Thus the rate of birth accumulation into death cohorts over thanatological age, Î¼â(x), is equal to the rate of attrition of birth cohorts over chronological age, Î¼(x) in stationary populations. Choose from 500 different sets of stationary population level flashcards on Quizlet. Total fertility rates in selected countries of Eastern Europe and the former Soviet Union. Scatterplots and coefficeints of determination (R2) for M, ÎA, F30, and F65+, coale-demeny female stable population, W.W. Kingkade, in International Encyclopedia of the Social & Behavioral Sciences, 2001. 8. The Romanian achievement owed much to the absence of modern methods of family planning coupled with a draconian ban on abortions, enforced to the extent of subjecting women to regular gynecological examinations to ensure that they were not employing any modern means of fertility limitation. By continuing you agree to the use of cookies. Age-specific fertility and mortality rates are constant, and their net effects "cancel out" â that is, for a given period there will always be equal numbers of births and deaths. p-value: 0.245586 Upper tail percentiles: 10% 5% 2.5% 1% Critical value 0.347 0.463 0.573 0.739. from above I infer level stationary at 5% as p â¦ Russia and Latvia, whose respective TFRs were 1.28 and 1.09 as of 1998, figure among the lowest-fertility countries in Europe. For the purposes of the present discussion, âlow fertilityâ is taken to refer to fertility at or below the âreplacement level,â which is the level of fertility that would be consistent with a stationary population in the long run; a TFR of 2.1 is considered conventionally to be equivalent to the replacement levelâ¦ Population Pyramids: Definition. The research paper by Rao and Carey (2015) presents a novel theorem in stationary populations that states under certain conditions, two graphs, one constructed based on captured data and other on age at capture, are equal. As Riffe notes, c(x) is proportional to the survival function, l(x), which, in turn, means that câ(y) is proportional to some thanatological function, lâ(y), a stationary thanatological cumulative increment function. If we change slightly the initial value in the domain of stability (for example, Î¼ = 2.7), the population converges to the same value of 0.6296. Only Romania achieved a major rise in TFR, which rose spectacularly from below replacement to over 3.5, then stabilized in the neighborhood of 2.5 for more than ten years. Using x to index chronological age and y to index thanatological age (time left until death), this equation becomes, where câ(y) refers to the stable thanatological age structure. The first measure, ÎA, is defined as the increase in mean age of the eventual stationary population from that of the initial population. The increase in the mean age of the population is simple and performs well, both theoretically and empirically. A Decrease font size. Y.J. (2017) whose age index is what they labeled thanatological age, i.e., time until death. 1. This is in large part attributable to the media’s repeated failures to put its fingers on the true and basic cause of this growth “malaise” facing our country. This includes the period after the dissolution of the Warsaw Pact and the breakup of the former Soviet Union. However, certain of these countries, such as Estonia, had actually achieved below-replacement fertility prior to World War II. For empirical relationships, two data sets are used. The inactive population within the age groups of 0 to 14 years and over 60 are all considered to be dependent. standard stationary population with replacement-level fertility (call that the RF stationary population), in which the population at age a is proportional to l(a). Srinivasa Rao, in, Integrated Population Biology and Modeling, Part B, Arni S.R. However, we have given a formula for computing probability of dying of an individual who is alive after truncation, by partitioning the number of individuals alive at the time of truncation. PoincarÃ© was the first to assume the possibility of âirregularâ or âchaoticâ behavior of solutions of differential equations. Population Composition: Definition. Copyright Â© 2020 Elsevier B.V. or its licensors or contributors. Not only would the total numbers be reduced, but we would no longer have to go through the agonies associated with sudden shifts in our demographic behavior (be it births, deaths, or moving) as we have with the baby boom that began in the late 1940s and with which we are still trying to cope as the “baby boomers” become “senior boomers” early in the twenty-first century. The vast majority of East European countries, together with the European republics of the former USSR, were in the intermediate stages of the transition prior to World War II, and completed the transition during the postwar period. No. Phone: 703-370-9510 | Email: npg@npg.org, Read the entire report by downloading the PDF, Activists’ Guide to Protecting America’s Future, Declining Snow in Northeast Region Will Have Lasting Effects. Individuals in the captive cohort are constructed by life lived and size of the population in each age-group. Once these restrictions were lifted, Romania's fertility declined to a level close to that of Russia, one of Europe's lowest. 