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A population is considered stationary if the growth rate is zero and the age structure is constant. It thus follows that a population is considered non-stationary if either its growth rate is non-zero and/or its age structure is non-constant. We propose three properties that are related to the stationary population identity (SPI) of population biology by connecting it with stationary populations and non-stationary populations which are approaching stationarity. One of these important properties is that SPI can be applied to partition a population into stationary and non-stationary components. These properties provide deeper insights into cohort formation in real-world populations and the length of the duration for which stationary and non-stationary conditions hold. The new concepts are based on the time gap between the occurrence of stationary and non-stationary populations within the SPI framework that we refer to as Oscillatory SPI and the Amplitude of SPI. This article will appear in Bulletin of Mathematical Biology (Springer), 26 pages

Spanish Abstract: El analisis de las dinamicas demograficas es uno de los campos con mayor nutricion dentro de la demografia teorica debido a su aplicacion en los modelos de estimacion y proyeccion de poblacion. Las poblaciones teoricas tuvieron su mayor desarrollo en la segunda mitad del siglo XX y sus supuestos en la actual transicion demografica mundial son aun de interes debido a su utilidad para el estudio en prospectiva de los fenomenos sociodemograficos en una poblacion. Esta investigacion representa una contribucion a este campo, estimando el modelo de poblaciones estables para Colombia en los anos 1950, 1980, 2000 y 2018, haciendo un analisis del ajuste de dicha teoria para las estructuras de poblacion del pais e identificar el acercamiento o distanciamiento frente a dicha poblacion teorica. English Abstract: The analysis of demographic dynamics is one of the fields with the most extended trajectory within theoretical demography due to its application in population estimation and projection models. Theoretical populations had their most considerable development in the second half of the 20th century, and its assumptions in the current world demographic transition are still of interest due to their utility for the prospective study of sociodemographic phenomena in a population. This investigation represents a contribution to this field, estimating the model of stable populations for Colombia in 1950, 1980, 2000 and 2018, making an analysis of the adjustment for the country's population structures and identifying the approach or distancing from the mention theoretical population.

Stationary population identity arises while studying the average time left to die after forming a captive cohort of individuals in a stationary population. We extend the ideas of stationary population identity, which arises in stationary populations to two dimensions with the aim to answer questions related to aging, age structure of the captive cohort. These ideas found to be useful for identifying individuals by age group in a captive cohort.

We are interested inferring rate processes on networks. In particular, given a network's topology, the stationary populations on its nodes, and a few global dynamical observables, can we infer all the transition rates between nodes? We draw inferences using the principle of maximum caliber (maximum path entropy). We have previously derived results for discrete-time Markov processes. Here, we treat continuous-time processes, such as dynamics among metastable states of proteins. The present work leads to a particularly important analytical result: namely, that when the network is constrained only by a mean jump rate, the rate matrix is given by a square-root dependence of the rate, kab ∝ (πb/πa)(1/2), on πa and πb, the stationary-state populations at nodes a and b. This leads to a fast way to estimate all of the microscopic rates in the system. As an illustration, we show that the method accurately predicts the nonequilibrium transition rates in an in silico gene expression network and transition probabilities among the metastable states of a small peptide at equilibrium. We note also that the method makes sensible predictions for so-called extra-thermodynamic relationships, such as those of Bronsted, Hammond, and others.

BACKGROUND The age distribution and remaining lifespan distribution are identical in stationary populations. The life table survival function is proportional to the age distribution in stationary populations. OBJECTIVE We provide an alternative interpretation of the life table when viewed by remaining years of life. CONCLUSIONS The functions describing the mortality of birth cohorts over age are identical to the functions describing the growth of death cohorts as time to death decreases in stationary populations.

