Counter-Examples about Lower- and Upper-Bounded Population Growth

ABSTRACT

For a unimodal growth function f having its maximum at a critical state x _c , the interval bounding the population size asymptotically is usually presented as being equal to [f ^○2(x _c ), f(x _c )]. This interval however does not represent the maximum range within which the population size can vary, even asymptotically. The actual invariant interval containing the population size is equal to: [min(x*, f ^○2(x _c )), f(x _c )], where x* denotes the non-zero fixed point, assumed to be unique, of the iteration of f.     

Calculating life tables by estimating Chiang’s a from observed rates

A simple, accurate method of life table construction is advanced based upon a new way to estimate Chiang’s _n a _x (the average number of years lived in the x to x + n age interval by those dying in the interval). The estimate for _n a _x leads immediately to an expression for l _x+n (the survivors to age x + n) in terms of l _x and the known mortality rates for the interval x to x+n and the two adjacent intervals. The complete solution for the basic life table is given. The proposed method and five other easily applied methods are then compared against the standard provided by the U.S. life tables for 1969–1971. The results attest to the excellent performance and high degree of accuracy of the proposed method. Finally, extensions of the method to multiple decrement and associated single decrement life tables are briefly described.     

Survival of related individuals: An extension of some fundamental results of heterogeneity analysis

Many ideas in the analysis of heterogeneous mortality are based on the relationship between individual and observed hazard rates. This connection is established with the help of conditional averaging procedure: The observed risk of death at age x is calculated among those who survive this age. The analogy of this result for bivariate survival model with correlated individual hazards is derived. In the case of correlated frailty model the parametric specification of the mean, variance and correlation coefficient of the bivariate frailty distribution among survivors is obtained. The relationship between local association measure and the characteristics of the bivariate frailty distribution among survivors is established.     

Sex-Structured Dynamic of Multi-Group Herpes Simplex Virus 2

A 3(n + l)-dimensional ordinary differential equation for HSV-2 includes l groups of men and n groups of women with different risks of infection. Global Lyapunov functions based on graph theory and on LaSalle invariance principle show that the model dynamics are completely determined by the basic reproduction number ?₀. The disease-free equilibrium is globally asymptotically stable when ?₀ ≤ 1; a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region when ?₀ > 1.     

Theoretical studies on the role of food exploitation for the competition of scaamble type

Summary A model was made to clarify the basic processes of competition to occur among larvae by the exploitation as defined byBakker (1969). It was found that this model is applicable to the experimental results on the food exploitation amongDroshophila larvae obtained byBakker (1961). In the model the preimaginal stage is divided into two periods;T _f which is the time that a group of larvae spends in exhausting the food after hatching, andT _s which is the duration of the starvation period afterT _f .T _f and thenW _l (larval body weight) just after the end ofT _f are decided byF _s (amount of food supplied per larva at larval hatching) andF _c (amount of food consumed per larva).T _f affects on the onset ofT _s as well asR _l (rate of decrease in the individual body weight duringT _s ).W _a (weight of emerging adults) is gotten by a subtraction ofR _l fromW _l just after the end ofT _f ,R _e is affected directly by these components ofW _l andR _l . As a result,W _a andR _e are expressed by functions ofF _s . This model confirmed that the food exploitation lead to the competition of scramble type. Finally it was suggested that there exist some strategies which prevent ill-effects owing to the food exploitation.     

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.     

Tuberculosis with relapse: A model

ABSTRACT

In a model of tuberculosis with relapse, the basic reproduction number R₀ includes new and relapse infections. Lyapunov functions help to prove that the global dynamic is completely determined by R₀. Replicated Latin hypercube sampling shows that early diagnosis and treatment are more efficient when relapse cases are considered.     

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.     

Prey capture tactics of spiders: An analysis based on a simulation model for spider's growth

Summary The prey capture tactics of spiders was analyzed, considering the energy gained by the capture of prey and that required for it. For the purpose of it, a growth model of spiders was constructed, expressing the flow rate of prey biomass to the spider's body by differential equations. Solving these equations under the differing values of three parameters, growth curves of spiders was obtained. These three parameters are the amount of prey biomass supplied daily to spiders,x ₀, the rate of prey capture of spiders, α, and a coefficient of the respiration rate required for the capture of prey,k. When the value ofk increased, spiders could grow only at high value ofx ₀. These results suggest that habitats with small prey biomass are preferred by spiders adopting a sit-and-wait tactics for prey capture, which requires small values ofk. Wolf spiders are one of these spiders showing that tactics. On the other hand, web-builders which require large amount of energy for spinning webs (namely, take large value ofk), are able to grow only in the habitats with large prey biomass. Each species of spiders are considered to locate in a certain point between both extremes of these tactics for the capture of prey.     

