An index of spatial distribution

Summary 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 varianceσ ², is proposed and a significance table provided.     

Spatial pattern indices based on distances between individuals on a line-segment with finite length

Summary Let us consider a strip-wise habitat of line-segment, like a corridor, to simplify the subject mathematically, and assume that the length of the habitat is γ and there aren individuals. Here, we assume that the spatial pattern of the individuals is random if then distances from the left end of the habitat to each individual follow a uniform distribution on the strip. Under such an assumption, the variance of the distances between any two neighbors is represented by the formulanθ ²(n+1)⁻²(n+2)⁻¹ and the variance betweenn+1 distances betweenn individuals from the left end to the right end to the strip, is represented by the formulanθ ²(n+1)⁻²(n+2)⁻¹. These two kinds of variances can be used for determining (1) the spatial pattern of a population on the strip and (2) the spatial structure within the population, by comparison with the variances calculated from the data. Two examples cited from the literature, a cattle population on a pasture and an aphid population on a sycamore leaf, are presented.     

The importance of non-linearity: A comment on the views of Taylor

Summary Taylor's (1984) view that his power law is better measurement of the distribution pattern of animals than 1/k, I _ρ and relation because almost all the data on means and variances of given species could be fitted by the power low, was critisized. Changes in values of 1/k, I _ρ and suggest the change of distribution pattern of the species, but the data can still be fitted by the power law because of the great linearizing power of log-log plots, and, using the power law only, we shall overlook the information on the meaningful changes in the distribution pattern.     

Binomial sampling plans for estimatingBemisia tabaci populations in cantaloupes

Early season infestations of the sweet potato whitefly,Bemisia tabaci (Gennadius), on cantaloupes,Cucumis melo L., were determined by counts of the number of adults per leaf in fields near Yuma, Arizona. We used these data to develop binomial sampling plans based on the relationship between mean densities of whiteflies per leaf,m, and proportion of leaves infested with more thanI whiteflies,P _I, according to the empirical model lnm=a′+b′ ln[−ln(1−P _I)]. The models were developed for the presence-absence approach (I=0) and for a cutoff value of three whiteflies per leaf (I=3). Four independent data sets were used to evaluate the models. Both methods yielded reliable predictions at low infestation levels, but some of the higherm values were overestimated. As the tentative economic threshold forB. tabaci is three adults per leaf, which corresponds to lowP _I values, results of the binomial sampling were satisfactory for pest management purposes.     

Analysis of spatial association between two species based on the interspecies mean crowding

Summary and Conclusion The measurement of spatial association between two species is considered on the basis of interspecies mean crowding. Two indices of overlapping, γ andC _p, are derived as geometric and weighted arithmetic means of the same component ratios related to inter-and intraspecies mean crowdings. Both indices behave in a similar way, ranging from 1 when the distributions of two species are completely overlapped to 0 when they are completely exclusive with each other. The former is essentially identical with indices proposed byKuno (1968) andPianka (1973), and the latter is a modified form ofMorisita's (1959)C _σ index. Indices to measure the degree of spatial correlation between species, Ω andR _μ, are then derived for both kinds of overlapping indices, which vary from 1 in complete overlapping, through 0 in independent occurrence, to −1 in complete exclusion. Various kinds of interspecies association are analyzed using these indices and an extended form of the regression graph which provides a convenient way of indicating the spatial interrelation between two species as well as distribution patterns of respective species. The method presented in this paper may also be applicable to compare temporal distribution patterns between species, similarity between communities, etc. For such a wider application which includes continuous as well as discrete distributions, the interpretation of intra-and interspecies mean crowdings is not necessarily appropriate, and hence the concept of mean concentration with the symbols and for intraspecies relation and and for interspecies relation is suggested. This study was supported by Science Research Fund (No. 148041) from the Ministry of Education.     