In this study the reproductive investment of six populations of Atlantic herring ( Clupea harengus ) in Norwegian waters was contrasted in relation to trade-offs with body growth (relatively slow-relatively fast) and migration distance (stationary-migratory). Down-regulation of fecundity through the process of atresia as well as standardisation of fecundity to the prespawning stage were included as process-oriented reproductive factors, applying both histological and image analysis techniques. The further analysis included historic information on body growth as well as published information on fecundity from several stocks in the North Atlantic. The Norwegian spring-spawning (NSS) herring could be split into three sub-components: migratory (oceanic), likely semi-stationary (coastal) and stationary. The latter one as well as three other populations were sampled in relatively isolated semi-enclosed areas (pond, “lake” or fjord). The study documented clear signs of trade-offs: migratory herring had a significantly higher growth rate and lower relative fecundity while stationary populations grew slower and presented higher values of relative fecundity. So these traits appeared highly plastic and for the first time explicitly demonstrated in the three types of NSS herring: stationary NSS herring had high fecundity and body condition while the truly migratory counterpart was low in both while the intermediate version was low in fecundity but high in condition. The literature-based analysis of other Atlantic spring-spawning herring populations seemed to corroborate the finding that slow-growing herring is relatively more fecund than the faster-growing populations.

First, we revisit the “life lived equals life left” property for stationary populations and discuss it from a more general perspective. Specifically, we show that identically distributed random age and the remaining lifetime in stationary populations have the same distribution as the equilibrium distribution of the renewal theory. Then we consider specific non-stationary populations that are closed to migration and have a constant birth rate. We obtain some useful inequalities between random age and remaining lifetime for different instants of calendar time. We also discuss the aging properties of populations using different stochastic orders.

In this article, we use mathematical modeling to study the impact of population dynamics on Y-chromosome STR-polymorphism accumulation in two independently evolving populations, namely, on the changes in genetic distance among the populations. Using two definitions of the genetic distance: (deltamu)2 and ASD, we carry out comparative research on the genetic distance changes and show that in contrast to (deltamu)2, ASD is characterized by a near-linear dependence on time. As the populations undergo oscillations, ASD is shown to be smaller than that in stationary populations. The linear dependence of ASD on time is shown to break down in relatively scanty stationary populations.

With constant immigration flows, there are an uncountably infinite number of stationary populations for any below-replacement fertility schedule f and life table l. Each corresponds to a different pattern of age-specific annual immigrant inflows. This brief paper derives a general formula for the age structure of these populations, provides numerical examples using contemporary Austrian demographic data, and illustrates an application of the main relationship to the study of stationary dependency ratios.

This note provides some earlier history of the relationship given in Formal Relationships 1, "Life left equals life lived in stationary populations," (Goldstein 2009) and shows that while the expectation of life at the mean age of the population is close to the mean age, this is not exactly so. (ProQuest: ... denotes formulae omitted.) 1. Earlier statements and proofs of the result The first paper in the "Formal Relationships," special collection, 'Life left equals life lived in stationary populations," (Goldstein 2009) did not contain any reference to the originator of the relationships. However, Lotka (1939) stated the result and makes it clear it was already part of the general actuarial knowledge of the time. Lotka1 wrote For a stationary population with an age distribution proportional to the life table, it is known that ["on sait que"] ... that is, the mean of the life expectancies of all the individuals in the popula- tion is equal to their mean age.* Lotka does not provide a derivation, but the source he cites (Spurgeon 1929) does pro- vide a full proof. The use of "for example" suggests that the result was well-established. Indeed, Spurgeon's text was an update of the original actuarial textbook for the Institute of Actuaries written by King (1st edition 1887, 2nd edition 1902), which contains nearly the same derivation, noting "the total future lifetime of a stationary population is exactly equal to its total past lifetime" (p. 67, 1902 edition). The "textbook" nature of this pub- lication suggests that the relationship was broadly known at the end of the 19th century. More extensive investigation of the historical actuarial literature may reveal earlier men- tions. 2. On Lotka's approximation In their translation of Lotka (1998, p. 82), Smith and Rossert provide the following "Translator's note," providing their own intuition on the result: The equality is explained by noting that the expected age of a random indi- vidual in the life table will be the mean age of the table, and the individual will have been sampled halfway through his or her lifetime. To say that the "individual" aged ... is halfway through his or her lifetime is approximately, but not exactly, true. It is equivalent to claiming, using parentheses rather than a subcript to indicate the age argument, (1) ... because then the expected age at death would be ... and the population age would be half of the expected life time length... Indeed, Lotka extensively discusses (1), which although he uses the "equals" sign, the context makes clear is meant to be read as an approximation (1939, pages 31-37). He first shows, in a number of then-available life tables, that one can well approximate the mean age by finding the age at which remaining life expectancy equals itself. …