Intermediate variables and educational differentials in fertility in Korea and the Philippines

This analysis compares the effects of contraceptive use and infant and fetal mortality on the pace offertility in Korea and the Philippines and explores the mediating effects of these intermediate variables on educational differentials in childspacing. For birth intervals initiated in a recent period before a sample survey, second, third and higher-order intervals are examined. Transitions within successive segments of interval exposure (q _xvalues) are examined rather than cumulative transitions (1 - l _xvalues). This methodological choice is substantively important because breastfeeding should primarily affect early segments of exposure and because it allows empirical examination of the timing of the effects of other variables such as contraceptive use. Further, this choice allows multivariate analysis within the structure of the life-table perspective. The results show substantial differences in patterns between Korea and the Philippines, indicate clearly the effect of each intermediate variable, and illustrate how educational differentials in fertility are affected by contraception and infant and fetal mortality.     

Population growth with variable family size

The Sharpe‐Lotka continuous time deterministic model of population growth is developed to take account of some possible forms of mother‐daughter fertility association, characterised here by a bivariable measure, A. This leads to a linear double integral equation for which, subject to certain conditions, a finite time solution can be found by Laplace transform methods and thus also model specific results relating the intergenerational fertility effect to the long term population growth rate and magnitude are established. The quantitative implications of the theory are illustrated by a consideration of a general bilinear form of A and in this context numerical results illustrating the finite time growth and also the long term distribution of fertility levels in the stable female population are obtained. In particular, it is shown that different fertility specific subpopulations can coexist indefinitely.     

Survival Convergence and the Preceding Mortality Crossover for Two Population Subgroups

In this paper, we present and develop the argument that if the survival functions for two population subgroups converge in later life, a mortality crossover must precede the occurrence of this convergence. Specifically, two survival curves, S ₁(x) and S ₂(x), associated with two distinct population subgroups, G₁ and G₂, tend to converge before all members die out, as often observed and anticipated. This convergence leads to an increased mortality acceleration for the “advantaged” group, and eventually fosters the occurrence of a mortality crossover. We present a mathematical proof for this relationship and offer several explanations for the mechanisms involved in the process of survival convergence and the preceding mortality crossover. This new presentation demonstrates that mortality crossover is a highly observable demographic event given the trend of survival convergence in later life.     

Adult population parameters and life tables of an epilachnine beetle (Coleoptera: Coccinellidae) feeding on bitter cucumber in Sumatra

Summary The population dynamics of an epilachnine beetle, which is closely related toEpilachna sparsa Dieke (henceforth called “sp. C”) and feeds on bitter cucumberMomordica charantia, was studied by mark-recapture of adults and the construction of life tables. The study was repeated three times, i.e., March–May, July–September and October–December in 1982, in Padang, Sumatra, Indonesia. After the establishment of the host plants, adults of “sp. C” soon colonized, and each study period ended in the death of the plants due to defoliation by the larvae and adults. The estimated mean length of residence of adults ranged from 6–11 days, but this was probably much shorter than the actual longevity, because the adults were so active that they flew away, or dropped off the plants, when they were approached or slightly disturbed. Life tables indicated that egg mortality ranged from 17.8–53.9%, and a parasitic waspTetrastichus sp. B made up 41.1–64.2% of egg mortality. Two wasps,Tetrastichus sp. C andPediobius foveolatus killed 1.2–19.4% (7.6–100%)^* of 4th instars and only the latter species attacked the pupae, killing 24.6–59.1% (45.1–72.4%). Parasitism and starvation by overcrowding contributed most to the total mortality from egg to adult emergence, which ranged from 89.4–99.5%. “Sp. C” had a higher diversity and level of parasitism than the Japanese species,E. vigintioctopunctata. The high dispersal power of “sp. C”, coupled with the prolongedl _x−m_x schedules shown under laboratory conditions, was advantageous for exploiting the food plant which was available throughout the year, but was rather patchily distributed in space.     