1 Questions & Answers Place. Intriguingly, the fertility levels of these countries converged to approximately the replacement level in the mid-1980s. The stationary population is no longer unique. States can become a true stationary population—one that is small enough to sustain life at a high level of quality. âUnder stationary growth conditions the level of Ï increases from almost undetectable levels to about one third the amount of Ï.â âAs an econometrician I have my doubts about the ratio of a non-stationary quantity and a stationary one.â These pronatalist policies were generally unsuccessful in bringing about increases in fertility to levels substantially above replacement. Kim, in International Encyclopedia of the Social & Behavioral Sciences, 2001. As Fig. This type of behavior is classified as âchaoticâ in the famous paper by Li and Yorke (1975): âperiod three implies chaosâ [13]. Test statistic: 0.1691746 . Again, luck allowed Robert May to discover a chaotic behavior in the numerical simulation of Verhulst's logistic equation. From very slightly different initial conditions, resulting from errors in experimental measurements for example, the solutions can exponentially deviate over time resulting in extreme sensitivity to the initial conditions. The converging process is limited by value, Î¼â = 3.5699456. The United States, as the millennium arrives, is overcrowded. We have not used any probabilistic assumptions on the remaining lifespans of the remaining individuals who are alive at the time of truncation in the captive cohort because such assumptions need to be justified and we have found no evidence in the published literature to start constructing partial age structure. Stationary definition, standing still; not moving. A Total Fertility Rate (TFR) of 2.1 represents the Replacement-Level Fertility: the average number of children per woman needed for each generation to exactly replace itself without needing international immigration.A value below 2.1 will cause the native population to decline Slovakia, Russia, Poland, Japan, Cuba etcâ¦ The countries I listed above (among others) have very small growths/declines in population each year. Demography (from prefix demo-from Ancient Greek Î´á¿Î¼Î¿Ï dÄmos meaning "the people", and -graphy from Î³ÏÎ¬ÏÏ graphÅ, ies "writing, description or measurement") is the statistical study of populations, especially human beings.. Demographic analysis can cover whole societies or groups defined by criteria such as education, nationality, religion, and ethnicity. The stable branch issued from the bifurcation at Î¼=1 produces three branches. The two quantities are separate aspects of the long-term change in size and age structure that results from a reduction in fertility to replacement level. More and more people are convinced that something must be done and this is especially evidenced by the current attention and concern given to the enormous and mostly unplanned suburban growth—or as it is generally called, “urban sprawl.” Our highways are overwhelmed; our water supplies are dangerously low in many areas; our schools are packed—the problems are endless. In general, all of the countries in question have experienced fertility decline during the postwar period. However, it is both surprising and disappointing that overpopulation, by itself, is seldom seen as the culprit lurking behind these countless problems. How do we reduce our population to a reasonable and sustainable level? Fertility levels in Eastern Europe and the former Soviet Union span most of the range of variation observed around the world (see Fig. An initial stationary population with r=0 bridges the negative and positive momentum seamlessly. The population momentum expresses population aging, and vice versa. At earlier dates (in the 1950s) this was also true of Albania and Azerbaijan. Stationary population level is when the national population stops growing. Arni S.R. Population evolution in stable (Î¼ = 2.7) and chaotic (Î¼ = 3.8) regimes. However, the ratio of the proportion under age 30 in the initial population to that in the ultimate population gives the population momentum directly. © 2020 Negative Population Growth All Rights Reserved. Table 1. and migration (the demographic variables) in various ways. While there has been a steady increase of population growth during the past two or three centuries, it has been especially rapid during the past 20 years. The theoretical linear relationships as well as some empirical results are presented in Kim and Schoen (1997). Stationary populations consist of two categories of cohorts, one of which is the birth cohortâall