Although new research suggests multi-directional trajectories in the development of the Zimbabwe Tradition (see Chirikure et al. 2016), regional population shifts need not be discounted, as some of these generated states (e.g. Vigneswaran & Quirk 2015). Oral-historical data from northern Zimbabwe counters persistent but often misleading views of pre-colonial states in south-central Africa as exercising power over static and stationary populations (Pikirayi 1993). Rather, human mobility shaped, among other things, the Zimbabwe Culture's spatial features, its strategies for accumulating power and managing resources, and the regional political, social and economic actors to which it was connected. This occurred with the demise of Great Zimbabwe from the second half of the fifteenth century and for much of the sixteenth. Ingombe Ilede attests to post mid fifteenth-century regional shifts in patterns of trade that would lure the Portuguese to south-central Africa from the early sixteenth century onwards. The Zambezi became the preferred inland route. Great Zimbabwe's expansionary thrusts to control this trade undermined its own political control over the southern Zimbabwe plateau, as this spawned new political formations like the Mwene Mutapa state and other polities, including Ingombe Ilede.

The average age of the individuals in a population is equal to the average remaining live expectancy when the population is stationary.

The first hypervariable region and several restriction fragment polymorphisms in mitochondrial DNA were investigated in 308 individuals belonging to 16 different populations from Southeast Africa, all of them Bantu speaking (see Fig. 1). A total of 30 available African populations (or populations with an important African influence) as well as other non-African ones were used for comparison. Preliminary results are shown in the present work. High diversity values were found for the samples analysed in this work, in comparison with other African samples. Nucleotide mismatch distributions are rugged and multimodal, which could reflect the scenario of stationary populations rather than one of expansion. Phylogenetic reconstruction allowed us to infer that the Southeast African populations are distant from other population groups, with the ancient ones (Pygmies and !Kung) more closely related. The results obtained are of special forensic interest because they produced new mtDNA patterns from human populations completely unknown (from a genetic point of view) until now, and to increment our mtDNA databases for forensic purposes. 2. Nucleotide and sequence diversity Sequence diversity and nucleotide diversity were computed for all the populations analysed and compared with other African ones. Almost all the populations analysed here showed similar values for these indices, the lowest values were in the Makonde population (p=0.0187 and d=6.830) and the highest in the Shona (p=0.0277 and d=10.177), and they were, on average, slightly higher than those found for other African populations. When

This article is a revision of the previous edition article by D. Courgeau, volume 15, pp. 10210–10214, © 2001, Elsevier Ltd.

International audience; We study the non-genetic inheritance of fertility from parents to offspring. For this purpose, we propose an exchangeable extension of the Wright-Fisher model. This extension allows us to introduce non-genetic fertility correlation in the forward in time process and to study its effects on the genealogies of individuals (or genes) samples. Since it is independent of the gene considered, this effect is uniform on the genome, even in diploid populations. For values of fertility correlation observed in human populations, we show that coalescence times are strongly but inhomogenously reduced and that the shape of gene genealogies is markedly unbalanced. Despite the fact that our simulations concern stationary populations, the former non-genetic effect is very similar to what has been described for populations of variable size such as populations passing through demographic bottleneck. However, additional strong tree imbalance due to non-genetic causes is reported here for the first time.

