution of the green peach aphid used in estimating the populations of gynoparae

Summary In 2 years, during the initial invasion of peach leaves by the green peach aphid,Myzus persicae (Sulzer), the number of gynoparae was low, and the distribution on leaves was random. Then as the mean number increased, the distribution became intermediate and could not be distinguished from either a Poisson or a negative binomial. Finally, as the mean continued to increase, the variance increased rapidly, and the population was found to fit a negative binomial distribution. Thus the aggregation response was verified because the dispersion pattern fitted a contagious distribution. A sampling plan was devised by which the dispersion parameterk was used to estimate the density of aphids per leaf based on the percentage of leaves infested. Sampling the third year of the study confirmed the validity of the sampling parameter that had been calculated from data for the 2 previous years.     

Preliminary survey on the population and distribution of gibbons in yunnan province

Of the gibbons in China,Hylobates hoolock, H. lar, andH. concolor leucogenys are distributed only in Yunnan Province, whileH. c. concolor occurs limitedly in Yunnan Province and Hainan Island. During the periods from February to November 1984, and March to August 1985, the authors undertook surveys of the population and distribution of gibbons in 56 counties of 10 prefectures in Yunnan Province, and obtained the first reliable population and distribution data on the gibbons of this Province. The findings serve to fill an important blank in gibbon research in Yunnan and should pave the way for further research in the future.     

Population dynamics of the chestnut gall-wasp,Dryocosmus kuriphilus Yasumatsu (Hymenoptera: Cynipidae) IV. Further analysis of the distribution of eggs and young larvae in buds using the truncated negative binomial series

Distribution pattern of eggs and the first instar larvae of the chestnut gall-wasp, Dryocosmus kuriphilus, per bud of the chestnut tree was re-examined using the truncatedPoisson and the truncated negative binomial series. The results are as follows:

1. About sixty percent of the distribution are approximated by the truncated negative binomial and another thirty percent by the truncatedPoisson . When the distribution has been approximated by the truncated negative binomial, the show scattered values, some of which are near thePoisson , but the mean value of is about 4 both in eggs and in the first instar larvae.
2. The number of buds which have not been infested by gall-wasps is resulted from various factors as follows: (a) Buds formed after the laying activity of gall-wasps has ceased; (b) Old and shrunk buds which were avoided in laying; (c) Buds in which eggs have died by the time of sampling; and (d) Buds escaping ovipositions by chance. Most of the gall-wasp-free buds located in the middle part of the shoot are accounted for the zero-class expected by the negative binomial series.
3. Brass (1958) moment method for estimating parameters of the truncated negative binomial give good precision within the range of from 1 to 10 and of from 1 to 6.
4. It is concluded that the truncated distributions used are useful for the purpose of analysis of the distribution pattern of the chestnut gall-wasp.

     

The spatial distribution of the wheat-bulb fly,Leptohylemyia coarctata (fall.) (Diptera: Anthomyiidae)

The spatial distribution of the eggs, larvae, pupae and adults of the wheat-bulb fly was investigated by fitting 42 sets of data comprising 1334 samples to the Poisson and negative binomial distributions, and by using the power law (S²=am^b). In general, the tests indicated that all stages were aggregated and fitted the negative binomial model.     

Preliminary studies for the population estimation of the cryptomeria red mite,Oligonychus hondoensis

This is the second paper dealing with techniques necessary to the population estimation of the Cryptomeria red mite feeding on a coniferous Cryptomeria tree, with the final object of determining the absolute population. Results are as follows:

1. The population can be defined as the whole of mites existing on branches with foliage younger than two years old, because more than 95% of individuals are always there.
2. Optimal sampling unit varies according to mite population. Unless a fixed unit is necessary, seasonal placement of units is recommended.
3. Since significant variance is associated with crown levels, representative sampling from each level is designed. These calculation should be based on the data transformed to logarithms, as the relation between mean and variance suggests that the population data are of contagious distribution.
4. A trial to calculate the total length of branches for each age, which is needed for the conversion to absolute population, is described.

     

Evaluation of statistical precision and design of efficient sampling for the population estimation based on frequency of occurrence

The binomial sampling to estimate population density of an organism based simply upon the frequency of its occurrence among sampled quadrats is a labour-saving technique which is potentially useful for small animals like insects and has actually been applied occasionally to studies of their populations. The present study provides a theoretical basis for this convenient technique, which makes it statistically reliable and tolerable for consistent use in intensive as well as preliminary population censuses. Firs, the magnitude of sampling error in relation to sample size is formulated mathematically for the estimate to be obtained by this indirect method of census, using either of the two popular models relating frequency of occurrence (p) to mean density (m), i.e. the negative binomial model, p=1−(1+m/k)^−k, and the empirical model, p=1−exp(−am^b). Then, the equations to calculate sample size and census cost that are necessary to attain a given desired level of precision in the estimation are derived for both models. A notable feature of the relationship of necessary sample size (or census cost) to mean density in the frequency method, in constrast to that in the ordinary census, is that it shows a concave curve which tends to rise sharply not only towards lower but also towards higher levels of density. These theoretical results make it also possible to design sequential estimation procedures based on this convenient census technique, which may enable us with the least necessary cost to get a series of population estimates with the desired precision level. Examples are presented to explain how to apply these programs to acutal censuses in the field.     

On Mohan's Property of the Poisson Distribution

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 (k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.     

On the estimation of insect population by time unit collecting

---

Contributions from the Entomological Laboratory, Kyoto University, No. 230.     

Spatial modelling of individual-level parasite counts using the negative binomial distribution

We present a spatial model for the mean and correlation of highly dispersed count data, and apply it to individual-level counts of the nematode Wuchereria bancrofti, a parasite of humans which causes the disease lymphatic filariasis. Our model uses the negative binomial distribution, whose shape parameter is a convenient index of over-dispersion. Spatial association is quantified in terms of a characteristic length, which has an intuitive interpretation as the distance over which correlation decreases by half. Demographic surveillance and mapping enable us to include individual-level covariates such as age and sex. We discuss the distinctive features of our model and interpret the results in terms of the epidemiology of lymphatic filariasis and possible implications for control programmes.     

Process generating the ditribution pattern of eggs of the common cabbage butterfly,Pieris rapae crucivora

The process generating the negative binomial in the distribution pattern of eggs of the common cabbage butterfly, Pieris rapae crucivora, was investigated by releasing the female adults in a net house where cabbages were planted. The distribution of butterflies visited and laid an egg or more per plant followed thePoisson series under the uniform light condition, while that of eggs laid per visit conformed to the logarithmic distribution. From these results, it may be concluded that the negative binomial arises from compounding of thePoisson and the logarithmic distribution. The observed frequency of eggs found per plant fitted to the negative binomial with parameter thus computed theoretically. The change in the degree of aggregation with the increase of the parental density was considered in connection with the above results.