Population dynamics of the chestnut gall-wasp,Dryocosmus kuriphilus Yasumatsu (Hymenoptera: Cynipidae)

Summary As a part of serial study on population dynamics of the chestnut gall-wasp,Dryocosmus kuriphilus, analyses of the distribution of eggs, gall-cells and emergent holes were made from the statistical point of view. Many of distributions of the eggs per bud could be described by the truncated Poisson, but some cases showed slight overdispersion than expected by chance. Because of no linear increase of with increasing (expected mean for complete sample), however, the truncated negative binomial seemed to be not so available for whole series. Distributions of the gall-cells and the emergent holes were, on the other hand, well described by the truncated Poisson distribution when the observed frequency was calculated for respective trees. Negative-binomial tendency found in distributions from some stations consisted of a certain number of trees would be resulted from mixture of Poisson populations with different means. Random or slightly concentrated oviposition and random mortality within galls was thus supposed for future study unless the proof favouring other distribution models would appear.     

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.     

I σ-Index,a measure of dispersion of individuals

Conclusion and summary As theI _σ-index is neither affected by the mean per sample unit except for regular distribution nor standing on the assumption of any definite type of contagious distribution, it may have the most wide range of application among the ones hitherto deviced for measuring the dispersion of individuals in a population. The relations of theI _σ-index to ,k of negative binomial distribution andN of binomial distribution as well as the new dispersion index,I _B , given in this paper may serve, if necessary, for the analysis of data in the ecological works. Aided by the Scientific Research Fund from the Ministry of Education.     

A comparative analysis on the distribution of nymphal populations of some leaf- and planthoppers on rice plant

The frequency distribution of the number of nymphs per hill of rice plant were analyzed for three species of rice leaf- and planthoppers, Nilaparvata lugens, Delphacodes striatella and Nephotettix cincticeps, based on the sampling data obtained during their last generations in the paddy field. For every species concerned, individual distributions were proved to be contagious and to fit well to the negative binomial distribution. Further, it was found that the value of negative binomial parameter k is so stable for same species that a single value of k is applicable for a series of counts with different means, whereas that k differs remarkably among different species: if the reciprocal of the weighted estimate of common k which is an adequate index measuring degree of contagiousness of the distribution, is compared among different species, it is higher in the order of Nilaparvata, Delphacodes and Nephotettix. The ecological and practical implication of constancy and heterogeneity within and among species was discussed respectively as to the value of parameter k.     

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.     

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.     

Multivariate Polya and Inverse Polya Distributions of Order k

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU , ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU , ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU , TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..     

The spatial distribution pattern of the brown planthopperNilaparvata lugens Stål (Homoptera: Delphacidae) in west Java,Indonesia

Spatial distribution pattern of the brown planthopper (BPH) was analyzed at 9 experimental fields in the northern part of West Java during two consecutive rice cropping seasons, i.e., wet and dry seasons. The population of each developmental stage and wing form of BPH at each location showed consistent departure from the random (Poisson) distribution, the variances of the densities in most cases exceeding their means. Namely, the distribution pattern of BPH per hill of rice plant was found to have a general tendency to be aggregated or contagious and to fit fairly well to the negative binomial model. The tendency for aggregation was further confirmed by both the β-values of -m regression being larger than unity and the C_A-values being larger than zero for each developmental stage. Although significant variations in the distribution pattern as measured by β- or C_A-value were observed between different developmental stages, between wing forms and among locations, the degree of aggregation for a given developmental stage at each experimental field remained fairly stable throughout the crop period, despite wide temporal changes in population density. Possible factors to explain these characteristics of the spatial distribution pattern of the BPH in West Java were discussed with reference to the process generating it.     

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.     

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.

     

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.     

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.

     

Cumulative Damage Survival Models for the Randomized Complete Block Design

A continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.     

Distribution of Polymorphus minutus among its intermediate hosts

Hirsch R. P. 1979. Distribution of Polymorphus minutus among its intermediate hosts. International journal for Parasitology10: 243–248. In 1971, Crofton investigated patterns of distribution of Polymorphus minutus in the intermediate host, Gammarus pulex. Among his conclusions were: (1) P. minutus populations occur in patterns similar to negative binomial distributions, and (2) parasite-induced host mortality results in patterns similar to truncated (high end) negative binomial distributions. Those conclusions, however, were not tested by statistical analyses. To test Crofton's observations, Chi-square goodness of fit tests were applied to data used by Crofton and an additional two stations sampled by Hynes & Nicholas in 1963. Analyses were expanded to include five theoretical distributions, four patterns of host mortality and various rates of host mortality. Truncated forms of negative binomial, positive binomial and Poisson distributions were also investigated where nontruncated distributions failed to fit observed distributions. It was found that negative binomial distributions most frequently describe patterns of P. minutus distribution with the exception of one population described by Poisson and another by positive binomial distributions. Crofton's assumption that truncated distributions result from parasite-induced host mortality seems unlikely in light of those analyses.     

Characterizations of Continuous Binomial and Negative Binomial Mixtures

Characterizations are given for binomial (n, ?) and negative binomial (k, ?) mixtures in terms of the regression function of the mixing random variable Y on the mixture variable X. In both cases Y is the parameter ? and the mixing distribution is a continuous distribution on (0, 1) and on (0, ∞) respectively. Biological applications are indicated.     

The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

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.     

Mixtures of Distributions by the Poisson Distribution of Order K

The Poisson-binomial, the Poisson-negative binomial (or Pascal) and Neyman's Type A distribution, which are Poisson mixtures of the binomial, the negative binomial and the Poisson distribution, respectively, have received a lot of attention in statistical literature [see e.g. Katti and Gurland (1961, 1962), Anscombe (1950), and Neyman (1939)]. In the present paper, their respective generalizations are introduced and briefly studied, when the Poisson distribution of order k [see Philippou (1983), Philippou, Georghiou and Philippou (1983), and Charalambides (1986)] replaces the Poisson distribution in its mixing role.     

An index of spatial distribution

The index , which takes the value of −1 for a perfectly regular distribution, +1 for a highly aggregated distribution and 0 for the distribution implied by the theoretical varianceσ², is proposed and a significance table provided.     

Point and Interval Estimation of the Population Size Using a Zero‐Truncated Negative Binomial Regression Model

This paper presents the zero‐truncated negative binomial regression model to estimate the population size in the presence of a single registration file. The model is an alternative to the zero‐truncated Poisson regression model and it may be useful if the data are overdispersed due to unobserved heterogeneity. Horvitz–Thompson point and interval estimates for the population size are derived, and the performance of these estimators is evaluated in a simulation study. To illustrate the model, the size of the population of opiate users in the city of Rotterdam is estimated. In comparison to the Poisson model, the zero‐truncated negative binomial regression model fits these data better and yields a substantially higher population size estimate. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)