A variety of open systems in nature and society exist under dynamic equilibrium, maintained by statistical counterbalance between the entering and leaving of individuals and the stationarity of the exchange processes. A network of functions characterizing the dynamics of such a stationary population is established and discussed, which allows the mutual transference of system properties without the need of any explicit information about the microdynamic processes. In order to illustrate the potential benefit of these interdependence relations, examples taken from diverse branches of research (adsorption and reaction kinetics, demographic analysis, and coronary blood flow diagnosis) are given.

The work presented considers a new coherent phenomenon, i.e. the influence of the phases of electromagnetic fields interacting with a multilevel system on the character of excitation of this system. On the basis of analysing the equation for an N-level quantum system density matrix, it was established that in multilevel systems containing a closed contour of transitions, resonantly interacting with the field, stationary populations always depend on the total interaction contour phase Phi . For the simplest multilevel systems, i.e. three-level, the authors have obtained analytical expressions, connecting populations with the relative field phase in such systems. Thus, for a Lambda system, interacting with three resonant electromagnetic fields, it is discovered that, depending on the total phase value of the transition contour, both destruction and successive restoration of a coherent trapping state can take place. The work provides qualitative explanations of phase effects and suggests some possibilities for their application.

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)|e(global) always exceeds P(global) unless all countries have the same life expectancy at birth in which case B(global)|e(global) = P(global). (authors)

The case-exposure study, described in 1983, is a modification of the case-control study that allows estimation of risk ratios without need for a rare-disease assumption. For a fixed population the approach is the same as the case-base or case-cohort study. Two sampling strategies have been described for extending application of the case-exposure approach to stationary populations, although the validity of the extension was not proven. As shown here, the two sampling strategies for stationary populations can provide valid estimates of the risk ratio, but restrictive assumptions related to stationarity may limit application of the strategy. We also show that the case-exposure study depends on the stationarity assumptions in a different, possibly more restrictive way than some other case-control approaches. We emphasize limitations of the case-exposure study that may preclude its use, particularly if the period of risk is long.

Adaptive (or stationary-phase) mutation is a group of phenomena in which mutations appear to occur more often when selected than when not. They may represent cellular responses to the environment in which the genome is altered to allow survival. The best-characterized assay system and mechanism is reversion of a lac allele on an F' sex plasmid in Escherichia coli, in which the stationary-phase mutability requires homologous recombination functions. A key issue has concerned whether the recombination-dependent mutation mechanism is F' specific or is general. Hypermutation of chromosomal genes occurs in association with adaptive Lac(+) mutation. Here we present evidence that the chromosomal hypermutation is promoted by recombination. Hyperrecombinagenic recD cells show elevated chromosomal hypermutation. Further, recG mutation, which promotes accumulation of recombination intermediates proposed to prime replication and mutation, also stimulates chromosomal hypermutation. The coincident mutations at lac (on the F') and chromosomal genes behave as independent events, whereas coincident mutations at lac and other F-linked sites do not. This implies that transient covalent linkage of F' and chromosomal DNA (Hfr formation) does not underlie chromosomal mutation. The data suggest that recombinational stationary-phase mutation occurs in the bacterial chromosome and thus can be a general strategy for programmed genetic change.

"This paper investigates the impact of preferences for male offspring to female offspring upon the sex ratio of the population. Asymmetric procreation behaviour of this kind is modelled by assuming that a female's procreation ceases only after at least one son or n daughters are born. It is shown that such asymmetric procreation behaviour has no effect on the sex ratio of the society, but influences rather the growth rate of the population. Finally, problems concerning the interrelationship between the sex ratio, the pattern of procreation, and the marriage regime in stationary populations are investigated."

SUMMARY A fixed ensemble of alternating renewal processes is proposed as a model for stationary populations. This model is applied to explore the relationship between survival times and forward recurrence times. In addition, it is shown that ergodic theory can be applied to the model. An example is presented.