Studies on the spatial distribution pattern of larvae of the mosquito,Anopheles sinensis, in rice fields

Summary The spatial distribution patterns of the population ofAnopheles sinensis larvae were studied in the rice field area in the suburb of Urawa city in Japan, during the summer seasons in 1973 and 1974. The distribution pattern of the larval population within the field, analysed by the m−m regression method, indicated that the basic component of larval distribution was not a group of individuals but a single individual and such components were distributed contagiously over the field. This basic pattern did not change significantly according to developmental stage, census date or field. Therefore, we could describe the distribution pattern of the population in a rice field by the single linear regression, x=0.021+1.339x(r²−0.912). Also, the relation for the whole population in the field area including the five fields could be shown by the linear regression, x=0.049+1.749x(r²−0.959). The value of α remained to be nearly equal to zero, but the value of β became larger than the value for the single-field relation. Such a change in distribution pattern seemed to reflect the greater heterogeneity in conditions among the fields than within individual field. Using the information on the distribution patterns mentioned above, some considerations were given on the sampling plans for mosquito larvae, including samplesize determination and application of sequential methods to estimate population size as well as to classify population level.     

Yearly variation in recruitment and its effect on population dynamics inYoldia notabilis (Mollusca: Bivalvia), analyzed using projection matrix model

Summary Long-term variation in recruitment was estimated by constructing projection matrices for a marine bivalve,Yoldia notabilis, at two stations in Otsuchi Bay, northeastern Japan, and the effects of its variation on population dynamics were examined using a simple matrix model. The matrix model was developed from the Leslie matrix, in which the population growth rate λ was expressed as a function of recruitment rater ₀. The equilibrium recruitment rater _s, or the recruitment rate required to maintain population at constant size (λ=1), was expressed by the reciprocal of the reproductive value of a newly recruited individual. The estimates ofr _s for the field population were lower at the shallower station than at the deeper station, reflecting higher survivorship and fecundity. Past recruitment rate estimated both by the field samplings for 3 years and by the back-calculation from the current age structure for over 10 years showed large yearly variation, ranging between 0 and 58.6×10⁻⁴. The estimates were larger thanr _s, and hence, large enough to increase population size (λ>1) only in approximately one-third of the estimated years. This suggests that the population has been maintained by occasional successful recruitment occurring once every few years.     

Measurement of non-randomness in spatial distributions

Summary The measurement of departure from randomness in spatial distributions has widespread application in ecological work. Several “indices of non-randomness” are compared with regard to their dependence on sample number, sample size and density. Criteria for the best choice of index for specific situations are discussed. A new coefficientC _x is proposed for use with positively contagious distributions and tests of significance are given. WhenC _x and another index (S ²/m−1) are used for positive and negative contagion respectively, values ranging from −1 through 0 (random) to +1 are obtained, regardless of sample number, sample size or density.     

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

Summary 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 theC _A -values being larger than zero for each developmental stage. Although significant variations in the distribution pattern as measured by β- orC _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.     

On the momentum of population growth

If age-specific birth rates drop immediately to the level of bare replacement the ultimate stationary number of a population will be given by (9): $$\left( {{\textstyle{{b\mathop e\limits^ \bullet {}_0} \over {r\mu }}}} \right)\left( {\frac{{R_0 - 1}}{{R_0 }}} \right)$$ multiplied by the present number, where b is the birth rate, r the rate of increase, \(\mathop e\limits^ \bullet _0 \) the expectation of life, and R ₀ the Net Reproduction Rate, all before the drop in fertility, and μ the mean age of childbearing afterwards. This expression is derived in the first place for females on the stable assumption; extension to both sexes is provided, and comparison with real populations shows the numerical error to be small where fertility has not yet started to drop. The result (9) tells how the lower limit of the ultimate population depends on parameters of the existing population, and for values typical of underdeveloped countries works out to about 1. 6. If a delay of 15 years occurs before the drop of the birth rate to replacement the population will multiply by over 2. 5 before attaining stationarity. The ultimate population actually reached will be higher insofar as death rates continue to improve. If stability cannot be assumed the ultimate stationary population is provided by the more general expression (7), which is still easier to calculate than a detailed projection.     