individuals who are born at the same time and thus who survive lockstep forward in time. The replacement level is generally associated with a total fertility rate of about 2.1. Time evolution in the logistic model for various values of Î¼. Find answers now! According to the values of Î¼, May observes various behaviors, both oscillatory and not, periodic and chaotic. Population evolution according to the values of Î¼. Both the low-fertility formerly Socialist countries of Eastern Europe and the republics of the former USSR, whose governments were pronatalist in orientation, experimented with policy measures to stimulate childbearing (David and McIntyre 1981, David and Skilogianis 1999). This paper also proposes mathematical conditions under which subpopulation growth and momentum to decide whether or not the total population remains stable. The system must âchooseâ between distinct branches and become increasingly sensitive to very weak perturbations. Stationary Population Level: the level in which a population has a basically even distribution of age groups and population ceases to grow. The trend in Albania's fertility level resembles closely that of Azerbaijan, although Albania's fertility has not fallen quite to the replacement level. The theoretical exposition reveals that, for this measure, the contribution from the quadratic term is not negligible, and that the coefficient of the linear term is affected by mortality levels, especially at the high end of the initial growth rate. See more. 2. The theory proposed is sensitive to the proportion of the population at each age and can be modified to adapt, even if there is no monotonic pattern for the two functions, H1 and H2 introduced. The state of a system becomes effectively impossible to predict or âchaoticâ. 2 illustrates, these linear relationships hold for negative momentum of population decline as well as for positive momentum of population growth. Drought Conditions Persist in Parts of the U.S. 10. Learn stationary population level with free interactive flashcards. The momentumâmean age linear fit is quite good, and the estimated slope (as well as the theoretical value) suggests that the mean age of the population increases by a little more than two years for every 10 percent increase in population momentum. A stationary population is a special example of a stable population with a zero growth rate, neither growing nor shrinking in size, and is equivalent to a life table population. The mathematical analysis (given in Appendix C) shows that we first find the bifurcation of McNeill and Walls at the point Î¼ = 1 (K = 0). Let us note only that the bifurcation between the situations of positive and negative values of K takes place at the value of Î¼ = 1. Population momentum and population aging during the transition to stationarity are two aspects of the same phenomenon. The fertility levels of most of the former Soviet republics discussed here are adjusted for underregistration of births according to the procedures of the International Programs Center of the US Bureau of the Census. They gave an innovative proof which uses sets and graphs and concepts of Carey's Equality. Figure 1. In Stage One the majority of death is concentrated in the first 5-10 years of life.Therefore, more than anything else, the decline in death rates in Stage Two entails the increasing survival of â¦ The active population is that group of population which is economically active and can be involved in any income generating activity. Stationary populations (revision) See PAPP101_S08 for an overview of a stationary population. This is an indication of the âbutterflyâ effect. patient, if we have the courage to adjust these demographic variables, especially immigration, then the United States can attain a smaller total population without enormous age bulgesâin other words, the United States can become a true stationary populationâone that is small enough to sustain life at a high level â¦ Members of the birth cohorts comprise the age structure of the population as measured from time since birth. The method proposed in this chapter can be also helpful in arranging age structure of a captive cohort on a continuous and flexible open intervals and geometric visualization of collective time lived and collective time remaining of a captive cohort. By increasing the value of the parameter Î¼, the four branches will generate eight branches (four stable and four unstable states), followed by sixteen branches (eight stable and eight unstable states) and so on (32, 64, 128, etc.). This stage is associated with undeveloped, low output and agricultural dominated conditions. Total Fertility Rate: the average number of children born to a woman during her lifetime. Fertility levels in the European former Soviet republics have fallen below replacement. This point acts as an attractor. The band formed by stable x* turns into a continuum; this is the beginning of the chaos. Table 2. Stationary population level is when the national population stops growing. The paper by Rao (2014) develops a new theory to understand the stability of populations. 