Loss of genetic variability in a fragmented continuously distributed population

An individual-based simulation model was used to examine the effect of population subdivision, dispersal distance of offspring, and migration rates between subpopulations on genetic variability(H ₁ H _S andH _T ) in a continuously distributed population. Some difficulties with mathematical models of a continuously distributed population have been pointed out. The individual-based model can avoid these difficulties and can be used to examine genetic variability in a population within which individuals are distributed continuously and in which the dispersal of individuals is disturbed by geographical or artificial barriers. The present simulation showed that the pattern of decrease inH ₁ had three stages. During the first stage,H ₁ decreased at the rates predicted by Wright’s neighborhood size. During the second stage,H ₁ decreased more rapidly when the migration rate decreased, while during the third stage, it decreased less rapidly when the migration rate decreased. Increasing the number of subdivisions increased the rate of decrease after the 200th generation. The pattern of decrease inH _T was classified into 2 stages. During the first stage, the rates of decrease corresponded with those of a randomly mating population. During the second stage, a decrease in the migration rates of the subpopulations slowed the rate of decrease inH _T . A uniform spatial distribution and a reduced total dispersal distance of offspring causedH ₁ H _S , andH _T to decrease more rapidly. Habitat fragmentation in a continuously distributed population usually was detrimental to the genetic variability in the early generations. Other implications of the results for conservation are discussed.     

Random walk Green kernels in the neutral Moran model conditioned on survivors at a random time to origin

In the theory of finite discrete-time birth and death chains with absorbing endpoint boundaries, the evaluation of both additive and multiplicative path functionals is made possible by their Green and λ–potential kernels. These computations are addressed in the context of such Markov chains. The application to the neutral Moran model of population genetics yields first hitting and return times. A neutral Moran bridge model, forward and backward in time, for a given total number x of survivors of a single common ancestor at some random time T to the origin of times, yields the age of a mutant allele currently observed to have x copies of itself. This forward theory of age, made possible by Green kernels, is comparable to Watterson’s backward theory of age, which makes use of the reversibility of the Moran chain.     

Impact of lek mating on the sterile insect technique: A modeling study

Summary Modelling studies are presented which describe the effect of lek mating on the control of a wild population by sterile male release. The mixed leks are assumed to follow a Poisson-binomial distribution and the system includes three parts: territory defense, matings inside a lek and matings outside a lek. The effects of parameters on the hatchability are discussed. Among the parameters, sterile type effect (W _s ), female choice (f _s ) and mating competitiveness (C _m ) are the most important. The application to determining the effects of sterile male release and on the proportion of sterile males required for eradication are also discussed.     

Life table technique for the working ages

Some results of efforts to design an integrated formula system for age-specific death rates, survivals and expectation of life are presented. The report deals only with the active ages 15–80. The system is based on the assumption that the age-specific central mortality rate (m _x ) can be expressed satisfactorily by means of a polynomial m = a ₀ + a ₁ x + a ₂ x ²+ ….     

Preliminary studies on the respiratory energy loss of a spider,Lycosa pseudoannulata

Summary To elucidate the basic food requirement of spiders, the important polyphagous predators of rice-plant insect pests, an attempt was made to measure the respiratory energy loss of fasting spiders,Lycosa pseudoannulata. Relationship between fresh (y) and dry (x) weights of spiders inhabiting the bottom layer of the rice-plant community was represented by the following allometric equation:y=0.428x ^0.872. The carbon dioxide production by previously fed and unfed females under the dark at 29°C 100% R. H. was measured by a titration technique. The relationship between fresh body weight and CO₂ production by unfed animals could be represented by the equationM=aW ^b, M being the CO₂ output per individual per day andW the fresh body weight. The constantb, which determines the slope of curve, was 0.808. Respiration of the adult female with 100 mg fresh weight was 1.155±0.250 mg CO₂/100 g fresh weight/day or 48.69 mg CO₂/g dry weight/day. This value corresponds to 35.81 cal/g fresh weight/day or 150.94 cal/g dry weight/day. Supposing the calorific content of spiders to be 5820 cal/g dry weight, rate of the respiratory energy loss to total energy of the body was estimated to be 2.60%. This rate did not strongly contradict with the loss of fresh body weight before and after the measurement. The metabolic rate showed remarkable fluctuation with changing food supply. The CO₂ production of starved individuals decreased to 83.63±16.34% as compared with individuals which were fed before the measurement.     

A stochastic computer model for simulating population growth

Summary A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model’s behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka’s Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values orr _m andK. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetleTribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.