World population in 2050: assessing the projections

This paper will review some population projections for the United States, the world, and selected major regions. The total population size, the youth dependency ratio, the elderly dependency ratio, and the total dependency ratio will receive most attention. The underlying assumptions regarding fertility, mortality, and migration will be reviewed. Projections from different sources will be compared where possible.Demography ; Economic conditions

Constant global population with demographic heterogeneity

To understand better a possible future constant global population that is demographically heterogeneous, this paper analyzes several models. Classical theory of stationary populations generally fails to apply. However, if constant global population size P(global) is the sum of all country population sizes, and if constant global annual number of births B(global) is the sum of the annual number of births of all countries, and if constant global life expectancy at birth e(global) is the population-weighted mean of the life expectancy at birth of all countries, then B(global) x e(global) always exceeds P(global) unless all countries have the same life expectancy at birth.average age, Cauchy-Schwarz inequality, constant population, heterogeneity, life expectancy, migration, pensions, population projection, stationary population, zero population growth

Life expectancy is the death-weighted average of the reciprocal of the survival-specific force of mortality

The hazard of mortality is usually presented as a function of age, but can be defined as a function of the fraction of survivors. This definition enables us to derive new relationships for life expectancy. Specifically, in a life-table population with a positive age-specific force of mortality at all ages, the expectation of life at age x is the average of the reciprocal of the survival-specific force of mortality at ages after x, weighted by life-table deaths at each age after x, as shown in (6). Equivalently, the expectation of life when the surviving fraction in the life table is s is the average of the reciprocal of the survival-specific force of mortality over surviving proportions less than s, weighted by life-table deaths at surviving proportions less than s, as shown in (8). Application of these concepts to the 2004 life tables of the United States population and eight subpopulations shows that usually the younger the age at which survival falls to half (the median life length), the longer the life expectancy at that age, contrary to what would be expected from a negative exponential life table.age-structured population, force of mortality, Jensen´s inequality, life expectancy, life table, longevity, negative exponential distribution, survival, USA

Life expectancy

We give simple upper and lower bounds on life expectancy. In a life-table population, if e(0) is the life expectancy at birth, M is the median length of life, and e(M) is the expected remaining life at age M, then (M+e(M))/2≤e(0)≤M+e(M)/2. In general, for any age x, if e(x) is the expected remaining life at age x, and ℓ(x) is the fraction of a cohort surviving to age x at least, then (x+e(x))≤l(x)≤e(0)≤x+l(x)∙e(x). For any two ages 0≤w≤x≤ω, (x-w+e(x))∙ℓ(x)/ℓ(w)≤e(w)≤x-w+e(x)∙ℓ(x)/ℓ(w) . These inequalities give bounds on e(0) without detailed knowledge of the course of mortality prior to age x, provided ℓ(x) can be estimated. Such bounds could be useful for estimating life expectancy where the input of eggs or neonates can be estimated but mortality cannot be observed before late juvenile or early adult ages.inequalities, life expectancy, life table, stationary population

Random evolutions in discrete and continuous time

AbstractThis paper points out a connection between random evolutions and products of random matrices. This connection is useful in predicting the long-run growth rate of a single-type, continuously changing population in randomly varying environments using only observations at discrete points in time. A scalar Markov random evolution is specified by the n×n irreducible intensity matrix or infinitesimal generator Q = (qij) of a time-homogeneous Markov chain and by n finite real growth rates (scalars) si. The scalar Markov random evolution is the quantity MC(t) = exp(Σnj=1sjgCj (t)), where gCj(t) is the occupancy times in state j up to time t. The scalar Markov product of random matrices induced by this scalar Markov random evolution is the quantity MD(t) = exp(Σnj=1 sjgDj (t)), where gDj(t) is the occupancy time in state j up to and including t of the discrete-time Markov chain with stochastic one-step transition matrix P = eQ. We show that limt→∞(1/t)E(logMD(t))=limt→∞(1/t)E(logMC(t)) but that in general limt→∞(1/t)logE(MC(t)) ≠ limt→∞(1/t)logE(MD(t)). Thus the mean Malthusian parameter of population biologists is invariant with respect to the choice of continuous or discrete time, but the rate of growth of average population size is not. By contrast with a single-type population, in multitype populations whose growth is governed by non-commuting operators, the mean Malthusian parameter may be destined for a less prominent role as a measure of long-run growth

Perturbation theory of completely mixed matrix games

AbstractHow do the value v and the solution x and y of a zero-sum two-person completely mixed game vary as the elements of the n×n payoff matrix A=(aij) are perturbed? Assuming v > 0, we show that, for i, j, k = 1, …,n, 0<dvdaij=x iyj<1(by a little-known theorem of Gross), that dxkdaij=x ivφjk, dykdaij=yjvφki and (for g, h=1, …, n) d2vdaghdaij=v[xiyhφjg+xgy jφhi], where φij is defined in terms of B=(bij)= A−1 by φij=(∑hbih)(∑gbgj)−bij∑g, hbgh. If A is a nonsingular M-matrix, then for i,j=1, …,n we have d2vda2ii<0, dxjdaij<0, and dyjdaji<0, but v is not concave as a function of the vector of diagonal elements (a11, …, ann)

Random Sampling of Skewed Distributions Implies Taylor’s Power Law of Fluctuation Scaling

Taylor’s law (TL), a widely verified quantitative pattern in ecology and other sciences, describes the variance in a species’ population density (or other nonnegative quantity) as a power-law function of the mean density (or other nonnegative quantity): Approximately, variance = a(mean)b, a \u3e 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analytically that observations randomly sampled in blocks from any skewed frequency distribution with four finite moments give rise to TL. We do not claim this is the only way TL arises. We give approximate formulae for the TL parameters and their uncertainty. In computer simulations and an empirical example using basal area densities of red oak trees from Black Rock Forest, our formulae agree with the estimates obtained by least-squares regression. Our results show that the correlated sampling variation of the mean and variance of skewed distributions is statistically sufficient to explain TL under random sampling, without the intervention of any biological or behavioral mechanisms. This finding connects TL with the underlying distribution of population density (or other nonnegative quantity) and provides a baseline against which more complex mechanisms of TL can be compared