1 The âreplacement levelâ of fertility is the level of fertility, which would, if maintained indefinitely in the absence of migration, ensure a stationary population in the long run. For the purposes of the present discussion, âlow fertilityâ is taken to refer to fertility at or below the âreplacement level,â which is the level of fertility that would be consistent with a stationary population in the long run; a TFR of 2.1 is considered conventionally to be equivalent to the replacement level. Definitions of Stationary Population, synonyms, antonyms, derivatives of Stationary Population, analogical dictionary of Stationary Population (English) Azerbaijan has completed the transition from high to replacement-level fertility since 1959. When 1 < Î¼ < 3, the populations tend by oscillations to a, Population Dynamics: Momentum of Population Growth, International Encyclopedia of the Social & Behavioral Sciences, , is defined as the increase in mean age of the eventual, ). By 1990, the TFRs of the European Soviet republics had returned to approximately their 1980 levels, while fertility decline had resumed in Soviet Central Asia. Although various legislative articles were implemented to enhance the effect of the pronatalist measures throughout the 1980s, they succumbed ultimately to inflation and to the collapse of the Soviet economy. Population Trend for Three Major Age Groups The East European countries were the first to introduce these measures, and did so in the late 1960s and the 1970s. We have constructed partial age structure in a deterministic way without any probabilistic assumptions. the level at which a national population ceases to grow: Term. Such considerations are new to the research in the stationary population identity which opened-up new doors in our theoretical understanding of the age structure of insect captive cohorts. Riffe notes that because l(x) is the sum of all deaths to the birth cohort at ages greater than x, it can be thought of as the deaccumulation of future deaths over age, where d(x) denotes deaths at age x which, in turn, gives the probability that a member of the birth cohort born at time t will die in the year tÂ +Â x. A Reset font size. High Stationary: High Birth Rate of High Death Rate: The first stage is characterised by high birth rate & high death rate, giving a low growth rate of population. Fig. If 0 < Î¼ < 1, the successive generations decrease in intensity to complete extinction. If 3 < Î¼ < 3.4495, he observes an oscillation between two values (doubling of the period). Chaotic time evolution in the logistic model. The study applied the ADF unit root test for stationary analysis on the variables to determine the right statistical model to use for this study. It is possible to group the countries under consideration according to their phase of the âDemographic Transitionâ from high, natural fertility to low, controlled fertility (Anichkin and Vishnevskiy 1994, Vishnevskiy and Volkov 1983). Stationary population can occur if there is no growth or declines in population. There is little evidence of any fundamental difference between the fertility levels and trends of the countries which comprised the former Soviet Union, and other East European countries (see Fertility Trends in the Formerly Socialist Countries of Europe). A. J.-M. AndrÃ©, in Theoretical and Computational Chemistry, 2004. 2861 Duke Street, Suite 36, Alexandria, VA 22314 They are: (a) the increase in the mean age of population (ÎA), (b) the relative size of the proportion under age 30 (F30), and (c) the relative size of the proportion above age 65 (F65+). In addition to the fertility level, the replacement of Some examples of successive generations are given in the next table. Stationary populations are the simplest form of stable populations. Subrata Lahiri, in Handbook of Statistics, 2018. Furthermore, with a smaller and stationary population, our fragile environment will be better protected. Finally, we will bequeath to our children a much more sustainable population whose members can feel secure in knowing that there is “enough for all of us.” This book concentrates on the following questions: How do we achieve these goals? Thus the Pitchfork bifurcation process leads to an infinite sequence of cycles of period 2n(n â â). The second category of cohort in stationary populations is the death cohort described by Riffe (2015) and Riffe et al. Defining deaths that occur together as a death or thanatological cohort, Riffe proved that chronological age structure equals thanatological age structure in a stationary population, where c(x) is the proportion age x in a stationary population and g(x) is the proportion of the population with x years remaining. The stationary oldest-old population is shaped by a constant number of persons annually reaching age 80 and then being subject to constant mortality. Srinivasa Rao, in Handbook of Statistics, 2018. For the purposes of the present discussion, âlow fertilityâ is taken to refer to fertility at or below the âreplacement level,â which is the level of fertility that would be consistent with a, Journal of Computational Design and Engineering. 