Stability Analysis and Dynamics Preserving Nonstandard Finite Difference Schemes for a Malaria Model

When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R ₀ is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough, R ₀ = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a specific algebraic decomposition into infective and non-infected humans and mosquitoes. Stability results are extended and the possibility of backward bifurcation is clarified. A dynamically consistent nonstandard finite difference scheme is designed.     

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.     

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.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Analysis and simulation modelling of population dynamics and bioenergetics ofCryptolestes ferrugineus (Coleoptera: Cucujidae) in stored wheat

Summary The consequences of infestation of stored wheat by the rusty grain beetle,Cryptolestes ferrugineus (Stephens) was determined for 222 d at 30°C in 70-1 drums containing wheat at 13.5% moisture content. Temperature, grain moisture, seed damage, germination and weight, dust weight, fat acidity values (FAV), published data on growth, reproduction, survival and cannibalism rates and energy budget were used to develop a computer simulation model to simulate the population dynamics ofC. ferrugineus at 30°C. In the insect-free control system, the fungi,Alternaria alternata decreased,Aspergillus glaucus group andPenicillium spp. increased, probably causing a rise in FAV of the grain. In the insect-infested system,C. ferrugineus could only eat the wheat germ of kernels that had a broken bran layer; 35.7% of the wheat germ or 914.6 J per 100 kernels was consumed. Within two generations after initial introduction,C. ferrugineus reached a peak in numbers and biomass polluting the ecosystem with excreta and remains, and accelerating the deteriorative process observed in the insect-free control system by increasing respiration temperature, FAV and reducing grain germination. After 87 d, the insect population declined to low levels. The simulation model provided a close match between the observed and predicted numbers of insect life stages and bioenergetic variables during the insect population growth phase. Simulation trials suggested that cannibalism of larger compared with smaller immature stages would be more wasteful of developmental time and energy, reducing the number of individuals reaching reproductive age, and that density-dependent fecundity was probably not an important regulatory mechanism ofC. ferrugineus population dynamics in this study. Contribution No. 1314 from Agriculture Canada Research Station, Winnipeg, Manitoba, R3T 2M9, Canada     

Dynamics of Mixed Coalitions Under Social Cohesion Constraints

ABSTRACT

Parameters for the birth and death diffusion life table model subject to downward jumps randomly occurring at a constant rate are estimated. The jump magnitudes have a beta distribution with support [0, l_x ], where l_x is the total number of survivors prior to the jump. The estimation method is maximum likelihood. The Cramer–Rao Lower bound and the asymptotic distribution for the MLE are derived. The model is applied to the U.S. men′s population from 1900 to 1999.     

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.     

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.     

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.     

Turbulent dynamics in a Xviith century population

A reconstruction of the population of the Pays de Caux (1589–1700) yields the time series of a fertility behavior indicator, the overall Coale index If. In spite of the noisy appearance of its evolution, the trajectory of If looks ordered, as if it were confined alternatively to two given zones, looping in each of them for a while, then suddenly jumping from the low one to the higher one, or slowly whirling down from the high to the low one.

An attempt is made to explain this general temporal structure by using a simulation model based on the autoregulation model (the so‐called European Marriage Pattern), putting into play a choice of the spouse function, a fertility function, modalities of marriage and remarriage, under the environmental forcing of the reconstructed mortality conditions.

The correspondence between reconstruction and simulation turns out to be quite good, not only for the population size or the Coale index, but also for the marriage series, quite independently of the reconstructioa

A second simulation with simulated mortality conditions shows a bifurcation point: as the mean frequency of crisis increases, the state of the system leaves the lower level and concentrates more and more in the higher level.

Thus, not only does the autoregulator model appear validated by empirical data, but its bi‐modal structure is revealed, depicting the dynamic response of a traditional community both to the environment and to the endogeneous demographic process.     

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.     

Note on the accuracy of local population estimates

Mr. Silcock's article will be of interest to all concerned with local population data. It may be useful to supplement it by a brief account of the fuller examination of the local population estimates made in 1951 by the General Register Office, since this covered all 1472 administrative areas in England and Wales and could be made in more detail than was possible for a private investigator.