Age distributions are given for four species in a virgin spruce—fir stand in New Hampshire together with comparative data from virgin northern hardwoods. Three distinct forms of age distribution were found: (1) essentially linear relationships of numbers of stems (log) vs. age (log) denoting stationary populations with a power function form of survivorship; (2) concave age distributions indicative of increasing populations; and (3) convex to bell—shaped age distributions reflecting declining populations with a low birth date. Picea rubens, Abies balsamea, Betula papyrifera var. cordifolia, and Sorbus americana all exhibited stationary age distributions in the virgin spruce—fir stand. In virgin northern hardwoods, Fagus grandifolia and Acer saccharum appeared to be stationary or increasing, respectively, while P. rubens and Betula alleghaniensis had age distributions typical of declining populations. See full-text article at JSTOR

The concepts of population production and biomass turnover are considered in terms of the mathematical formalism for stage-structured populations in continuous time. The aim of this analysis is to provide a single framework for evaluating the various approaches and methods currently employed for estimating population production. The production rate of a stationary population can be formulated in several equivalent ways by applying change of variable and integration by parts to the expression defining production. One of the formulations states that the production rate (in weight/time) equals the recruitment rate times the mean weight at death. The mean weight at death can be decomposed into the weight at recruitment and the mean weight added by growth during life. The turnover of biomass equals the turnover of individuals, corrected for the ratio of mean weight at death to mean weight of live individuals. It is shown that production estimation methods of Allen (1951) and Ricker (1946) underestimate production in that they ignore the weight at recruitment relative to the mean weight at death. With reference to a numerical example, the removal-summation, increment summation, growth-survivorship, and size-frequency methods are discussed and modifications proposed. The analysis of stationary populations is extended by discussing time-invariant cohorts and generations, for which an equivalent average stationary population can be defined. Finally, both differences and correspondences are discussed that relate production on the individual level to population production.

Darwinian fitness with respect to the age of maturity (α) in stationary populations can be written as the product of two functions: pre-α survival times the Fisherian reproductive value of an age α (a just mature) individual. This reduces normalizing selection on α to the maximization of a simple product,a twodimensional problem (Charnov in press). I apply this products theorem to α in Pandalid shrimp and compare the results to previous analysis (Roff, 1986) of α in fish.

The effect of low temperature storage and milk quality on cheese yield was investigated. Raw milk was stored at 5, 7.5, and 10°C; grade A milk was stored for 6 to 12 d and manufacturing grade milk for up to 6 d. At 1-d (manufacturing grade) and 2-d (grade A) intervals, aliquots from each treatment were pasteurized and manufactured into cheese, and the resulting yield was compared with that obtained on d 0 for manufacturing and grade A milks, respectively. Each treatment was terminated when stored milk coagulated when pasteurized. Increase in total bacterial count ranged from one log cycle for high count manufacturing grade milk to more than three log cycles for low count grade A milk. Similar changes were observed for proteolytic and psychrotrophic populations. As stationary populations were approached, essentially all organisms were psychrotrophic, and most were proteolytic and lipolytic. Cheese yield was affected by initial psychrotrophic populations and length of time raw milk was stored. Recovery of cheese solids decreased by approximately .5% for manufacturing grade milk per day of storage up to 4 d. Further storage increased the loss of cheese solids, and loss of cheese yield correlated with increase in bacterial population. Yield loss appeared to be due to exocellular enzymes causing breakdown of proteins and fats. Protein degradation was accompanied by increased moisture in the curd. Cheese quality decreased as psychrotrophic populations increased. After pasteurization high count stored raw milk contained enough psychrotrophs to dominate the aging process, producing gassy unclean cheese. Stored milk cheese had a weak body with bitter flavors typical of that produced by heat-stable proteolytic enzymes.

In a large population of constant size, there is a unique equilibrium distribution for every deleterious autosomal dominant or deleterious X-linked gene. The purpose of this paper is to determine the mean vector and covariance matrix for such an equilibrium distribution. The theory of branching processes with immigration provides the framework for our investigation. Autosomal dominants can be treated using single-type branching processes; X-linked genes, using two-type branching processes. Application is made to Huntington's chorea and Becker's muscular dystrophy.