1). The following calculations are based on an annual contingent of 10,000 persons of whom 3500 are male and 6500 female - the average proportions observed in the group of thirteen countries in 1950-1990. The first is a set of Coale-Demeny West model stable populations (see Demographic Models) that includes 39 stable populations with the intrinsic growth rate r ranging from â0.01 to 0.05 in increments of 0.005 at three mortality levels (e0 values of 60, 70, and 80). ScienceDirect Â® is a registered trademark of Elsevier B.V. ScienceDirect Â® is a registered trademark of Elsevier B.V. URL:Â https://www.sciencedirect.com/science/article/pii/S0169716118300294, URL:Â https://www.sciencedirect.com/science/article/pii/S0169716118301056, URL:Â https://www.sciencedirect.com/science/article/pii/S0169716118300269, URL:Â https://www.sciencedirect.com/science/article/pii/S1380732304800153, URL:Â https://www.sciencedirect.com/science/article/pii/B0080430767021069, URL:Â https://www.sciencedirect.com/science/article/pii/B0080430767021823, Integrated Population Biology and Modeling, Part A, James R. Carey, ... Arni S.R. By gradually increasing the value of Î¼ beyond 3, May discovered a new phenomenon with regard to the behavior of bifurcations as previously described. In the chapters that follow, we will illustrate several population scenarios by manipulating fertility, mortality. The first plot (from left to right) shows the relationship between M and ÎA, the second M's relationship with F30, and the third with F65+. Srinivasa Rao, James R. Carey, in, In continuation of the publications of âCarey's Equality and Fundamental Theorem on, http://mbi.osu.edu/resources/success-stories/, http://esciencenews.com/articles/2014/10/06/new.theorem.determines.age.distribution.populations.fruit.flies.humans/, ) shows that we first find the bifurcation of McNeill and Walls at the point Î¼ = 1 (K = 0). Thus lâ(y) can be defined as the members born up to year t that die in exactly y years, The rate at which new births accumulate to death cohorts over thanatological age, Î¼â(y), is given by, and since l(x)Â =Â lâ(y) for xÂ =Â y, then. Figure 2 presents the empirical two-way scatterplots for the momentum, M, and each of the three aging measures generated by the first data set. The dependent variable in this study was the unemployment rate and independent variables were SMEs growth, Population growth and education level. Contents Defintion of Population Stable and Stationary Population Calculation of Person-years lived Conclusion 3. Above a critical point, there exists an infinite amount of overlapping bifurcations and behaviors of varying periodicity and as well as an infinite amount of initial points x0 which lead to completely aperiodic trajectories. Stationary population can occur if there is no growth or declines in population. How does that population’s distribution attain relative efficiency (where there are no surging “bulges” in certain age groups)? Another characteristic of Stage Two of the demographic transition is a change in the age structure of the population. 6. These populations remain in the early phases of the demographic transition, exhibiting sustained fertility decline. 40. stationary population level: A point in growth, in which a region stops growing in population. The concepts and ideas of the two research publications, mentioned above, may be followed up for further in demographic research. Stationary population identity with two variables is structured in this chapter to suit analysis required to perform on a captive cohort when there exist very large number of individuals in each age-group of a population. If the population is classified according to sex, it is called composition of population by sex. The only major difference between these regions (former USSR and East Europe) at the time of writing is that the former Soviet Central Asian populations, in which fertility decline is a comparatively recent phenomenon, have not yet attained the low fertility levels which typify the European populations. Truncation from right is taken as a variable to construct partial age structure of the population. Sketch of bifurcations in May's numerical logistic model. We use cookies to help provide and enhance our service and tailor content and ads. Our quality of life, however defined, will improve. At the opposite extreme, in the indigenous population of former Soviet Central Asia, fertility decline did not take hold until the mid-1970s. The second data set is comprised of 41 observed populations in 1985 (Keyfitz and Flieger 1990). Under