Any census, of course, provides information not available, at least in such detail, at other times or from other sources, and also serves as a base from which estimates for succeeding years can be derived. In addition, however, the General Register Office takes the opportunity of a census to try and assess the accuracy of the various types of current population estimates made by the Department. In the case of local administrative areas the comparison of actual and expected populations made after the 1931 Census is discussed in the Text Volume of the Registrar General's Statistical Review for 1930 (pages 100-102).     

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.     

SHORT REVIEWS

Books reviewed in this issue. Juha M. Alho , Svend E. Hougaard Jensen, and Jukka Lassila (eds .) Uncertain Demographics and Fiscal Sustainability Simone M. Caron Who Chooses? American Reproductive History Since 1830 Ian Dowbiggin The Sterilization Movement and Global Fertility in the Twentieth Century Duncan Green From Poverty to Power: How Active Citizens and Effective States Can Change the World World Health Organization World Malaria Report 2008     

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.     

The species-area relation II. A second model for continuous sampling

Summary A second mathematical model describing the species-area relation was proposed for continuous expanding of sample area. This model is expressed asS=λ ln(1+x/E) whereS is the number of species occurring in an areax, and λ andE are the constants termedspecific diversity andelemental area respectively. As a result of testing the validity of the model for several sets of data, it was shown that the above equation would provide an adequate fit to a group of species belonging to a single synusia which exists in an open habitat. The ecological implications of parameters involved were discussed and the characteristic area presented previously (Kobayashi, 1974) was defined in terms ofE. The relation between results obtained by discrete sampling and continuous sampling was examined and the possibility of converting one to another was suggested. Contribution from the Laboratory of Applied Zoology, Yamagata University, No. 79.     

Age patterns of mortality for older women: An analysis using the age‐specific rate of mortality change with age

The age‐specific rate of mortality change with age, defined by k(x) = d Inμ(x)/dx, where μ(x) is the age‐specific death rate at exact age x, is estimated for middle and old ages in ten selected populations that are considered to have relatively accurate age data. For females in each of the study populations, k(x) follows a bell‐shaped curve that usually peaks around age 75. In some of the populations, the age pattern of k(x) for males is confounded with substantial cohort variations, which seem to reflect long‐term impacts of their World War I experiences.

Among the mathematical models proposed by Gompertz, Makeham, Perks and Beard, only the Perks model is consistent with the bell‐shaped pattern of k(x). It is shown that, if the risk of death for every individual follows the Makeham equation and if the individual frailty is gamma‐distributed, then the age‐specific death rate follows the Perks equation.     

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.     

Estimating the goodman,keyfitz, pullum kinship equations: An alternative procedure∗

As is often the case in demography, Goodman, Keyfitz and Pullum (1974) developed their theory of the interrelationships of fertility, mortality and kinship numbers by means of continuous mathematics [integrals], but resorted to ad hoc finite approximations for calculating results in concrete empirical cases. Their reason: ‘Ordinarily, we cannot evaluate the l(x) and m(x) functions for arbitrary values of x, since the data are usually collected for five‐year age intervals’ [p. 24]. Recent developments in computer software now provide an alternative, two‐step procedure that avoids extensive programming of finite approximation algorithms: 1) using a popular scientific curve‐fitting package, functions are found to represent particular sets of discrete data on fertility and mortality, 2) the resulting functions and parameter estimates are then inserted directly into the kinship equations, and the integrals evaluated numerically using readily available mathematics software. This procedure has potentially wide application in other areas of population mathematics where theory is given by integrals and other continuous expressions, but data are for discrete age groups.     

Effects of host and parasitoid age on search behaviour and oviposition rates inLariophagus distinguendusF?rster (Hymenoptera: Pteromalidae)

Summary Oviposition rates and related behaviours were quantified forLariophagus distinguendus F?rster attackingCalosobruchus chinensis (L.) andC. maculatus (F.). Oviposition rates varied with parasitoid age; parasitoids aged 1–7 days laid approximately twice as many eggs per day as those aged 8–14 days. Similar differences were noted in search rates and handling times; younger parasitoids had higher attack rates and lower handling times than older parasitoids. Search rates and handling times also varied with the host stage available for attack. Search rates were higher and handling times were lower on larger stages. The results are discussed with reference to their impact on the dynamical behavior of insect parasitoid-host populations.