Relationship between surplus energy and body weight in fish populations

Summary This paper theoretically analyses the relationship between surplus energy, which is available for either somatic growth or reproduction, and body weight. From the data of metabolism and growth of the biwamasu,Oncorhynchus rhodurus, obtained by Miura et al., a Bernoulli's differential equation is induced to represent the relationship between body weight and the sum of surplus energy and active metabolic rate. Solving this equation gives the amount of surplus energy,f(W_x), as follows:f(W_x) = (αW _x ^1−γ +β^1−γ)^1/(1−γ)−W_x, in which α, β and γ are constants andW _x is body weight at agex. The function is applied to ten fish populations and consequently it is found to be useful for a wider age range and a wider variety of fishes than the conventional function.     

A probabilistic interpretation of the United Nations’ 1995–2005 population projections

I develop probabilistic interpretations for the United Nations’ 10-year population forecasts by comparing 1995 projections for 212 countries to the population sizes reported for 2005. Errors in the estimation of the intrinsic rate of increase, presumably caused by erroneous assumptions about birth, death and/or immigration rates, appear to be more consequential than errors based on inaccurate estimation of the starting, or ‘jump-off’, population size. For only about 20% of the countries did the ‘actual’ 2005 population size fall between the United Nations’ low- and high-variant projections. I propose prediction intervals for country-specific population sizes 10 years in the future of the form [ N_i^￠ (t+10) / k ,  k ·N_i^￠ (t+10) ],[ N_i^{\prime} (t+10) / k , \, k \cdot N_i^{\prime} (t+10) ], where N _i ′(t + 10) is the medium-variant prediction for year t + 10 made in year t, and k is a number that varies with starting population size. Based on the 1995–2005 United Nations’ data, values of k giving 95% coverage range from 1.11 for countries with a population on the order of 10⁹, to 1.45 for countries with a population of 10⁵.     

Some extensions of the keyfitz momentum relationship

A stable population, such that the total birthrateB(t) =B _o e ^ro^t, is abruptly altered by modifying the age-specific birth rate,m(x). The survivor function remains unaltered. The modified population ultimately settles down to a stable behavior, such thatB(t) =B ₁ e ^r ₁ ^t . It is shown thatB ₁/B ₀ = (R ₀ ?R ₁)/[(r ₀ ?r ₁)R ₀ Z ₁], whereR ₀,R ₁ are the net reproduction rates before and after the change, and \(\bar Z_1 \) expected age giving birth for the stable population after the change. The age structure and transients resulting from the change are also described. The effect of an abrupt change in the survivor functionl(x) is also investigated for the simple case where the change is caused by alteringl(x) toe ^?λx l(x). It is shown that the above ratio becomes \(B_1 /B_0 = N_1 /N_0 = [1 - \smallint _0^\infty e^{ - kx} g(x)dx]/\bar Z_1 \lambda \) , whereN refers to the numbers in the population,k =r ₀ + λ, andg(x) =m(x)l(x), the value before the change. A measure for the reproductive worth of the population is also established.     

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

Summary 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.     

Larval growth dynamics ofHemipyrellia ligurriens (Calliphoridae) andBoettcherisca formosensis (Sarcophagidae) in crowded and uncrowded cultures

Summary Larval growth patterns ofHemipyrellia ligurriens (Calliphoridae) andBoettcherisca formosensis (Sacophagidae) in crowded and uncrowded cultures were compared. Growth of the larvae followed a sigmoid curve. The highest larval growth rates were 0.33 and 0.36 mg h⁻¹ for uncrowded and crowdedH. ligurriens respectively. The corresponding figures were 2.38 and 1.23 mg h⁻¹ forB. formosensis. Larvae of both species attained maximum weight earlier in crowded cultures than in uncrowded cultures, although the final weights attained in crowded cultures were less. The earlier period of most rapid growth in both species was interpreted as a result of intraspecific facilitation at higher larval densities. UncrowdedB. formosensis had a shorter larval to pupal development and an earlier period of most rapid growth than uncrowdedH. ligurriens, suggesting the former may be superior in exploiting carcasses with limiting food.     

Some strange properties of the logistic equation defined withr andK: Inherent defects or artifacts?