such conditions, the population will eventually stabilize into a stationary population, with no year-to-year changes in age-specific rates or in total population. Beyond this value of Î¼, the overlap of various periods is so great that we observe a chaotic behavior. Unlike, in one-dimensional stationary population identity (Srinivasa Rao and Carey, 2015), in this chapter, the collective lengths of remaining lifespans associated with the age at death during follow-up (through the construction of two functions, H1 and H2) and number surviving at the time of follow-up (as explained through Theorem 2 and Corollary 1) are simultaneously considered. patient, if we have the courage to adjust these demographic variables, especially immigration, then the United States can attain a smaller total population without enormous age bulges—in other words, the United Peter Turchin, in Population Dynamics, 1995. Srinivasa Rao, James R. Carey, in Handbook of Statistics, 2019. The Soviet Union introduced a major set of pronatalist incentives in 1981. The mathematical analysis shows a pitchfork bifurcation. Fig. Fertility in Australia. We have used distinct parts of the partition to suit monotonic nature of the age structure and one can consider all the parts of the partition function when there is no significant pattern of the age structure. Other articles where Stationary phase is discussed: bacteria: Growth of bacterial populations: â¦growth is followed by the stationary phase, in which the size of a population of bacteria remains constant, even though some cells continue to divide and others begin to die. The middle plot shows that the relative size of the proportions under age 30 is not only linear, but is identical to the momentum. I run the KPSS test with trend and level on same time series and the results are: H0: level stationary vs. H1: Unit root. One primary motivation for developing nonlinear methods for time-series analysis is to be able to determine the qualitative type of stationary population dynamics that may be characterizing the studied system. The two branches observed between Î¼=1 and Î¼=3 are now four branches (two stable and two instable). As a consequence, the values for a given step, xn, are never reproduced. 2. Fig. Thus, although the proportion over age 65 has great appeal as a conventional index of aging, it performs the poorest among the three measures of aging considered. Population Dynamics in the Noisy World. The fit is not as good as with the other two measures. As shown in Appendix C, May proposed a numerical equivalent of the logistic equation where the population of a given generation (n+1) is related to the population of the previous generation (n) by the iterative formula: May's conclusions were published in a reference paper entitled âSimple Mathematical Models with very Complicated Dynamicsâ [12]. This was the result observed in 1963 by Lorenz. The second measure, F30, is defined as the ratio of the population under age 30 in the initial population to that in the ultimate stationary population, and the third measure, F65+, is defined as the ratio of the proportion over age 65 in the initial population to that in the ultimate stationary population. The argument is that when subpopulations will have presence of population momentum, then the local stability of the total population could become unstable. 7. The behavior of the trajectories is never the same even if the measurement times are infinitely large. We consider these goals to be not only ideal but necessary if the United States is to maintain anything close to our current quality of life and sustainability. Presentation on Stable and Stationary Population Presented By Aminul Islam ID:14115570 Department of Population Science and Human Resource Development, University of Rajshahi E-mail:amin_pshrd@yahoo.com 2. James R. Carey, ... Arni S.R. A parameter Î¼ replaces the parameter K of the original equation. A Increase font size. Fertility in the World. Stationary Population Level: Definition. One of them is the unstable continuation of the stable branch and two new stable branches are produced. Shortly afterwards, fertility began to rise in the European republics of the USSR, together with Armenia and Georgia. The reader can check this behavior in May's numerical equation of the logistic model. However, the vast majority of the countries in question qualify as low-fertility countries, having experienced TFRs on the order of two to three for most of the period since World War II. This is a direct consequence of the fact that the population size under age 30 remains constant before and after the transition, as discussed in Sect. When 1 < Î¼ < 3, the populations tend by oscillations to a stationary population. Death rate is high during this stage for a â¦ 1. The stationary phase is followed by the death phase, in which the death of cells in the population exceeds theâ¦ Fig. As recently as the mid-1970s, Uzbekistan, Turkmenistan, and