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 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.     

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.     

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.     

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.     

Influence of instantaneous fertility decline to replacement level on population growth: An alternative model

If age-specific birth rates m, of a stable population drop abruptly tom _x/R ₀, whereR ₀ is the net reproduction rate, then, according to Keyfitz, the size of the ultimate stationary population relative to.that at the beginning of the process is given byI =be ⁰ ₀ R ₀ ? 1)/(rμR ₀, whereb andr are the birth rate and the rate of growth, respectively, of the stable population,e ₀ ⁰ the life expectancy at birth, andμ the average age at childbirth in the resulting stationary population. Noting that the decline inm _x need not necessarily be uniform, investigation has been carried out to examine the effect on Iwhen fertility decline is more rapid at higher ages. In particular, the effect of the reduced age-specific rates such asm _xe^? rx (which also produces a stationary population) has been analyzed, and simplifications of the results carried out separately for three different models of the net maternity function. It has also been shown that when m, drops abruptly to somem _x ^*, where the form ofm _x ^* need not be specified except for the restriction that the resulting population will be stationary, the value of the index can be approximately obtained fromI ^* =be ₀ ⁰ (1 -rμ/2), whereμ is the average age at childbearing of the initial stable population.     

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.     

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.     

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.     

Determining the birth function for an age structured population

This paper deals with an inverse problem in age‐structured population dynamics, namely the recovery of the unknown birth function from the additional or overposed data consisting of the total population over a time interval equal to the maximum life span of the species. Conditions on the data are given to guarantee the existence and uniqueness of a solution, and the question of continuous dependence of the birth function on the data is addressed. Some numerical simulations are presented to indicate that one can in fact use the methods of the paper to reconstruct the birth function.     

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.     

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.     

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.     

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.     

On a Pandemic Threshold Theorem of the Early Kermack–McKendrick Model with Individual Heterogeneity

A pandemic threshold theorem of the Kermack–McKendrick epidemic system with individual heterogeneity is proved from the definition of R ₀ by Diekmann, Heesterbeek, and Metz. The early Kermack–McKendrick epidemic model is extended to recognize individual heterogeneity, where the state variable indicates an epidemiological state or genetic, physiological, or behavioral characteristics such as risk of infection. With the basic reproduction number R ₀ for the heterogeneous population, the final size equation of the limit epidemic starting from a completely susceptible steady state at t = ?∞ has a unique positive solution if and only if R ₀  > 1. The main result is that the positive solution of the final size equation gives the lower bound of the intensity of any epidemic starting from a host population composed of susceptible and a few infected individuals who spread on a noncompact domain of the trait variable.     

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.     

The solution of time‐dependent population models

We analyze the dynamics of age‐structured population renewal when vital rates make a transition in a finite time interval from arbitrary initial values to any specified final values. The general solution to the renewal equation in such cases is obtained. This solution describes the birth sequence explicitly, and also leads to a general formula for population momentum. We show that the duration of the transition determines the complexity of the solution for the birth sequence. For transitions that are completed in a time smaller than the maximum age of reproduction, we show that the classical Lotka solution found in every textbook also applies, with a small modification, to the time‐dependent case. Our results substantially extend previous work that has often focused on instantaneous transitions or on slow and infinitely persistent change in vital rates.     

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.     

Persistence of a fig wasp population and evolution of dioecy in figs: a simulation study

The relationship between fig trees and their pollinator wasps is a well-known example of species-specific obligate mutualism. In this article we present a stochastic model of this mutualistic system, referring to data on a dioecious fig (Ficus schwarzii) in Borneo, and examine the conditions for the persistence of a wasp population for a given period. (1) When the average duration of the flowering interval of fig trees is short, even a small fig population can sustain a wasp population successfully. A population whose average period of flowering cycle is half that of another population can sustain a wasp population with a number of trees less than half of the other population. (2) The wasp survival rate (WSR) is higher when (a) the variation of the interval periods of fig flowering is smaller, (b) the fig population size is larger, and (c) figs can prolong their receptivity to wait for a wasp if no wasps are available. (3) WSR is predictable from the average proportion of the fig's receptive phases, in which wasps are available, to their total receptive phases. (4) The persistence period of a wasp population increases exponentially with the number of fig trees. Based on these results we propose a new hypothesis, as a possible scenario, on the evolution of dioecy from monoecy in Ficus. Received: November 13, 1998 / Accepted: July 14, 1999     

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.