Phage λ · C that has been propagated in Escherichia coli C plates with very low efficiency on E. coli K. The probability that such restricting cells in log-phase or stationary cultures will become phage-yielders rises sharply as the m.o.i. by λ · C increases. The process, which has been termed cooperative infection, is detectable at an m.o.i. of 2 in this system. Both λ · C and λ · K plate with very low efficiency on E. coli C (P1) and on E. coli K (P1). The question of whether these restricting hosts can support cooperative infection has been examined. Neither λ · C nor λ · K could cooperatively infect E. coli C (P1). In E. coli K (P1), λ · K barely overcame the restriction barrier in stationary cultures only, and only at an m.o.i. of 15 and greater. The efficiency was very much lower than that observed in the case of cooperative infection of E. coli K by λ · C. It was concluded that λ · K cannot cooperatively infect E. coli K (P1). In E. coli K (P1), λ · C cooperatively infected only a small fraction of log-phase or stationary populations. It appeared that this fraction was composed of those cells which could support phage growth when singly infected with λ · K. These cells retained the P1-directed mechanism which modifies λ DNA, but probably lost the P1-directed mechanism which restricts λ DNA. The behavior of restricted λ in the P1-lysogens contrasts sharply with their efficient cooperative infection by restricted phage T1 described by other workers. The biochemical data available on the degradation of restricted phage DNA and the properties so far described for isolated restriction enzymes cannot yet explain why cooperative infection takes place in one system but not in another.

This paper investigates the influences of death rates on average income in stationary populations, through changes in the age distribution and changes in the upper age limit of the population taking part in production. It is assumed that decreases in death rates are followed by a decrease in the number of births to make the total population remain constant. This may reduce the loss in labour input due to the births. The formal analysis introduces a productive ages a function of expectation of life at birth. We further introduce a consumption unit scale to be used in defining average income. The formulae derived are in principle applicable to any given pattern of change in death rates. The method is illustrated numerically on the basis of changes in Norwegian death rates between 1901–11 and 1921–31. Some tentative consclusions are drawn. The main result seems to be that it is a necessary condition for achieving any appreciable increase in average income through a decline in death rates, that the up...

The aim of this note is to show the conditions under which mean ages and life expectancies at birth in stationary populations stay in different order. Lets assume that the mean age of a stationary population A is greater than that of another stationary population B. Does it mean that the life expectancy at birth for A is also greater than that for B? The aim of this note is to show the conditions under which the above quesitons would have a negative answer. In 1977 Stephen W. Salant demonstrated with a mathematical model concepts concerning unemployment spells in a situation of stationary unemployment (what he calls "stable economic conditions." The model implies that E(Sa) < E(Sb) (page 43 Note 10) where T is the duration of the interruped spells of unemployment and S of the complete spells. In the demographic chart T corresponds to the age of individuals at a given moment (census) and S to the age at death of those individuals. E(T) would then be the mean age of the population E(S) the life expectancy at birth and Var(S) the variance corresponding to the ages at the time of death. If the mean age in A is greater than that in B and the difference in the variation coefficients at the time of death in stationary populations A and B is more than double the differnce of the age averages the order of life expectancy at birth would then be reversed. Salants analysis is a far cry from the techniques and the language usually employed in demography. However I believe that this result could be of some interest to demographers. (full text)

Study of movement, whether in migratory or relatively stationary populations of turtles, can provide insight into habitat utilization by a species. Undisturbed movement in pond or lake populations of Chrysemys picta has been discussed in a number of studies (Gibbons, 1968; Mallet, 1975; McAuliffe, 1978; Pearse, 1923; Sexton, 1959). Movement after displacement in this species has been examined by Ernst (1970), Ream (1967) and Williams (1952). Movement of river populations has not been studied in C. picta, but has been in Pseudemys scripta (Moll and Legler, 1971), Trionyx muticus (Plummer and Shirer, 1975) and Graptemys pseudogeographica and G. ouachitensis (Vogt, 1980). Most studies of the movement of individuals or