Tajikistan registered total fertility rates (TFRs) from six to seven, well within the range of high-fertility Third World countries. Furthermore, if the domain of variation of K was - â < K < + â, that of Î¼ is restricted to the values 0 < Î¼ < 4. What are some examples of stationary population level? If 0 < Î¼ < 1, the successive generations decrease in intensity to complete extinction. In Azerbaijan, fertility decline was halted temporarily in the mid-1980s. The bottom line is positive: if we are In continuation of the publications of âCarey's Equality and Fundamental Theorem on Stationary Population Modelsâ (vide http://mbi.osu.edu/resources/success-stories/) and âNew theorem determines the age distribution of population from fruit flies to humansâ (vide http://esciencenews.com/articles/2014/10/06/new.theorem.determines.age.distribution.populations.fruit.flies.humans/), Rao (2014) and Rao and Carey (2015) (see also Rao, 2012) have shown very interesting and new ideas and concepts on population stationary as well as on population stability and momentum. Bifurcations in May's numerical logistic model. Structure of a population in terms of age, sex and other properties such as marital status and education: Term. 9. X-concentration as a function of time (first line) and of Y-concentration (second line) (a) in a stationary state, (b) in a doubling period regime, (c) in a quadrupling period regime d. in a chaotic regime. A smaller and stationary (that is, having no further growth or decline) population is in the best interest of the United States. Close to the bifurcation points, natural fluctuations or very slight changes in initial conditions play a role of increasing importance. The reader can easily reproduce these results by means of a simple pocket calculator. To appreciate the pace of population growth we should recall that world population doubled in about 1,700 years from the timâ¦ To calculate population momentum for population A, a theoretical population is constructed in which the birth rate for population A immediately becomes replacement level. The stability domains of each cycle are progressively reduced: Fig. We are fast approaching 300 million inhabitants. In most cases (e.g., Bulgaria, Hungary, Poland), these measures involved financial incentives such as extended maternity leave, cash awards, and housing subsidies. A Total Fertility Rate (TFR) of 2.1 represents the Replacement-Level Fertility: the average number of children per woman needed for each generation to exactly replace itself without needing international immigration.A value below 2.1 will cause the native population to decline The three measures of aging are highly correlated with each other and with the population momentum, for these four quantities are linear functions of the intrinsic growth rate, r, of the initial population. After the breakup of the USSR, fertility levels continued to decline in all former Soviet republics (Vishnevskiy 1999). Since tÂ +Â x refers to the future, we use index y for thanatological age and thus deaths that occur at time t comprise a death cohort. Stable populations are theoretical models widely used by demographers to represent and understand the structure, growth and evolution of human populations. Typically, the introduction of such measures was followed by a modest rise in fertility of relatively brief duration, and a longer-term slackening or cessation of fertility decline. The population of the world, now somewhat in excess of three billion persons, is growing at about two per cent a year, or faster than at any other period in mans history. Concurrently, levels of fertility rose somewhat in Tajikistan, and to a lesser extent in Turkmenistan and Uzbekistan. the stationary population (population replacement level, total fertility rate requires approximately 2.08) since the mid-70s, together with the low-fertility rate trend continuing for a quarter-century, make the de-population which start at the beginning of this century almost inevitable. Population in OECD countries is predicted to increase from the current level of 1.25 billion people to 1.3 billion in 2020 and further increase to 1.39 billion by 2050. In the chaotic region, a similar weak variation gives way to completely different successive evolutions. The third plot presents the relationship between the momentum and the relative size of the proportion above age 65. To make the relationship between the dual consequences of fertility declineâpopulation momentum and population agingâmore specific, three measures of aging were considered (Kim and Schoen 1997). Figure 2. Thanatological age equates individuals that share a common terminal state rather than a common origin state. This series of period doublings also makes it impossible in practice to obtain analytical solutions, while numerical solutions are very easily obtained on a pocket calculator.

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