Summary In some situations the logistic equation in the usual expression, dN/dt=r(1−N/K)N, exhibits properties that are biologically unrealistic. For example, whenr≦0 the population can no longer show any normal, negative response in per-capita growth rate to increasing density. Also, when the equation is employed in the Volterra's competition model, a familiar but incredible conclusion is derived which says that the outcome of competition is entirely independent of the reproductive potentialr of each species. It is shown that all such strange properties are mere artifacts arising peculiarly in thisr-K model from its misleading implicit supposition thatK could be independent ofr, and they can be readily removed by alternative use of a plainer, classical form of the model, dN/dt=(r−hN)N.     

Body Image and Quality of Life in a Group of African American Women

African American (AA) women’s preference for a larger body size and underestimation of their body weight may affect the relationship between their body weight and weight-related quality of life (QOL). We wanted to examine the relationship between weight-related QOL and body mass index (BMI) in a sample of overweight AA women. Thirty-three overweight AA women completed a clinic visit to measure height, weight, and complete surveys including the Impact of Weight on Quality of Life-Lite (IWQOL-Lite) and the Stunkard Figure Rating Scale. BMI was calculated using measured height and weight. Correlations and linear regression models were estimated using SAS v 9.1. In this sample, the mean total quality of life score was 78.00 ± 17.68 on a 100 point scale. There was a modest correlation between BMI and total weight-related QOL (r = −0.034, p = 0.053). Body image dissatisfaction was the strongest predictor of total quality of life score (p = 0.04). African American women’s unique cultural perception of body image may play a key role in weight-related QOL.     

Spatial pattern analysis of the incidence of aster yellows disease in lettuce

The degree of aggregation of lettuce plants infected by aster yellows phytoplasma (AYP) was investigated in 12 fields from three experiments. Position of diseased and healthy plants was mapped in a 6–9×12-m section of each field; for most analyses, fields were divided into 10-plant quadrats. Mean disease incidence (p) ranged from 0.01 to 0.30. The frequency of diseased plants was described by the beta-binomial distribution, with an index of aggregation (θ) ranging from 0 to 0.17, positively correlated withp, and generally increasing over time within a field. Distance-class analysis revealed a core-cluster size of only a few plants. However, spatial autocorrelations ofp between quadrats were not significant, indicating that the scale of spatial pattern was small, generally less than 10 plants. An overall measure of aggregation was given by the slope parameter of the binary form of the power law, in which the log of the calculated variance is regressed on the log of the theoretical variance for a binomial distribution. The slope was 1.18 and significantly different from 1. Results for this “simple-interest” disease are interpreted in relation to the persistent transmission of AYP by its aster leafhopper vector.     

Analysis of changes of insect-plant ratio-improvement of key-factor analysis for evaluating bottom-up effects in insect population dynamics

A revised key-factor analysis was presented for analyzing the temporal changes in the ratio of insect absolute number to plant resource. Ten data sets for 5 insect species were then analyzed. In this key-factor analysis, the key factor is defined as the factor contributing highly to between-year variation inR _r , the log rate of the inter-year change of the insect-plant ratio. The yearly change of plant resource was handled as a separate factor, expressed byr _pl , log ratio of plant resource in yearn to plant resource in yearn+1. The following was revealed: 1) In 7 of the 10 data sets examined,r _pl influenced variations ofR _r ; in particular in 3 casesr _pl was the main key factor. 2) Generation-to-generation fluctuations of absolute insect densities showed density dependence in 4 cases, while those of insect-plant ratios, in 8 cases. 3) The Royama model or a linear model, explained well the relationship between log insect-plant ratio (X _r ) andR _r and the relationship betweenX _r and log yearly change rate of absolute insect density (R _abs ). However, in the 7 cases in whichr _pl was a critical factor for variations ofR _r , with, increase ofX _r ,R _r showed a steeper, decrease around the equilibrium point (the point for whichR _r is 0) thanR _abs . This occurred becauser _pl tended to be negatively correlated withX _r . Consequently, in two casesX _r fluctuated cyclicly or chaotically although without the changes in plant resource, fluctuations ofX _r would be damped oscillations approaching equilibrium.